hot beats and tune outs: atom interferometry with laser-cooled lithium · 2018. 10. 10. · doctor...

Hot Beats and Tune Outs: Atom Interferometry with Laser-cooled Lithium

by

Kayleigh Cassella

A dissertation submitted in partial satisfaction of therequirements for the degree of

Doctor of Philosophy

in

Physics

in the

Graduate Division

of the

University of California, Berkeley

Committee in charge:

Professor Holger Muller, ChairProfessor Dan Stamper-Kurn

Professor Jeffrey Bokor

Spring 2018


Copyright 2018by

Kayleigh Cassella

1

Abstract


by

Kayleigh Cassella

Doctor of Philosophy in Physics

University of California, Berkeley

Professor Holger Muller, Chair

Ushered forth by advances in time and frequency metrology, atom interferometry remainsan indispensable measurement tool in atomic physics due to its precision and versatility. Asequence of four π/2 beam splitter pulses can create either an interferometer sensitive tothe atom’s recoil frequency when the momentum imparted by the light reverses directionbetween pulse pairs or, when constructed from pulses without such reversal, sensitive to theperturbing potential from an external optical field. Here, we demonstrate the first atominterferometer with laser-cooled lithium, advantageous for its low mass and simple atomicstructure. We study both a recoil-sensitive Ramsey-Borde interferometer and interferometrysensitive to the dynamic polarizability of the ground state of lithium.

Recoil-sensitive Ramsey-Borde interferometry benefits from lithium’s high recoil fre-quency, a consequence of its low mass. At an interrogation time of 10 ms, a Ramsey-Bordelithium interferometer could achieve sensitivities comparable to those realized at much longertimes with heavier alkali atoms. However, in contrast with other atoms that are used foratom interferometry, lithium’s unresolved excited-state hyperfine structure precludes the thecycling transition necessary for efficient cooling. Without sub-Doppler cooling techniques.As as result, a lithium atomic gas is typically laser cooled to temperatures around 300 µK,above the Doppler limit, and well above the recoil temperature of 6 µK. This higher tem-perature gas expands rapidly during the operation of an atom interferometer, limiting theexperimental interrogation time and preventing spatially resolved detection.

In this work, a light-pulse lithium matter-wave interferometer is demonstrated in spiteof these limitation. Two-photon Raman interferometer pulses coherently couple the atom’sspin and momentum and are thus able to spectrally resolve the outputs. These fast pulsesdrive conjugate interferometers simultaneously which beat with a fast frequency componentproportional to the atomic recoil frequency and an envelope modulated by the two-photondetuning of the Raman transition. We detect the summed signal at short experimentaltimes, preventing perturbation of the signal from vibration noise. This demonstration of asub-recoil measurement with a super-recoil sample opens the door to similar scheme withother particles that are difficult to trap and cool well, like electrons.

2

An interferometer instead composed of π/2-pulses with a single direction of momentumtransfer, can be sensitive to the dynamic polarizability of the atomic ground state. Byscanning the frequency of an external driving field, such a measurement can be used todetermine the atom’s tune-out wavelength. This is the wavelength at which the frequency-dependent polarizability vanishes due to compensating ac-Stark shifts from other atomicstates. Lithium’s simple atomic structure allows for a precise computation of properties withonly ab initio wave functions and spectroscopic data. A direct interferometric measurementof lithium’s red tune-out wavelength at 670.971626(1) nm, is a precise comparison to existing‘all-order’ atomic theory computations. It also provides another way to experimentallydetermine the S− to P− transitions matrix elements, for which large correlations and smallvalues complicate computations. Finally, a future measurement of lithium’s ultraviolet tune-out wavelength of at 324.192(2) nm would be sensitive to relativistic approximations in theatomic structure description.

Atom interferometry simultaneously verifies existing atomic theory with measurements ofatomic properties and searches for exotic physics lurking in plain sight. The techniques devel-oped here broaden the applicability of interferometry and increase measurement sensitivityby simplifying cooling, increasing atom number and reducing the cycle time. Overcom-ing the current experimental limitations on interrogation time would allow for ultra-precisemeasurements of both the tune-out wavelength and the fine structure constant.

i

To my mom and step-dad, who filled me with enough resolve to do hard things.To my sisters, who grew with me and tethered me to real things.

To my husband, who unfolded all the crumpled parts of me, again and again.To my children, my greatest teachers, who sprinkled light in all the dark places.

I dedicate this work to you.

ii

Contents

Contents ii

List of Figures vi

List of Tables viii

1 Outward bound 11.1 Corpuscular and undulatory . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 Waves of matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 α, the fine structure constant . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.1 hM

measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3 α, the polarizability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3.1 Dynamic polarizability . . . . . . . . . . . . . . . . . . . . . . . . . . 101.4 Previous measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.4.1 λto measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.5 Overview of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2 Atom interferometry 162.1 Light off . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.1.1 The free evolution phase . . . . . . . . . . . . . . . . . . . . . . . . . 222.2 Light on . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2.1 Raman scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.2.1.1 Dressed states . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.2.2 The interaction phase . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.2.3 The separation phase . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.3 The total phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.4 Conjugate interferometers with the π

2-π

2-π

2-π

2. . . . . . . . . . . . . . . . . . 33

2.5 The Ramsey-Borde interferometer . . . . . . . . . . . . . . . . . . . . . . . . 332.5.1 cRBI phase computation . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.6 The copropagating interferometer . . . . . . . . . . . . . . . . . . . . . . . . 382.6.1 cCPI phase computation . . . . . . . . . . . . . . . . . . . . . . . . . 41

iii

3 Lithium, the smallest alkali 423.1 Lithium, the lightest alkali . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.2 Lithium, the simplest alkali . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.2.1 The Hylleraas basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.3 Dynamic polarizability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.4 Lukewarm Lithium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.4.1 Lithium atom interferometry in the space domain . . . . . . . . . . . 533.5 Advantages of light-pulsed interferometry with lithium . . . . . . . . . . . . 54

4 Experimental Methods 564.1 Lithium Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.1.1 Modulation Transfer Spectroscopy . . . . . . . . . . . . . . . . . . . . 594.1.2 The cascade of frequency generation . . . . . . . . . . . . . . . . . . 62

4.1.2.1 Tapered amplifiers . . . . . . . . . . . . . . . . . . . . . . . 634.2 Cooling and trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.2.1 2D MOT frequency generation . . . . . . . . . . . . . . . . . . . . . . 674.2.2 3D MOT frequency generation . . . . . . . . . . . . . . . . . . . . . . 674.2.3 Vacuum system and optics . . . . . . . . . . . . . . . . . . . . . . . . 68

4.2.3.1 2D MOT chamber . . . . . . . . . . . . . . . . . . . . . . . 704.2.3.2 3D MOT chamber . . . . . . . . . . . . . . . . . . . . . . . 72

4.2.4 Experimental sequence . . . . . . . . . . . . . . . . . . . . . . . . . . 734.3 State preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.3.1 Frequency generation for optical pumping light . . . . . . . . . . . . 764.3.2 Optical pumping optics . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.3.2.1 Quantization axis . . . . . . . . . . . . . . . . . . . . . . . . 784.4 Interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.4.1 Frequency generation for Raman beams . . . . . . . . . . . . . . . . . 784.4.2 Raman optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.5 Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.5.1 Absorption imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.5.2 Wollaston prism technique . . . . . . . . . . . . . . . . . . . . . . . . 844.5.3 Time-of-flight imaging . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5 Hot Beats 875.1 Super-recoil lithium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.1.1 Large bandwidth pulses . . . . . . . . . . . . . . . . . . . . . . . . . 885.1.2 k-reversal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.2 Simultaneous and conjugate . . . . . . . . . . . . . . . . . . . . . . . . . . . 895.3 Overlapped, simultaneous and conjugate . . . . . . . . . . . . . . . . . . . . 91

5.3.1 Hot beats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 925.3.2 Time-domain fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . 925.3.3 Frequency-domain fitting . . . . . . . . . . . . . . . . . . . . . . . . . 94

iv

5.4 Phase noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955.5 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.5.1 Vibration immunity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

6 Tune-outs 1006.1 Previous polarizability measurements . . . . . . . . . . . . . . . . . . . . . . 101

6.1.1 The differential Stark shift . . . . . . . . . . . . . . . . . . . . . . . . 1016.1.2 Space-domain atom interferometry . . . . . . . . . . . . . . . . . . . 102

6.2 Light-pulsed interferometric lithium tune outs . . . . . . . . . . . . . . . . . 1036.2.1 φto, the tune-out phase . . . . . . . . . . . . . . . . . . . . . . . . . . 1036.2.2 The tune-out beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1066.2.3 Experimental Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . 1066.2.4 Detection & Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

6.2.4.1 Principal component analysis . . . . . . . . . . . . . . . . . 1106.3 Towards tune-out . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

6.3.1 Precision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1116.3.1.1 Single-photon scattering . . . . . . . . . . . . . . . . . . . . 1126.3.1.2 Beam shaping . . . . . . . . . . . . . . . . . . . . . . . . . . 113

6.3.2 Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1136.4 Hyperfine dynamic polarizabilities . . . . . . . . . . . . . . . . . . . . . . . . 113

7 Conclusion 1177.1 Outlook for recoil-sensitive interferometry with super-recoil samples . . . . . 117

7.1.1 h/me measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1177.2 Outlook for tune-out interferometric measurements in lithium . . . . . . . . 119

7.2.1 Beyond the red . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1197.2.2 Investigation of nuclear structure between isotopes . . . . . . . . . . 121

7.3 Atom interferometry with lukewarm lithium . . . . . . . . . . . . . . . . . . 1227.3.1 Sisyphus cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1237.3.2 Gray molasses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

7.4 Onward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

A Properties of lithium 125A.1 The level spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129A.2 Interaction with static fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 129A.3 Interaction with dynamic fields . . . . . . . . . . . . . . . . . . . . . . . . . 131

A.3.1 Reduced Matrix Elements in Atomic Transitions . . . . . . . . . . . . 131A.4 Clebsch-Gordan coefficients for D–line transitions . . . . . . . . . . . . . . . 132

B Two-Level System 136B.1 Flip-flop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

B.1.1 On resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

v

B.1.2 Almost on resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

C Bloch sphere 141C.1 Simulations of interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . 144

C.1.1 Mathematica code . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

D Magneto-optical traps 147D.0.1 Optical molasses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147D.0.2 Magnetic trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

E α0, the static polarizability 151E.1 Nonrelativistic α(0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

F Hyperpolarizability 154F.1 Positive and negative frequency components . . . . . . . . . . . . . . . . . . 156

G Matlab simulation of thermal cloud 158G.0.1 Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158G.0.2 Matlab functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

G.0.2.1 Preprecompute.m . . . . . . . . . . . . . . . . . . . . . . . . 161G.0.2.2 simulatef0is1.m . . . . . . . . . . . . . . . . . . . . . . . . . 162

Bibliography 166

vi

List of Figures

1.1 Optical Michelson-Morley and Mach-Zehnder configurations . . . . . . . . . . . 31.2 Plot of dynamic polarizability for 7Li’s 2S2 level. . . . . . . . . . . . . . . . . . . 14

2.1 Recombination of the superposition at the last π/2-pulse results in interference . 182.2 Interferometers in the π/2-π/2-π/2-π/2 geometry . . . . . . . . . . . . . . . . . 192.3 Trajectories of atom in configuration space . . . . . . . . . . . . . . . . . . . . . 222.4 Effective wave vector and momentum coupling . . . . . . . . . . . . . . . . . . . 242.5 Three-level system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.6 Conjugate interferometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.7 Interaction geometry for the lower Ramsey-Borde interferometer . . . . . . . . . 342.8 Interaction geometry for the upper Ramsey-Borde interferometer . . . . . . . . 362.9 Interaction geometry for the lower copropagating interferometer . . . . . . . . . 392.10 Interaction geometry for the upper copropagating interferometer . . . . . . . . . 40

3.1 Comparison of RBI for 7Li and 133Cs . . . . . . . . . . . . . . . . . . . . . . . . 433.2 The computed dynamic polarizability for the lithium ground state . . . . . . . . 513.3 The Maxwell-Boltzmann distributions for atoms at the recoil temperature (blue)

and at 300 µK (red). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.1 Experimental sequence and settings . . . . . . . . . . . . . . . . . . . . . . . . . 574.2 Lithium spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.3 Circuit schematic of master ECDL frequency lockbox . . . . . . . . . . . . . . . 624.4 Experimental frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.5 Frequency generation for the 2D MOT. . . . . . . . . . . . . . . . . . . . . . . . 684.6 Frequency generation for the 3D MOT and pusher beam . . . . . . . . . . . . . 694.7 2D MOT chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714.8 2D MOT optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.9 3D MOT chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.10 Microwave spectrum of the |F = 2,mF 〉 ground state. . . . . . . . . . . . . . . . 754.11 Magnetic field gradient decay. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764.12 Energy level diagram showing the frequencies for optical pumping. . . . . . . . . 774.13 Schematic of optical pumping frequency offset lock electronics . . . . . . . . . . 77

vii

4.14 Optical pumping optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.15 Two-photon Raman transition level diagram . . . . . . . . . . . . . . . . . . . . 804.16 Optical set-up for generating the Raman frequencies. . . . . . . . . . . . . . . . 814.17 Optics set-up for Raman beams . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.18 The beam path for the imaging light as it transverse the vacuum apparatus. . . 85

5.1 Comparison of pulse bandwidth to temperature of atom cloud . . . . . . . . . . 895.2 Vacuum tube switch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 905.3 Spatially overlapped interferometer outputs . . . . . . . . . . . . . . . . . . . . 915.4 Data and fits for a range of two-photon detunings δ . . . . . . . . . . . . . . . . 935.5 Fit data showing both the amplitude modulation as well as the fast frequency

component, the recoil frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . 945.6 Fast fourier transform of beating interferometers . . . . . . . . . . . . . . . . . . 955.7 Fourier transformed data for various δ’s. . . . . . . . . . . . . . . . . . . . . . . 965.8 A plot of the standard deviation resulting from fits of the Fourier-transformed

data in the time- and frequency-domain at different two-photon detunings. . . . 98

6.1 Scheme for tune-out measurement in thermal atom interferometer . . . . . . . . 1026.2 Orientation of the tune-out beam with respect to the atom cloud . . . . . . . . 1046.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1056.4 A gaussian beam’s spatial dependence . . . . . . . . . . . . . . . . . . . . . . . 1056.5 Tune-out frequency generation and optics . . . . . . . . . . . . . . . . . . . . . 1076.6 Tune-out measurement sequence and settings . . . . . . . . . . . . . . . . . . . 1086.7 A comparison between the analysis performed without (top) and with (bottom)

the tune-out pulse. The lower principal component analysis only contains thatdependent upon the extra light, switched on during the T ′ time step in the inter-ferometer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

6.8 Premliminary tune-out plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1126.9 Anamorphic prism pair transforms beam shapes between circular and elliptical. 1136.10 PC breakdown with elliptical beam. . . . . . . . . . . . . . . . . . . . . . . . . . 1146.11 Scalar and tensor dynamic polarizabilities for hyperfine ground state levels in 7Li. 1156.12 Comparison of dynamic polarizability between lithium’s hyperfine ground states 116

7.1 Ramsey-Borde interferometer for electrons . . . . . . . . . . . . . . . . . . . . . 1187.2 Dynamic polarizability of lithium’s ground level . . . . . . . . . . . . . . . . . . 1207.3 Lithium’s UV tune-out . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1207.4 Sisyphus cooling frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

A.1 Vapor pressure of lithium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

C.1 Model of trajectory of state on the Bloch sphere. . . . . . . . . . . . . . . . . . 146

D.1 Magneto-optical trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

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List of Tables

1.1 Tune-out measurements to-date, method of measurement and reference. . . . . . 12

2.1 Trajectories for lower Ramsey-Borde interferometer . . . . . . . . . . . . . . . . 352.2 Trajectories for upper Ramsey-Borde interferometer . . . . . . . . . . . . . . . . 372.3 Trajectories for lower copropagating interferometer . . . . . . . . . . . . . . . . 392.4 Trajectories for upper copropagating interferometer . . . . . . . . . . . . . . . . 40

3.1 Comparison of mass and single photon recoil velocity (frequency) for lithium,rubidium and cesium’s D2-lines . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.2 Angular momentum configurations for the S, P , D states of lithium. . . . . . . 483.3 Scalar polarizabilities and Stark shift values for 7Li . . . . . . . . . . . . . . . . 51

4.1 Experimental detunings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.1 Fitting parameters for Fig. 5.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

A.1 Physical properties of lithium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125A.2 Physical properties of lithium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127A.3 7Li D2 (2S1/2 → 2P3/2) Transition Properties . . . . . . . . . . . . . . . . . . . . 128A.4 7Li D1 (2S1/2 → 2P1/2) Transition Properties . . . . . . . . . . . . . . . . . . . . 128A.5 Clebsch-Gordan coefficients for the D2-line transition with σ+-polarized light such

that m′F = mF + 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133A.6 Clebsch-Gordan coefficients for the D2-line transition with σ−-polarized light such

that m′F = mF − 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133A.7 Clebsch-Gordan coefficients for the D2-line transition with π-polarized light such

that m′F = mF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134A.8 Clebsch-Gordan coefficients for the D1-line transition with σ+-polarized light such

that m′F = mF + 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134A.9 Clebsch-Gordan coefficients for the D1-line transition with σ−-polarized light such

that m′F = mF − 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135A.10 Clebsch-Gordan coefficients for the D1-line transition with π-polarized light such

that m′F = mF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

ix

E.1 Scalar polarizability differences α0(nPJ)− α0(nS) in a.u. for 7Li. . . . . . . . . 151

x

Acknowledgments

The work presented here on atom interferometry with laser-cooled lithium is the culmina-tion of the efforts of many. I want to acknowledge those who fought or continue to fight ‘thered devil’: Eric Copenhaver, Robert Berghaus, Geena Kim, Paul Hamilton, Chen Lai, Pro-fessor Yanying Feng, Quinn Simmons, Simon Budker, Hunter Akins, Biswaroop Mukherjee,Dennis Schlippert, Daniel Tiarks and Trinity Joshi. I am grateful to have had the oppor-tunity to learn from and work alongside Eric Copenhaver most recently, during which theprojects presented in Chapters 5 and 6 were born.

The constant amongst these generations of the lithium project is my advisor HolgerMuller. Holger continues to fearlessly lead all of us through extraordinary part per billionexperimental endeavors - whether by aligning the tricky double-pass AOM, using aluminumfoil to successfully impedance match everything or helping you navigate the subtleties ofnoise, sensitivity and precision - Holger has been instrumental on many occasions in helpingme find that epsilon. I am deeply grateful to Holger and an OK experimentalist because ofhis guidance, support and patience.

I would have been lost without the group members of the early days including: JustinBrown, Brian Estey, Paul Hamilton, and Geena Kim. I am grateful to have worked withsome of best interferometeers around: Phillip Haslinger, Chenghui Yu, Weicheng Zhong,Richard Parker, Xuejian Wu, Osip Schwartz, Jeremy Axelrod, Matt Jaffe, Victoria Xu, andJordan Dudley. To the next generation going forth in Physics - Zachary Pagel, Joyce Kwan,Robert Berghaus, Randy Putnam, Ryan Bilotta, Dalila Robledo, and Bola Malek - mayentropy be on your side. I look forward to learning of all that you discover.

I would like to thank my committee members, Jeffrey Bokor and Dan Stamper-Kurn,who patiently nudged me towards scientific maturity but all the while requiring I stand onmy own.

I am grateful administrators in the department who have helped me navigate the fineprint, find my lost child on Cal-Day, advocate for my needs: Ann Takizawa, Claudia Tru-jillo, Donna Sakima, Joelle Miles, Amanda Dillon, Eleanor Crump, Anthony Vitan, CarlosBustamante, Amin Jazaeri, and Rachel Winheld.

The faculty in the Biology, Chemistry and Physics departments at my undergraduatecollege, Indiana University South Bend were the first faces of academia I saw many yearsago and sent me off to Berkeley with enough momentum to get me through. Thank you toAnn Grens, Monika Lynker, Bill Feighery, Doug McMillen, Matthew Marmorino, Jerry Hin-nefeld, Ilan Levine, and Henry Scott, who taught and inspired me. In building a communityaround the undergraduates at IUSB, they created a space that protected and cultivated mycuriosity. I have only the utmost gratitude for Rolf Schimmrigk, who not only challengedme to understand my own questions, but to reach. He also instilled in me a love for good,consistent notation which I have tried to implement here.

My journey to this point has been because of invaluable friends supporting me throughoutand often telling me exactly what I needed to hear, even though I often could not listen. Tomy friends and in particular: Austin Hedeman, Hilary Jacks, Trinity Joshi, Kate Kamdin,

xi

Ming Yi, Shun Wu for being the quantization axis when things got muddled and pointingme forward. My deepest gratitude to Emily Grace whose advice and support has helpedme (and continues ) to navigate the most difficult days. Thanks to our family’s UC Villagecomrades and close friends for the being a constant source of support, friendship, and drinkingsolidarity.

My family’s love and support enabled me to strive towards what I needed; to my par-ents, Debbie, Don, Kirsten, Bruce and Frank, my sisters, Taryn, Jordan and Isla, and mynephew Lucian, thank you. I am deeply grateful to our co-parents, Jenni and Frank Almeidaand Meryl and Rob McCarthy, and to their parents, who are the village for my children.Thank you for your understanding, encouragement, friendship and for supporting me in thisendeavor.

None of this work would have been possible without Arran, who has kept me wholethroughout graduate school, and the children I partly call mine, Finn, Emily, Maggie, Alex,to whom I have dedicated this work. Thank you for your patience and love.

1

Chapter 1

Outward bound

Today, Physics is faced with many known unknowns. The Standard Model of particle physicsprovides a theoretical framework for the electromagnetic, weak and strong forces. It suc-cessfully incorporates experimental data for the fundamental particles composing the knownmatter in our universe [1] and is even able to predict the electron magnetic moment to apart per trillion [2]. However, its shortcomings are glaring. It neglects the major knownconstituents of the universe - dark matter [3] and dark energy [4, 5] - and fails to providean explanation for the observed baryon asymmetry [6] or the fine-tuning of θ in quantumchromodynamics [7] and completely omits the last fundamental interaction, gravity.

While the Standard Model (SM) is too incomplete to be the final unifying theory ofphysics, a consensus as to what higher energy theory must underly it has yet to be reached [8–12]. This realization motivates some to look outward, away from the high energy, subatomicregime of particle physics, and instead to lower energy atomic systems. In these systems, anunprecedented level of experimental control [13–15] and sensitivity [16, 17] has been realizedonly over the last few decades.

Massive particle colliders, such as the Large Hadron Collider (LHC) at CERN, maybe able to search for yet undiscovered particles but such particles often are anticipated toproduce observable effects in low-energy precision measurements [7]. For example, time-reversal symmetry violation could explain the origin of baryon asymmetry [6] and wouldmanifest as a measured electric dipole moment (EDM). Extensions made to the SM for thepurposes of incorporating gravity into its theoretical framework often result in violationsto Lorentz symmetry and CPT invariance, both which could be observed in a low-energysystem [18–20].

Such effects arising from whatever exotic physics lay lurking beyond the SM needs to bemeasurable in the atomic systems we are championing, comparable to theoretical calculationssuch that a ‘sufficiently significant and robust discrepancy’ [7] can be demonstrated. Ameasurement needs a ruler, a reference with the same dimensions allowing for a comparisonbetween two systems. Fundamental constants become a focus of many because their role iscentral in physical theories. Physical constants have an intrinsic theory-dependent existenceand determine the magnitude of physical processes but simultaneously have values that

CHAPTER 1. OUTWARD BOUND 2

cannot be predicted theoretically.

(They are) constants whose value we cannot calculate with precision in termsof more fundamental constants, not just because the calculation is too compli-cated but because we do not know of anything more fundamental.

Steven Weinberg [21]

Another focus of metrology is the determination of atomic properties. A comparison ofexperimentally determined atomic properties with those that have been theoretically com-puted can inform modern computational methods. Demanding atomic structure calculationscan be made more tolerable with experimentally measured values, such as energies or tran-sition matrix elements, as input.

1.1 Corpuscular and undulatory

When two propagating waves originating from the same source are incident at the same pointin space, the resultant amplitude is the sum of the individual waves’ amplitudes. Each wave,characterized by crests and troughs or portions of positive and negative amplitude, will have afinal amplitude ranging in magnitude from the sum to difference of the individual amplitudes.This superposition of waves which can be either constructive (sum) or destructive (difference)is known as the wave-like phenomenon of interference.

In the early 19th century, interference fringes of light were demonstrated Thomas Young’sdouble slit interferometer in which a sunlit small hole illuminated two subsequent smallholes. From the spatial separation of the observed fringes, Young was able to estimate thewavelength of different colors in the spectrum as mentioned below in a quotation from thechapter Experiments and Calculations Relative to Physics Optics of Ref. [22].

In making some experiments on the fringes of colors accompanying shadows,I have found so simple and so demonstrative a proof of the general law of theinterference of two portions of light, which I have already endeavored to establish,that I think it right to lay before the Royal Society a short statement of the factswhich appear to me so decisive.

The proposition on which I mean to insist at present is simply this - thatfringes of colors are produced by the interference of two portions of light; and Ithink it will not be denied by the most prejudiced that the assertion is provedby the experiments I am about to relate, which may be repeated with great easewhenever the sun shines, and without any other apparatus than is at hand toeveryone.

Thomas Young, 1804


beam splitter

mirror

mirror

source

detector

xIn

ten

sity

beam splitter

d

beam splitter

mirror

mirror

source

detector

xIn

ten

sity

Mach-Zehnder

Michelson-Moreley

Figure 1.1: (Left) An optical interferometer in the Michelson-Morley geometry, in whichthe distance d transversed by some movable mirror will be detectable as a phase shift in theoverall signal detected at the output. (Right) An optical interferometer in the Mach-Zehndergeometry, in which the differential phase shift between two arms φ is made to vary.

Young’s experiment played a major role in the acceptance for the wave-like nature oflight, which at the time was contrary to Newton’s corpuscular theory. Following Young’sdouble slit interferometer, optical interferometry became and still remains an indispensabletool. Such optical interferometers, like the Michelson and Mach-Zehnder interferometersdepicted in Fig. 1.1, utilize a ‘beam splitter’ or half-silvered mirror to split the initial waveinto two beams

E1,2 = E1,2 cos(φ1,2 − ωt

). (1.1)

These beams propagate along different paths and are eventually directed back to one an-other with mirrors and recombined with the same (Michelson) or additional (Mach-Zehnder)beam splitter. The detected light has an intensity given by

I ∝ E21 + E2

2 + 2E1E2 cos(φ2 − φ1). (1.2)

A difference in path length between the wave functions produces a difference in phase inthe interference fringes at the output. In the Michelson interferometer depicted in Fig. 1.1,


the change in phase difference is

δ(φ2 − φ1) =2π

λ2d, (1.3)

for light of wavelength λ. This relation reduces to an equation for measurement displacementd

d = Nλ

2(1.4)

where N is the number of interference fringes.Being able to discern minute changes, on the order of a fraction of a wavelength, over

the much larger distance of the beams path translates into extraordinary measurement sen-sitivity and precision. This demonstrates the power of interferometry as a tool for precisionmeasurement; such qualities poise interferometry as an indispensable tool for metrology.

1.1.1 Waves of matter

The resolution of the “ dispute over two viewpoints on the nature of light: corpuscular andundulatory” [23] with the development of quantum theory had implications that extendedbeyond ‘light atoms’ and into matter. In the early 20th century, quantum mechanics cat-alyzed experimental explorations into the wave-like nature of matter and an acceptance forthe wave-particle duality. In the non-relativistic limit, the dynamics of matter waves aredescribed by the time-dependent Schrodinger equation(

− ~2

2M∇2 + V (r, t)

)ψ(r, t) = i~

∂ψ(r, t)

∂t(1.5)

or for the simplified case of a time-independent potential, V (r)(∇2 +

2M

~2(E − V (r))

)ψ(r) = 0. (1.6)

Hence, for a particle with mass M and total energy E, its local wavenumber k (themagnitude of its wave vector) in a potential V (r) is

k(r) =1

~√

2M(E − V (r)). (1.7)

Quantum mechanically, such a particle with velocity v, can be characterized by a wave withmean de Broglie wavelength of λdB = 2π/k(r) = 2π~/Mv, where ~ is the reduced Planckconstant.

Control over the atom’s velocity or momentum translates directly to control of the meande Broglie wavelength. The coherent atom optics designed to prolong the quantum coherenceof atomic beams [24] and isolate the atom source from the environment laid a foundationof techniques instrumental to the realization of interference and eventually interferometers


with atoms. Gratings of light akin to mechanical gratings imparted momentum upon theatomic wave packets [25], putting the atoms into superposition of momentum states. Ex-ploiting light’s complementary role as a refractive, reflective, absorptive structure to matterand the coupling of such interaction to the atom’s internal energy state was crucial to theadvancement of atomic physics; from frequency standards [16], quantum information scienceto atom interferometry [26, 27].

Analogous to optical interferometry, in an atom interferometer atoms are coherentlyexcited into a superposition of quantum states and allowed to propagate along alternativepaths in either the space- or time-domain. The interference that results after recombiningthe wave function reveals a phase shift arising from a difference between paths as experiencedby the wave packet of the evolving superposition. Since this relative phase translates intothe detection probability for the atom in a particular interferometer output port, the phasedifference is evident in a measurement of atom flux at the end of the interferometer.

Despite matter’s tendency to interact strongly with other matter and its short coherencelength, atom interferometry is a crucial tool in searches for beyond the Standard Modelphysics due to its versatility and sensitivity. Compared to light, matter is susceptible to alarger range of phenomena, including gravity, and offer advantages stemming from the wideselection of available atomic properties. Applications of atom interferometers include: ac-celerometry [28–31] , gravity gradiometry [32–35], rotation sensing [36, 37], fifth force searchesconnected to dark energy [38] and dark matter [39], and measurements of fundamental con-stants [17, 40–42] to measurements of atomic properties, like the static and dynamic po-larizability [43, 44]. The continued advances in time and frequency metrology [16] and theconsequently extraordinary accuracy with which laser frequencies can be measured meansthat manipulating atoms with optics built from light offers a route to even higher sensitivityand precision than that obtainable with crystal structures.

Here, we demonstrate the first atom interferometer with laser-cooled lithium. Lithium isadvantageous to metrology because of both its low mass and low electron number. Our atominterferometer utilizes coherent atom optics that entangle the internal and external atomicstates, coupling the atom’s internal energy state to its momentum. We study recoil-sensitiveinterferometry, relevant in determinations of the fine structure constant with a measurementof h/M , and interferometry sensitive to the dynamic polarizability of lithium’s ground state2S1/2.

1.2 α, the fine structure constant

In 1916, Sommerfeld introduced the constant α to quantify the relativistic correction, knownas ‘fine structure’, to the spectrum of the Bohr atom []. Sommerfeld’s original interpretationof this dimensionless number, approximately equal to 1

137, was that it quantified the ratio of

the velocity of the electron in a Bohr atom to the speed of light [45],


α =1

4πε0

q2e

~c(1.8)

where qe is the charge of an electron, ε0 is the permittivity of the vacuum, ~ = h/2π is thereduced Planck constant, and c is the speed of light. While the fine structure constant isprevalent throughout many various subfields of physics [46, 47], from the Josephson-junctionoscillations in condensed matter [48] to the spectrum of muonium [49] to the Lamb shiftin atomic physics [50], it can be fundamentally defined as the coupling constant for theelectromagnetic force or the affinity for which charged particles couple to electromagneticfields at low energies. Additionally, in particle physics the fine structure constant can berelated to the magnetic moment of the electron µ = µS. This value has been measured asµ/µB = -1.001 159 652 180 73 (28) [51]. The numerical value of the electron’s magneticmoment can be calculated from the Standard Model. In this calculation, µ/µB, whereµB = qe~/2m, acquires contributions from several aspects of the Standard Model, expressedas follows:

−µµB

= 1 + aQED + aQCD + aweak. (1.9)

The quantum chromodynamic correction aQCD contributes at two parts per trillion andquantifies the electron’s interaction with hadron-antihadron pairs. The electroweak in-teraction correction aweak is smaller than the measurement precision of 2.8 parts in 1013.Both these contributions are determined from measured values. The largest factor in theabove expansion of the the electron’s anomalous magnetic moment is the QED contributionaQED ≡ ge−2

2, at 0.1%. This term comes from contributions to the moment arising from

loops made of virtual photons and leptons.The QED contribution aQED can be expanded perturbatively in α

aQEDe =

∑n

C2n

(α

π

)n= C2

(απ

)+ C4

(απ

)2+ C6

(απ

)3+ C8

(απ

)4+ C10

(απ

)5+ ...

(1.10)

The renormalizability of QED ensures that the factors C2n in the above expansion convergeto a finite value. Redefining these factors as follows [2]

C2n = A2n1 + A2n

2 (me/mµ) + A2n2 (me/mτ )

+A2n3 (me/mµ,me/mτ ), (1.11)

where me/mµ and me/mτ are the electron-muon and electron-tau mass ratios, allows theexpansion to be re-expressed as a sum of like-contributions for a given order

aQEDe =

∑n≥1

Ak1(απ

)n+∑n≥2

A2n2 (me/m`′)

(απ

)n+∑n≥3

A2n3 (me/m`′ ,me/m`′′)

(απ

)n.


(1.12)

At increasing order, computing these corrections requires increasingly complicated appli-cations of quantum electrodynamics. For example, even at third order in α, O(α3), thereare 72 diagrams to consider. Most recently, the tenth-order QED contribution to anomalousmoment was determined [2] and with the previously mentioned measurement yielded a valueat the 0.25 ppb level for the fine structure constant of

α−1 = 137.035999173(35). (1.13)

Other determinations of α, particularly those independent from the QED and StandardModel framework, are needed to test its theoretical formalism. Because α is defined interms of parameters that cannot yet be independently calculated, a determination of thefine structure constant must be pursued through either a direct measurement or determinedindirectly through the measurement of other quantities.

We can define the fine structure constant not in terms of an expansion in QED butinstead relative to other fundamental constants [52]

α2 =2hR∞mec

, (1.14)

where me is the mass of the electron and R∞ is the Rydberg constant [53, 54]. Historically,the Rydberg constant arises in the context of the Bohr atom and is defined as

1

λ= R∞

(1

n21

− 1

n22

)=

meq4e

8ε20h

3c

(1

n21

− 1

n22

), (1.15)

and has already been determined spectroscopically to great accuracy, with a relative uncer-tainty of 7× 10−12. Therefore, a determination of α is possible via a measurement of h/me.Recognizing that free electrons are difficult to trap and coherently manipulate, we expandh/me again, redefining α now as

α2 =2R∞c

h

me

=2R∞c

u

me

M

u

h

M(1.16)

where u is the atomic mass unit. The relative mass of the electron me/u [55, 56] and therelative mass of an atom M/u [57] can both be determined extraordinarily precisely. Re-ported values are known to better than 2.0 × 10−10. Therefore, for the above definition ofα, what remains left to be determined is the quantity h

M. As discussed in the next section,

atom interferometry is able to determine this ratio independently of the SM, from a measure-ment of the atom’s recoil splitting and knowledge of the photon frequency [17, 40, 41, 58, 59].Comparing the results obtained for α with atom interferometry to that determined throughQED computations, in conjunction with the measurement of the electron’s anomalous mag-netic moment, is a consistency check of the Standard Model in particle physics but from ameasurement made at the resolution of an atom!


1.2.1 hM measurement

The ratio of Planck’s constant to the mass of an atom, hM

, is intimately connected to thekinetic energy imparted to an atom by a photon. For an atom with initial ground internalenergy state |a〉 and velocity v0 in the presence of an external resonant field with frequency,ωL = ωe − ωa where ωe − ωa is the energy splitting the ground and excited state |e〉, theinitial total energy and momentum for the system is

E0 =Mv2

0

2+ ~ωa + ~ωL

p0 = Mv0 + ~k. (1.17)

Absorbing a photon from the field (λ) consequently imparts momentum pγ = h/λ to theatom, in the direction of the photon, and will recoil with a velocity of ∆v = pγ/M = h/λM =~k/M . The atomic resonances will also be Doppler-shifted as ∆ω/2π = ∆v/λ = 1/λ2(h/m).

The final energy and momentum of this atom-light system are

Ef =M

2|v0 + δv|2 + ~ωe

pf = M(v0 + δv). (1.18)

The resonant laser frequency for the field is imposed via energy conservation as

ωL = v0 · k +~k2

2M+ (ωe − ωa) (1.19)

which depends upon the energy level difference between the atomic states, the first-orderDoppler shift, v0 ·k, and a similar-in-spirit recoil shift arising from the change in momentumaccompanying photon absorption.

An accurate knowledge of the laser wavelength in combination with a measurement ofthe frequency shift leads to a measurement of h/M and therefore a determination of α. Aswill be discussed in Chapter 2 the sensitivity of an interferometer depends upon severalthings but a particular interferometer geometry that is sensitive to the recoil frequency ofthe atoms in the Ramsey-Borde interferometer which consists of four beam splitter withthe second pair having a reverse effective wave-vector compared to the first two pulses.An atom interferometer becomes sensitive to the recoil frequency of the atom when duringinterferometry, a difference in energy exists for the coherent superposition resulting fromthe kinetic energy obtained after absorption (and emission) of a photon. The coherentsuperposition spends an unequal amount of time recoiling or with a kinetic energy impartedto it by the absorption and emission of a photon. This difference in energy will show up inthe quantum phase difference read out at the end of interferometry.

1.3 α, the polarizability

The field of cold-atomic physics exists due to the ability to manipulate atoms (trapping,laser cooling) with electromagnetic fields. The first order response of an atom to an applied


electric field is its polarizability. Classically, when an external electric field is applied tomatter, the charged particles in the object are rearranged. The polarizability characterizesthe response of the atomic charge cloud to this perturbing field [60].

For a perfectly conducting sphere of radius r0 in a uniform electric field E , the resultantelectric field at a position in space given by r > r0 is

E −∇(E · rr30/r

3). (1.20)

This expression is equivalent to replacing the perfectly conducting sphere by an electricdipole with a dipole moment given by

d = αE (1.21)

where α = r30 is the polarizability of the sphere.

Quantum mechanically, a system of particles with positions ri and electric charges qiexposed to an applied, uniform electric field (E = E ε) can be described by following Hamil-tonian

H = H0 +H ′ = H0 − E ε · d (1.22)

where the electric dipole d is a sum over the individual particle dipoles

P =∑i

qiri. (1.23)

For an atom, this summation is over all charged particles of the atom, including the nucleus.Treating the field strength E as a perturbation parameter and expanding the energy and

wave function leads to the following:

E = E0 + EE1 + E2E2 + ... (1.24)

|Ψ〉 = |Ψ0〉+ E|Ψ1〉+ E2|Ψ2〉+ ... (1.25)

such thatH0Ψ0 = E0Ψ0.

The atomic polarizability α is identified from the energy-level shifts for the state |Ψ〉, whichup to the second order energy correction via perturbation theory is

δE = 〈Ψ|E · d|Ψ〉+∑k

|〈Ψ|E · d|Ψk〉|2

E0 − Ek= −1

2E2α (1.26)

where |Ψk〉 label all other atomic states. Assuming |Ψ0〉 is an eigenfunction of parity, thenthe first-order shift vanishes.


1.3.1 Dynamic polarizability

The frequency-dependent or dynamic polarizability quantifies the response of an atom tothe presence of an external off-resonant optical field. As derived previously, the atom-fieldinteraction energy or termed ac Stark shift, is given to first order as

U = −1

2α〈E2〉 (1.27)

with electric field intensity, |E(ω)|2 and the kets indicate a time average.This energy shift of an atomic energy level can be written explicitly in terms of the

electric dipole transition matrix elements with initial state |i〉 and excited states |k〉 in thepresence of a monochromatic field E as [61]:

δEi = −∑k

2ωik|〈i|ε · d|k〉|2|E|2

~(ω2ik − ω2)

(1.28)

where ωik := (Ek − Ei)/~.The dynamic polarizability of the atom is defined as follows

α(ω) =∑k

2ωik|〈i|ε · d|k〉|2

~(ω2ik − ω2)

(1.29)

and can be broken down into contributions from core electrons, αc, a modification resultingfrom core-valence interactions, αvc, and a contribution from the valence electron, αv

α(ω) = αv + αc + αvc. (1.30)

The contribution from the valence electron dominates the above sum, especially in thecase presented here: a measurement of the tune-out wavelength between the D1 and D2 linesin lithium. The expression can be reduced into a sum only over the excited electronic states|k〉 coupled to the initial (ground) state of the atom |i〉 by the off-resonant external opticalfield.

The reduced dipole matrix elements in the definition of the polarizability can be obtainedfrom oscillator strengths fgk, transition probability coefficients Akg and line strengths Sgk

which lead to the following alternative definitions for the polarizability.

α(ω) =q2e

M

∑k 6=g

fgk

ω2gk − ω2

(1.31)

α(ω) = 2πε0c3∑k 6=g

Akgω−2gk

ω2gk − ω2

× gkgg

(1.32)

α(ω) =1

3~∑k 6=g

Sgkωgk

ω2gk − ω2

(1.33)


Considering an initial state given in terms of the hyperfine-basis, |i〉 = |nLFmF 〉, thesum then is over final states |k〉 = |nL′F ′m′F 〉 and the polarizability is defined as

αµν(ω) =∑F ′m′F

2ωF ′F 〈FmF |dν |F ′m′F 〉〈F ′m′F |dν |FmF 〉~(ω2

F ′F − ω2). (1.34)

Decomposing the polarizability into irreducible tensors leads to the following expressionfor the ac Stark shift

δE(F,mF ;ω) = −1

4|E|2α(0) − iα(1) (ε∗ × ε) · F

2F

+α(2) 3[(ε∗ · F)(ε · F) + (ε · F)(ε∗ · F)

]− 2F2

2F (2F − 1)

(1.35)

in terms of the scalar, vector and tensor polarizabilities defined as

α(0)(F ;ω) =2

3

∑F ′

ωF ′F |〈F‖d‖F ′〉|2

~(ωF ′F − ω2)

α(1)(n, J, F ) =∑F ′

(−1)F+F ′+1

√6F (2F + 1)

F + 1

1 1 1F F F ′

ωF ′F |〈F‖d‖F ′〉|2

~(ωF ′F − ω2)

α(2)(n, J, F ) =∑F ′

(−1)F+F ′

√40F (2F + 1)(2F − 1)

3(F + 1)(2F + 3)

1 1 2F F F ′

ωF ′F |〈F‖d‖F ′〉|2

~(ωF ′F − ω2).

(1.36)

Precise calculations of atomic polarizabilities have implications in many areas of physics,from fundamental searches to quantum information processes [62, 63], and also in opticalcooling and trapping [64]. For instance, parity nonconservation experiments (PNC) in heavyatoms search for new physics beyond the electroweak sector of the standard model throughthe precise evaluation of the weak charge or parity violation in the nucleus with nuclearanapole moment evaluations [65, 66]. These experiments require detailed studies of parity-conserving quantities, like atomic polarizabilities, to accurately determine the uncertaintyin the theoretical value [67, 68].

As experimental capabilities continue to grow, the requirements for greater precision andaccuracy have necessitated a greater understanding of the corrections for the effects of theelectromagnetic fields used to manipulate the atoms. This is evident in the ‘next-generation’of atomic clocks which have recently renewed an interest in polarizability [69]. These stan-dards are significantly impacted by a displacement of the atom’s energy levels resulting fromblackbody radiation shifts (BBR) [70–72]. BBR shifts are the universal ambient thermalfluctuations of the electromagnetic field, given by

∆E = − 2

15(απ)3α0(0)T 4(1 + η) (1.37)


λto [n]m Method ReferenceK 768.9712(15) AIFM [43]Rb 789.85(1) BEC diffraction [76]Rb 790.032388(32) BEC AIFM [77]Rb 790.01858(23) BEC scattering in OL [78]Rb 423.018(7), 421.075(2) BEC diffraction [79]He∗ 413.0938(9)(20) Trapping potential [80]

Table 1.1: Tune-out measurements to-date, method of measurement and reference.

where α is the fine structure constant, α0(0) is the static scalar polarizability, T is the temper-ature, and η is a correction factor containing the frequency dependence of the polarizability[73]

η ≈ − 40π2T 2

21αd(0)S(−4) (1.38)

with the following sum rule:

S(−4) =∑n

f(1)gk

(∆Egk)4. (1.39)

The differential Stark shift caused by external electromagnetic fields leads to a temperature-dependent shift in the transition frequency of the two states involved in the clock transition.

1.4 Previous measurements

The tune-out wavelengths for several of the alkali-atoms have been measured, see Fig. 1.1,but measurements of polarizabilities are less abundant than theoretical determinations. Forlithium, there are currently only indirect Stark shift measurements between the groundand excited states [74, 75] and a static polarizability measurement made with thermal atominterferometry [44], as discussed in Chapter 3. A direct measurement of lithium’s tune-out wavelength between the 2P1/2 and 2P3/2 excited levels with atom interferometry is thepursuit of ongoing work here. The status and project outlook is the focus of Chapter 6.

Presently, many of the best estimates of atomic polarizabilities are derived from a compos-ite analysis which blends first principle calculations of atomic properties with experimentalmeasurements. This sum-over states method is widely applied to systems with one or twovalence electrons and can be combined with oscillator strengths or matrix elements derivedfrom experimentally measured values [81]. While ideally the total uncertainty of the theoret-ical value should provide an estimate as to how far away an observed value is from the actualexact result, without knowledge of the exact value the evaluation of the complete theoreticaluncertainty is non-trivial. It ultimately requires the knowledge of a quantity that is notknown beforehand nor can be determined by the adopted methodology. The most common


numerical uncertainties are associated with the choice of basis sets, configuration space, ra-dial grid or termination of iterative procedures after achieving some convergence tolerance.For example, in a Hylleraas calculation, expectation values are expected to converge as 1/Ωp

whereΩ = j1 + j2 + j3 + j12 + j13 + j23 (1.40)

is the summed polynomial power for the correlated wave function. Varying appropriateparameters and tabulating the results may allow for an estimate on uncertainties in vari-ous atomic properties [82, 83] but numerical constraints resulting from measured values canensure that these intense computations be performed within a reasonable amount of time.

A second class of uncertainties are those associated with the particular theoretical com-putation methodology, such as the uncertainty associated with halting a perturbation theorytreatment. Developing hybrid theoretical approaches may be the key towards better com-putations of atomic structure properties. While directly incorporating the Dirac Hamilto-nian into orbital-based calculations is standard, this is not the norm for calculations withcorrelated basis sets. Correlated basis set computations are unmatched in their realizedaccuracies. Comparing such calculations with both relativistic and non-relativistic orbital-based ones could be used to estimate relativistic corrections to the Hylleraas calculationsand greatly increase the obtained precision [84, 85].

Lithium’s simple atomic structure allows for a precise computation of properties withonly ab initio wave functions, those derived from first principles in quantum mechanics, andspectroscopic data. Lithium’s polarizability could be pivotal in metrology [81]; a measuredvalue would constrain the calculated dynamic polarizabilities and thereby refine the methodof computation. Furthermore, a multispecies interferometer with lithium would be capableof normalizing another atom’s polarizability to that of lithium’s [24]. This could lead toa new accuracy benchmark for many elements in conjunction with a definitive calculationof α0. Hylleraas polarizability calculations could serve as standard for coupled-cluster typecalculations applied to larger atoms, like cesium.

1.4.1 λto measurement

The tune-out wavelength, λto, is defined as the wavelength at which the dynamic polariz-ability vanishes,

α(ωto) = 0. (1.41)

Contrary to conventional spectroscopic methods which measure the energy of a particularatomic electronic states indicated by poles in the frequency response of the atom to theexternal optical field [66], the tune-out is a zero in the atom’s frequency response.

The dynamic polarizability is made up largely by contributions from the valence electron.This work aims to measure the red tune-out wavelength in lithium, located between the2P1/2 and 2P3/2 levels. Between a nearby pair of such dipole allowed transitions, a zero inthe dynamic polarizability will occur where the opposite signs of detuning of the light with


670.955 670.960 670.965 670.970 670.975 670.980

-2000

-1000

1000

2000

wavelength [nm]

pola

riza

bil

ity [

10

4 a

.u.]

Dynamic polarizability of 2S2 state in lithium

670.9714 670.9718 670.9724 670.9728

10

5

-5

-10

Figure 1.2: A plot of the dynamic polarizability α(ω) for 7Li’s ground state 2S2.

respect to the atomic transition will perfectly cancel out. Due to this cancellation, no netenergy shift will be experienced by the ground level.

α(ωto) =S1

3~

(ωD1

ω2D1− ω2

to

+RωD2

ω2D2− ω2

to

)+ αrem(ωto) = 0 (1.42)

where αrem accounts for remaining contributions to the dynamic polarizability (core, core-valence) and R = S2/S1 is a ratio of the line strengths for the D1- and D2-lines. The locationof the zero depends primarily on the ratio of the matrix elements of the two states.

An interferometer composed of π/2-pulses, all with a single direction of momentum trans-fer, is no longer sensitive to the recoil frequency of the atoms. This interferometer geometrycan be made sensitive to an external optical potential [86] and thus used instead to determinethe atom’s tune-out wavelength. By pulsing on light during the atom’s free evolution, theatom’s response as the frequency is swept over the anticipated tune-out point can be tracked.As the dynamic polarizability goes to zero, the effect on the atoms from the additional pulseof light will diminish as well. As the frequency is moved away from tune-out, the atoms willagain be perturbed by the interaction with the light.

A direct interferometric measurement of lithium’s red tune-out wavelength at 670.971626(1)nm, is a precise comparison to existing ‘all-order’ atomic theory computations. The locationof the zero in the atom’s frequency response depends primarily on the ratio of the matrix ele-ments of the two states, providing a route toward a precise determination of the S− and P−transitions matrix elements, providing independent verification of QED predictions for suchtransition rate ratios [87]. A precise determination of matrix elements is also necessary in ex-perimental endeavors including: measurements of parity violation [88], the characterizationof Feshbach resonances [89, 90], and estimation of blackbody shifts [69, 91].


Perturbations to the core or higher level contributions is the polarizability could beobservable at experimental precisions of 0.1 ppb [92]. Furthermore, a future measurement oflithium’s ultraviolet tune-out wavelength, predicted to occur at 324.192(2) nm, would haveincreased sensitivity to relativistic approximations made in the atomic structure description.

1.5 Overview of this thesis

Here, the versatility and utility of atom interferometry, both in verifying existing atomic the-ories with measurements of atomic properties, like the tune-out wavelength, and in searchesfor exotic physics, like with a recoil-sensitive interferometer is demonstrated. Chapter 2 dis-cusses the theory behind the coherent atom-light interactions comprising the beam splittersand mirrors used in interferometry as well as the phase calculation for the recoil-sensitiveRamsey-Borde and tune-out sensitive schemes. Chapter outlines the advantages of usinglithium, the smallest alkali in both mass and electron number, in atom interferometry. Italso reviews previous interferometry measurements with lithium relying on thermal atomicbeams and diffraction gratings of standing light.

A summary of the experimental details required to prepare the atom source for interfer-ometry is included in Chapter 4. Previous work on this experiment, as detailed in Ref. [93]resulted in a new cooling technique for atom interferometry with lithium [94]. However, sincejoining the Muller group in the Fall of 2013, the focus shifted to interferometry and withoutsuch additional cooling. I, Professor Yanying Feng and Chen Lai first explored inertially-sensitive interferometry with a Mach-Zehnder interaction geometry. From mid-2015 through2016, Eric Copenhaver and I have been focused instead on recoil-sensitive interferometrywith lithium, discussed in Chapter 5. Following that demonstration and a diversion intomodeling the system’s intrinsic and mysterious imperfection, we shifted toward a similarinterferometer scheme as detailed in Chapter 6. This second project exploits the simplicityof lithium’s electronic structure, the last nicety of lithium. This work is ongoing presentlyand Chapter 6 discusses the set-up thus far and also exhibits preliminary data. The outlookand future of atom interferometry with laser-cooled lithium is explored in the final chapter.

16

Chapter 2

Atom interferometry

This chapter describes in detail the theoretical underpinnings behind the manifestation ofinterference between the components of the atom’s wave function as it transverses an in-terferometer. Phase is accrued both in the absence and presence of the external light fieldand if a spatial separation exists between wave packets at recombination. Details of theatom-light interaction for the case of two-photon Raman transitions is outlined here. In thelast sections, the total phase difference between upper and lower trajectories for the fourπ/2-pulse configurations is derived algebraically.

A discussion of interferometry begins at the experimental end, with interference. As seenin the optical interferometers discussed in Chapter 1, an atom interferometer proceeds by‘beam splitting’ an initial matter-wave |Ψ〉 either via slit or grating, optically or mechanically,such that the total wave function is now defined as a linear combination of two or moredifferent states

|Ψ〉 =∑n

|ψn〉. (2.1)

The split matter-wave propagates freely, eventually interacting with a second beam split-ter fracturing each of the wave packets of the superposition. For the optical interferometersdiscussed previously, a second beam splitter recombined the light which had propagated alongdifferent paths, yielding two output ports. Detection after recombination projects the finalmatter-wave probabilistically determined by the phase difference accrued along the pathsof propagation. Each component is traveling along a different path in spacetime and hasaccumulated quantum phase from period of evolution and as with optically interferometry,a difference in the final phases will be heralded by interference at the output.

Consider the simplified circumstance that the interaction turned on to create this super-position couples the initial state to only one other state |ψ2〉. The atom after a first ‘beamsplitter’ is given by

1√2|ψ1〉+

i√2|ψ2〉. (2.2)

After the matter-wave has evolved for a time T and each component has accrued a phaseφi, a second beam splitter is used to recombine the wave function. At the interferometer’s

CHAPTER 2. ATOM INTERFEROMETRY 17

output the total wave function is given by

|ψout〉 =1

2

(eiφ1 − eiφ2

)|ψ1〉+

i

2

(eiφ1 + eiφ2

)|ψ2〉. (2.3)

A measurement of the |ψ1〉 population results in interference as seen in the followingdetected intensity I

I = |〈ψ1|ψout〉|2 =1

4|〈ψ1|eiφ1 − eiφ2|ψ1〉|2 = sin2

(φ1 − φ2

2

).

(2.4)

The above consideration proceeded assuming perfect splitting, but under a more realisticapproximation, the amplitudes of the states after recombination are scaled by c1 and c2 withthe interferometer’s contrast C defined as

C =Imax − Imin

Imax + Imin=

2c1c2

c21 + c2

2

. (2.5)

The phase difference accumulated over the course of the atom interferometer is a sum ofcontributions arising from a free evolution phase, an atom-light interaction or ‘laser’ phase,and a separation phase that occurs if the wave packets of the superposition are separatedspatially at the last pulse:

φtot = φfree + φγ + φsep. (2.6)

In this chapter, the origin of these phases will be discussed. An understanding of thesephases arises out of an understanding of the dynamics of the quantum system, particularlyone in which the total Hamiltonian has a contribution from an interaction Hamiltonian whichcharacterizes a perturbation to the atom from an external electromagnetic field.

For an atom with internal energy states |a〉 and |b〉 such that Ea < Eb, the dynamics aredescribed by the following Hamiltonian

Htot = H(0) + H(1) =p2

2M+ Hint − d · E + H(1) (2.7)

where p is the momentum operator in the center-of-mass frame. The internal degrees of freeof the atom are described by the operator Hint defined explicitly as

Hint|a〉 = Ea|a〉 = ~ωa|a〉 and Hint|b〉 = Eb|b〉 = ~ωb|b〉. (2.8)

The coupling to the electromagnetic field is given by d ·E , the projection of the dipole op-erator d along the direction of the external electric field. The additional term H(1) representsany additional perturbative interactions such as those resulting from an external disturbanceor potential present during interferometry.

The sequence of pulses for a particular interferometer geometry results in a unique totalphase difference between wave packets of the superposition. In this chapter, the phase for


beam splitter

mirror

mirror

source

detector

x

Inte

nsi

ty

beam splitter

Mach-Zehnder atom interferometer

Figure 2.1: The coherently split wave function transverses an upper and lower trajectorysimultaneously. Upon recombination at a final pulse τπ/2, with effective wave vector k,interference occurs with a phase given by the phase difference between paths traveled by thesuperpositon wave packets, φu.


τπ/2

z

|b > |b >|a >

|a >

τπ/2

τπ/2

τπ/2

|b >

|a >

|b > |b >

|b >

τπ/2

|b >

z

|a >

|b >

|b > |b > |b >|a >

|a >

|a >τπ/2

τπ/2

τπ/2

Figure 2.2: Two interferometer geometries can be realized when utilizing four sequentialπ/2-pulses. When the effective k wave vector of the light is maintained throughout thepulses the parallelogram geometry is realized (left). or (reversed) as shown here. Reversingthe momentum direction creates the trapezoidal geometry (right) and has the added featureof building in a sensitivity to the recoil frequency ωr of the atoms.

an interferometer comprised of four beam splitter pulses will be derived, in the cases withand without a k-reversal mid-interferometer. It will be shown that reversing the direction ofmomentum transfer between the pulses pairs builds in a sensitivity to the recoil frequency,called a Ramsey-Borde interferometer. Without such reversal, the phase resulting from theinterferometer is independent of the recoil frequency but by turning on an additional externaloptical field between the pulse pairs, the interferometer’s phase will have a dependence onthe dynamic polarizability of the atom.

2.1 Light off

Quantum mechanics allows for the state of an atom at time t to be described by its state atan earlier time t0 < t with the evolution operator U(t, t0),

|ψ(t)〉 = U(t, t0)|ψ(t0)〉. (2.9)

Projecting this state vector onto the position basis leads to the following configuration-space representation or wave function of the atom at (t, z(t))

ψ(z(t), t) = 〈z|ψ(t)〉 = 〈z|U(t, t0)|ψ(t0)〉

=

∫dz0〈z|U(t, t0)|z0〉〈z0|ψ(t0)〉, (2.10)

in terms of the wave function at an earlier time t0, considering all possible starting points.The probability amplitude that the atom, starting at (t0, z0), will be found at z after a timet − t0 is known as the ‘conventional’ quantum propagator K. The quantum propagator isdefined as

K(zt, z0t0) ≡ 〈z|U(t, t0)|z0〉. (2.11)


Furthermore, the probability amplitude associated with a even later time tf , such thatz(tf ) = zf , can be determined from the previous amplitude, with it now as the startingpoint. The atom evolves from (t, z) to this later position,

ψ(zf , tf ) = 〈zf |ψ(tf )〉 = 〈zf |U(tf , t)|ψ(t)〉

=

∫dz 〈zf |U(tf , t)|z〉〈z|ψ(t)〉

=

∫dz

∫dz0 〈zf |U(tf , t)|z〉〈z|U(t, t0)|z0〉〈z0|ψ(t0)〉

=

∫dz

∫dz0 K(zf tf , zt) K(zt, z0t0) 〈z0|ψ(t0)〉. (2.12)

Therefore, the total amplitude for a particular evolution between two points in space-timecan also be calculated by considering the amplitude for a trajectory first to an intermediateposition z(t)(t ∈ [t0, tf ]), followed by the evolution from this intermediate time z(t) to thefinal time tf . This is known as the composition property of the quantum propagator.

An evolving atom has infinitely many possible paths between its initial (t0, z0) and a finalposition (tf , zf ). Feynman defined the quantum propagator as a sum of contributions fromall possible paths connecting the initial and final points expressed as

K(zf tf , z0t0) = N∑

Γ

eiSΓ/~. (2.13)

Expressing the quantum propagator as a functional integral over all possible paths Γ,

K(zf tf , z0t0) =

∫ zf

z0

Dz(t) eiSΓ/~, (2.14)

connects the initial (t0, z0) and final (tf , zf ) positions of the state. The path integral methodlinks the traditional formulations of quantum mechanics to the more intuitive principles ofwave mechanics.

The phase factor, here equal to the action SΓ scaled by ~, can be rewritten in terms ofthe integral of the system’s Lagrangian L(z, z). The Lagrangian is a function of position zand velocity z over a path Γ = z(t) from initial point (z0, t0) to final point (zf tf ) given by

SΓ =

∫ tf

t0

dt L[z(t), z(t)]. (2.15)

In the limit that the phase evolution of the atomic wave function is

SΓ/~ 1, (2.16)

as is often the case in atom interferometers, the rapidly oscillating phases of neighboringpaths will destructively interfere. This will not occur if the paths are close to the extremaof the action.


For paths close to which the action is extremal, rather than conspiring to cancel, theslowly varying terms will be the dominant contribution to the integral. In this limit calledthe ‘classical’ limit, only paths close to the classical one Γcl contribute to the phase. Theclassical path is the one for which the action is extremal, given by

δS =

∫ tf

t0

[∂L∂zδz(t) +

∂L∂zδz(t)

]= 0

=∂L∂zδz(t)

∣∣∣∣tft0

+

∫ tf

t0

[∂L∂z− d

dt

∂L∂z

]δz(t)dt = 0. (2.17)

Above, the first term vanishes due to the boundary conditions

δz(t0) = δz(tf ) = 0. (2.18)

Imposing that the second term must also equal zero as well yields the classical equations ofmotion, the Euler-Lagrange equations:

∂L∂z− d

dt

∂L∂z

= 0. (2.19)

In the absence of an electromagnetic field, the Lagrangian for an atom with mass M andinternal energy ~ωi in a uniform potential V (z) is a quadratic function of position z andvelocity z

L =

(Mz)2

2M− V (z)− ~ωi (2.20)

with the following (classical) solutions

zcl(t) = v0 − g(t− t0) (2.21)

zcl(t) = z0 + v0(t− t0)− g

2(t− t0)2. (2.22)

Therefore, in an atom interferometer the above path integral for the accrued quantumphase during free evolution will be dominated by the atom’s classical trajectories. An ar-bitrary path z(t) can be parameterized in terms of its deviation, ξ(t), from the classicalone

z(t) = zcl(t) + ξ(t). (2.23)

The action is

S[zcl(t) + ξ(t)] = Scl(zf tf , z0t0) +

∫ tf

t0

dt

[M

2ξ2

]. (2.24)

and the quantum propagator expressed in terms of the deviation ξ, is

K(zf tf , z0t0) =

∫ ξ(tf )

ξ(t0)

Dξ(t) exp

i

~S[zcl(t) + ξ(t)]


(z0,t0)

(zf,tf)

zcl(t)

z cl(t)+ξ

(t)

Figure 2.3: The quantum propagator is the sum of contributions from all possible pathsconnecting the initial and final points. The atom’s path in configuration-space is dominatedby the classical action, shown in red. An arbitrary path, shown in blue, can be parameterizedin terms of its deviation ξ(t), from the classical path.

=

∫ 0

0

Dξ(t) exp

i

~S[zcl(t) + ξ(t)]

= exp

i

~Scl(zf tf , z0t0)

×∫ 0

0

Dξ(t) exp

i

~

∫ tf

t0

dt

[M

2ξ2

].

(2.25)

2.1.1 The free evolution phase

The functional integral in Eq. 2.25 is independent of both z0 and zf and is denoted in theliterature as F (tf , t0) [95]. The propagator simplifies to

K(zf tf , z0t0) = F (tf , t0) exp

i

~Scl(zf tf , z0t0)

. (2.26)


The wave function after some time tf − t0 of free evolution is

ψ(tf , zf ) = F (tf , t0)

∫dz0 exp

i

~Scl(zf tf , z0t0)

ψ(t0, z0) (2.27)

where the classical action Scl(zf tf , z0t0) for the Lagrangian in Eq. 2.1 is explicitly determinedto be

Scl(zf tf , z0t0) =

∫ tf

t0

dt

[Mv(t)2

2−Mgz(t)− ~ωi

]=

M

2

(zf − z0)2

tf − t0− Mg

2(zf + z0)(tf − t0)

−Mg2

24(tf − t0)3 − ~ωi(tf − t0) (2.28)

and reduces to a function only of the path’s endpoints Scl ≡ Scl(zf tf , z0t0).Therefore, a determination of the initial and final points for the particular step of free

evolution is all that is needed in order to calculate the phase acquired by the atom as itevolves freely!

2.2 Light on

The atom also acquires a phase shift from the laser pulses used to coherently split, manipulateand recombine the matter-waves. Two diffraction mechanisms that are commonly used inlight-pulse atom interferometry are Bragg and Raman scattering. In Bragg scattering, apair of counter-propagating laser beams induce a two-photon transition between momentumstates, leaving the atom’s internal state unchanged. In Raman scattering, a pair of laserbeams interact with the atom but the two-photon transition is accompanied by a transition toanother internal state. Here, two-photon Raman transitions are discussed in more detail sincethe work in this thesis utilizes Raman scattering as the mechanism behind interferometrywith laser-cooled lithium.

2.2.1 Raman scattering

A stimulated Raman transition couples the atom’s momentum to its internal energy state.It can be formally defined as a two-photon transition from one ground state to another andbetween different motional states, mediated by an excited state. Appendix B reviews theexample of the two-level system.

Consider a three-level atomic system with ground states |a〉 and |b〉, separated by anenergy difference of ~ωba = ~(ωb − ωa), and excited state |e〉, at energy ~ωe.

The general state vector of the atom at a time t is

|ψ〉 = ca(t)|a〉+ cb(t)|b〉+ ce(t)|e〉. (2.29)


E + p2/2m

p

p

na = 0

a

b

na = -2na = 2

nb = -1 nb = 1

Figure 2.4: For an atom starting in state |a, p = 0〉 absorption of ω1 (blue) followed byemission of ω2 (red) will move the atom up or down in momentum given the direction ofkeff = k1 − k2, the effective wave vector of the light.

The atom is irradiated by two monochromatic fields given by

E(r, t) = ε1

(E1e

+i(k1·r−ω1t) + E∗1e−i(k1·r−ω1t)

)+ ε2

(E2e

+i(k2·r−ω2t) + E∗2e−i(k2·r−ω2t)

)(2.30)

where E1,2 = ε1,2E1,2 with ε1,2 representing the unit field polarization vectors and k1,2 andω1,2 are the wave vector and frequency for the external fields.

The dynamics of the system is described by the time-dependent Schrodinger equation,

i~∂

∂t|ψ(t)〉 = H|ψ(t)〉, (2.31)

where the Hamiltonian H = HA + HAF is the sum of the free atomic Hamiltonian

HA =p2

2M+ ~ωa|a〉〈a|+ ~ωb|b〉〈b|+ ~ωe|e〉〈e| (2.32)


Ωa2

| e , p + ħk1>

| a , p >

| b , p + ħ (k1- k

1) >

ω1

ω2

Ωa1

Ωb1

Ωb2

ωa

ωb

ωe

ωba

∆ = ∆ a1= ∆ b2

∆ a2

∆ b2

δ

Figure 2.5: A three level atomic system with two ground- (|a〉 and |b〉) and one excited-state (|e〉) are coupled by frequencies of light at ω1 and ω2. The external field is detuned fromthe excited by a single-photon detuning ∆ and a two-photon detuning given by δ quantifiesthe difference between (ωb − ωa) the frequency splitting between the two ground states and(ω1 − ω2) the frequency difference of the light.


and atom-field interaction Hamiltonians with interaction terms arising from the atom-fieldelectric dipole interaction V = d · E .

The dipole operator is defined as

d =

[〈a|d|e〉σa + 〈b|d|e〉σb

]+

[〈a|d|e〉σ†a + 〈b|d|e〉σ†b

]. (2.33)

with σn := |n〉〈e| or defined instead in terms of matrix elements of the dipole interaction as

d =

[µae|a〉〈e|+ µbe|b〉〈e|

]+

[µ∗ae|e〉〈a|+ µ∗be|e〉〈b|

]. (2.34)

The strength of the coupling of level |n〉 to the excited level |e〉 through the field Ei isdescribed by the Rabi frequency given by

Ωni := −〈n|d · Ei|e〉~

= − µne · Ei~

. (2.35)

Rewriting the interaction Hamiltonian in terms of the Rabi frequency Ωni yields

HAF = −~2

(Ωa1|a〉〈e|+ Ω∗a1|e〉〈a|+ Ωb2|b〉〈e|+ Ω∗b2|e〉〈b|

)−~

2

(Ωa2|a〉〈e|+ Ω∗a2|e〉〈a|+ Ωb1|b〉〈e|+ Ω∗b1|e〉〈b|

). (2.36)

The Hamiltonian in the rotating frame is

H = −~(

∆1|e〉〈e|+ δ|b〉〈b|+(

Ω1|e〉〈a|+ Ω2|e〉〈b|+ Ω∗1|a〉〈e|+ Ω∗2|b〉〈e|))

(2.37)

with ∆1,2 equal to the single photon detunings from excited state |e〉, the two-photon detun-ing is given by δ = ∆1 −∆2 and the energy of |a〉 and the excited state is set to zero. Thetwo beams are detuned by many line widths from single photon resonance and therefore thedetuned resonant excitation to |b〉 will dominate over incoherent spontaneous emission fromthe excited state |e〉.

The Rabi frequency determines the time period at which the atoms will ‘flop’ betweenthe two ground states due to the interaction with the driving field. By the dipole interaction,a laser with frequency ωn will couple a state n (|n〉) to the intermediate or excited state (|e〉).If the beam is detuned by a largely sufficient amount ∆n Γ from single photon resonancethen the occupation of |e〉 will be negligible. A pair of frequencies with a common detuning∆1 = ∆2 ≡ ∆ induces a coherent transfer between the ground states |a〉 and |b〉. This typeof transition defines a two-photon Raman transition. Because a photon posses momentumdictated by its wave vector, ~k, it is possible to define a momentum basis for this particularatomic transition.


The rotating frame is defined by the following state vector

|ψ〉 = ca|a〉+ cb|b〉+ ce|e〉 (2.38)

for which the ground states |n〉 have been effectively boosted by energy ~ωn. The dynamicsof the system is now described by the rotating-frame free atomic Hamiltonian,

HA =p2

2m+ ~∆|a〉〈a|+ ~∆|b〉〈b| (2.39)

and the rotating-frame interaction Hamiltonian,

HAF =~2

(Ωa1

[|a〉〈e|e−ik1z + |e〉〈a|eik1z

]+ Ωb2

[|b〉〈e|e−ik2z + |e〉〈b|eik2z

]),

(2.40)

which does not yet consider any ac-Stark related energy shifts that may be present in thesystem. In reality, each optical field couples to all energy levels.

The atomic state vector defined in terms of internal energy and momenta states is

|ψ〉 = ca|a, p〉+ cb|b, p+ ~(k1 − k2)〉+ ce|e, p+ ~k1〉. (2.41)

Projecting the state vectors onto the configuration basis and boosting all energies by−~∆ produces the following equations of motion:

i~∂tψe =p2

2mψe +

~Ωa1

2eik1zψa +

~Ωb2

2eik2zψb − ~∆ψe (2.42)

i~∂tψa =p2

2mψa +

~Ωa1

2e−ik1zψe + ~(∆1 −∆)ψa (2.43)

i~∂tψb =p2

2mψb +

~Ωb2

2e−ik2zψe + ~(∆2 −∆)ψb. (2.44)

This two-photon process can be viewed as an effective one-photon process with effectivek-vector given by

keff = k1 − k2 = 2k (2.45)

and an effective frequency that is twice what was realized with the one photon process.Adiabatically eliminating the excited state gives

ψe =Ωa1

2∆eik1zψa +

Ωb2

2∆eik2zψb (2.46)

for the excited state wave function and two coupled equations of motion for the ground states

i~∂tψa =p2

2mψa +

~ΩR

2ei(k2−k1)zψb + ~(∆1 + ΩAC

a )ψa (2.47)


i~∂tψb =p2

2mψb +

~ΩR

2ei(k1−k2)zψa + ~(∆1 + ΩAC

b )ψb (2.48)

where

ΩR :=Ωa1Ωb2

2∆(2.49)

is the two-photon or Raman Rabi frequency and the ac Starks shifts are given by

ΩACn :=

∑i=1,2

|Ωni|2

4∆ni

, n = a, b. (2.50)

An atom initially in state |a〉 will be excited by a rightward traveling photon (~k1) withfrequency ω1 to a virtual level, linewidths away from the excited state |e〉. The likelihood ofspontaneous emission from the excited state is small and the atom is stimulated down to |b〉by a leftward traveling photon (~k2) of frequency ω2. The following is the Raman resonancecondition, conserving energy and momentum, for such a process

(ω1 − ω2)− ωab = v · (k1 − k2)± ~2m

(k1 − k2)2. (2.51)

2.2.1.1 Dressed states

The effective two level system that falls out of the three level system in the ‘adiabatic limit’,for which the fast dynamics of the excited state average to zero, has the following simpleHamiltonian

H =

[ΩAC

1ΩR2ei(δt−φL)

ΩR∗2e−i(δt−φL) ΩAC

2

]. (2.52)

By transforming the wave function as

|ψ′〉 = e−i(ΩAC1 +ΩAC2 )t/2I|ψ〉, (2.53)

the energies are shifted by −~(ΩAC1 + ΩAC

2 )/2. Then, rotating the wave function by thedetuning from two-photon Raman resonance δ given by

δ =

((p + ~(k1 − k))2

2m~+ ωb

)−(

(p)2

2m~+ ωa

)− (ω2 − ω1) (2.54)

through the operator R

R = eiσzδt/2

[e−iδt/2 0

0 eiδt/2

](2.55)


yields the (time-independent effective) Hamiltonian for the system

Heff = −~

[(δAC−δ)

2ΩR2e−iφL

ΩR∗2eiφL − (δAC−δ)

2

](2.56)

with eigenvalues ±~2

√|ΩR|2 + (δAC − δ)2 and wave function

|ψ(t0)〉R = ca(t0)e−i(ΩAC1 +ΩAC2 )t/2e−iδt0/2|a〉+ cb(t0)e−i(Ω

AC1 +ΩAC2 )t/2eiδt0/2|b〉. (2.57)

In the presence of the external field for an interaction duration τ = t− t0, the atom willevolve according to the time-dependent Schrodinger equation but with Hamiltonian Heff andtime-evolution operator

U(t0, t) = e−iHR(τ)/~ = eiβ(τ)/~|β〉〈β|+ eiα(τ)/~|α〉〈α| (2.58)

which has been expanded in the two-dimensional basis of its eigenvectors

|β〉 = cosΘ

2|a〉Re−iφL/2 − sin

Θ

2|b〉ReiφL/2 (2.59)

|α〉 = sinΘ

2|a〉Re−iφL/2 + cos

Θ

2|b〉ReiφL/2 (2.60)

with

tan Θ =−ΩR

(δAC − δ)cos Θ = −(δAC−δ)

ΩRsin Θ =

ΩR

ΩR

. (2.61)

The probability amplitudes ca(t0 + τ) and cb(t0 + τ) are

ca(t0 + τ) = ei(ΩAC1 +ΩAC2 )τ/2eiδτ/2

(cb(t0)ei(δt0−φL)

[i sin Θ sin

(ΩRτ

2

)]

+ ca(t0)

[cos

(ΩRτ

2

)− i cos Θ sin

(ΩRτ

2

)])(2.62)

cb(t0 + τ) = ei(ΩAC1 +ΩAC2 )τ/2e−iδτ/2

(ca(t0)e−i(δt0−φL)

[i sin Θ sin

(ΩRτ

2

)]

+ cb(t0)

[cos

(ΩRτ

2

)]+ i cos Θ sin

(ΩRτ

2

)). (2.63)

The squared amplitudes yield the probabilistic populations of atoms in the correspondingstate for a pulse of time τ . A π/2-pulse satisfies ΩRτ = π/2 and yields the followingamplitudes

ca(t0+τ) =eiπ(ΩAC1 +ΩAC2 )/4ΩReiπδ/4ΩR

√2


(i sin Θ

)+ ca(t0)

(1− i cos Θ

))


cb(t0 + τ) =eiπ(ΩAC1 +ΩAC2 )/4ΩRe−iπδ/4ΩR

√2


(i sin Θ

)+ cb(t0)

(1 + i cos Θ

))and a ‘mirror’ pulse or π-pulse satisfies ΩRτπ/2 = π with the following amplitudes:

ca(t0 + τ) = eiπ(ΩAC1 +ΩAC2 )/2ΩReiπδ/2ΩR


(i sin Θ

)− ca(t0)

(cos Θ

))

cb(t0 + τ) = eiπ(ΩAC1 +ΩAC2 )/2ΩRe−iπδ/2ΩR


(i sin Θ

)+ ca(t0)

(i cos Θ

)).

In the absence of the external optical field coupling internal atomic states, the Rabifrequencies which are proportional to the transition dipole matrix elements will vanish Ω1 =Ω2 = ΩR = 0, as will all the terms resulting from the light shift. Therefore, when the atomis in the dark the generalized Rabi frequencies and Θ parameter becomes

ΩR = −δ cos Θ = −1 sin Θ = 0

and the amplitudes corresponding to this free evolution are as follows:

ca(τ + T ) = eiδτ/2ca(τ)

[cos(−δT/2

)+ i sin

(−δT/2

)]= ca(τ)

cb(τ + T ) = e−iδτ/2cb(τ)

[cos(−δT/2

)− i sin

(−δT/2

)]= cb(τ).

2.2.2 The interaction phase

From the equations given in the previous section, for an atom-light interaction of time τ , aphase ±φ is imprinted onto the matter-wave. The probability amplitude is a function of thesystem’s parameters as well as the initial amplitude prior to turning on the light.

The driving fields for the three-level atomic system are

E(r, t) = ε1E1ei(k1·r−ω1t+φ1) + ε2E2e

i(k2·r−ω2t+φ2) (2.64)

which for the initial condition φ1(t = 0)− φ2(t = 0) = φ0, an effective phase φ is a functionof the atom’s position in spacetime given by

φ(r, t) = (k1 − k2) · r− (ω1 − ω2)t+ φ0 (2.65)

The phase is evaluated at each position and the corresponding time is referenced withinthe overall pulse sequence. Setting t = 0 at the beginning of the first pulse in the interfer-ometer implies that t corresponds to the absolute time.


2.2.3 The separation phase

The superposition of states at the interferometer’s end may not overlap perfectly in position-or momentum-space at the last π/2–pulse. This discrepancy may arise due to perturbationsin either the atom-light interaction or potential during the sequence. This results in anadditional phase shift of the final wave function given by

∆φsep = p · δr (2.66)

where p is the average canonical momentum of the detected atomic wave function and δr isthe separation between the two wave packets. From here on out, we neglect this contributionto the phase.

2.3 The total phase

An interferometer’s geometry or sequence of pulses will determine the final phase differencebetween the atomic superposition. Consider the interferometer shown in Figure 2.7 withperiods of evolution, T , T ′ and T . At the last pulse, the amplitude cb of the final statevector at time tf = (tf−τ) + τ is a function of the amplitudes for each initial state withrespect to the last pulse τ given by

cb(tf)

=eiφ

ACτ/2e−iδτ/2√2

(ca(tf−τ)e−iδ(tf−τ)+φ

(4)L

[i sin Θ

]+ cb(tf−τ)

[1 + i cos Θ

]).

Consider the probability for an atom to be found in the |b〉 state after the pulse sequence,in the approximation where the laser is close to resonance, the following approximations canbe made

δAC − δΩR

1 → cos Θ ≈ 0 and sin Θ ≈ 1 (2.67)

yielding

|cb(tf )|2 =1

2

[|ca|2 + |cb|2 + icacb

(e−iφ − eiφ

)]=

1

2

[|ca|2 + |cb|2 − 2cacb sinφ

].

The wave packets interfere and an atom that completes the interferometer will probabilis-tically be projected into a state that is a function of the phase difference acquired betweenthe interferometer arms

P|b〉 =1

4|eiΦu + eiΦ` |2. (2.68)

The total phase difference is computed by considering the total trajectory in spacetimeas a piecewise evolution of phase differences. The periods of free evolution and the periodsduring which the atom-light interaction is turned on are considered separately.


Φu,`a,b =

∑i

(1

~S[zi, pi, Ti] + φγ[zi]

)∣∣∣∣u,`a,b

(2.69)

where S[zi, pi, Ti] is the action along the classical path beginning at zi with momentum pifor a time of evolution Ti and φγ[zi] is the laser phase imprinted on the atom during the ithpulse.

The free evolution phase is determined by evaluating the action for the time of evolutionTi

S[z(t0), p(t0), Ti] =

∫ Ti

t0

L[z(t), p(t)]dt (2.70)

where z(t) and p(t) are solutions to the Euler-Lagrange equation with initial position z(t0)and momentum p(t0). The Lagrangian for an atom in state ~ωn moving in a gravitationalpotential is

L =p2

2M−Mgy +

1

2Mγy2 − ~ωn. (2.71)

Here, g is the gravitational acceleration along the y-axis and γ = 2GMe/R3e is the gravity

gradient. Going forward the gradient is set to zero and in subsequent chapters gravity willalso be neglected due to the orientation of the interferometer.

Therefore, the free evolution phase with respect to the motion of the particle along thedirection set by the interferometry beam z is

φf(z(t), p(t), Ti

)=

1

~∑i

(∫ Ti

0

L[z(t), p(t)]dt

)=

1

~∑i

(∫ Ti

0

[M

2

(∂z(t)

∂t

)2

−Mg cos θz(t)− ~ωn]dt

)

=1

~∑i

[M

2

(z(Ti)− z0

)2

Ti−Mg cos θ

(z(Ti) + z0

)Ti

2− Mg2T 3

i

24− ~ωnTi

](2.72)

with cos θ quantifying the projection of gravity along the interferometry axis z. The initialtime has been set to zero, t0 = 0. This computation is performed for each segment of thetrajectory, using the final position and velocity for the previous path as the starting pointof the atom’s motion entering the current segment.

The laser phase φ(zi)γ is evaluated similarly, as a piecewise function along the atom’strajectory given by

φ(zi)γ = (k1 − k2)× z − (ω1 − ω2)× ti + φ0, (2.73)

where ti corresponds to the absolute time for the atom at position zi in the interferometer.For the counter-propagating Raman scheme employed here, the wave vector of the externalfield are for our purposes are equal in magnitude and oppositely directed k1 = −k2.


T

|a , p + ħkeff

⟩

TT’

|b , p ⟩t

z

|a , p - ħkeff ⟩

|b, p

+2 ħ

k eff ⟩

|b, p

+b,

p2

+ +ħk

eff

ħkħk⟩

T

|a , p

+ ħk eff

⟩

TT’

|b , p ⟩t

z

Figure 2.6: (Left) The conjugate interferometer geometry for the Ramsey-Borde four π/2-pulse sequence. (Right) The conjugate interferometer geometry for the copropagating fourπ/2-pulse sequence.

2.4 Conjugate interferometers with the π2-π2-π2-π2

A consequence of the π2-π

2-π

2-π

2is that the outputs of the second pulse not contributing

to the lower interferometer may, if addressed at the third and forth pulse, close a secondinterferometer. This second interferometer is referred to as the conjugate in a double schemeand is realized with and without k-reversal. Phase calculations for both the lower and upperinterferometers with and without flipped the effective wave vector of the Raman light forthe second pulse pair, is detailed in the last sections of this chapter.

2.5 The Ramsey-Borde interferometer

An iconic paper by Borde explored an atom interferometer described as an ‘optical Ramsey’interferometer [96], which consisted of four (optical) beam splitter pulses such that the initialstate was split similarly as in a Ramsey interferometer but the momentum was also addressed,put into a superposition given by

|a, p〉 → sin θ|a, p〉+ cos θ|b, p+ 2~k〉

where p is the initial momentum and k is the wave vector of the light.In a Ramsey-Borde interferometer (RBI), an equal time of evolution denoted by T sepa-

rates the pulses in each pair and a second time of duration (not necessarily equal to the first)T ′ separates the two pairs of pulses. The effective wave vector which determines the directionof momentum transfer is reversed for the second pulse pair and a final phase difference isproduced that is proportional to the recoil frequency of the atom ωr = ~k2/(2m).


T

|a , p +

ħkeff ⟩

TT’

|b , p ⟩

|b , p ⟩

|a , p - ħkeff ⟩

τπ2

t

z

1

2

1

2 1

2

1

2

Figure 2.7: A Ramsey-Borde interferometer’s interaction geometry is shown here. Thisinterferometer is built from four π/2-pulses (beam splitters) separated by three periods offree evolution: T , T ′, and T , respectively. the effective k-vector of the light is switchedbetween the pulse pairs, reversing the direction of momentum imparted to the atoms andbuilding sensitivity to the kinetic energy of recoiling that is imparted to the atom afterabsorption and emission in the two-photon transition. Prior to the last recombination pulsein this geometry, the interferometer arms are in states |b, p〉 and |a, p− ~keff〉.

For an interferometer with pulses that induce a transition between internal states of theatom, the lifetimes need to be at least comparable to the transit time of the atom through theinterferometer. Otherwise, spontaneous decay will destroy coherence. Such interferometershistorically have operated in the space-domain, utilizing single photon transitions in atomssuch as magnesium or calcium that have such long-lived metastable states. The spatialresolution of the two paths enjoyed by such schemes allows for sensitivity to field gradientsas well as inertial displacements. Here, we use two-photon transitions for the interferometrypulses. Transitioning the atoms between hyperfine ground states, allows us to completelydisregard any spontaneous decay. Both states are for our experimental purposes consideredto be infinitely long-lived with lifetimes much longer than the interrogation time in theinterferometer.

The relative phase shift between the two paths is the total difference in phase acquiredalong each path given by

∆Φf = Φuf − Φ`

f =1

~[∆S12 + ∆S23 + ∆S34

]where the time ti is constant between the upper and lower trajectories but the position ziand momentum pi may differ.

Here, the difference between actions for the paths at a particular time interval ∆Snm =Sunm − S`nm and the differences in phase for each segment are computed below.


Table 2.1: Positions and velocities for trajectories in the lower RBI.

Time segment Lower path, (zi`, vi`) Upper path, (ziu, viu)

[t1, t2] = T z0 + v0T − g2T 2 z0 + (v0 + vr)T − g

2T 2

v0 − gT v0 − gT + vr

[t2, t3] = T ′ z2` + v0T′ − g

2

[(T + T ′)2 − T 2

]z2u + v0T

′ − g2

[(T + T ′)2 − T 2

]v0 − g(T + T ′) v0 − g(T + T ′)

[t3, t4] = T z3` + v0T − g2

[(2T + T ′)2 − (T + T ′)2

]z3u + (v0 − vr)T − g

2

[(2T + T ′)2 − (T + T ′)2

]v0 − g(2T + T ′) v0 − g(2T + T ′)− vr

1

~(Su12 − S`12

)= kT

(2v0 + vr − 2gT

)− ωabT (2.74)

1

~(Su23 − S`23

)= −2gkTT ′ (2.75)

1

~(Su34 − S`34

)= kT

(− 2v0 + vr + 2g(T + T ′)

)− ωabT (2.76)

A total phase difference from the free evolution is given by the following expression:

∆Φf = 2kvrT − 2ωabT. (2.77)

The lower arm or path in a normal Ramsey-Borde interferometer never changes state noracquires momentum and thus will receive no phase contribution from the interaction withthe laser. The upper arm acquires a laser phase given by

∆Φγ = φγ1(t0)− φγ2(t0 + T ) + φγ3(t0 + T + T ′)− φγ4(t0 + 2T + T ′)

= (2kz0 − ω12t0)− (2kz2 − ω12t2) + ((−2k)z3 − ω12t3)− ((−2k)z4 − ω12t4)

= −4kvrT − kg(2T 2 + 2TT ′

)+ 2ω12T

Therefore, the total phase difference is

Φtot = −2kvrT − 2kg(T + T ′

)T − 2

(ωab − ω12

)T. (2.78)

The recoil velocity vr results from the transfer of 2~k of momentum. Inputting thedefinition of the recoil frequency into the above expression ωr = ~k2/2m leads to

Φtot = −4~k2T

m− 2kg

(T + T ′

)T − 2

(ωab − ω12

)T

= −8ωrT − 2kg(T + T ′

)T − 2δT (2.79)

where the definition for the two-photon detuning is used to rewrite ωab − ω12 = δ.


T

|a , p +

ħkeff ⟩

TT’

|b , p ⟩

τπ2

t

z

1

2

1

2

1

2

1

2

|a , p +

ħkeff ⟩

|b , p

+ 2ħk

eff ⟩

Figure 2.8: A (conjugate) Ramsey-Borde interferometer’s interaction geometry is shownhere. This interferometer is built from four π/2-pulses (beam splitters) separated by threeperiods of free evolution: T , T ′, and T , respectively. the effective k-vector of the light isswitched between the pulse pairs, reversing the direction of momentum imparted to theatoms and building sensitivity to the kinetic energy of recoiling that is imparted to the atomafter absorption and emission in the two-photon transition. Prior to the last recombinationpulse in this geometry, the interferometer arms are in states |b, p+ 2~keff〉 and |a, p+ ~keff〉.

2.5.1 cRBI phase computation

The phase difference between the upper and lower trajectories can be computed by consider-ing again a piecewise evolution for the atom during interferometry. These phase differencesare the same as in the normal RBI for the first two periods of free evolution T and T ′ butdiffer for the final T . In each interferometer, the probability of detection at a output depends


Table 2.2: Positions and velocities for trajectories in a conjugate RBI.


[t1, t2] = T z2` = z0 + v0T − g2T 2 z2u = z0 + (v0 + vr)T − g

2T 2


[t2, t3] = T ′ z3` = z2` + (v0 + vr)T′ − g

2

[2TT ′ + T ′2

]z3u = z2u + (v0 + vr)T

′ − g2

[2TT ′ + T ′2

]v0 − g(T + T ′) + vr v0 − g(T + T ′) + vr

[t3, t4] = T z4` = z3` + (v0 + 2vr)T − g2

[3T 2 + 2TT ′

]z4u = z3u + (v0 + vr)T − g

2

[3T 2 + 2TT ′

]v0 − g(2T + T ′) + 2vr v0 − g(2T + T ′) + vr

on the phase difference between the arms of the interferometer. A nicety of the simultaneousconjugate RBI scheme is that accelerations add equally to the phase shifts for the normaland upper interferometer phases. Phase fluctuations of the lasers or vibrations of the systemcan be canceled by considering both interferometers.

Repeating the procedure outlined above but now for the upper conjugate Ramsey-Bordeinterferometer yields

1

~(Su12 − S`12

)= kT

(2v0 + vr − 2gT

)− ωabT (2.80)

1

~(Su23 − S`23

)= −2gkTT ′ (2.81)

1

~(Su34 − S`34

)= kT

(− 2v0 − 3vr + 2g(T + T ′)

)− ωabT (2.82)

and the total phase difference from the free evolution of the coherent superposition is givenby

∆Φf = −2kvrT − 2ωabT. (2.83)

For the upper RBI, assume that during the first pulse, the upper arm has been imprintedwith the laser phase during the atom-field interaction but then that all subsequent pulsesonly affect the lower arm of the upper interferometer:

∆Φγ = φγ1(t0)−(φγ2(t0 + T )− φγ3(t0 + T + T ′) + φγ4(t0 + 2T + T ′)

)= (2kz0 − ω12t0)− (2kz2` − ω12t2) + (−2kz3` − ω12t3)− (−2kz4` − ω12t4)

= 2k(z4` − z3` − z2`)− ω12(t0 − t2 + t3 − t4)

= 2k(2vrT − g(T 2 + TT ′)

)− ω12

(− T + T + T ′ − 2T − T ′

)= 4kvrT − 2kg(T + T ′)T + 2ω12T (2.84)


Therefore, the cRBI has a total phase difference given by

Φctot = 2kvrT − 2kg(T + T ′)T − 2δT = +8ωrT − 2kg(T + T ′)T − 2δT. (2.85)

2.6 The copropagating interferometer

If instead of reversing the effective k-vector between the pulse pairs, the direction is main-tained then the copropagating scheme (CPI) depicted in Fig. 2.9 will be realized. The phaseshift of this interferometer, contrary to a Ramsey-Borde interferometer, will not depend onthe frequency of the laser generating the pulses and therefore be less sensitive to frequencyfluctuations. This loss of dependence arises because each component of the superpositionwill spend an equal amount of time, T , recoiling.

The copropagating interferometer has the same dependence in sensitivity as a Mach-Zehnder configuration characterized by a π

2− π− π

2pulse sequence, as can be seen from the

above computation. If an additional potential term is turned on in the total Hamiltonianduring the T ′ period of free evolution, a resulting phase difference will be realized. Whenmeasuring the tune-out wavelength here, we are unable to spatially resolve the arms of theinterferometer as has been done to-date. We focus on the spatial gradient of the interactioninstead and the implement detection and analysis are discussed more thoroughly in Chapter6.

The total phase in the presence of a ‘tune-out’ beam switched on during the evolutiontime T ′ is given by

φto =α(ω)sT ′

2ε0c~× ∂I

∂z(2.86)

where s = 2vrT is the separation between the arms of the interferometer, which can bedefined in terms of the recoil velocity of the atom and the intensity of the beam with waistw is given by

I(r, ω0) = I0e−2r2/w2

(2.87)

where I0 = 2P/(πw20).

The phase difference between the upper and lower trajectories during free evolution is

∆Φf =1

~

(kT (2v0 + vr − 2gT )− ωabT − 2gkTT ′ − kT (2v0 + vr − 2g(T + T ′)) + ωabT

)= 0.

The phase difference between trajectories resulting from the atom-light interaction is

∆Φγ = +φγ1(t0)− φγ2(t2)−(φγ3(t3)− φγ4(t4)

)= 2k

(z0−z2u−z3`+z4`

)− ω12

(t0−t2−t3 + t4

)= −2kg

(T + T ′

)T. (2.88)


T

|a , p +

ħkeff ⟩

TT’

|b , p ⟩

|b , p ⟩τ

π2

t

z

1

2

1

2

1

2

1

2

Figure 2.9: A (lower) copropagating interferometer’s interaction geometry is shown here.This interferometer is built from four π/2-pulses (beam splitters) separated by three periodsof free evolution: T , T ′, and T , respectively. The effective k-vector of the light is maintainedbetween the pulse pairs and the lack of k-reversal is evident in that both the upper and lowerarms spend equivalent durations of time in each momentum state (compared to each other).Prior to the last recombination pulse in this geometry, the interferometer arms are in states|b, p〉 and |a, p+ ~keff〉.

Table 2.3: Positions and velocities for the trajectories in the lower copropagating interfer-ometer scheme.



2T 2


[t2, t3] = T ′ z3` = z2` + v0T′ − g

2

[2TT ′ + T ′2

]z3u = z2u + v0T

′ − g2

[2TT ′ + T ′2

]v0 − g(T + T ′) v0 − g(T + T ′)

[t3, t4] = T z4` = z3` + (v0 + vr)T − g2

[3T 2 + 2TT ′

]z4u = z3u + v0T − g

2

[3T 2 + 2TT ′

]v0 − g(2T + T ′) + vr v0 − g(2T + T ′)


2

T

|a , p +

ħkeff ⟩

TT’

|b , p ⟩

|b , p ⟩

τπ2

t

z

1

2

1

2

2

1

2

1

2

|a , p +

ħkeff ⟩

Figure 2.10: A (upper) copropagating interferometer’s interaction geometry is shown here.This interferometer is built from four π/2-pulses (beam splitters) separated by three periodsof free evolution: T , T ′, and T , respectively. The effective k-vector of the light is maintainedbetween the pulse pairs and the lack of k-reversal is evident in that both the upper andlower arms spend equivalent durations of time in each momentum state (compared to eachother). Similar to the lower interferometer discussed in the previous section, prior to thelast recombination pulse in this geometry, the interferometer arms are in states |b, p〉 and|a, p+ ~keff〉.

Table 2.4: Positions and velocities for the trajectories of the upper copropagating interfer-ometer



2T 2


[t2, t3] = T ′ z3` = z2` + (v0 + vr)T′ − g

2

[2TT ′ + T ′2

]z3u = z2u + (v0 + vr)T

′ − g2

[2TT ′ + T ′2

]v0 − g(T + T ′) + vr v0 − g(T + T ′) + vr

[t3, t4] = T z4` = z3` + (v0 + vr)T − g2

[3T 2 + 2TT ′

]z4u = z3u + v0T − g

2

[3T 2 + 2TT ′

]v0 − g(2T + T ′) + vr v0 − g(2T + T ′)


2.6.1 cCPI phase computation

An additional (conjugate) interferometer is also formed in the co-propagating geometry, sim-ply resulting from the use of π/2-pulses. The lower and upper interferometers interferometersshare similarities in the phase accrued during the T evolution time even though the upperinterferometer is offset in space to the lower one. Details for the free evolution phase can befound in Table 2.4.

These interferometers differ as to the state of the atom during the T ′ evolution period.This becomes important because it is precisely during this time in which the perturbingtune-out light is flashed on the atoms. The hyperfine ground states of lithium have dynamicpolarizabilities that differ by almost 1 GHz, a result of differing transition matrix elementClebsh-Gordan coefficients. See Chapter 6 for more details.

42

Chapter 3

Lithium, the smallest alkali

Lithium is the third element (Z = 3) of the periodic table and the lightest of the alkali atoms.Its smallness make it an advantageous choice for atom interferometry, particularly its smallmass and low electron number, as will be discussed more explicitly. Refer to Appendix Afor the general physical and optical properties of lithium.

3.1 Lithium, the lightest alkali

Table 3.1 shows a comparison of lithium to the heavier alkalis, rubidium and cesium, bothused atom interferometry determinations of the fine structure constant α through an h/Mmeasurement [17, 40].

As discussed in Chapter 1, a measurement of h/M in combination with an accuratedetermination the k-vector or frequency of the light, leads to a non-QED determination ofthe fine structure constant α (independently of g − 2) by way of the following relation

α2 =2R∞c

u

me

M

u

h

M.

(3.1)

The quantity h/M , is proportional to the recoil frequency ωr of the atom after interacting

Element Z Mass Recoil velocity Recoil frequencyLithium 3 7.0160 u [57] 8.5682 cm/s 2π × 63.8498 kHz

Rubidium 37 86.909 u [97] 5.8845 mm/s 2π × 3.7710 kHzCesium 55 132.905 u [97] 3.5225 mm/s 2π × 2.0663 kHz

Table 3.1: Comparison of mass and single photon recoil velocity (frequency) for lithium,rubidium and cesium’s D2-lines

CHAPTER 3. LITHIUM, THE SMALLEST ALKALI 43

Cs

55 132.9054

Cesium

Li3 6.941

Lithium

t

z

Figure 3.1: Lithium’s light mass and consequently higher recoil velocity means that forthe same amount of time, a lithium atom will transverse a larger spacetime area duringinterferometry than will a heavier alkali atom, like cesium, depicted here.

with a photon of wave vector k,

ωr =~k2

2m. (3.2)

For the two-photon Raman transitions utilized as the beam splitters and mirror in thiswork, two driving fields are present. The atom undergoes stimulated emission from |e〉(following absorption to the excited state via the ‘first’ photon) and transition to anotherstate |b〉. This leads to the following equations for the momentum ‘kick’ and resonantphoton frequency, given conservation of momentum and energy in the atom-light system foran atom of mass M , with velocity v0 in the presence of external driving fields characterizedby frequencies ω1 = c/k1 and ω2 = c/k2:

δv =~M

keff =~M

(k1 − k2) (3.3)

ωL = (ωb − ωa) + v0 · k± ωr. (3.4)

Above, a positive recoil shift, ωr = ~k2

2M, corresponds to absorption and a negative shift

corresponds to emission.A Ramsey-Borde interferometer, as discussed in Chapter 2, is sensitive to the kinetic

energy obtained by the atom from interaction with a photon. It realizes a phase difference


∆φ± given by

∆φ± = ±8ωrT + 2kazT (T + T ′) + 2δT (3.5)

where here ± denotes either the upper (normal) or lower (conjugate) interferometer, k isthe effective wave vector, az denotes accelerations along the interferometry axis and δ is thetwo-photon detuning. The atomic recoil frequency given by

ωr =~k2

M. (3.6)

It is no surprise that this imparted energy of the recoiling atom is inversely proportional toits mass. Therefore, lithium’s lightness corresponds to a higher phase accrued than comparedto heavier atoms in recoil-sensitive Ramsey-Borde interferometers.

Assuming a fixed phase uncertainty given by shot-noise

δφ± =1√N

(3.7)

where N is the total number of atoms participating in the interferometer per experimentalshot. The overall sensitivity to the recoil frequency δωr is

δωr =δφ±

8ωrT× ωr (3.8)

Comparing to state-of-the-art h/M measurement, a cesium Ramsey-Borde interferometerutilizing Bragg diffraction and Bloch oscillations with a phase difference at its output of

δφ± = ±(8n(n+N)ωrT − nωmT

)+ nkgT (T + T ′) (3.9)

where n is the Bragg diffraction order, counting the number of 2n~k kicks the atom receivesand N is the number of Bloch oscillations.

Neglecting large momentum transfer techniques like Bloch oscillations, a cesium interfer-ometer with n = 5, T = 100 ms, and N = 105 due to a reduction from velocity selection,has a project sensitivity to ωr of

δωCsr =1/√N

8n2ωrT× ωr ≈ 4× 10−8ωr. (3.10)

At an interrogation time of only 10−ms, a Ramsey-Borde lithium interferometer couldachieve comparable sensitivities

δωLir =1/√N

8nωrT× ωr ≈ 6× 10−8ωr, (3.11)

where N = 107 since we forgo velocity selection and address the majority of the trappedatoms via fast interferometry pulses.


3.2 Lithium, the simplest alkali

Moving away from hydrogenic systems, atomic structure computations increase in complex-ity and decrease in accuracy due to the presence of electron correlations and the difficulty inapproximating relativistic corrections. The Dirac equation looses utility after one electron,with Dirac-like methods lacking the computational precision needed to be compared with ex-perimental measurements [98–101]. Computations based on the relativistic coupled-clustermethod and relativistic many-body perturbation theory do not include the electron corre-lations accurately enough to be competitive with methods based on the NRQED approachand explicitly correlated functions [102]. Ultimately, quantum electrodynamics is neededto consistently describe these relativistic, correlated electrons in a many-electron system.Perturbation theory affords a description in an extension of QED through a perturbativeexpansion in powers of the fine structure constant α referred to as nonrelativistic quan-tum electrodynamics (NRQED). Based upon the foundational nonrelativistic, Schrodingerpicture, NRQED allows for the determination of relativistic and QED corrections perturba-tively. For example in NRQED, relativistic and QED corrections to the energy levels areaccounted for in the following expansion:

E(α) = Mα2ε(2) +Mα4ε(4) +Mα5ε(5) +Mα6ε(6) + ... (3.12)

where the expansion coefficients ε(i) are expressed in terms of the first- and second-ordermatrix elements of the nonrelativistic wave function Ψ

ε(i) = 〈Ψ|Oi|Ψ〉+∑j

|〈Ψ|Oi|φj〉|2

EΨ − Eφj. (3.13)

The wave function Ψ is determined from the time-independent Schr odinger equation

HΨ = EΨ

where H is the nonrelativistic Hamiltonian (infinite mass system) given by

H =∑i

p2i

2M−∑i

Zqieri

+∑i>j

qieqje

rij(3.14)

where ri is the position of electron i with charge qie from the nucleus (Z).The electron-electronreplusion is given by the term 1/rij with the inter-electronic separation rij = |ri − rj|.

The quality of Ψ dictates the accuracy of the numerical calculations. The presence of therepulsion term 1/rij precludes separability for the Hamiltonian and consequently an exactsolution to the Schr odinger equation. However, in 1929 Hylleraas proposed when consideringhelium, to expand the wave function in terms of an explicitly correlated variational basis set[103],

Ψ∞(r1, r2) =∑i,j,k

aijkri1rj2rk12e−αr1−βr2YMl1l2L(r1, r2) (3.15)


where YMl1l2L is a vector-coupled product of spherical harmonics with total angular momentumL and magnetic quantum number M . The coefficients aijk are linear variational parameterswhereas α and β are nonlinear parameters, imposing the distance scale for the wave function.

For up to three-electrons there exists such an explicitly correlated variational basis forwhich analytical computational methods have been developed [104]. This so-called Hylleraasbasis has been fully implemented in the two- and three-electron case [105–107] is precludedfor use in systems with more than three electrons due to the difficulties of performing theHylleraas multi-center integrals.

Therefore, lithium’s three electron positions it right at the threshold of accessibility, forwhich these explicitly correlated variational wavefunctions in Hylleraas coordinates can beapplied. Calculations for the lithium atom can be done very accurately within NRQEDbecause numerical solutions of the Schrodinger equation can be obtained accurately for thelow-lying states of three-electron atoms.The high accuracy of these calculations can be usedin conjunction with the high precision experiments to build unique measurement tools suchas the testing of higher-order relativistic and QED corrections to the transition frequencies.These first principle calculations can serve as atomic-based standards for quantities that arenot amenable to precision measurement. Measurements of the polarizability ratio betweenan atom and lithium can be measured to a higher degree of precision than can be done solelywith the individual atom. Measurements of the ratio in conjunction with high-precision abinitio calculations could lead to a new level of accuracy in polarizability measurements foratomic species commonly used in cold-atom physics.

3.2.1 The Hylleraas basis

The internal dynamics of lithium can be described by (3.14) for three-electrons. Transformingto the center-of-mass frame gives

H0 = − 1

2µ

3∑i=1

∇2i −

1

m0

3∑i>j≥1

∇i · ∇j + Z3∑i=1

qiri

+3∑

i>j≥1

qiqjrij

(3.16)

with reduced atom mass µ = mm0/(m + m0), electron mass m, and nuclear mass m0. Thecharge of the electron’s is now denoted by q. Variationally solving the energy eigenvalueequation or stationary Schr odinger equation

H0Ψ0(r1, r2, r3) = E0Ψ0(r1, r2, r3) (3.17)

yields a variational wave function as a linear combination of the following terms

Ψ0 = A(φ(r1, r2, r3)) (3.18)

in terms of the explicitly correlated basis functions in Hylleraas coordinates

φ(r1, r2, r3) = rj11 rj22 r

j33 r

j12

12 rj23

23 rj31

31 e−αr1−βr2−γr3YLM(`1`2)`12,`3

(r1, r2, r3)χ1 (3.19)


and the three-particle anti-symmetrizing operator A,

A = (1)− (12)− (13)− (23) + (123) + (132), (3.20)

which ensures that the wave function satisfies the Pauli principle.The three-electron-spin wave function is denoted by

χ1 = α(1)β(2)α(3)− β(1)α(2)α(3) (3.21)

with spin angular momentum equal to 12. This can be interpretted as that electrons 1 and

2 first couple to a spin singlet state with S12 = 0 and then this state is coupled to the finalelectron to produce a final state with total spin of 1

2[107].

The product of spherical harmonics for the three-electron case, YLML

(`1`2)`12,`3, is a given by

YLML

(`1`2)`12,`3(r1, r2, r3) =

∑mi

⟨`1m1`2m2

∣∣`12m12

⟩⟨`12m12`3m3

∣∣LML

⟩×Y`1m1(r1)Y`2m2(r2)Y`3m3(r3). (3.22)

The angular momentum coupling schemes for a three-body system∣∣(`1, `2)`12`3;LM⟩,∣∣`1(`2, `3)`23;LM

⟩,∣∣(`1, `3)`13`2;LM

⟩(3.23)

such as lithium are not unique but should be physically equivalent since unitary transforma-tion exist among them.∣∣`1(`2, `3)`23;LM

⟩= (−1)`1+`2+`3+L

∑`′12

√(2`′12 + 1)(2`23 + 1)

×

`1 `2 `′12

`3 L `23

∣∣(`1, `2)`′12`3;LM⟩. (3.24)

Only a state that satisfies the following conditions

`1 + `2 + `3 = L for parity (−1)L,`1 + `2 + `3 = L+ 1 for parity (−1)L+1,

needs to be included as part of the Hylleraas bases.The outline of the procedure for such a computation can be found in Ref. [107–109].

To summarize, in the Hylleraas method a series of calculations are performed of increasingdimension while keeping non-linear parameters (α, β, γ) constant. All terms are includedsuch that they satisfy

j1 + j2 + j3 + j12 + j23 + j31 ≤ Ω (3.25)

where here Ω is an integer that is progressively increased until the energy eigenvalue convergesin value [87]. For a particular state, including additional angular momentum configurations,which albeit redundant physically, has been shown to increase the rate of convergence[107].


Table 3.2: Angular momentum configurations for the S, P , D states of lithium.

state no. `1 `2 `12 `3 L MS 1 0 0 0 0 0 0

P1 0 0 0 1 1 02 0 1 1 0 1 03 1 0 1 0 1 0

D

1 0 0 0 2 2 02 0 1 1 1 2 03 0 2 2 0 2 04 1 0 1 1 2 05 2 0 2 0 2 06 1 1 2 0 2 0

3.3 Dynamic polarizability

In Chapter 1, the ac-Stark of an initial state |i〉 is defined as a sum over all other states |k〉given by

∆E|i〉 = −∑k

2ωki|〈i|ε · d|k〉|2|E (+)0 (r)|2

~(ω2ki − ω2)

(3.26)

with ωki := (Ek − Ei)/~.From this expression, for a sum over hyperfine states, |k〉 = |F ′m′F 〉, the polarizability

can be defined in terms of the following two-tensor

αµν(ω) =∑F ′m′F

2ωF ′F 〈FmF |dν |F ′m′F 〉〈F ′m′F |dν |FmF 〉~(ω2

F ′F − ω2). (3.27)

A decomposition of the polarizability leads to an expression for the ac-Stark shift of theenergy for hyperfine state |FmF 〉:

∆E(F,mF ;ω) = −α(0)(F ;ω)|E (+)0 |2 − α(1)(F ;ω)(iE (−)

0 × E (+)0 )z

mF

F

−α(2)(F ;ω)

(3|E (+)

0z |2 − |E(+)0 |2

)2

(3m2

F − F (F + 1)

F (2F − 1)

)(3.28)

with the following scalar, vector and tensor polarizabilities:

α(0)(F ;ω) =∑F ′

2ωF ′F |〈F‖d‖F ′〉|2

3~(ω2F ′F − ω2)

(3.29)


α(1)(F ;ω) =∑F ′

(−1)F+F ′+1

√6F (2F + 1)

F + 1

1 1 1F F F’

ωF ′F |〈F‖d‖F ′〉|2

~(ω2F ′F − ω2)

(3.30)

α(2)(F ;ω) =∑F ′

(−1)F+F ′+1

√40F (2F + 1)(2F − 1)

3(F + 1)(2F + 3)

1 1 2F F F’

ωF ′F |〈F‖d‖F ′〉|2

~(ω2F ′F − ω2)

(3.31)Ultimately, when we think about an atom’s polarizability, we think about its interac-

tion with an external field. Such an atom-field interaction corresponds to the interactionHamiltonian

Hint = HStark = −d · E (3.32)

with energy eigenvalue arising from the expectation value of the effective Hamiltonian appliedto the |FmF 〉 state

∆E|FmF 〉(ω) = 〈FmF |HStark|FmF 〉 (3.33)

with HStark given by

HStark = −α(0)(F ;ω)|E (+)0 |2 − α(1)(F ;ω)(iE (−)

0 × E (+)0 )z

FzF

−α(2)(F ;ω)

(3|E (+)

0z |2 − |E(+)0 |2

)2

(3F 2

z − F2

F (2F − 1)

)(3.34)

with the scalar α(0), vector α(1) and tensor α(2) polarizabilities defined as above.In Ref. [61] a basis-independent description for this shift is given by way of the tensor

polarizability operator αµν

αµν(ω) = α(0)(ω)δµν + α(1)(ω)iεσµνFσF

+3α(2)(ω)

F (2F − 1)

[1

2(FµFν + FνFµ)− F2δµν

3

](3.35)

with effective HamiltonianHStark = −αµν(ω)E

(−)0µ E

(+)0ν . (3.36)

The Kramers-Heisenberg formula expresses the polarizability in terms of the dipole matrixelements as

αµν(ω) =∑i

2ωi0〈g|dµ|i〉〈i|dν |g〉~(ω2

i0 − ω2)(3.37)

which for a spherically symmetric atom, possessing identical dipole components dµ, the scalarpolarizability becomes

α(ω) =∑i

2ωi0|〈g|dµ|i〉|2

~(ω2i0 − ω2)

. (3.38)


A variety of definitions exist for the dynamic polarizability in terms of atomic properties

α(ω) = 2πε0c3∑k 6=i

Akiω−2ik

ω2ik − ω2

gkgi

(3.39)

α(ω) =2

3~∑k 6=i

|〈k|qer|i〉|2ωikω2ik − ω2

(3.40)

α(ω) =1

3~∑k 6=i

Sikωikω2ik − ω2

. (3.41)

(3.42)

A number of different computational methods for atomic polarizability exist since mostthat are used to determine the atomic wave function and energy levels can be adapted to apolarizability computation. An overview but not exhaustive list of such approaches can befound in Ref. [81]; their list emphasizes methods which have achieved the highest accuracyor utility. The relativistic single-double all-order many-body perturbation theory (MBPT-SD) calculation of the dynamic polarizability of lithium [98] is fully relativistic and treatscorrelations effects to a high level of accuracy. However, it does not achieve the same levelof precision and has a less exact treatment compared to the variational calculation [110] nordoes the MBPT-SD calculation consider finite mass effects. In this particular methodology,the computational uncertainty related to the convergence of the basis set can be determined.The difference between calculated and experimental binding energies yields an estimate forthe size of the relativistic correction to the polarizability.

Rewriting Eq. 3.41 in terms of the dipole matrix elements and oscillator strengths fikyields

α(ω) =2

3~∑k 6=i

|〈k|d|i〉|2ωikω2ik − ω2

=∑k 6=i

fik

~((ωk − ωi)2 − ω2

) . (3.43)

This description is in the spirit of Ref. [92] which derives the polarizabilities for the2S level of lithium based upon another calculation, both utilizing the variational Hylleraasmethod. Uncertainty in the ground state polarizability creeps in primarily due to the un-certainty in the 〈2s‖d‖2p〉 matrix element. A comparison to this matrix element derivedfrom an analysis of the ro-vibrational spectrum of the Li2 dimer still yields an overestimationaccording to the authors due to the complexity of the spectrum.

The work presented in this thesis focuses on the dynamic polarizability for lithium’sground state, 2S. Highly accurate results for lithium’s atomic properties can be obtainedwith only ab initio wavefunctions due to its simple electron structure.

3.4 Lukewarm Lithium

In lithium, the hyperfine splitting between the F ′ = 3 and F ′ = 2 excited states of the 2P3/2

state is only 1.6Γ, where natural linewidth (of the D lines) is Γ = 2π × 5.9 MHz. Despite


0.05 0.10 0.15

-3000

-2000

-1000

1000

2000

3000

Hylleraas

MBPT-SD

TDGI

CI-Hylleraas

ω [a.u.]

dynamic polarizability [a.u.]

Computed dynamic polarizability for the Li ground state

Figure 3.2: A plot of the results from various computational methods for the dynamicpolarizability in the Li ground state.

Method Year Value [a.u.]Expt. 164.2(11)Hylleraas [∞Li] 2010 164.112(1)Finite mass [7Li] 2010 164.161(1)Relativistic [7Li] 2010 164.11(3)Expt. (2s− 2p1/2) -37.14(2)Hylleraas [7Li] 2013 -37.14(4)RLCCSDT -37.104

Table 3.3: Computed and measured scalar polarizabilities and Stark shift values for lithium.


speed [m/s]

Maxwell-Boltzmann Speed Distribution for Super- and Sub-recoil Atoms

0.5 1.0 1.5 2.0

Pro

bab

ilit

y d

en

sity

0

6 μK (Trec

)300 μK

Figure 3.3: The Maxwell-Boltzmann distributions for atoms at the recoil temperature (blue)and at 300 µK (red).

the inverted structure of the level, this small excited state splitting leads to off-resonantexcitations of the F = 2 → F ′ = 2 transition and hence subsequent, frequent decay downinto the F = 1 lower energy ground state. The presence of repumping light, light from theF = 1 lower energy ground state to the F ′ = 2 excited state, is therefore crucial to trappingand cooling lithium; atoms must be pumped back into F = 2 the state being addressed bythe cooling light. The small separation between hyperfine states of the excited 2P3/2 state inlithium thwarts efficient polarization gradient cooling ultimately resulting in an atom source,prior to interferometry, that is at approximately 300 µK.

The inability to cool further given our experimental constraints means that not only arethe atoms on average at a higher velocity but the velocity width of the sample is muchbroader, as can be seen in Figure 3.3. While some experiments employ a velocity selectionto reduce this width, doing so would result in a dramatic loss of atoms participating ininterferometry. From the Maxwell-Boltzmann probability density distribution for atoms ata temperature T

fv(T ) = 4πv2

√M

2πkBT

3

e−Mv2/2kBT , (3.44)

the probability of finding an atom at a velocity up to the recoil atomic velocity at a tem-perature of 300 µK can be determined by integrating the density distribution up to this


bound ∫ vr

0

fv(300 µK)dv. (3.45)

Doing so yields that about 0.1% of atoms have a velocity below or equal to the recoil velocityof lithium. If we were to use this as a cut-off in velocity selection, we would exclude 103 atomsof the 107 atoms initially trapped. This translates into almost a reduction in experimentalsensitivity of two orders of magnitude!

3.4.1 Lithium atom interferometry in the space domain

Previously, atom interferometry with lithium had been performed with a supersonic beamatomic beam seeded in argon gas (contrary to the light-pulsed atom interferometry discussedin Chapter 2). In this instance, the split atomic wave function is allowed to propagate overthe space-domain, contrary to the time-domain utilized in light-pulsed atom interferometryschemes. For an atomic beam with average velocity vz, diffraction is generated transverse tothe propagation direction of the atomic beam by a periodic diffraction grating, potentiallygenerating many momentum components for the scattered wave. In the interferometry ex-periments with lithium, elastic Bragg diffraction on standing waves of light (λ = 671 nm)performed the function of a beam splitters and mirror. A matterwave given by

ψ(r) = exp[ik · r] (3.46)

incident on the light-grating Gj, with diffraction period d and wave vector kj = 2π/d, willproduce a diffraction order p given by

ψp(r) = cj(p) exp[ik · r + ipkj · (r− rj)] (3.47)

where rj is the position of the grating and cj(p) is the amplitude of order diffracted wave.For the n-th momentum component, the average momentum transferred to the waves ischaracterized by a diffraction angle in the far field given by

θn ≈δpnpbeam

=λdBd. (3.48)

In the far field, the resolution of different diffraction orders becomes contingent on thetransverse momentum distribution of the incoming beam being smaller than the transversemomentum imparted to it by the grating, δpn. This is the same as requiring that the trans-verse coherence length be larger than a few grating periods (accomplished via collimatingthe beam).

A Mach-Zehnder configuration utilized in the thermal atom interferometry experimentswith lithium was realized via an output with an upper path created by orders p, −p, and 0and lower path created by orders 0, p, and −p, as shown in Fig. These trajectories interfereto create a resultant wave function given by

ψu/`(r) = cu/` exp[i(k · r + φu/`)] (3.49)


where cu/` is the product of the amplitudes at each grating and φu/` is the laser phase. Fringevisibility of V = 84.5± 1% was observed with the first diffraction order.

The ability to interrogate one arm of the interferometer, which are spatially resolved andin the same internal energy state, is a benefit of performing atom interferometry in this way.However, when a static perturbation is applied to one beam for interferometry performed inthe space domain, typically the phase shift resulting will depend upon the atom velocity, v.For an atom entering a region with a nonzero electric field E, the phase shift of the atomicwave is

φ=2πε0α

~v

∫E2(z)dz (3.50)

where α is the static polarizability of the atom. A measurement of the static polarizabilityof lithium, determined to be

α = (24.33± 0.16)× 10−30m3

= 164.19± 1.08au, (3.51)

performed by applying an electric field to one of the trajectories in via an inserted septum [44],confirms that the velocity is a significant systematic in these experiments. The uncertainty,equal to 0.66%, is dominated by the uncertainty in the most probable velocity. Only a theAharonov-Casher or He-McKellar-Wilkens phase shift is an exception to this dependency,being independent of the atom velocity in the beam. However, measurements of the ACphase, while confirming the velocity independence, only reached statistical uncertainties ofapproximately 3% [111]. This is a result of the phase’s dependence on the hyperfine Zeemansublevel of the atom. The experimental challenges in optically pumping a supersonic atomicbeam, particularly in controlling residual magnetic fields, are nontrivial [112].

3.5 Advantages of light-pulsed interferometry with

lithium

The benefit afforded with a thermal atom interferometer in the enjoyed spatial separationbetween the interferometer arms is countered by a reduction in experimental sensitivity thatresults from a necessary velocity selective process of the atom source. The coherence lengthof an atomic beam is given by

`coh =1

σk=〈λdB〉

2π

〈p〉σp

(3.52)

where σp is the width of the beam’s momentum distribution. The size of the optical pathdifference between interferometer arms or realized contrast of the detected flux is dependentupon the longitudinal coherence length of the interferometer. Additionally, a highly colli-mated atomic beam is necessary with thermal atom interferometry since the atom opticssuperimpose only momentum states. This velocity selectivity strongly reduces the flux ofatoms and the sensitivity of the experiment.


Light-pulsed atom interferometry offer advantages stemming from the ease at whichatoms can be manipulated by light but also the extent at which the interaction of atomsand light are understood. In such interferometry experiments, tagging the momentum stateswith the internal energy states of the atom is a consequence of two photon Raman tran-sitions but such a coupling then allows for state selective detection of the outputs. Forinterferometry with a thermal atomic beam, such a coupling can be achieved albeit onlywith a single-photon process. Atoms like calcium and magnesium have been used in thermalRamsey-Borde interferometry to realize the same configurations as mentioned in Chapter2. Such experiments are limited by the lifetime of the atom’s metastable state however. Incomparison, the hyperfine ground state which lithium is transitioned to (or any alkali forthat matter) is considered infinitely long-lived. There is not really a branching for decayback into the other hyperfine ground state and thus interferometry done with these atomswill not be limited by the lifetime of the second state. This coupling or momentum to atomicspin which is discernible with absorption imaging allows us to refrain from the lossy velocityselection required with lithium for atom interferometry in the space domain. Light-pulsedatom interferometry in general allows for greater experimental sensitivity to be achieved inrecoil-sensitive interferometry since two-photon transitions are possible.

56

Chapter 4

Experimental Methods

This chapter describes experimental methods used to prepare the atom sample for inter-ferometry as well as the procedures, techniques, and hardware common to both recoil- andpolarizability-sensitive measurements detailed in Chapters 5 and 6, respectively. In light-pulse atom interferometry, the sensitivity scales with the pulse separation time during in-terferometry, the time during which the atom exists in a superposition of states accruinga phase difference projected at detection. Thus, an experiment’s precision is increased asthe duration of this time is increase, contrary to space-domain atom interferometers whomrequire a lengthening in space and consequently of the actual experimental apparatus. Lowertemperatures allow these experimental times to be extended by hindering the ballistic ex-pansion of the atoms out of the interferometry beam. However, it is important to avoidadditional measurement systematics and maintain a dilute, noninteracting sample. Theexperimental techniques of laser cooling and trapping are commonplace in modern atomicphysics experiments and here, I will only discuss details pertinent to this experiment whichimplements Doppler cooling with magneto-optical trapping. A more substantial explanationcan be found in Appendix D and in many other papers [113–115] and textbooks [116–118].

Prior to performing interferometry with lithium, the atom source must be carefully andprecisely created. An experiment cycle consists of the following steps:

1. Atomic beam creation.

2. Two-dimensional cooling and trapping in a magneto-optical trap.

3. Loading and subsequent cooling and trapping in a three-dimensional magneto-opticaltrap.

4. Further cooling by ramping settings to and holding in a compressed magneto-opticaltrap.

5. Confinement of atoms in optical molasses while experimental (magnetic) fields stabilize.

6. Preparation of atoms into the magnetically insensitive |F = 2,mF = 0〉 ground stateby optically pumping off the D1-line.

CHAPTER 4. EXPERIMENTAL METHODS 57

dwell

10 ms

MOT load

0.8 s

cMOT ramp

5 ms

cMOT hold

50 ms

OM hold

2 ms

pump

70 μs

Ramanpulses

160 μs

repump

20 μs

image atoms

150 ms

image background

150 ms

BMOT

2D shutter

pusher shutter

imaging AOMs

3D trap AOM

3D repump AOM

3D TA switch

Raman pulse gen.

OPEN

opt. pumping AOM

ftrap

3D

frepump

3D

Ptrap

3D

Bbias

τimg

τimg

τOP

Prepump

3D

OPEN

OPEN

OPEN

OPEN

OPEN

Experimental Sequence and Settings

dig

ital

con

trol

an

alo

g c

on

trol

Figure 4.1: The time sequence and analog and digital settings for an experimental run areshown here with respective time durations (not to scale). The majority of experimental timeis devoted to obtaining and preparing the atom source.

7. Interferometry with counter-propagating two-photon Raman transitions pulsed on ina π

2− π

2− π

2− π

2sequence. Momentum reversal between pulse pairs is optional.

8. Absorption imaging of the |F = 1〉 hyperfine ground state.

Lithium’s low vapor pressure at room temperature requires that the atoms be heated inan oven in order to realize an atomic beam. A first stage of cooling in a two-dimensionalmagneto-optical trap catches atoms from the effusive oven and pushes them down a differ-ential pumping stage into a three-dimensional trap. Once trapped, the atoms are opticallypumped into the magnetically insensitive mF = 0 sublevel of the F = 2 ground hyperfinestate. Then, interferometry follows, during which we flash on four π/2-pulses, either with orwithout a reversal of momentum between the two pairs. An image of the atoms is taken ofthose atoms in the |F = 1〉 ground state following this sequence. A Wollaston prism placedimmediately before our camera allows for two images to be made during a single exposure of


the camera. From these absorption images, a picture of the cloud is discerned and the nor-malized atom number, spatial extent, position and density of the cloud can be determined.Interference fringes can be traced out in time by varying a parameter, like the duration of afree evolution time-step or frequency difference of the Raman laser, and plotting the atompopulation.

4.1 Lithium Spectroscopy

The experiment begins with a homemade vapor cell of lithium. This 60-cm long pipe iscapped with two AR-coated windows, wound in heating tape and thermal insulation, andfilled with inert argon gas to a pressure of 200 mTorr. Inside, is a chunk of lithium hasbeen placed, which when heated to 350 C, performs the crucial function of the frequencyreference in the experiment. The frequency of a first ‘master’ external cavity diode laser(ECDL) is referenced to this spectroscopy source. All other lasers in the experiment arereferenced to the master ECDL either directly through injection locking or indirectly byinjection or frequency-offset locking to a directly referenced laser.

The lithium’s low vapor pressure requires atoms to be heated considerably above roomtemperature, to approximately 350C, in order to create an appropriate atom vapor for thespectroscopy reference. At such a high temperature, atom speeds can be as fast as a fewkilometers per second, resulting in spectral line broadening of a transition by as much asa few GHz. The comparatively small hyperfine splitting of the ground state at just under1 GHz( 0.803 GHz) necessitates that Doppler-free spectroscopy be employed. Doppler-freespectroscopic methods interact only with a single velocity class in the sample and hence, areimmune to the unavoidable broadening of the hot sample.

Many Doppler-free spectroscopic techniques utilize methods based upon the saturationof absorption, a two beam pump-probe scheme. In saturated absorption spectroscopy, thepopulation difference between an upper (N2) and lower (N1) energy level, connected by anoptical transition, is modified a strong ‘pump’ laser beam. For beams with frequency ω andcounter-propagating through the vapor cell, atoms with velocity v = (ω−ω0)/k that Dopplershifts them into resonance with the pump light will thus be excited to the upper level ofthe reference transition. This reduces the population difference of the lower energy, groundstate effectively ‘burning a hole’ in the level. This ‘hole burning’ or reduction in population,caused by a beam with intensity I, has a line width given by

∆ωhole = Γ

(1 +

I

Isat

)1/2

(4.1)

where Γ and Isat are the line width and saturation intensity of the transition. Close toresonance ω ω0 the pump reduces the absorption of a probe beam and thus results in anarrow peak of the transmitted probe intensity.

In this experiment, the ‘master’ ECDL if frequency referenced to the cross-over (cross-over resonance) between the transitions to the 2P3/2 excited level from the ground hyperfine


states, |2S1/2, F = 1〉 and |2S1/2, F = 2〉. Cross-over resonances occur when the pumpbeam burns two holes in the velocity distribution, producing two peaks in the spectrum forwhich the laser frequency corresponds to the two transition frequencies. An additional peakappears if the hole burnt by one transition reduces the absorption for the other transition.

4.1.1 Modulation Transfer Spectroscopy

Modulation transfer spectroscopy (MTS) is a Doppler-free spectroscopic method that achievessub-Doppler resolution, a requirement for laser frequency-locking. Like in standard satura-tion absorption or hyperfine pumping spectroscopy, MTS employs two counter-propagatingpump and probe laser beams, but with powers set to be approximately equal. While MTSis able to obtain high sub-Doppler resolution, it does so at the expense of a limited capturerange (< 100 MHz) or the system’s tolerable frequency excursion.Single beam techniques,while hindered by the Doppler-broadening of features, are however able to tolerate a muchlarger excursion, often 100s of MHz.

Two clear advantages do set MTS apart from other frequency locking methods. First,MTS generates dispersive-like line shapes atop a zero background, removing the extra com-plex and costly demodulation step required in frequency modulation spectroscopy (FM). Thebackground free signal also means that the zero-crossings of the MTS signals are accuratelycentered on the corresponding atomic transitions and the error signal is immune to lockingfrequency drifts potentially caused by reference level fluctuations [119]. Secondly, the signalsare dominated by contributions from closed atomic transitions, especially beneficial whenthe atomic spectrum contains several closely spaced transitions.

In MTS, an intense monochromatic pump beam with frequency ωc passes through anelectric-optical modulator (EOM), driven at a frequency ωm. The transmitted light getsphase-modulated in this process [120],

E = E0 sin(ωct+ δωmt) (4.2)

and when written more explicitly in terms of the modulation index δ and the nth orderBessel function Jn(δ) is given by

E = E0

∞∑n=0

[Jn(δ) sin(ωc + nωm)t+ (−1)n+1Jn+1(δ) sin(ωc − (n+ 1)ωm)t

]. (4.3)

The pump beam has a carrier wave (n = 0) at ωc flanked by first-order sidebands (n = ±1)at frequencies ωc±ωm. The modulation index is typically less than one and thus the energycarried in sidebands at higher order is negligible compared to that of the carrier and first-orders and will not be considered here.

The probe beam does not transverse the EOM but passes through the lithium vapor celldirectly. The phase-modulated pump beam, after it passes through the EOM as discussed,is aligned co-linearly to the probe beam. If the sub-Doppler resonance condition is satis-fied, then sufficiently nonlinear interactions between these two beams result in a four-wave


mixing, transferring modulation to the probe beam. Four-wave mixing occurs when the non-linearity or third order susceptibility (χ(3)) of the hot lithium gas (the absorber) mediatesa combination of the two frequency components of the pump with the counter propagatingprobe. This mixing generates a fourth wave, a sideband on the probe beam, and occurs foreach sideband of the pump beam.

Because this transfer of modulation occurs only if the sub-Doppler resonance conditionis satisfied, the error signal generated by MTS is stable against fluctuations in polarization,temperature and beam intensity. Additionally, the position of the zero-crossing remains onthe center of resonance, unaffected by many systematics that plague other techniques likemagnetic field or polarization shifts.

A photodetector detects the probe beam after it passes through the vapor cell. Thesidebands produced in the atomic vapor beat with the probe carrier frequency and producealternating signals at the modulation frequency ωm.

The beat signal is given by

S(ωm) =C√

Γ2 + ω2m

∞∑n=−∞

Jn(δ)Jn−1(δ)[L(n+1)/2 + L(n−2)/2 cos(ωmt+ φ) (4.4)

×(D(n+1)/2 +D(n−2)/2) sin(ωmt+ φ)] (4.5)

with

Ln =Γ2

Γ2 + (∆− nωm)2(4.6)

and

Dn =Γ(∆− nωm)

Γ2 + (∆− nωm)2(4.7)

and where Γ is the natural line width, ∆ is the detuning from the line’s center, φ is thedetector’s phase with respect to the applied modulation field and C is a constant representingall other properties of the atom medium and probe beam.

For a strong carrier wave, the above equation simplifies to

S(ωm) =C√

Γ2 + ω2m

J0(δ)J1(δ)[(L−1 − L−1/2 + L1/2 − L1) cos(ωmt+ φ)

×(D1 −D1/2 −D−1/2 +D−1) sin(ωmt+ φ)]. (4.8)

As shown in Figure , light out of the ‘master’ ECDL is split with a series of half-waveplates and polarizing beam splitting cubes (PBS). The first PBS reflect a majority of thelaser power to a mirror and second PBS which then send the beam through an acoustic-optical modulator. The beam is doubled passed and shifted down in frequency by 400 MHz.The beam passes through a quarter wave plate both exiting the AOM and again after beingreflected by the retro-mirror. This results in a λ/2 rotation in total such that when againencountering the PBS at the beginning of the optical path, the beam in now transmittedthrough. It is then coupled into a fiber and sent to the spectroscopy board such that the


hwp

PBS prism pair

master ECDL

EOM

PBS

hwp

hwphwp

hwp

hwp

PBS

PBS

PBS

PBS

3D injection lock

2D injection lock

qwp

pinhole

pinhole

200 MHz AOM

lithium vapor cell

Figure 4.2: Modulation transfer spectroscopy set-up. Light originates from a commercialToptica external cavity diode laser and is sent through a series of half-wave plates and beamsplitters. Light split off here is used to injection lock the FP diode laser further downstreamin the optical path. The majority of the beam is doubled passed through an 200 MHz acousticoptical modulator. This frequency shifted light is fiber-coupled and sent to the spectroscopyset-up, as shown. Locking the down-shifted light to the 2S, F = 2 to 2P3/2 state results inlocking this master laser to the crossover between the ground states.


OP27

IN

10k

1M

10k

OP27OP27

OP27

OP27

OP27

OP27

OP27

MON

10k

20k

1k

200k

470pF 20k

10k

10k

10k

100

To Current

100k

1k

10k

100k

10k

470pF1M10k

20k

1k

30k

To PZT

OP27

-12V +12V

Figure 4.3: Circuit schematic of master ECDL frequency lockbox.

resulting pump and probe are approximately equal in power; 0.5 mW is realized in eachbeam with waists of 1.4 mm. The set-up is similar to that of Ref. [121]. The pump beamis first steered through the electric-optical modulator, modulated at a frequency of 13 MHz,before passing through the lithium vapor cell. After the EOM, the pump beam consists ofa carrier wave flanked by first-order sidebands at the modulation frequency. The initiallyunmodulated probe beam is sent directly into the vapor cell, with the beams aligned so as tooverlap centrally in the cell but remain spatially separated at the mirrors immediately afterthe windows. After the vapor cell, the probe passes through another PBS and is detectedvia a photodiode as shown.

A homemade feedback circuit is connected to the current and piezo of the master ECDL.A Schmitt trigger inside the circuit generates an internal ramp, sweeping the diode currentand thereby scanning the laser frequency. There are PI feedback loops for the diode currentand piezo which function to fix the laser frequency to the desired locking point. From theerror signal generated, we lock the frequency-shifted light to the |F = 2〉 to D2-line whichdue to the initial down shifting of its frequency by approximately 400 MHz translates intoultimately locking the master ECDL to the cross-over between |F = 2〉 and |F = 1〉 to 2P3/2

state. The four parameters responsible for realizing the correct, mode-hop free wavelengthin the master ECDL are: temperature, current, grating angle, and piezoelectric actuatorvoltage.

4.1.2 The cascade of frequency generation

The small proportion of light picked off from the master ECDL injection locks two semi-conductor Fabry-Perot (FP) laser diodes. These lasers essentially are intermediate optical


amplifiers of the well-defined, but weak, reference light from the initial ECDL. Light pro-duced by these FP lasers is frequency shifted further to generate the desired frequencies fortrapping, repumping, pushing and imaging in both the 2D and 3D cooling set-ups. Thelight from the master ECDL is aligned to the FP laser diodes so as to ‘seed’ or be injectedinto the diode’s cavity. If the injected photons dominate over the other frequency modes inthe resonator, the seed beam will be amplified resulting in the FP laser lases at the samefrequency as the reference light. The maximum power of commercially available FP laserdiodes varies for different wavelengths and unfortunately, both lithium’s D1- and D2-linesrequire light for which commercially available diodes only are rated to output maximumpowers of 20 mW. Therefore, heated 660 nm FP diodes are used in the experiment. Heatingthese commercially available diodes to 60-70C allows us to achieve the wavelengths (671nm) required for lithium.

In order to realized thermal stability at such high temperatures, two layers of boxessurround the FP laser, both designed to have separable walls in order to facilitate easieraccess to the diode. The laser diode is mounted on the typical f = 4.5 mm collimationtube and underneath the tube holder a piece of indium foil is placed for thermal conductionand then a thermoelectric cooler (TEC) for heating. Indium foil is used in place of thermalpaste which has been observed to evaporate at high temperatures and condense on the viewport of the housing. This assembly is enclosed with a black Delrin box with an AR-coatedwindow on the front face for the laser beam output and a laser diode protection circuit onthe back wall. This enclosure is then enclosed in an aluminum box, also made of separablefaces. As reported in Ref. [93], the double box design has demonstrated greater stability incomparison to a single layer enclosure. A thermistor is used in conjunction with the TECto inform a commercial temperature controller (Wavelength electronics PTCxK-CH Series)and stabilize the temperature of the diode to the desired value.

4.1.2.1 Tapered amplifiers

The FP laser diodes typically output less than 100 mW of power in a single Gaussian spatialmode when heated. After the maze of optics needed to shift the reference frequency forcooling and trapping, we realize only at most 10 mW split between repumping and coolingfrequencies. This necessitates the use of a tapered amplifier chip to amplify the reducedlaser power, to realize the appropriate intensities for employing optical molasses. Contraryto the FP diodes, a tapered amplifier is a broad area laser diode in front of a straight narrowwaveguide acting as a modal filter at the back.

The wide output facet of the TA, 100s of µm wide and a few µm thin, means that thebeam output from the TA diverges faster in the vertical compared to the horizontal axis.To mitigate this and collimate the vertical axis of the beam, a short focal-length asphericlens is mounted directly after the TA. A cylindrical lens is then placed further along in theoptical path, and outside the TA housing, to collimate the beam horizontally. Unfortunately,the short focal lengths of the collimation lenses used here produce a resulting beam shapethat is very sensitive to the placement (position and angle) of the aforementioned lenses.


Tapered amplifier

output facet

gain region

entrance facet

injected beam

This precise lens placement after replacement of a TA chip (necessary after wear, tear andheating to a balmy temperature) requires a 3D-translation stage and constant monitoring ofthe output power while the glue or epoxy is drying. We have found that when replacing theTA chip, coupling the beam into an optical fiber after all of the collimation optics allows fornot only power monitoring during gluing and drying but assesses the stability of the beamshape as well. Expansion or contraction of the epoxy used when gluing these optics shouldbe considered in the replacement process.

In our experiment, we seed tapered amplifiers with a range of powers, anywhere between5 - 20 mW. Depending on the particular TA chip, we measure approximately 200 to 500mW of amplified output power. When operating a TA at a high current, care is required toprevent the diode chip from becoming unseeded for extended periods of time. Doing so, canresult in degradation and damage to the laser. Observed symptoms of degradation includedecreased output power, shittier output beam shape, and fast oscillations or repeated spikesand dips in output power at particular current settings. Such fast spikes average to a lowerpower when observed with a less-responsive commercial power meter compared to a fastphotodiode.

4.2 Cooling and trapping

Following the oven, first a two- and then three-dimensional stage of cooling via a magneto-optical trap (MOT) are used to first catch and cool down the hot atoms. A MOT is a hybridtrap combining the position-dependent absorption cross section generated by a spatially-dependent magnetic field B = B(z) =≡ Az with a velocity-dependent radiation scatteringforce resulting from a red-detuned optical field. Optical molasses is the Doppler-coolingtechnique characterizing the process implemented via the optical field; orthogonal pairs ofcounter-propagating laser beams red-detuned from an atomic transition result in a force


803.5 MHz

91.8 MHz

18.1 MHz2P3/2

2P1/2

2S1/2

F= 2

F= 1

F= 2

F= 1

10

.05

6 G

Hz

F= 2

F= 3

F= 1F= 0

67

0.9

61

56

1 n

m6

70

.97

6 6

58

nm

(a)

(b)

(c)

(d)

(e)

(f)

Figure 4.4: Experimental frequencies broadly referenced to the fine-structure splitting inlithium for (b) MOT cooling and (a) repumping, (c) optical pumping, (d) imaging, andRaman (e) ω1 and (f) ω2.


damping the motion of the atoms [113, 122].Introducing a spatially varying magnetic quadrupole field to the aforementioned optical

molasses with a particular orientation of polarization (σ+/σ−), is all that is needed to makea MOT. Trapping happens here because as the atom moves away from the center of the trapand the magnetic field zero, the opposing beam will still be Doppler-shifted but now theenergy levels will get Zeeman-shifted.

For the simplified case of the two-level transition, such as |F = 0〉 → |F ′ = 1〉 shown inAppendix D, as the atom moves rightward away from the trap center, the excited m′F = −1level is Zeeman-shifted into resonance. The optical molasses beam impinging on the atomsfrom the right is (conveniently!) of σ− polarization and thus the atom is allowed to and willabsorb a photon from this direction, transitioning to the |F ′ = 1,m′F = −1〉 excited state. Itwill also get a ‘kick’ of photon momentum, opposing its motion and directed back to the trapcenter. At low intensity, the total force on the atoms in terms of the ratio of the incident tosaturation intensity I/I0, detuning from resonance ∆ and line width Γ becomes

F± = ±~kΓ

2

I/Isat

1 + I/Isat +[2(∆±)/Γ

]2 (4.9)

where the detuning for each beam, defined in terms of the effective magnetic moment µ′ ≡(geme − ggmg)µB is given by

∆± = ∆∓ k · v ± µ′B

~. (4.10)

Depending upon the branching ratio of the excited state, an atom may absorb a photonbut then decay to an energy level other than the ground state of the atomic transition. Ifthis occurs, the atom will no longer be addressed by the optical molasses. The full scatteringpotential will not be realized by the atom; it will be cooled only to the point at whichits leaves the two-level system. This hurdle to cooling can be overcome by incorporatingadditional frequencies of light to address atoms which have fallen into these other energylevels, not of the cooling transition. This ‘repumping’ light re-excites atoms back to theupper energy level and in doing so, in repumping them, the undesired decay is undone.The atom now has another chance to behave and decay appropriately and thus maintain itparticipation in cooling.

Repumping is crucial to laser-cooling lithium. Unlike its sister alkali’s, rubidium andcesium, which both enjoy a spectrally well-resolved cycling transition on the D2 line, thehyperfine splittings of the 2P3/2 state in lithium are comparable to its natural line width ofΓ/2π = 5.8 MHz. Even with an inverted level structure, this small excited state splittingresults in off-resonant excitations of the |2S1/2, F = 2〉 → |2P3/2, F

′ = 2〉 transition leadingto subsequent (and frequent) decay down into the |2S1/2, F = 1〉 ground state, almost 1 GHzbelow that addressed by the laser-cooling light. The lack of a closed cooling transition forlithium requires that an appreciable amount of repumping light be included in the MOTs,compensating for the atom leakage to the dark state manifold.


4.2.1 2D MOT frequency generation

Half-wave plates and PBS’s, shown in Fig. 4.2, divide light originating from the masterECDL amongst the spectroscopy set-up and FP laser diodes for eventual 2D and 3D MOTfrequency generation. At the second PBS, the reflected beam is fiber coupled after which it isthen steered backwards through the rejected port of an optical isolator (after the appropriatepolarization alignment) so as to injection lock the FP laser used in the generation of the 2DMOT cooling and repumping frequencies. This optical scheme is shown in Fig. 4.5. Prior topassing through the AOMs, a portion of the laser’s light is picked off for frequency referencinganother FP laser diode and second homemade ECDL used in the optical pumping scheme.

The remainder of power is split, each portion sent to a 200 MHz acoustic-optical modu-lator. Light used for cooling (repumping) is frequency-shifted approximately -(+) 400 MHzby double-passing the AOMs in a ‘cat’s eye’ configuration, necessary for robust alignmentthat remains independent of the beam diffraction angle and actual frequency shift or sweepimposed by the modulator [123]. The detuning used for an experimental sequence are givenin Table 4.1. The AOMs are driven by amplified voltage controlled oscillators, controlledwith the Cicero and Atticus duo. After double-passing the AOMs, the outgoing light isorthogonally polarized relative to its initial state and is picked out of the incoming beampath at the PBS adjacent to the mirrors steering light through the AOMs. The beams,which are also orthogonally polarized relative to the other, are overlapped at another PBS,sent through a half-wave plate and final PBS such that both frequencies are of the samepolarization upon entering the optical fiber seeding the 2D tapered amplifier.

4.2.2 3D MOT frequency generation

Returning once more the the master ECDL, the remaining power transmitted through thesecond PBS is sent in free space and back coupled through the rejected port of anotheroptical isolator (after aligning its polarization appropriately) to injection lock the FP laserused in the frequency generation of the 3D MOT cooling and trapping stage, as well as thepusher beam and the light for absorption imaging. Light from this FP laser passes throughan isolator and telescope prior to being split at a PBS with a majority fiber-coupled and sentto the optics used in generating the frequencies for the pusher beam and 3D MOT opticalmolasses. The set-up is shown in Fig. 4.6. The remaining power is reflected at the PBS andcoupled into a fiber sent to the imaging AOMs. The details of the imaging optics will bedescribed in more detail later in the chapter.

Fig. 4.6 shows the optical set-up used for generating the cooling and repumping 3D MOTfrequencies as well as the pusher and imaging beams. As in the case of the 2D MOT, the totalpower (now out of a fiber) is split by a series of half-wave plates and PBSs and double-passedin the ‘cat’s eye’ geometry. Contrary to what was discussed in the context of the 2D MOT,this is done through a series of three 200 MHz AOMs, shifting the light either up (repumping)or down (cooling) by approximately 400 MHz (see Table 4.1 for detuning details). Powerbalances can be tuned by adjusting the half-wave plate prior to the PBSs feeding the AOMs.


hwp

hwp

hwp

PBS

hwp

PBS

PBS

hwp

PBS

hwp

PBS

hwp

PBS

qwp

200 MHz AOM

200 MHz AOM

pinhole

pinhole

qwp

150 mm

150 mm

100 mm100 mm

hwphwp

to OP ECDLto Raman

diode laser

to 2D TA

from

Master ECDL

Figure 4.5: The set-up for the frequency generation for the 2D MOT optical molasses isshown. The frequencies for cooling and repumping light are generated with two double-passed 200 MHz AOMs, shifting either up or down, following initial optics separating aportion of the light to serve as a frequency reference for both interferometry and opticalpumping. The frequency-shifted light exiting the double-pass is orthogonally-polarized withrespect to the incoming light and the other beam. The beams are spatially overlapped at aPBS and fiber-coupled to a tapered amplifier which further increase the light intensity forcooling and trapping in the 2D MOT.

After transversing these optics, the light from the repumping and cooling paths is combinedby overlapping the orthogonally polarized beams at a PBS and fiber-coupled, sent to seedanother tapered amplifier. After amplification in the TA, the bichromatic light is againfiber-coupled and sent to a homemade 1:6 fiber-splitter, eventually becoming the six armsof the 3D MOT.

4.2.3 Vacuum system and optics

This section describes in detail the experiment’s vacuum system set-up as well as the opticsused for the 2D and 3D MOT cooling and trapping stages, including the set-ups for beamintensity amplification. The 2D MOT is based off a previous design [124], also with lithium.This is contrary to other cold atoms experiments with lithium employing instead a Zeemanslower as an intermediate between the hot atom source and the three-dimensional trap [125–128]. Advantages of a 2D MOT over a Zeeman slower include: compactness, large flux,


PBS

qwp

pinhole

200 MHz AOM

PBShwp

PBS PBS

hwp

hwp

PBShwp

PBShwp

hwp

200 MHz AOM

200 MHz AOM

pinhole

pinhole

qwp

qwp

hwp

hwp

hwp

to 2D MOT

to imaging

to 3D TA

100 mm100 mm

150 mm

150 mm

150 mm150 mm

from

Master ECDL 75 mm -25 mm

Figure 4.6: The set-up for the frequency generation for the 3D MOT optical molasses andpusher beam is shown. These frequencies for cooling, repumping and pushing are gener-ated with three double-passed 200 MHz AOMs, shifting either up or down as indicated.The frequency-shifted cooling and repumping light exiting the double-pass is orthogonally-polarized with respect to the incoming light and the other beam. The beams are spatiallyoverlapped at a PBS and fiber-coupled to a tapered amplifier which further increase thelight intensity for cooling and trapping in the 3D MOT. The pushing light is transmittedthrough the last PBS, sent to a mirror and then periscope to be aligned through the 2DMOT chamber, along the symmetry axis of the magnetic field.


Beam Transition Detuning [Γ]2D MOT trap F = 2→ F ′ = 3 -1.892D MOT repump F = 1→ F ′ = 2 0.52pusher F = 2→ F ′ = 3 0.163D MOT trap F = 2→ F ′ = 3 -3.873D MOT repump F = 1→ F ′ = 2 0.92cMOT trap F = 2→ F ′ = 3 -1.88cMOT repump F = 1→ F ′ = 2 -0.22imaging F = 1→ F ′ = 2 -3.55molasses trap F = 2→ F ′ = 3 -1.88molasses repump F = 2→ F ′ = 2 -0.22

Table 4.1: Frequencies detunings in units of line width Γ for both the 2D and 3D MOT,compressed MOT, molasses hold and imaging. The hyperfine states for the transition|2S1/2〉 → |2P3/2〉 are indicated.

and the absence of additional residual magnetic fields present which may perturb the atoms(quantization axis) during state selection or interferometry.

4.2.3.1 2D MOT chamber

A cylindrical lithium oven (51 cm3), wrapped in heating tape, is heated to just shy of400C, is the starting point for the high flux, cold atom source. The oven is made primarilyfrom stainless steel (304) except at the flange where a nickel gasket is used in conjunctionwith a variation of stainless steel more robust to corrosion (316LN), both a consequence oflithium’s reactivity. It 6 cm hangs below the two-dimensional MOT vacuum apparatus, astainless steel (304) six-way cross. Four ConFlat flanges supporting AR-coated view portsare configured at 45 while two more (one with a view port for the pusher beam) are alignedto the symmetry axis of a 2D quadrupole magnetic field. The magnetic field is created bytwo sets of permanent bar magnets (Nd2Fe14B) with gradient of 50 G/cm. This region ofthe experimental apparatus is separated from an ultra-high vacuum portion by a differentialpumping tube with inner diameter 4.5 mm, emerging off the skewed ‘X’ of 2D MOT-chamberand extending 6 inches towards the larger side of the vacuum chamber, ending at a gate valvejust prior to the chamber. A differential pumping stage is required to restrict the conductanceof particles from the hot atom-vapor thereby maintaining low pressures in the larger trappingregion with pressures observed to typically differ by more than 500:1 [93]. A flange cross alsoextends outward, perpendicular to the differential pumping stage, and is connected to anion gauge, metal-valve and turbo pump, allowing for the 2D MOT chamber to be efficientlypumped while baking.

Two pairs of retro-reflected laser beams with σ+/σ− polarization and red-detuned tolithium’s 2P3/2 excited state are steered into the chamber as shown in Fig. 4.7. They crossorthogonally above the neck of the oven, capturing atoms in a column along the symmetry


qwp qwp

oven

bar

magnet

retro-mirror retro-mirror

Figure 4.7: The portion of the vacuum chamber that is the 2D MOT is shown here as wellas the path of the beams into the chamber. A cylindrical lithium oven hangs below thesix-way cross, which is capped with AR-coated view ports held by 2.75 inch stainless steelConflat flanges. A 50 G/cm magnetic field gradient is generated with sets of permanentmagnets (Nd2Fe14B); quadrupole field lines shown in drawing. The beams are steered tomirrors placed above the upper view ports which direct the light through the chamber,through a quarter-wave place and to a mirror. This mirror then retro-reflects the light, backthrough the chamber, now of opposite circular polarization given its double pass throughthe quarter-wave plate.


axis. The 2D MOT beams originate from a commercial tapered amplifier (TOptica BoosTA),which when seeded with 2.2 mW of cooling and 5.0 mW of repumping light produces approx-imately 200 mW of total bichromatic laser power as measured prior to the shutter shown inFig. 4.7. Following the shutter, a telescope expands the beam to a 1/e2 5-mm sized waist. Amirror the directs the beam to a 50/50 beam splitter cube which reflects the light upwards,to a mirror placed above the closest view port, and transmits under the oven and 2D MOTchamber to another mirror which also directs the light to an upper mirror positioned abovethe second upper view port. The two beams are steered into the chamber and through thelower opposite view port on the other side of the symmetry axis. After exiting the cham-ber, the light passes through a quarter-wave plate, hits a mirror and is retro-reflected backthrough the chamber. Optimum trapping defined by the number of atoms captured in the3D MOT either with real-time fluorescence detection or absorption imaging occurs when theretro-reflected beam is not overlapped with the incoming beam.

Atoms are trapped and cooled radially along the symmetry axis of the 2D quadrupolefield. A pusher beam with approximately 1 mW of power in a waist of 1.2 mm and red-detuned from the 2P3/2 level kicks atoms down the differential pumping tube into 3D MOTchamber. It is aligned so as to more effectively load atoms but also not disturb the real timetrapping in the 3D chamber.

4.2.3.2 3D MOT chamber

The three-dimensional magneto-optical trap (3D MOT) is a spherical steel octagon (r = 8inches), capped by two large (r = 8 inch) view ports with six smaller AR-coated ones forbeam access positioned as shown in Fig. 4.9. This part of the vacuum chamber is connectedto the differential pumping stage through a gate valve, open during normal operation buthelpful in modular baking of the chamber. An ion pump (Varian StarCell; 40 L/s), ion gauge,and titanium sublimation pump (TSP) are connected to the chamber on the side oppositethe 2D MOT chamber via flange crosses, as shown. For 30 A of current, hollow rectangularmagnetic coils wound 64 times per side in an anti-Helmholtz configuration generate a 20G/cm magnetic field gradient at the center of the chamber.

Approximately 1.2 mW of cooling and 2 mW of repumping light seed a commercialtapered amplifier (TOptical TApro), coupled from the optical scheme shown in Fig. 4.6. Theamplified output at approximately 150 mW, passes through an optical isolator, telescope forbeam shaping and PBS which sends a small fraction of the light to a cavity for monitoring.The remainder of the beam is coupled into an optical fiber and sent into a homemade fibersplitter. The 1:6 fiber splitter distributes the total power evenly amongst six optical fibers.The light out of these fibers, 10 mW of power in each, is telescoped to achieve a 6.5-mmbeam waist for each MOT arm and then directed to the atoms.


50 mm

-25 mm

500 mm

BS

hwp

hwp

shutter

optical

isolator

2D TA

2D FP laser

2D MOT

S to N arm

2D MOT

N to S arm

100 mm

Figure 4.8: The optical scheme intended to amplify the intensity of the 2D MOT coolingand repumping frequencies. Light fiber-coupled after being frequency shifted in the AOMmaze of Fig. 4.5 seeds a tapered amplifier as shown. Its steered through a telescope, opticalisolator and shutter used to control 3D MOT loading. The light passes through anothertelescope that expands it to a 5 mm Gaussian beam waist. It is directed to a 50/50 beamsplitter which reflects one portion of the beam upwards and transmits the second portionunder the oven to a mirror which then directs it upwards.

4.2.4 Experimental sequence

Once atoms are loaded into the 3D MOT, they are further cooled in a compressed magneto-optical trap (cMOT) stage. This additional step is necessary because the experimentalsettings optimized for loading the maximum number of atoms are not optimal for coolingthe collective; both the number density and temperature dependent independently on severalparameters such as the magnetic field gradient, light intensity, detunings, and number ofatoms [129]. For a single atom, a change in the radiation forces arises from the presence ofother atoms which create a background field of scattered photons [130].

Therefore, in order to obtain lower temperatures while capturing as many atoms aspossible for interferometry, the 3D MOT cooling and repumping light are ramped closerto resonance with the 2P3/2 state over a 5-ms duration. This is accomplished by changingthe frequencies of the AOMs through the control voltages sent to the driving VCOs. Theintensity in each frequency is simultaneously reduced by ramping up the attenuation of theVCO output and the magnetic field is lowered. After ramping the frequencies, attenuationsand magnetic field, a ‘cMOT hold’ time-step of 500 ms is implemented. Time-of-flightimages taken immediately following loading of the 3D MOT indicate that the atoms are at


TSP

ion pump

oven

permanet magnet

pusher

2D MOT

3D MOT

Figure 4.9: The 3D MOT chamber as part of the larger vacuum apparatus which consistsadditionally of a lithium oven, 2D MOT chamber, differential pumping stage, an ion pumpand TSP pump. The mirror and 50/50 BS of the 2D MOT optics, the periscope for thepusher beam and the steering mirrors for one of the six 3D MOT arms are shown.

temperatures of 1 mK whereas after compression, taking a temperature has shown the atomsto be at approximately 200-300 µK, almost at the Doppler cooling limit.

A ‘molasses hold’ time-step follows the compressed MOT, during which the 3D MOTmagnetic field is completely shut off while the bias magnetic fields are maintained on. Thepurpose of this step is to create a delay between the slow decay of the eddy currents induced inthe vacuum chamber and the beginning of optical pumping, for which a defined quantizationaxis needs to be set. An unexpected consequence is a launch of the atoms in the z direction,resulting from both the presence of the laser light and bias magnetic fields. It has yet to bedetermined whether this due an imbalance of intensity resulting from alignment.

4.3 State preparation

Figure 4.10 shows a microwave spectrum taken of the atoms with and without opticalpumping. Following the loading and cooling of atoms in the 3D MOT and the opticalmolasses hold, atoms can be found distributed among magnetic sublevels in the |F = 2〉ground state. Scanning a microwave pulse through the a frequency range centered on the|F = 2,mF = 0〉 → |F = 1,mF = 0〉 transition around 803.5 MHz and imaging the F = 1population allows us to determine the approximate distribution of atoms. It confirms thatfollowing the molasses timestep, atoms are distributed among the magnetic sublevels of the


800 801 802 803 804 805 806 807

0

0.1

0.2

0.3

0.4

0.5

Frequency [MHz]

No

rmal

ized

po

pu

lati

on

MW spectrum with and without optical pumping

with pumping

without pumping

Figure 4.10: Microwave spectrum of the |F = 2,mF 〉 ground state.

|F = 2〉 hyperfine ground state.Preparing the atoms into the mF = 0 magnetic sublevel is crucial. Eddy currents in the

steel vacuum chamber persist for 10s of milliseconds even after the magnetic field coils havebeen shut off. If the interferometer phase has a dependence on the difference in internalenergies, as ours does when employing the Ramsey-Borde geometry, then the gradient pro-duced by these Eddy currents will Zeeman shift the magnetic sublevels by varying amount,dephasing our interferometers, and reducing contrast. Preparing the atoms into a particularmagnetic sublevel prior to interferometry can mitigate this effect. Particularly, opticallypumping the atoms into the mF = 0 magnetic sub-level which is insensitive to the first orderZeeman shift, allows us to overcome the limitations of these persistent Eddy currents. Westill observe an effect from the quadratic Zeeman shift.

After the optical molasses step lasting on the order of a few -ms, during which thequadrupole decays, the atoms are found distributed among all five magnetic sublevels of the|F = 2〉 ground-state manifold. After 1.5 ms of optical molasses, when the magnetic fieldpersists from the quadrupole field eddy currents has decayed to below 1 G/cm.


500 1000 1500 2000

0.5

1.0

1.5

2.0

2.5

3.0

0

Time [µs]

Gra

die

nt

[G/c

m]

Magnetic Field Gradient Decay

Figure 4.11: The decay of the magnetic field gradient over a 2 ms duration shows that thegradient from eddy currents induced by the quadrupole field persists well into the experi-mental stages following the compressed MOT.

We optically pump the atoms with linearly polarized light from the ground state F = 2to excited 2P1/2 level. We optically pump the atoms on the D1–line because the D2–line inlithium is unresolved and consequently more lossy. Selection rules forbid mF = 0→ m′F = 0for a F →′ F = F transition making the mF = 0 ground sublevel dark to the opticalpumping light.

However, excited m′F magnetic sublevels can decay into the mF = 0 ground state by σ±

transitions. The absence of an excitation path out of the |F = 2, 0〉 state means that atomswill build up in this magnetically insensitive ground state since atoms that decay into it willbe uncoupled from the excited level. In each of the six 3D MOT beams, we use 1.5 mW ofD2 MOT repumping light, detuned from |F = 1〉 → F ′ = 2〉 of the 2P3/2 state, to recoveratoms that decay to F = 1, the lower hyperfine ground state of lithium.

4.3.1 Frequency generation for optical pumping light

The frequency of the reference ECDL is resonant with the crossover to the 2P3/2 state inlithium. In order to generate the frequency need for optical pumping on the D1 line, anexternal cavity diode laser is frequency offset locked to the 2D MOT FP diode laser priorto any frequency shifting for trapping or repumping light, so resonant essentially with thecrossover. The offset lock needs to be at approximately 10 GHz, the fine-structure splitting


2P3/2

2P1/2

2S1/2

-2 +1 mF

0

2π × 92 MHz

2π × 10 GHz

-3 +3

+2

2π ×803.5 MHz

F= 2

F= 1

F= 2

F= 1

-1

Figure 4.12: Energy level diagram showing the frequencies for optical pumping.

Vbias

LO

delay line

servo

spectrum analyzer

Figure 4.13: Light from the lasers’ beat note is incident on a fast photodetector and mixedwith a local oscillator at 9894 MHz. The signal is filtered and amplified in various stages,finally power split prior to a frequency mixer. One part of the signal goes directly intothe mixer and the other transverses a delay line first, set so as to induce a phase shift andconsequently error signal for the servo.


in lithium. The optical pumping laser is beat against the reference laser and sent to afast photodetector (GaAs MSM Photodetector G4176) as shown in note is subsequentlymixed with a local oscillator (Vaunix LabBrick LMS) at 9894 MHz and 10 dB of power. Aschematic of the locking electronics is shown in Fig. 4.13. A 600 MHz delay-line (trombonelock) generates the error signal sent to the locking electronics that subsequently stabilizesthe laser current and piezo voltage.

4.3.2 Optical pumping optics

This light directly out of the optical pumping ECDL is coupled into a fiber and 3 mW oflight tuned to within a line width (Γ = 2π× 5.87 MHz) of the |F = 2〉 to |F ′ = 2〉 transitionon the well-resolved D1 line (2P1/2 state) is sent to the 3D MOT chamber. The light exitingthe fiber passes through an AOM which acts as a switch, through a telescope expanding thebeam to a 3.6 -mm Gaussian waist, periscope, polarizing beam splitter cube and waveplate.The beam is polarized along z realize a π-polarization relative to the set quantization axis,required in our pumping scheme. After 50 s of optical pumping, greater than 80% of theatoms occupy the dark state.

4.3.2.1 Quantization axis

The quantization axis is defined by three pairs of bias magnetic field coils which frame theanti-Helmholtz coils of our trap, generating a zero field to within 10 mG along x and y andapproximately 1 G along the z. The quadrupole field remains appreciable for millisecondsdue to eddy currents in the steel vacuum chamber, despite the relatively quick 250 µs decayof the current in the MOT coils after the current is shut off. To limit the thermal expansionof the atom cloud while the eddy currents decay prior to interferometry, the 3D MOT beamsare left on as optical molasses. Unfortunately, the small detuning of these beams from theunresolved excited 2P3/2 state in lithium thwarts polarization gradient cooling during thisstep.

4.4 Interferometry

Two-photon Raman transitions comprise the atom optics utilized here for interferometrywith laser-cooled lithium. Fig. 4.15 depicts the frequencies we are using for interferometry,including the single- and two-photon detunings, ∆ and δ, respectively.

4.4.1 Frequency generation for Raman beams

Light resonant with the crossover to excited 2P3/2 state injection, picked off from the 2DFP laser diode, is sent to injection lock a third FP laser. This light is single passed throughan acoustic-optical modulator which shifts the light down in frequency by an additional 200


PBS

hwp

hwp

pinhole

reference

hwp

beat note detector

BS

80 MHz AOM

hwp

hwp

optical isolator

optical isolator

hwp

quantizaton axis

retro-mirrorPBS

π-polarization

Figure 4.14: The optical pumping beam optics and frequency-offset locking scheme. An ex-ternal cavity diode laser is beat against reference light, here light resonant with the crossoverto which spectroscopy is referenced. The majority of the light is coupled into an optical fiberand sent to a fiber port placed near the 3D MOT portion of the vacuum chamber. The beampasses through an AOM, used to control the switching or duration of the pulse sent to theatoms, telescope, to a periscope and through a PBS which helps ensure that the polarizationof the light is along the quantization axis (π), crucial for our optical pumping scheme.


ω1

ω2

ωba

∆

δ

|b, p + ħ (k1- k

2)>

|a, p >

|e p + ħ k1>

E

Figure 4.15: The two-photon Raman transitions that are the optics for the atoms duringinterferometry consist of two frequencies ω1 and ω2 red-detuned from the excited state by∆. The two-photon detuning is δ and quantifies how off-resonance these two frequencies arefrom the red-detuned frequency value.

MHz (will become our single-photon detuning). Prior to passing through the AOM, a smallfraction of this beam is picked off via a polarizing beam splitter cube and sent to the tune-outECDL, eventually becoming the reference light for our second frequency offset.

After being frequency shifted, this beam is fiber coupled and sent to a homemade taperedamplifier (EYP-TAP-0670-00500) for amplification to approximately 400 mW. We realizebetween 180 - 200 mW prior to an optical fiber at the end of this optical path, downstreamfrom the TA optics. This light is then sent to the two 400 MHz AOMs depicted in Fig. 4.16.

A serial controlled four-channel Direct Digital Synthesizer (DDS) sources the AOMsused to frequency shift the light, 200 MHz red-detuned to the crossover of lithium’s groundstate. Output frequencies are set and controlled via Cicero and Atticus, the word generatorand server respectively, which are responsible for executing the commands sent to various


EOM

Raman beam, k2

Raman beam, k1

400 MHz AOM

PBS

Raman TA

PBS

PBS

to FP cavity

200 MHz

300 MHz

11 dBm

250 MHz600 MHz200 MHz

13 dBm

7 dBm

2 f0

f1

f2

to the wall

hwp hwp

hwp

400 MHz AOM

hwp

hwphwphwp

Driving Electronics for 400 MHz Raman AOMs

Figure 4.16: Optical set-up for generating the Raman frequencies. Light red-detuned fromthe cross-over is amplified in a homemade tapered-amplifier and sent to the above set-up.Two 400 MHz AOMs are utilized to achieve the 800 MHz splitting needed. The zero orderexiting the first AOM is picked off and sent to the second as shown. The polarizationsare orthogonal at the recombination PBS and through the EOM. The frequencies are splitat the PBS following the EOM and sent to two optical fibers which will take the light toopposite sides of the vacuum chamber, creating the counter-propagating Raman beams forinterferometry.


bias coils

Top View

MOT coil

viewport

2D MOT

atom cloud

Raman beam, k1

Raman beam, k2

PBS

hwp

hwp

Figure 4.17: The Raman beams horizontally transverse the 3D MOT portion of the vacuumchamber after being expanded with 1:4 telescopes. The beams pass through a PBS and waveplate combination to clean up their polarization so as to be lin⊥lin.

hardware (GPIB, digital, analog, RS232) during an experimental run.

4.4.2 Raman optics

The optical scheme for the interferometry beams is shown in Fig. 4.17. The single-photondetuning of our Raman pair is ∆ = 2π × 210 MHz red-detuned from the D2 line. Asdiscussed in the prior section, a pair of 400 MHz acoustic-optical modulators tunes thefrequency difference of the two interferometry beams to be ωA− ωB + δ. At the 2 mm atomcloud, the pair coincides in beams of 2.1 mm Gaussian waists. A lin⊥lin configuration inimplemented such that one beam is polarized along x and the second beam polarized alongy. Since the two-photon Rabi frequency scales as 1/∆, one of the single-photon detuning, weachieve a high Rabi frequency ΩR ∼ 2π × 1.6 MHz or larger and thus require a short pulseduration in the experiment. For beam powers of approximately 30 mW of ω1 and 15 mW ofω2, we drive a π pulse in 320 ns with about 30% efficiency, addressing a considerable fractionof the atoms whose two-photon resonance conditions are Doppler broadened through thethermal velocity spread.


The interferometry axis, z, is almost completely perpendicular to the force of gravity, gy.The beams comprising the counter-propagating Raman transitions horizontally transversethe vacuum apparatus as shown in Fig. 4.17. We have measured the effect of gravity on aninertially sensitive Mach-Zehnder interferometer.

4.5 Detection

The atom number, temperature, and cloud density can be determined via absorption imagingthe atom cloud at the end of an experimental cycle. During such imaging, the shadow cast bythe cloud, generated by sending light resonant to the |F = 1〉 → 2P3/2 transition, is imagedonto a CCD camera (Pixelfly PCO). The number density of the atoms can be measuredby the observing the absorption of a resonant probe beam with the input of the averagescattering cross-section of photons by the atoms.

4.5.1 Absorption imaging

Consider a laser with intensity I0(x, z) propagating along the y-axis in the 3D MOT chambersuch that it is aligned to intersect the atom clouds at the end of the experimental sequence.The intensity at a particular position in space, transmitted through the atom cloud, is givenby the following expression

I(x, z) = I0(x, z)e−OD(x,z) (4.11)

where the optical depth profile of the atomic sample, OD(x,z), is equivalent to the columndensity of the sample at position (x, z) multiplied by the absorption cross section of thetransition σtot

σtot =σ0

1 + 4∆2/Γ2 + I0/Isat. (4.12)

Here, ∆ is the detuning from resonance, Γ is the natural line width of the transition, Isat

is the saturation intensity, and σ0 is the resonant cross section given by

σ0 =Γ~ω0

Isat=

3λ2

2π(4.13)

where ω0 and λ are the resonant frequency and wavelength, respectively.Comparing the probe intensity both with and without the atoms present allows the

transmittance or absorption to be determine and consequently the optical depth or thenumber density profile can be computed,

OD(x, z) = n(x, z)× σ = ln

(I0(x, z)

I(x, z)

). (4.14)

Integrating over the optical depth produces the atom number N

N =

∫ ∫n(x, z)dxdz =

1

σ0

∫ ∫OD(x, z)dxdz. (4.15)


4.5.2 Wollaston prism technique

Normalizing our detection scheme is crucial for eliminating the observed fluctuations of thetotal atom number per experimental cycle. To do this, a novel technique was developedwhere we take two absorption images of the atom cloud during a single exposure of the CCDcamera (PCO pixelfly). The two images are state selectively detected using two overlappedbut orthogonally polarized and independently switched beams with 9-mm waists tuned to theD2 transition for |F = 1〉 atoms. The first image measures the population in the |F = 1〉state and the second measures the total population in both the hyperfine ground states,|F = 1〉 and |F = 2〉 states.

The imaging scheme is simple: after illuminating the atoms with either of the imagingbeams, the light passes through a Wollaston prism positioned before the camera. Lightincident on the Wollaston prism will be deflected afterwards based upon the polarizationof the beam; orthogonal polarizations will be sent to opposite sides of the CCD array.Therefore, each imaging beam is coupled to a particular image generated, based on thedesignated polarization.

During the first 90 s of the exposure, we illuminate the atoms with one beam and imageonly the population in the |F = 1〉 state. After a 10-s delay, we switch on the orthogonallypolarized beam of the same frequency for 90 s while simultaneously turning on the 3D MOTcooling light. The cooling light depumps the atoms from |F = 2〉 to |F = 1〉 via the D2 lineand therefore we detect the sum of their populations or all atoms present regardless of theirinternal energy state. Due to its deflection at the Wollaston prism, this second image formson the other side of the CCD. Finally, we allow the atoms to disperse and take a second(background) exposure with the same pulse sequence to generate side-by-side absorptionimages of the entire cloud and the |F = 1〉 population. The ratio of the two absorptionimaging signals gives PF = 1〉, or the normalized population in the |F = 1〉 state.

4.5.3 Time-of-flight imaging

Releasing the atoms from optical molasses allows the cloud to ballistically expand. The atomsfly away from each other since the sample has a (very) non-zero thermal velocity. Assumingthe atoms are thermalized during cooling, the velocity of the atoms can be described by aMaxwell-Boltzmann distribution given by

f(v) =

√M

2πkBTe−Mv2/2kBT (4.16)

where kB is the Boltzmann constant and T is the temperature of the atom ensemble.The spatial distribution of atoms at a time after release, referred to as the ‘time of flight’,

is determined by considering the overlap of the atom’s initial density profile ρ0(z) with thevelocity distribution

f(z, v, t) =

∫ ∞−∞

f(v)ρ0(z′)δ(z − z′ − vt)dz′ (4.17)


200 MHz AOM

200 MHz AOM

hwpPBS

hwp

hwp

PBS

qwp

qwp

PCO.pixelfly

from 3D FP diode laser

Wollaston prism

Figure 4.18: The beam path for the imaging light as it transverse the vacuum apparatus.


where δ(z) is the Dirac delta function. Assuming an initial Gaussion density given by

ρ0 = N (2πσ2z)−1/2ez

2/2σz (4.18)

simplifies the integral to the following expression

f ′(z, v, t) =N

2πσ

√M

kBTexp

(−Mv2

2kBT− (z − vt)2

2σ2

)(4.19)

where N is a normalization factor. The atom number density is found by integrating withrespect to velocity

ρ(z, t) =

∫ ∞−∞

f ′(z, v, t)dv =N√

2π(σ2 + kBTt2/M)exp

(− z2

2(σ2 + kBTt2/M)

).(4.20)

The distribution of the atom cloud’s density in two dimensions is

ρ(x, z, t) =1

2πσ2(t)exp

[−(z2 + x2

)2σ2(t)

](4.21)

with

σ(t) =√σ2

0 + (kBT/M)t2 (4.22)

Comparing images taken at variable durations of the time of flight allows for the deter-mination of the temperature of the atoms. Plotting the final density with respect to time t,allows for the temperature to be determined

T =Mv2(σ2

f − σ20)

(t2f − t20). (4.23)

87

Chapter 5

Hot Beats

The scaling of the atom’s recoil frequency ωr with the inverse of its mass positions atoms, likelithium, as advantages candidates in recoil-sensitive interferometers, offering an automaticincrease in sensitivity and measurement precision for the same interrogation time comparedto a heavier atom. In addition to our measurement sensitivity benefiting from lithium’s highrecoil frequency of ωr = 2π × 63 kHz, the lack of both additional time-consuming coolingand lossy velocity selection steps [131] boost our measurement sensitivity.

However, further cooling beyond the standard Doppler temperature, obtained in a magneto-optical trap, offer advantages in detection at the interferometer outputs. In this chapter, thedetails of recoil-sensitive interferometry with a super-recoil sample of lithium is discussedincluding how the lack of spatial resolution in detection can be overcome.

5.1 Super-recoil lithium

While this is the first demonstration of atom interferometry with laser-cooled lithium, orany atom lighter than sodium-23, lithium has been used in supersonic atomic-beam interfer-ometers. Cooling lithium below the recoil temperature Tr, where the average thermal speedequals the recoil velocity, is difficult due to its small excited state 2P3/2 hyperfine splitting,as discussed in Chapter 3. In lithium, the realization of Sisyphus cooling is contingent onthat the light is detuned by an amount sufficiently greater than the hyperfine structure.This rather large detuning ensures that the fine structure of the atomic energy levels, ratherthan the hyperfine structure, dominate the atom’s response to the radiation. Achievingsub-Doppler temperatures directly in standard ‘D2’ optical molasses cooling is impossible.

Without employing any additional sub-Doppler cooling techniques [94, 132], the atomsprior to interferometry are limited in temperature to the Doppler temperature of 140 µKset by the cooling transition to the 2P3/2 state. Previously, our experimental set-up did suc-cessfully implemented Sisyphus cooling, achieving 1D temperatures of 40 µ K [94]. Unfor-tunately, due to the limited availability of laser power, we cannot simultaneously implementboth Sisyphus cooling and interferometry. Prior to interferometry, as measured with time-

CHAPTER 5. HOT BEATS 88

of-flight imaging techniques, the atoms are at super-recoil temperatures of approximately300 µK, fifty times the recoil limit 50Tr.

After the 2D and 3D MOT trapping and cooling stages, 10 million lithium atoms areoptically pumped into the magnetically insensitive sublevel, mF = 0, of the |F = 2〉 groundstate which is required to mitigate dephasing effects arising from persistent Eddy currentsin the vacuum chamber even after the quadrupole coils have long ceased. Interferometryfollows; in a Ramsey-Borde interferometer, four sequential π/2 pulses split, redirect andrecombine the atom wavefunction. The sequence π

2− π

2− π

2− π

2, consisting of three periods

of free evolution (T , T ′ and T respectively), reveals a phase difference Φ+(−) for the upper(lower) conjugate interferometer given by

Φ± = ±8ωrT − 2kazT (T + T ′)− 2δT. (5.1)

From the above expression, it is evident that the phase difference is proportional to therecoil frequency of the atom ωr = ~k2/(2m), the amount of kinetic energy the atom gainsafter recoiling from the emission and absorption of a photon. Here, the second term arisesfrom any accelerations az, like gravity or vibrations, along the axis of the laser beam withk = (k1 + k2)/2 being the average wavenumber of the counter-propagating beams, and thethird term from the detuning of the laser beams from two-photon resonance in the absenceof ac-Stark shifts, δ = ω1 − ω2 − (ωA − ωB).

5.1.1 Large bandwidth pulses

At the atom cloud, 30 mW of ω1 and 15 mW of ω2 red-detuned by ∆ = 2π×200 MHz intersectin waists of 2.1-mm. The Raman Rabi frequency for the two-photon process is measuredto be ΩR ∼ 2π × 1.6 MHz or larger during interferometry, corresponding to π-pulses beingdriven in 320 ns (with approximately 30% transfer efficiency).

A plot of the probability of transfer as a function of momentum for these pulse parametersis shown in Fig. 5.1 and compared to the Maxwell-Boltzmann velocity distribution along theinterferometry axis. Even while broadly spread in terms of velocity, the majority of the atomcloud is addressed or has an appreciable probability of transfer at the short interferometerpulse durations.

5.1.2 k-reversal

As seen in the previous section, time is of the essence. Short pulse durations are necessaryin order to address an appreciate fraction of the trapped population and the overall timeduration of interferometry is limited by speed at which atoms are exiting the interferometerbeams. Since the sensitivity of this particular measurement scales with T , the first and lastfree evolution times, it is ideal to reduce the length of T ′ as much as possible.

A distinguishing factor of the Ramsey-Borde interferometer compared to the alternativeco-propagating scheme discussed in the next chapter is that a reversal of the effective wave


tran

sfer

pro

bab

ilit

yM

B

−2 −1 0 1 2

vz [m/s]

-vz +3v

z

Maxwell-Boltzmann distribution vs. pulse bandwidth

Figure 5.1: A comparison of the bandwidth of the interferometry pulses to the Maxwell-Boltzmann velocity spread along the axis of interferometry. Raman resonances for the lower(red) and upper (blue) conjugate Ramsey-Borde interferometers are indicated.

vector direction of the light builds in the sensitivity to the atom’s recoil frequency. This flipmust occur between the pulse pairs during the 10 µs free evolution time step, T ′.

Experimentally, this is achieved by first orthogonally polarizing and then spatially over-lapping the two Raman frequencies ω1 and ω2 and passing them through an electro-opticalmodulator (EOM). The EOM sits prior to two optical fibers into which the frequencies arecoupled, ending on either side of the 3D MOT portion of the vacuum chamber. The EOMacts as a voltage-controlled wave plate which when quickly switched from 0 V to 215 V dur-ing T ′ (switch shown in Figure 5.2), the polarizations of the frequencies is rotated by 90. Apolarizing beam splitter separates the two frequencies according to their polarization beforethe fibers and thus switching the polarization by 90 reverses the Raman wave-vectors keff.Therefore, switching the voltage on the EOM allows us to alternate which frequency is sentto a particular fiber, flipping keff and effectively closing the Ramsey-Borde interferometer.

5.2 Simultaneous and conjugate

In chapter 2, it was shown that a consequence of the four π/2-pulse interferometer scheme isthe creation of a second conjugate interferometer. The normal and conjugate interferometersshare the first and second beam splitter pulses; the trajectories emerging from these transitioncomprise both the upper and lower interferometers. However for the third and fourth pulse,the lower interferometer requires a transition coupling |F = 2, p = 0〉 → |F = 1, p = −2~k〉while the upper interferometer requires the coupling |F = 1, p = +2~k〉 → |F = 2, p =


+6.3V

+214V

+6.3V

1k

18k, 10W

EO (100pF)

Figure 5.2: Circuit schematic of the vacuum tube switch needed to quickly switch theelectric-optical modulator between pulse pairs in a Ramsey-Borde interferometer. A pulse issent to the gate of a field effect transistor in series with a vacuum tube as shown. Openingor closing the gate results in a voltage drop across the 18k sense resistor connected to thetop of the 100 pF electric-optical modulator.

+4~k. In principle, a Doppler shift resulting from the difference in speed between lowerand upper interferometers distinguishes the Raman resonance conditions by 8ωr. Low inbandwidth pulses typical to other experiments resolve this frequency difference and onlyexcite a single transition. Hence, only one of the interferometers is closed unless an additionalfrequency component is incorporated into the third and fourth pulses to address the secondinterferometer.

Our high bandwidth pulses at π/2-pulse durations of 160 ns, which are required to addressthe large velocity width of the thermal cloud, also drive the transitions in both conjugateinterferometers. Without incorporating an additional frequency component, both the |F =2, p = 0〉 → |F = 1, p = −2~k〉 and |F = 1, p = +2~k〉 → |F = 2, p = +4~k transitions arecoupled and we unavoidably close both conjugate interferometers simultaneously.


T TT’

170 μm

Figure 5.3: The output ports of the upper and lower conjugate interferometer are onlyseparated by approximately 170 µm at experimental time scales.

5.3 Overlapped, simultaneous and conjugate

Given the common and differential phase components between conjugate interferometers,comparison of the two can allow for the phase contribution arising from unwanted accelera-tions or even gravity to be discerned [133, 134], leaving just to phase resulting from the recoilfrequency

Φ± = ±φr − φg − φdetuning. (5.2)

Phase extraction methods for direct rejection of such common-mode inertial signals likegravity and vibrations rely on being able to discriminate between the interferometer pair indetection [135]. Here, the thermal cloud of atoms expands ballistically during interferometry.This is responsible for smearing out the positions of the interferometer outputs with one andalso between the upper and lower conjugates. Four interferometer outputs are overlappedand spatially unresolved when imaging.

The probability for atoms to detected in the imaged ground state, denoted here by |b〉,after the interferometer sequence is

P|b〉 = D(1− C− cos(Φ−)− C+ cos(Φ+)

)(5.3)


where C± are the fringe contrasts for each interferometer and D is the overall offset in atompopulation. If the contrasts for each interferometer are approximately equal, C− = C+ ≡C/2, the signal can be simplified to the following

P|b〉 =D

2

[2− C

(cos Φ− + cos Φ+

)]=D

2

[2− C

(cos(−φr + φc) + cos(φr + φc)

)]=

D

2

[2− C

(cos(φr) cos(φc)− sin(φc) sin(φc) + cos(φr) cos(φc) + sin(φc) sin(φc)

)]= D

[1− C

(cos(8ωrT ) cos(2δT + 2kazT (T + T ′))

)]. (5.4)

For near equal contrasts, the phase component φc common to both Ramsey-Borde inter-ferometers is separated from the differential phase component φr, proportional to the recoilfrequency. Because we perform interferometry horizontally through our vacuum chamberand are perpendicular to gravity, at short interrogation times the δ-term contribution to theinterferometer phase φc dominates over that resulting from gravity.

5.3.1 Hot beats

Interferometry pulses, which couple the momentum state to an internal energy state, allowsfor us to discern the signal of interest even though we create and close both the normal andconjugate interferometers simultaneously and spatially overlapped. By varying the separa-tion time T (and maintaining that T ′ = 10 µs), we trace out the interference fringes. Thetwo-photon detuning δ, a consequence of the Raman interferometer pulses, is kept constantand small compared to the recoil frequency ωr. Therefore, the summed signals in the limitthat the contrasts from each is approximately equal, can be described by a fast oscillationfrequency at 8ωr within an envelope function oscillating more slowly at a frequency set bythe Raman detuning, δ, and accelerations present on the atoms az. Even without the ex-clusion of gravity, the effect of vibrations and a nonzero two-photon detuning will act tomodulate the amplitude of the interference fringe signal whereas the recoil frequency of theatoms will act as the fast the frequency component and remain essentially untouched bythese perturbations to the signal.

5.3.2 Time-domain fitting

As mentioned in chapter 4, the Wollaston prism normalization technique in our detectionscheme allows two images to be captured during a single exposure. We detected the summedinterference fringes from the simultaneous conjugate interferometers, taken over 100s of µsof T -evolution. The data presented in Figure 5.4 shows the summed interference signal forsetting of the two-photon detuning, fit with a least squares functional form given by

Fit = D

(1−exp(−T/τ)

[C− cos

((−8ωr + 2δ)T +φ−

)+C+ cos

((8ωr + 2δ)T +φ+

)])(5.5)


ω1-ω

2=803.506 MHz

ω1-ω

2=803.508 MHz

ω1-ω

2=803.510 MHz

ω1-ω

2=803.512 MHz

ω1-ω

2=803.514 MHz

ω1-ω

2=803.516 MHz

ω1-ω

2=803.518 MHz

50 100 150 200 250 300

T [μs]

Norm

aliz

ed p

opula

tion i

n F

=1 g

round s

tate

Figure 5.4: Data and fits for a range of two-photon detunings δ.


50 100 150 200 250 3000.085

0.09

0.095

0.1

0.105

T (μs)

PF

=1

100 105 110 115 120 125 130

0.09

0.095

0.1

0.105

T (μs)

PF

=1

Figure 5.5: The probability of detecting atoms in the |F = 1〉 state oscillates, beating dueto a nonzero two-photon detuning, δ = 2π × 4.3 kHz. Each point is the average of fiveexperimental shots and error bars have been omitted for clarity. Fitting (in green) yieldsωr = 2× 63.165± 0.002 kHz. Closer inspection of the long time scans reveals the fast recoilcomponent of the fringes. Table 5.1 shows results of the fit with 1− σ precision.

Values and 1-σ uncertainties are shown below in Table 5.1, resulting from the fit data inFigure 5.5. This fit yields a confidence interval that constitutes a 32 ppm recoil measurementperformed over only 2 hours. Averaging across 10 such data sets, each with a different δresults in an attained precision of 10 ppm. This phase sensitivity of the fit corresponds to50 times larger than the shot-noise limit.

5.3.3 Frequency-domain fitting

In Figure 5.6, the fast Fourier transform (FFT) of the same data fit in Figure 5.5 hasbeen taken. From the FFT of the averaged data, we are able to resolve the two frequencycomponents, ω = 8ωr2δ and ω+ = 8ωr+2δ, that constitute the sum of conjugate interferencefringes as considered previously. The two peaks differ by 4δ.


Table 5.1: Fitting parameters for Fig. 5.5.

Fit = D

(1− e−T/τ

[C− cos(−8ωrT + 2δT + φ−) + C+ cos(8ωrT + 2δT + φ+)

])D τ C− C+ ωr/2π δ/2π φ− φ+

0.09595(2) 297(8) µs 0.069(1) 0.067(1) 63.165(2) kHz -4.312(8) kHz -0.72(2) 0.37(2)

400 450 500 550 600

0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

1010-8

Frequency [kHz]

Figure 5.6: A fast Fourier transform (blue) of the data from Fig. 5.5. A fit given by thesum of two Lorentzians is shown in pink.

The Fourier transforms were fitted with a sum of two Lorentzians

Fit =A+Γ+

(ω − ω+)2 + Γ2+

+A−Γ−

(ω − ω−)2 + Γ2−

(5.6)

and such fits for a range of two-photon detunings are shown in Fig.5.7 with the indicatedδ’s. As the the two-photon detuning is tuned through zero, the peaks of the FFT convergeand then once again separate in frequency space.

5.4 Phase noise

In the simplest case the interferometer phase will have statistical variance denoted by σ2Φ

due to shot noise or Poissonian counting statistics in detection

σ2Φ ≡ 〈(Φ− 〈Φ〉2)〉 =

1

C2N(5.7)


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

460 480 500 520 540

Po

wer

[×

10

7]

Frequency [kHz]

Comparison of FFTs for Various δ

δ/2π =8.28(3) kHz

δ/2π =4.35(2) kHz

δ/2π =0(15) kHz

δ/2π =12.10(4) kHz

δ/2π =3.49(3) kHz

Figure 5.7: Plot of the Fourier transformed data for a range of two-photon detuning δ asindicated. The central peak, at which δ ≈ 0 yields a large amount of variation in the fittedparameters.


where N is the total number of counted atoms and C is the interferometer’s contrast givenby

C =Imax − Imin

Imax + Imin=ψ∗aψ

∗b + ψ∗bψ

∗a

|ψa|2 + |ψb|2(5.8)

for a mean interferometer detection intensity 〈I〉 = |ψa|2 + |ψb|2 with superposition wavepackets ψa and ψb along trajectories a and b, respectively.

Phase noise σΦ in an interferometer can be a result of noise arising from any of theparameters on which the phase depends

σΦ = σP ×∂Φ

∂P,

where P = P (Φ) = I0

(1 − V cos(Φ)

)is the probability amplitude with visibility V and

intensity I0 can be determined from the interferometer’s sensitivity function g(t). Thisfunction quantities the dependence of the transition probability δP at a time t on the relativephase shift of the laser φ

g(t) ≈ limδφ→0

δP (δφ, t)

δφ

Rewriting Eq. 5.9 making explicit the dependence upon the single photon detuning ∆for the Raman transition with Rabi frequency Ω and pulse time τ yields

σΦ = σ∆ ×∂Φ

∂P× ∂P

∂∆= σ∆ ×

−τΩ sin(Ωτ/2) cos(Ωτ/2)

∆(I0V ) sin Φ=π

V× d∆

∆(5.9)

We suspect that noise observed in the data is mostly a result of laser noise. We haveconfirmed this via numerical simulation which were adapted from previous studies in theresearch group of noise in Ramsey-Borde interferometers. The line width of the Ramanlaser is appreciable compared to its single-photon detuning, measured with an optical fiberinterferometer to have a FWHM of approximately 2 MHz.

5.5 Outlook

The coherence time of the interferometer is not yet limited by the thermal expansion outof the Raman beam but instead by magnetic dephasing of the mF = 0 atoms due to thequadratic Zeeman shift [136]

δEqzs =(gJ − gI)µ2

BB2

6hAs, (5.10)

where gJ and gI are the electronic and nuclear g-factors respectively, µB is the Bohr magne-ton, B is the amplitude of the magnetic field and As [137] is the hyperfine constant. Evenwhen performing interferometry 2 ms after the compressed, the magnetic field gradient thatsurvives produces inhomogeneous quadratic Zeeman shifts of the atoms in the ‘magnetically-insensitive’ sublevel. Therefore, the interferometer’s phase depends upon the atoms spatial


-2 0 2 4 6 8 10 12 1463.12

63.13

63.14

63.15

63.16

63.17

63.18

63.19

63.2

Two-photon detuning, δ/2π

Rec

oil

fre

quen

cy, ω

rec/

2π

Uncertainty in ωrec

/2π for various detunings δ/2π

Two-photon detuning, δ/2π

Rec

oil

fre

qu

ency

, ω

rec/

2π

Uncertainty in ωrec

/2π for various detunings δ/2π

0 2 4 6 8 10 12 1463

63.05

63.1

63.15

63.2

63.25

63.3

Figure 5.8: (Left) A plot of the standard deviation resulting from fits of the data over thetime-domain at different two-photon detunings. (Right) A plot of the standard deviationresulting from fits of the Fourier-transformed data in the frequency-domain at different two-photon detunings.

position in the vacuum chamber. Extending the optical molasses hold step to 5 ms reducesthis effect; it has been observed that the gradient reduces by 50% compared to the 2 ms holdduration. The interferometer contrast indeed decays at half the rate. Magnetic gradientcompensation would lead to longer coherence times and improved sensitivity.

At a conservatively projected T = 1 ms, we estimate the shot-noise-limited sensitivitywith 107 atoms to be 100 ppb/

√Hz. Sub-Doppler cooling would reduce the temperature

of the atoms to approximately 40 µK or 8Tr [94, 132] and would improve the experimentalsensitivity by

√50/8 3, the techniques presented here and in [138] would still be required.

5.5.1 Vibration immunity

Lithiums high recoil frequency allows us to take sensitive data at T < 10 ms, and thereforeto make full use of the common-mode rejection of vibration-induced signals. Phase shiftsfrom vibrations cancel when the fringes are summed in our detection scheme since they enter


the conjugate interferometers with opposite sign. The only effect of vibrations is then anamplitude modulation of the fringes.

Consider phase equation for a Ramsey-Borde interferometer with a stochastic, Gaussian-distributed acceleration along the quantization axis az, with a zero mean and a standarddeviation σ. Such vibrations modulate the interference contrast for

φvib = 2kσT (T + T )π, (5.11)

which decreases proportionally to az due to aliasing. Other interferometers in the groupoperating on a similar optical tables without vibration isolation [38, 139] accrue phase shiftsmuch less than π due to vibrations, even at T = 10 ms.

100

Chapter 6

Tune-outs

The tune-out wavelength (λto) is the wavelength at which an atom’s dynamic polarizabilitygoes vanishes. Alternatively, it can be thought of as the point at which the energy shiftresulting from the presence of an external optical field is zero. Between atomic resonances,where light is red-detuned from one level and blue from the other, these opposing contri-butions conspire to produce a root in the energy shift spectrum. This can be seen moreexplicitly by writing out the polarizability for an alkali as a sum primarily of contributions

670.955 670.960 670.965 670.970 670.975 670.980 670.985

-2×107

-1×107

1×107

2×107

wavelength [nm]

polarizability [AU]

Dynamic polarizability of 2S2 state in lithium

D1-lineD

2-line

CHAPTER 6. TUNE-OUTS 101

from the D1- and D2-lines

α(ω) =1

3~SD1

(ωD1

ω2D1 − ω2

+RωD2

ω2D2 − ω2

)+ αrem (6.1)

where R is the ratio of the line strengths of the D2- and D1-lines, SD2and SD1

, given by

R =SD2

SD1=

∣∣〈ψ2S‖d‖ψ2P3/2〉∣∣2∣∣〈ψ2S‖d‖ψ2P1/2〉∣∣2 (6.2)

and αrem represents the contributions from core electron excitations, higher energy transi-tions of the valence electrons, and the coupling of the valence and core electrons.

6.1 Previous polarizability measurements

The increasing precision and accuracy attained and required in experimental atomic physicsnecessitates a better understanding of the interactions between the atoms and external op-tical fields. For lithium, there are currently only indirect Stark shift measurements for thedynamic polarizability between the ground and excited state [74, 75] and a static polarizabil-ity determination made with thermal atom interferometry [44, 140], as discussed in Chapter3. A direct measurement of lithium’s tune-out wavelength between the 2P1/2 and 2P3/2 withatom interferometry is the pursuit of ongoing work here. The status and project outlook isthe focus of this Chapter.

6.1.1 The differential Stark shift

The Stark shift predates the modern formalism of quantum mechanics [60] but does allowfor measurements of polarizability differences. In such a measurement, the shift in frequencyof an atomic spectral line is measured as a function of either the static or dynamic electricfield strength, effectively measuring the polarizability difference between the the two atomicstates involved in the transition.

Of the few experimental measurements of the ac-Stark shift at optical frequencies, arecent experiment was actually performed with lithium. Varying laser intensity and trackingthe transition frequency dependence allowed a determination of the ac-Stark shift [74, 75]. Adifficulty exists in the interpretation of the ac-Stark shift experiments and a lack of preciseknowledge about the overlap of the laser beam with atoms in the interaction region. A high-precision measurement of the 2s − 3s transition in lithium has uncertainty resulting fromac-Stark shifts at the frequencies of the pump and probe laser of a two-photon resonancetransition between the states.

A composite method involving the high-precision measurement of the (2s − 3s) transi-tion in conjunction with a Hylleraas method determination of lithium’s ground state 2S1/2

dynamic polarizability and a configuration interaction plus core polarization CICP method


atom beam

nanograting

TO laser

z

x

Figure 6.1: In an atom interferometer uses a atomic beam as its source, the wave functionsmust be split by an appreciable amount in the x-domain to make a measurement. Due tothis, it is possible to incorporate a laser experimentally such that it will irradiates only oneof the interferometer arms.

calculation of lithium’s excited 3S1/2 state dynamic polarizability was performed to assess thereliability of the experiment’s determination. This method obtained an overall uncertaintyof only slightly better than 1% [92].

6.1.2 Space-domain atom interferometry

Lithium’s static polarizability α0 was measured with a thermal Mach-Zehnder atom interfer-ometer [44, 140]. A uniform electric field was applied to one of the interferometer’s separatedarms, shifting its energy by the Stark potential

U = −α0E2

2. (6.3)

The phase shift observed in the output depends upon this added potential energy. A de-termination of the static polarizability was made by studying the dependence of the resultingphase on the voltage V applied to generate the external electric field

α0 =

(φ

V 2

)(D2

Leff

)(2~v) (6.4)

where D is the distance between electrode and septum, v is the mean velocity of the atombeam, and Leff is the effective length of the interaction region.

An accurate determination of the polarizability requires precise knowledge of not onlythe phase shift as a function of the applied voltage V but also the geometry of the interactionregion and the velocity of the atomic beam.


A nicety of a tune-out interferometry measurement of the dynamic polarizability ratherthan the static polarizability is that while the phase φ depends on the frequency of thetune-out beam given for a thermal atom interferometer as follows:

φ(ω) =α(ω)

2ε0c~v

∫sd

dxI(ω, z)dz (6.5)

where I(ω, z) is the intensity of the ‘tune-out’ beam, v is the atom’s velocity, and s is thespatial separation of the arms, the frequency at which the polarizability vanishes or φ(ω) = 0will be independent of both the laser and atomic beam properties [141].

6.2 Light-pulsed interferometric lithium tune outs

To measure lithium’s tune-out wavelength, we employ an interferometer scheme consisting offour beam-splitter (π/2) pulses with three periods of free evolution T , T ′, and T . Contraryto a Ramsey-Borde scheme, here this interferometer is without the reversal of the momentumtransfer for the last two pulses. Hence, the electric-optical modulator implemented to achievethis purpose, has been removed from the Raman board.

6.2.1 φto, the tune-out phase

As discussed in Chapter 2, an interferometer composed of four co-propagating beam splitterpulses is no longer sensitive to the recoil frequency of the atoms but can be made sensitiveto a phase induced by the interaction of the atom’s wave function with potential of a drivingfield turned on during the T ′ free evolution time. During this time of free evolution, thewave functions of the coherent superposition are in the same internal energy state, allowingfor a determination not of a differential tune-out between hyperfine levels, but instead thetune-out or for a particular state.

This interferometer geometry [142] has been utilized previously with an atomic beam ofcalcium to make a measurement of the static polarizability [143]. It is similar to a Mach-Zehnder configuration except with the π-pulse split into two π/2-pulses as evident by theinterferometer’s phase difference.

The phase of the aforementioned interferometer in the presence of a ‘tune-out’ beampulsed on during the free evolution time T ′ is given by

φto =α(ω)sT ′

2ε0c~× ∂I

∂z(6.6)

where s = 2vrT is the separation between the arms of the interferometer, which can bedefined in terms of the atomic recoil velocity vr and the intensity of the beam with waist wis given by

I(r, ω0) = I0e−2r2/w2

(6.7)


s

s

t

z

z

x

Figure 6.2: The tune-out beam is aligned centrally to the cloud of atoms. Atoms on eitherside undergoing interferometry will see an intensity gradient with opposite sign over the armseparation distance of s.

where I0 = 2Pπw2

0.

Unlike in thermal atom interferometers, the lack of spatial separation between the in-terferometer arms precludes the ability to interrogate only a single arm with the tune-outlight. However, by positioning the beam such that it is central to the cloud of lithiumatoms, atoms on either side will ‘see’ an opposite relative gradient over the superposition’strajectories during interferometry. This results in atoms that accrue a phase shift equal inmagnitude for the same starting positions (aside from position) but opposite in sign.

By measuring the phase accrued on either side of the tune-out frequency, the slopethrough zero is determined and thus the tune-out frequency to within a given precision. Thegiven precision is set by the sensitivity of the experiment, dependent on the experimentalparameters of the tune-out beam and of the atom cloud, particularly in the thermal dephasingof the atoms during interferometry.

Varying the frequency of the tune-out laser allows us to track the dependence on the


T

|a , p + ħkeff

⟩

TT’

|b , p ⟩

t

z

|a , p + ħkeff

⟩

|b , p ⟩

Figure 6.3

z [w]

No

rma

lize

d A

mp

litu

de

[I 0]

Spatial Dependence of Tune-out Beam Intensity

-2 -1 1 2

1.0

0.5

0.5

1.0

00

∂zI(x,z)

I(x,z)

Figure 6.4: The spatial dependence of the intensity of a Gaussian beam varies in units ofbeam waist. For positions on either side of the center, movement in the same direction willbe at equal and opposite intensity gradients.


phase variation across the cloud as a function of frequency. For a particular beam waistwhich corresponds to a particular intensity in the chamber as seen by the atoms, we predicta given contrast slope of the tune-out signal.

A further complication occurs due to thermal dephasing that occurs resulting from thesignificant spread of the atoms in momentum space. An atomic cloud at a temperatureof 300 µK corresponds to a average thermal speed of approximately 0.6 m/s. Therefore,it is possible for an atom to transverse a distance of vth

(2T + T ′

)during the time of the

interferometer.

6.2.2 The tune-out beam

The tune-out beam originates from a commercial Topitca ECDL () that is frequency-offsetlocked from the cross-over frequency between the hyperfine ground states to the 2P3/2 state.Light from the diode laser seeding the tapered-amplifier for interferometry is the referencelight in this locking scheme. The local oscillator is a frequency-doubled Agilent functiongenerator which is mixed with the laser beat note and sent to a servo similar to what isutilized in both our spectroscopy and optical pumping schemes.

The majority of the light from the tune-out ECDL is sent through an AOM after whichthe -1 order is coupled into an optical fiber and sent to below the vacuum chamber of the 3DMOT. Here, it is telescoped to achieve a 150 µm waist at the atoms. The waist is measuredvia single-photon scattering by moving the frequency of the beam closer to resonance witheither the D1- or D2-line. The beam passes through a half-wave plate and PBS before it issteered with two mirrors such that it propagates upwards, along the same path as the imaginglight. After transversing the chamber, the beam is picked out of the imaging path with a‘D’-shaped mirror, sent to another mirror mounted above the chamber and retro-reflected.Retro-reflecting increases the intensity at the atoms.

6.2.3 Experimental Sequence

During the second free evolution time step T ′, if considering a single interferometer thenboth components of the superposition are in the same momentum and internal energy state.Because each hyperfine has a different tune-out wavelength resulting from varying transitionmatrix elements, pulsing on the tune-out beam during this time step is crucial. The tune-outbeam is switched on in the middle of this time step as shown in Fig. 6.6. We also turn onimaging light for a pulse duration of 60 µs to decohere the upper interferometer unavoidablygenerated in the conjugate scheme due to the fast beamsplitter pulses.

Rather than scanning the free evolution time between pulses and tracing out interferencefringes over an extended duration, for a fixed time of evolution, the frequencies from theDDS function generator driving the Raman AOMs is stepped between the pulse pairs. Ex-perimentally, by switching which output port from the DDS is mixed with a third frequencyalso originating from the DDS, the result is a modulation inducing an interferometer phaseduring the experimental cycle.


hwp

PBS

hwp

hwp

pinhole

reference light

hwp

hwp

beat note detector

BS

80 MHz AOM

hwpPBS

Figure 6.5: The optics used to frequency-offset lock and steer the tune-out beam and thepath transversed through the vacuum chamber.


dwell

10 ms

MOT load

0.8 s

cMOT ramp

5 ms

cMOT hold

50 ms

OM hold

2 ms

pump

70 μs

Ramanpulses

160 μs

repump

20 μs

image atoms

150 ms

image background

150 ms

BMOT

2D shutter

pusher shutter

imaging AOMs

3D trap AOM

3D repump AOM

3D TA switch

Raman pulse gen.

OPEN

τLKB

Raman pulse gen.Raman pulse gen.Raman pulse gen.

opt. pumping

tune-out AOM

ftrap

3D

frepump

3D

Ptrap

3D

Bbias

τimg

τimg

τto

τOP

Prepump

3D

OPEN

OPEN

OPEN

OPEN

OPEN

Experimental Sequence and Settings for Tune-out

dig

ita

l co

ntr

ol

an

alo

g c

on

tro

l

Figure 6.6: The experimental sequence and settings utilized for a measurement of 7Li’stune-out wavelength is shown here. Two additional pulses, that of the tune-out light beingswitched but also a pulse of imaging light differ from the recoil-sensitive interferometrydetailed in the previous chapter.

Data is taken for the tune-out measurement at the point for which the signal slope is thehighest and the interferometer is thus most sensitive. For a given frequency of the tune-outbeam, an experimental cycle is run with a pulse of tune-out light and without such a pulse.While there are always two runs for every frequency of the beam, data has been taken bothat a fixed frequency for a length of time as well as while scanning the frequency many times.Optimization of data taking is still currently being pursued.

6.2.4 Detection & Analysis

The atoms are detected with absorption imaging again implementing the Wollaston cubewhich allows for two images to be taken during one exposure. A series of images at aparticular frequency of the tune-out beam are taken, alternating between switching the


Figure 6.7: A comparison between the analysis performed without (top) and with (bottom)the tune-out pulse. The lower principal component analysis only contains that dependentupon the extra light, switched on during the T ′ time step in the interferometer.

tune-out light either on or off. This allows us to normalize to the observed drift in atomcloud position, seen over the course of the experiment, as well as other experimental defects.

Both the pulsed and unpulsed data are analyzed with a method of statistical image anal-ysis called principal component analysis. Doing so allows us to discern only the significantfeatures different between the sets which is being experimentally imposed to be the pres-ence (or lack of) tune-out light. Normalizing the pulsed to the unpulsed in the final pulsedimage results in a principal component corresponding to the spatial effect of the tune-outlight. A plot of the projection of the pulsed to unpulsed basis determines the ‘magnitude’or significance of the tune-out signal and for a scan of frequencies is used to determine theexperimental precision; we expect at tune-out that no difference we be apparent betweenbases.


6.2.4.1 Principal component analysis

The goal of principal component analysis (PCA) is to represent an image Xi as a linearcombination of a set of basis images

Xi = M + Y1iu1 + Y2iu2 + ...+ Ypiup (6.8)

where M is the mean image for the set given by

M =1

N

(X1 + X2 + ...+ XN

). (6.9)

Each image Xi is a p-dimensional vector, where p is the number of pixels. Given a set ofN images, PCA constructs a new matrix of dimension p×N such that each column of thismatrix is a vector given by the difference between a particular image of the N size set andthe mean image

X =(X1 −M X2 −M ... XN −M

)=(X1 X2 ... XN

). (6.10)

From this matrix in mean-deviation form, a covariance matrix (p× p) is calculated

S =1

N − 1XXT (6.11)

where the diagonal element Sjj, for an arbitrary pixel j, is the sample variance

Sjj = σ2jj =

1

N − 1

N∑i=1

X2ji (6.12)

and the total variance is found by computing the trace of the matrix, tr(S). The off-diagonalelements are the covariances between pixels. In the instance that σ2

jk = 0, then the pixelsare said to be uncorrelated.

This covariance matrix is used to determine the new basis for the set of N images

X = PY (6.13)

where P is the orthonormal matrix composed of principal components

P =(u1 u2 ... up

)(6.14)

and the matrix Y is an N ×N matrix of coefficients or weights. The mean-subtracted imagecan be represented as a linear combination in terms of the new PCA basis as

Xi = Y1iu1 + Y2iu2 + ...+ YNiuN . (6.15)


PCA calculates the basis images such that they are orthonormal with the goal to find anew better basis for the data using a limited number of principal components. PCA requiresthat for a set of images, the coefficients for a particular image vector basis j be statisticallyuncorrelated with the set of coefficients of any other basis image k. This condition is met ifand only if the sample covariance matrix of the PC basis D is diagonal

D =1

N − 1YYT . (6.16)

This is the same as ensuring that the PCs are unit eigenvectors of S.Therefore, given this analysis of principal components an image of the set i can be written

in the new basis asXi = M + Y1iu1 + Y2iu2 + ...+ YNiuN . (6.17)

When performing this analysis on similar images, the most significant features of an imagecan be reconstructed with a PC basis of reduced dimensionality q < N ,

Xi ≈M + Y1iu1 + Y2iu2 + ...+ Yqiuq. (6.18)

The dimensionality of the image is reduced by p pixels to q strongest PC weights. Thisreductions allows specific experimental parameters to be correlated with the image weights.

6.3 Towards tune-out

The flip side to the nicety of lithium’s simple electron structure allowing for its atomicproperties to be computed with out standard quantum mechanical techniques, it that suchcomputations have reached an extraordinary level of precision from the perspective of aprecision metrologist. Furthermore, at this resolution of optical frequencies, one must nowuse a frequency comb as a reference; the intrinsic accuracies of frequencies used experimentalfor trapping, pumping, and interferometry are known to values at least an order of magnitudeabove the intended tune-out measurement.

6.3.1 Precision

While large-scale calculations done with correlated Hylleraas basis sets can attain a degree ofprecision not possible for calculations based on orbital basis sets. Such a computation [92],utilizing solutions to the norelativistic Schrodinger equation along with experimental energyvalues computed the tune-out wavelength for lithium’s ground state to be at 670.971 626(1)nm. An inclusion to the dynamic polarizability of an remainder term αrem(ω) estimated tobe 2.333 824 a3

0 or likewise an perturbation to the ratio of reduced dipole matrix elementsR defined as

|〈ψ2S(r)‖d‖ψ2P3/2〉|2

|〈ψ2S(r)‖d‖ψ2P1/2〉|2

= (2 +R) (6.19)


3300 3320 3340 3360 3380 3400 3420 3440 3460 3480 3500

-5.5

-5.0

-4.5

-4.0

-3.5

Frequency [MHz]

PC

to p

roje

ctio

n [

×10

15]

Magnitude of Tune-out PC vs. Frequency

Figure 6.8: A plot of the magnitude of the projection on the TO principal component overa span of frequencies.

results in the above uncertainty to the wavelength and a needed precision of approxi-mately 500 kHz. Currently, we are measuring 20 MHz within the predicted tune-out fre-quency within a period of hours.

A plot of the overlap onto the principal component correlated with the tune-out pulsefor various frequencies is shown in Fig. 6.8.

6.3.1.1 Single-photon scattering

Increasing the intensity of the tune-out beam is a route towards higher experimental sensitiv-ity however an issue arises due to the single-photon scattering limit in lithium at detuningswhich correspond to the tune-out wavelength frequency.

In the case of π polarization, the scattering rate for the hyperfine state F is given by [61]

Rsc =1

4

∑qF ′′

[∑F ′

ΩFF ′〈FmF |F ′mF ; 1 0〉√

ΓF ′F ′′〈F ′mF |F ′′mF − q; 1q〉

×(

ω

ωF ′F

)3/2(1

ω − ωF ′F− 1

ω + ωF ′F

)]2

(6.20)


Figure 6.9

with a decay-rate for small hyperfine splittings given by the following in terms of total decayrate of level J ′ denoted by ΓJ ′ ,√

ΓF ′F ′′ =√

ΓJ ′(−1)F′′+J ′+1+I

√(2F ′′ + 1)(2J ′ + 1)

J ′ J ′′ 1F ′′ F ′ I

. (6.21)

6.3.1.2 Beam shaping

Altering the profile of the beam will increase or decrease the sensitivity of the measurementby altering the intensity and thereby phase scaling

δφ ≈ ITT ′

w0

. (6.22)

An anamorphic prism pair inserted after the beam transforms the profile from circularto elliptical as shown in Figure 6.9. Elongating the beam allows us to reduce the area butmaintain the same intensity, advantageous to avoid the single-photon scattering limit at therequired detunings. A plot of the overlap on the tune-out PC for various Agilent frequenciesis shown in Fig. with the fitting parameters resulting from a linear fit applied to the presenteddata.

6.3.2 Accuracy

An optical frequency comb is necessary to determine the wavelength for tune-out with therequired accuracy. We plan to use a commercial Menlo Systems (FC8004) optical frequencysynthesizer that is based on a femtosecond laser frequency comb, with a specified combfrequency spacing of approximately 200 MHz and accuracy of 10−14.

This system consists of a fs-laser, a nonlinear photonic crystal fiber and nonlinear inter-ferometer intended to measure the offset frequency of the comb by interfering the spectralparts around 532 nm and 1064 nm. By beating the tune-out beam against the comb, onecan accuracy determine the wavelength of the light.

6.4 Hyperfine dynamic polarizabilities

The hyperfine ground states of lithium do have different dynamic polarizabilities resultingfrom differing values of dipole matrix elements. In order to measure the absolute tune-out


PC 1, max=0.158, min=-0.183

5

10

15

20

50

100

150

200

250

PC 2, max=0.146, min=-0.141 PC 3, max=0.214, min=-0.395

PC 4, max=0.180, min=-0.329

5 10 15 20

5

10

15

20

PC 5, max=0.127, min=-0.191

5 10 15 20

PC 6, max=0.375, min=-0.234

5 10 15 20

Figure 6.10: Image taken with elliptical beam, after broken down into principal components.

Frequency [MHz]

PC

to p

roje

ctio

n [

×10

16]

Magnitude of Tune-out PC vs. Frequency

2350 2400 2450 2500 2550 2600

-1

0

1

2

3

4

5

6

7

m = 324215719246765.4; σm = 5982647790464.469;

xint

= 2396.0814; σXint

= 1.9603;


state F α1 (a.u.) αT1 (a.u.)7Li 2S1/2 1 161.983 -9.78[-7]7Li 2S1/2 2 161.984 5.94[-6]

Figure 6.11: Scalar and tensor dynamic polarizabilities for hyperfine ground state levels in7Li.

wavelength for only the F = 2〉 ground state, imaging light resonant with the F = 1to2P3/2

is flashed on during the T ′ free evolution step of the interferometer. This light decoheres theconjugate interferometer that is created in this scheme.

To compute the scalar the dynamic hyperfine polarizability, first the dipole matrix ele-ments are determined with respect to the different hyperfine levels. Using the Wigner-Eckarttheorem, the transition amplitude is rewritten in terms of a reduced matrix element

〈F‖er‖F ′〉 ≡ 〈JIF‖er‖J ′I ′F ′〉

= 〈J‖er‖J ′〉(−1)F′+J+1+I

√(2F ′ + 1)(2F + 1)(2J + 1)

J J ′ 1F ′ F I

(6.23)

depending only upon L, S, and J quantum numbers.The oscillator strength fkgi for a dipole transition (k = 1) is given by

fkgi =2|〈JIF‖er‖J ′I ′F ′〉|2ωgi

3(2F + 1)(6.24)

where ~ωgi is the energy required to excite the |g〉 → |i〉 transition between states denotedhere |g〉 and |i〉.

From the oscillator strength, the dynamic polarizability is determined with the oscillator-strength sum rules,

α(ω) =∑i

f(1)gi

ω2gi − ω2

. (6.25)

Here, the sum over intermediate states includes all allowed fine-structure and hyperfine-structure allowed transitions. In the limit that the frequency goes to zero, the static polar-izability should be recovered.

The tensor component of the dipole polarizability (for states in which F > 1/2) is givenby

αT1 (ω) = 6

(5F (2F − 1)(2F + 1)

6(F + 1)(2F + 3)

)1/2

×∑i

(−1)F+F ′F 1 F ′

1 F 2

f

(1)gi

ω2gi − ω2

.

(6.26)


Figure 6.12: This plot shows how the dynamic polarizability of each hyperfine ground state(orange is the F=1 state and green is the F=2 state) and the difference in α1(ω) (blue) varieswith frequency.

Plotting the dynamic polarizabiity, α(ω), for each state and solving for the roots of theexpression or when α(ω) = 0, we find that the |F = 1〉 ground state has a tune-out at670.961 nm (446.8039412 THz) while the |F = 2〉 ground state has a tune-out at 670.962nm (446.8031617 THz). These differ from the 2S1/2 tune-out wavelength (670.972 nm) by2π × 488 MHz and −2π × 292 MHz, respectively.

117

Chapter 7

Conclusion

This chapter explores implications and potential applications of the techniques demonstratedint the preceding two chapters, as well as the outlook for atom interferometry with lukewarmlithium.

7.1 Outlook for recoil-sensitive interferometry with

super-recoil samples

With non-zero two-photon detuning for the Ramsey-Borde interferometer, the interferencefringes allow for the determination of the recoil frequency independent of two-photon detun-ing and vibrations. Our results relax cooling requirements for recoil interferometry, allowingfor increased precision through high experimental repetition rates [144]. This demonstrationof interferometry with a sample of atoms at ‘super-recoil’ temperatures opens the door torecoil measurements with other particles that are difficult to cool to subrecoil temperatures,such as electrons.

7.1.1 h/me measurement

Electrons have recoil frequencies on the order of GHz. They are susceptible to relativisticeffects [145] and consequently a recoil-sensitive measurement can be used to measure Lorentzcontraction [146].

A measurement of h/me also allows for a more direct determination of the fine structureconstant α; allowing one to neglect measurements of reduced electron and atom masses [2].

While Kapitza-Dirac scattering has been proposed to realize matterwave beam splittersfor electrons in a Ramsey-Borde interferometer [147], any vibrations or nonzero two-photondetuning will modify the phase ∆φ− for a single Ramsey-Borde,

∆ϕ =4

m~k2

LT′ − 2akLTT

′. (7.1)

CHAPTER 7. CONCLUSION 118

z

- rec

- rec

+ rec

+ rec

T TT’

Δz

Figure 7.1: Ramsey-Borde interferometer for electrons. Four bichromatic laser pulses,composed of two counterpropagating waves detuned by multiples of the recoil shift ωrec.These waves are used to split and recombine the electron beam as shown, producing a totalof eight partial beams.

As we have shown, the inclusion of the simultaneous conjugate interferometer (∆φ+)recovers the recoil phase independently of a two-photon detuning even when the outputs ofconjugate interferometers are spatially unresolved, as would the case for electron plasmas ina Penning-Malmberg trap [148].

The required spectral resolution for detection could be achieved with bichromatic Kapitza-Dirac pulses. Bichromatic pulses with very large intensity have been proposed to impartmomentum to an electron while inducing a spin flip [149] and hence couple the electrons ex-ternal and internal degrees of freedom. With such beam splitters acting on a spin-polarizedsample and spin-dependent detection, the techniques we demonstrate in this work pave theway for a recoil-sensitive electron interferometer.


7.2 Outlook for tune-out interferometric

measurements in lithium

A direct interferometric measurement of lithium’s red tune-out wavelength at 670.971626(1)nm, is a precision comparison to existing ‘all-order’ atomic theory computations. A deter-mination of lithium’s dynamic polarizability would be pivotal in metrology [81]. In additionto informing existing computational methods, a next generation experiment, in which an-other atomic species would be included in the same apparatus as lithium, would allow fora direct comparison and normalization, overcoming systematics resulting from different ex-perimental environments. This could lead to a new accuracy benchmark for many elements.Hylleraas polarizability calculations could serve as standard for coupled-cluster type calcu-lations applied to atoms larger than lithium, like cesium. Additionally, it provides anotherway to determine the S− to P− transitions matrix elements for which large correlationsand small values complicate computation. Aside from informing atomic structure calcula-tions, applications of tune-out wavelength measurements include making state-selective andmulti-species traps for ultra-cold atomic physics experiments and quantum computing, spin-dependent dispersion compensation for an atom interferometer gyroscope, and can be usefulPNC experiments.

The lack of spatial resolution during interferometry to measure lithium’s dynamic polar-izability is overcome by aligning the tune-out beam to the center of the atom cloud so thatfor atoms on either side, an equal and opposite ac-Stark shift will be induced. This resultsin atoms on opposite side of the beam center accruing opposite phase shifts, evident in thenormalized atom number (corresponds to probability amplitude for state) during imagingas a corresponding excess or absence of population. An imaging technique called principalcomponent analysis allows for this variation through the tune-out frequency to be observedby splitting up the obtained image into a simplified basis and then allowing images takenwith the tune-out light to be projected onto images taken without the tune-out light. Thisdifferential imaging scheme, while not simultaneous, does allow for the determination in themidst of varying atom cloud position.

7.2.1 Beyond the red

A second tune-out frequency in lithium occurs right before the 3S → 3P1/2, 3P3/2 excitations,at 324.192(2) nm [98]. The attained uncertainty in its computed value is much larger than thefirst tune-out frequency in the red part of the spectrum, in part due to a greater dependenceupon the details of the atomic structure description [92]. Recalling the definition of thedynamic polarizability in terms of oscillator strengths

α(ω) =∑k

fgk

(Ek − Eg)2 − ω2(7.2)


0.04 0.06 0.08 0.10 0.12

-1000

-500

500

1000

Dynamic polarizability of 7Li atom in its ground state (2S1/2

)

α [au]

ω [au]

ωuv

0ω

red

0

Figure 7.2: Dynamic polarizability of the 2S1/2 level in 7Li. Tune-out wavelengths havebeen indicated at values of 0.067 906 526 572 and 0.140 907 329.

Dynamic polarizability of 7Li atom in its ground state (2S1/2

)

324.16 324.18 324.20 324.22 324.24

-4

-2

2

4

324.20 α [au]

ω [au]

Figure 7.3: Lithium’s second tune-out wavelength at 324.192(2) nm has a larger computationuncertainty due to the greater impact of the atomic structure description on the value.


or more explicitly for the UV tune-out

α(ω) =f2p1/2

∆E22p1/2− ω2

+f2p1/2

(2 +R)

∆E22p3/2− ω2

+f3p1/2

∆E23p1/2− ω2

+f3p1/2

(2 +R)

∆E23p3/2− ω2

+ αrem(ω)

(7.3)

where R is defined as

|〈ψ2s(r)‖d‖ψ2p3/2(r)〉|2 = |〈ψ2s(r)‖d‖ψ2p1/2

(r)〉|2(2 +R). (7.4)

The energies for the excited states needed in the computation can be referenced to exper-imental data. However, the value of R which was found to be R = 0.00096199, the scaling ofthe matrix elements, must be determined from a relativistic model potential calculation [92].The inclusion of R impacted the determined value at the part per million level contrary to thedependence of the red wavelength which has little to dependence on R. However, inclusionof αrem, referred to in the literature as the background polarizability, impacted the value forthe UV tune-out wavelength at the forth significant digit. A future measurement of lithium’sultraviolet tune-out wavelength would be more sensitive to relativistic approximations in theatomic structure description.

7.2.2 Investigation of nuclear structure between isotopes

Improved computation methods, guided by a comparison of measured and calculated atomicproperties can inform theories beyond even that surrounding atom-light interaction. Therelativistic and QED corrections have been calculated for the hyperfine splitting of the 2S1/2

ground state of lithium [150], exactly accounting numerically for electronic correlations.This theoretical computation can be compared with experimental values determine atomicnuclear properties such as nuclear charge radius between isotopes. A significant difference inmagnetic moment distributions has been computed between lithium’s isotopes. It is claimedthis may signal the existence of some unknown spin-dependent short-range force betweenhadrons and the lepton.

In an atom, the magnetic moment of the nucleus interacting with the magnetic momentof the electrons, produces the hyperfine splitting of energy levels. In a many electron atom,theoretical predictions are limited by correlations between the multiple electrons which aredifficult to account for using relativistic formalism based on the Dirac Hamiltonian. Explic-itly correlated basis sets are able to accurately account for electron correlations, achievinga few ppm accuracy. A nonrelativistic QED approach perturbatively treats both relativisticand QED effects [150]. All corrections are expanded in powers of the fine structure constantand expressed in terms of the effective Hamiltonian.


7.3 Atom interferometry with lukewarm lithium

The limiting factor in the current experiment is time. The inability to wait for the Eddycurrents induced in the vacuum chamber to decay hinders the interferometer’s obtainablecontrast. Extending the interferometer’s interrogation time to one at which a competitiveh/M measurement would be possible is also not allowed at the current temperature of theatoms. Further cooling, beyond that achievable with only optical molasses, is a route towardsboth decreased thermal dephasing and resulting systematics and increased experimentalsensitivity and hence precision in measurement [151].

803.5 MHz

91.8 MHz2P1/2

2S1/2

F= 2

F= 1

F= 2

F= 1

δ

δ2

δ1

670.9

76 6

58 n

m

δ

δ2

δ1

2P3/2

10.0

56 G

Hz

670.9

61 5

61 n

m

-1/2 +1/2mJ -3/2 +3/2

Sisyphus cooliing Gray molasses cooling

Figure 7.4: Laser frequencies for the implementation of sub-Doppler (left) Sisyphus coolingand (right) gray molasses cooling in lithium.


Two sub-Doppler cooling techniques that would allow for an order of magnitude reductionin temperature are mentioned in the following sections. To experimentally pursue either,an additional FP diode laser and tapered amplifier would be needed at minimum. Eachscheme utilizes light close to lithium’s D1 line, already generated experimentally for opticallypumping and tune-out light. The additional lasers would allow for more power aroundD1 would could then be split among optical pumping, tune-out, and sub-Doppler cooling.Frequency shifting further in each of these set-ups could be achieved with either acoustic-optical modulators and electric-optical modulators.

7.3.1 Sisyphus cooling

A simple route to sub-Doppler cooling would be to implement on the experiment the previ-ously demonstrated Sisyphus cooling [94]. Sisyphus cooling achieved temperatures as low as40 µK in one dimension for up to 45% of the cooled atom fraction. This method uses polar-ization gradient cooling, but detuned from the 2P3/2 excited state by 19 GHz; such detuningsare still produced using in the experiment for both optical pumping and the tune-out. Thecooling process operates on a timescale of milliseconds, requiring 180 mW of total powersplit into three retro-reflected 0.7-mm waist beams, one sent along the axis of the MOT coilsand two overlapped with the MOT beams in the plane of the MOT coils. The cooling beamsare linearly polarized and reflected through a quarter-wave plate. This cooling scheme isrobust against ambient background magnetic fields of up to 100 mG.

7.3.2 Gray molasses

A sub-Doppler cooling technique using three-dimensional bichromatic laser cooling on the D1

line has been implemented in lithium, obtaining temperatures of 60 µK with upwards of 108

trapped atoms after only 1.5 - 2.0 ms of cooling [132]. This method combines a gray molassescooling scheme on the same transition we use to optically pump, the |F = 2〉to|F ′ = 2〉transition while simultaneously addressing the |F = 2〉to|F ′ = 1〉 transition, albeit phase-coherently such that a velocity selective coherent population trapping of atoms is realized[152]. The beams have intensities of approximately I ≥ 45Isat, created with a total ofapproximately 150 mW in 3.4-mm waists of σ+ − σ− counterpropagating pairs.

7.4 Onward

In conclusion, recoil-sensitive Ramsey-Borde interferometry and interferometry sensitive tothe atomic dynamic polarizability has been demonstrated with laser-cooled lithium-7 at alukewarm 300 µK (50 Tr). The large Doppler spread of the sample is addressed with fastpulses, driving simultaneous conjugate interferometers in both instances. We suppress first-order magnetic dephasing and extend the coherence time by optically pumping the atoms tothe magnetically insensitive |F = 2,mF = 0〉 state using lithiums well-resolved D1-line. We


overcome the necessity of spatial resolution by instead spectrally resolving the interferometerarms via two-photon Raman transitions. In the case of the tune-out measurement, ananalysis of image data such that cloud movement and variations are normalized to canallow for an identification of the phase shift from the tune-out beam.

125

Appendix A

Properties of lithium

Lithium is the lightest alkali atom and the third element (Z = 3) of the periodic table. As asolid, lithium is a soft, shimmery metal. However, moisture and nitrogen in the air quicklycorrode lithium to a dull gray and eventually black tarnish. Due to this reactivity, in naturelithium can only be found in compounds and never freely.

When in its ground state, lithium’s three electrons are found in the configuration 1s22s1.Two electrons occupy the lowest s-orbital and a lone valence electron sits outside the closedshell, characteristic of the alkali atoms located in the first column of the periodic table. Thephysical properties of bulk lithium are listed in the table below.

Lithium’s vapor pressure as a function of temperature is given with Antoine parametersfrom NIST’s tabulated atomic data

log10(P ) = 4.98260− 7918.984

(T − 9.52)(A.1)

where the pressure calculated is in the units of bar and here T is in units of K. These

Table A.1: Physical properties of lithium

Property Symbol Value ReferenceAtomic number Z 3 [153]Nuclear lifetime τn stable [154]Atomic weight Ar,std 6.941 [153]

Density (300 K) ρm 0.534g·cm−3 [154]Melting point TM 453.69 K [154]Heat of Fusion QF 2.99 kJ·mol−1 [154]Boiling point TB 1615 K [154]

Heat of Vaporization QV 134.7 kJ·mol−1 [154]Vapor pressure (300 K) Pv [155]

Ionization limit EI 5.391 7149 5(4) eV [156]

APPENDIX A. PROPERTIES OF LITHIUM 126

76.068 MHz

152.137 MHz

17.402 MHz

8.701 MHz

1.733 MHz

1.162 MHz

2.875 MHz

502.190 MHz

301.314 MHz

57.424 MHz

34.454 MHz

6.955 MHz

2.516 MHz

11.097 MHz

2P3/2

2P1/22P3/2

2P1/2

2S1/2 2S1/2

F=1/2

F=3/2

F=5/2

F=3/2

F=5/2

F=3/2

F=1/2

F= 2

F= 3

F= 2

F= 1

F= 2

F= 1

F= 1

F= 0

8.335 MHz

10.050 GHz

10.056 GHz

6Li 7Li

coefficients were calculated by NIST from the author’s data taken over temperatures from298.14 to 1599.99 K.

Gehm’s data sheet on lithium reports functions of pressure with respect to temperaturefor both the solid and liquid phase as

log10 PV = −54.87864− 6450.944

T− 0.01487480T + 24.82251 log10 T (solid)

log10 PV = 10.34540− 8345.574

T− 0.00008840T − 0.68106 log10 T (liquid)

(A.2)

where here contrary to the aforementioned vapor pressure function, pressure in in units ofTorr (mmHg) and again temperature is in K.

Lithium has two naturally abundant and stable isotopes, 7Li and 6Li with four andthree neutrons respectively. General isotopic properties can be found in the table but theexperiments presented in this thesis only use the bosonic isotope, lithium-7.


200 250 300 350 40010

-10

10-9

10-8

10-7

10-6

10-5

10-4

Temperature (°C)

Vap

or

Pre

ssu

re (

Torr

)

Figure A.1: Vapor pressure of lithium.

Table A.2: Physical properties of lithium

Property Symbol 7Li 6Li ReferenceAtomic number Z 3 3 [153]Total nucleons Z+N 7 6

Natural abundance η 92.5% 7.5% [154]Atomic mass m 7.016003 amu 6.0151214 amu [153]Nuclear spin I 3/2 1 [157]

Magnetic moment µ +3.25644 +0.822056 [157]


Table A.3: 7Li D2 (2S1/2 → 2P3/2) Transition Properties

Property Symbol Value ReferenceFrequency ω0 2π × 446.810183 THz [158]

Transition energy ~ω0 2.94128× 10−7 eVWavelength (vacuum) λ 670.961561 nm

Wave number k 9.36445× 104 cm−1

Lifetime τ 27.1 ns [155]Natural line width Γ = 1/τ 2π × 5.87 MHzOscillator strength f [108]Recoil frequency ωr 2π × 63.312 kHz

Recoil energy vr 8.49594 cm/sRecoil temperature Tr 6.07695 µK

Doppler temperature TD 142 µK

Table A.4: 7Li D1 (2S1/2 → 2P1/2) Transition Properties

Property Symbol Value ReferenceFrequency ω0 2π × 446.800130 THz [158]

Transition energy ~ω0 2.94122× 10−7 eVWavelength (vacuum) λ 670.976658 nm

Wave number k 9.36424× 104 cm−1

Lifetime τ 27.1 ns [155]Natural line width Γ = 1/τ 2π × 5.87 MHzOscillator strength f [108]Recoil frequency ωr 2π × 63.309 kHz

Recoil energy vr 8.49574 cm/sRecoil temperature Tr 6.07668 µK

Doppler temperature TD 142 µK

Below are tabulated values for 7Li optical properties for the D-line transitions. Therecoil velocity vr = ~k

Mand frequency ωr = ~k2

2Mfor the transition corresponds to one ~k

of photon momentum being imparted to the atom. The recoil temperature is defined asTr = 2Erec/kB = ~2k2

MkBwhere kB is the Boltzmann constant. The Doppler temperature is the

theoretical minimum temperature for a given transition, determined by the line width of thetransition: TD = ~Γ

2.


A.1 The level spectrum

In an atom, the effects of relativity and spin on the dynamics of the electron results in finestructure, defined as the coupling between the electron’s orbital angular momentum, L, andits spin angular momentum, S. Fine structure enlarges the Hilbert space of the system,

ε = εorb ⊗ εspin (A.3)

and the total electron angular momentum, J , is a sum of its spin and orbital momentumgiven by

J = L + S (A.4)

with |J| =√J(J + 1)~, bounded by

|L− S| ≤ J ≥ L+ S. (A.5)

In an alkali atom, the transition in which the valence s-orbital electron is excited tothe p-orbital (L = 0 → L = 1), is fractured by fine structure into a doublet of spectrallines, the D1 and D2 lines from 2S1/2 − 2P1/2 and 2S1/2 − 2P3/2, respectively. The nonzeronuclear angular momentum I of lithium (3/2 for lithium-7), couples to the electron’s totalangular momentum J and each of these fine structure levels is further split and has additionalhyperfine splitting. The total atomic angular momentum F is

F = J + I (A.6)

and where F can have values|J − I| ≤ F ≥ J + I. (A.7)

There exist 2F + 1 magnetic sublevels for each hyperfine level F in the the atom. Thesesublevels determine the angular distribution of the electron and furthermore, in the presenceof an external magnetic field, break the degeneracy of the hyperfine energy level. The statesare labeled by the quantum numbers (mF ), associated with the Fz operator.

A.2 Interaction with static fields

The hyperfine magnetic quantum numbers mF , which satisfy −F ≤ mF ≤ F , are degeneratein the absence of an external field. An externally applied magnetic field couples to themagnetic moments and if the energy shift due to this field is small compared to the finestructure splitting then the interaction Hamiltonian is the fine-structure interaction plus themagnetic-dipole interaction of the nuclear magnetic moment with the magnetic field.

H(hfs)B = H

(fs)B − µI ·B =

µB~

(gJJz + gIIz)Bz (A.8)

and first order perturbation theory leads to the following energy shift

∆E(hfs)B = µBgFmFB (A.9)


and the Lande gF factor (making the approximation that gI gJ) is

gF ∼ gJF (F + 1)− I(I + 1) + J(J + 1)

2F (F + 1)(A.10)

Static electric fields also shift the fine- and hyperfine-structure in an atom, much like howa static magnetic field shifts the energy levels. Consider the following atom-field interactionHamiltonian

HE = d · E (A.11)

where E is a static electric field and d is the atomic dipole operator.The energy level shift for an arbitrary state |α〉 is

∆Eα = 〈α|HE|α〉+∑j

|〈α|HE|βj〉|2

Eα − Eβj(A.12)

with |βj〉 labeling all other atomic states and Eα and Eβj representing the correspondingenergy levels. Because the dipole operator can only couple states of opposite parity, thefirst-order shift to the energy vanishes leaving only the second-order term. The effect istherefore second order in E and is called the quadratic Stark effect.

Defining an ‘effective’ Stark interaction Hamiltonian as

HStark :=∑j

HE|βj〉〈βj|HE

Eα − Eβj=∑j

dµ|βj〉〈βj|dνEα − Eβj

EµEν (A.13)

makes the energy shift look ‘first-order’ (albeit it is really a second order effect from secondorder perturbation theory). The Stark Hamiltonian has the form of a rank-2 tensor operator,contracted twice with the electric field vector.

HStark = SµνEµEν (A.14)

and

Sµν =∑j

dµ|βj〉〈βj|dνEα − Eβj

. (A.15)

Decomposing the tensor form of the Stark shift into its respective parts (the first being asvia Steck the orientation-independent part and the second being the anisotropic part) yields

∆Eα = 〈α|Sµν |α〉EµEν =1

3〈α|S(0)|α〉E2 + 〈α|S(2)

µν |α〉EµEν (A.16)

The fine-structure scalar polarizability is

α(0)(J) := −2

3

∑J ′

|〈J‖d‖J ′〉|2

EJ − EJ ′(A.17)


with the energy shift given by

∆E(0)J = −1

2α(0)(J)E2. (A.18)

The tensor polarizability is defined as

α(2)(J) := −〈J‖S(2)q ‖J〉

√8J(2J − 1)

3(J + 1)(2J + 2)(A.19)

with the energy shift given by

∆E(2)|JmJ 〉 = −1

4α(2)(J)(3E2

z − E2)(3m2

J − J(J + 1)

J(2J − 1)

). (A.20)

When the atom has hyperfine structure, the effective hyperfine Stark Hamiltonian is

HStark(F ) = −1

2α(0)(F )E2

z −1

2(F )(3E2

z − E2)(3F 2

z /~2 − F (F + 1)

F (2F − 1)

)(A.21)

and the quadratic Stark shift in energy is

∆E|FmF 〉 = −1

2α(0)(F )E2

z −1

4α(2)(F )(3E2

z − E2)(3m2

F − F (F + 1)

F (2F − 1)

). (A.22)

A.3 Interaction with dynamic fields

A.3.1 Reduced Matrix Elements in Atomic Transitions

The electric-dipole transition matrix elements quantify the interaction between internalatomic states and an external optical field that is nearly resonant with an energy split-ting of the states. While magnetic and even higher order multipole transitions are possible,these higher-order effects are much less influential and not considered here.

The dipole operator, a rank k = 1 tensor, governs atomic electric-dipole transitions.Fora transition between hyperfine states |FmF 〉 → |F ′m′F 〉 the probability amplitude is relatedto the Rabi frequency by

Ω(i)FmFF ′mF ′

= −1

~〈α′F ′m′F |d · Ei|αFmF 〉 (A.23)

where Ei is the electric component for a particular driving field in a Raman system andd = er is the atom’s dipole moment. Decomposing the field and position vectors intoirreducible tensor operators yields

d · Ei = eEi∑q

(−1)qrqε−q (A.24)


with εi being the polarization vector for the electric field and q ∈ −1, 0, 1.Applying the Wigner-Eckart theorem gives the following result

〈FmF |dq|F ′m′F 〉 = 〈F‖d‖F ′〉(−1)F′−F+m′F−mF

√2F + 1

2F ′ + 1〈F ′m′F |FmF ; 1− q〉.

(A.25)

From the above expression, it is apparent that when considering the dependence on twomagnetic sublevels of a matrix element, the entirety of the angular dependence is givensimply by a Clebsch-Gordan coefficient.

The reduced hyperfine matrix element can be decomposed in terms of the fine-structurereduced matrix element

〈F‖d‖F ′〉 = 〈JIF‖d‖J ′I ′F ′〉

= 〈J‖d‖J ′〉(−1)F′+J+1+I

√(2F ′ + 1)(2J + 1)

J J’ 1F’ F I

. (A.26)

Furthermore, the fine-structure reduced matrix element can be factored into a reducedmatrix element depending only upon the atom’s orbital angular momentum, quantum num-ber L, as

〈J‖d‖J ′〉 = 〈JSJ‖d‖L′S ′J ′〉

= 〈L‖d‖L′〉(−1)J′+L+1+S

√(2J ′ + 1)(2L+ 1)

L L’ 1J’ J S

. (A.27)

A.4 Clebsch-Gordan coefficients for D–line transitions


σ+-polarization |F ′ = F + 1〉 |F ′ = F 〉 |F ′ = F − 1〉

|F = 2,mF = −2〉√

1/30√

1/12√

1/20

|F = 2,mF = −1〉√

1/10√

1/8√

1/40

|F = 2,mF = 0〉√

1/5√

1/8√

1/120

|F = 2,mF = 1〉√

1/3√

1/12

|F = 2,mF = 1〉√

1/12

|F = 1,mF = −1〉√

1/24√

5/24√

1/6

|F = 1,mF = 0〉√

1/8√

5/24

|F = 1,mF = 1〉√

1/4

Table A.5: Clebsch-Gordan coefficients for the D2-line transition with σ+-polarized lightsuch that m′F = mF + 1.

σ−-polarization |F ′ = F + 1〉 |F ′ = F 〉 |F ′ = F − 1〉

|F = 2,mF = −2〉√

1/2

|F = 2,mF = −1〉√

1/3 −√

1/12

|F = 2,mF = 0〉√

1/5 −√

1/8√

1/120

|F = 2,mF = 1〉√

1/10 −√

1/8√

1/40

|F = 2,mF = 1〉√

1/30 −√

1/12√

1/20

|F = 1,mF = −1〉√

1/4

|F = 1,mF = 0〉√

1/8 −√

1/24

|F = 1,mF = 1〉√

1/24 −√

5/24√

1/6

Table A.6: Clebsch-Gordan coefficients for the D2-line transition with σ−-polarized lightsuch that m′F = mF − 1.


π-polarization |F ′ = F + 1〉 |F ′ = F 〉 |F ′ = F − 1〉

|F = 2,mF = −2〉 −√

1/6 −√

1/6

|F = 2,mF = −1〉 −√

4/15 −√

1/24√

1/40

|F = 2,mF = 0〉 −√

3/10 0√

1/30

|F = 2,mF = 1〉 −√

4/15√

1/24√

1/40

|F = 2,mF = 1〉 −√

1/6√

1/6

|F = 1,mF = −1〉 −√

1/8 −√

5/24

|F = 1,mF = 0〉 −√

1/6 0√

1/6

|F = 1,mF = 1〉 −√

1/8√

5/24

Table A.7: Clebsch-Gordan coefficients for the D2-line transition with π-polarized light suchthat m′F = mF .


|F = 2,mF = −2〉√

1/6√

1/2

|F = 2,mF = −1〉√

1/4√

1/4

|F = 2,mF = 0〉√

1/4√

1/12

|F = 2,mF = 1〉√

1/6

|F = 2,mF = 2〉

|F = 1,mF = −1〉 −√

1/12 −√

1/12

|F = 1,mF = 0〉 −√

1/4 −√

1/12

|F = 1,mF = 1〉 −√

1/2

Table A.8: Clebsch-Gordan coefficients for the D1-line transition with σ+-polarized lightsuch that m′F = mF + 1.



|F = 2,mF = −2〉

|F = 2,mF = −1〉 −√

1/6

|F = 2,mF = 0〉 −√

1/4√

1/12

|F = 2,mF = 1〉 −√

1/4√

1/4

|F = 2,mF = 2〉 −√

1/6√

1/2

|F = 1,mF = −1〉 −√

1/2

|F = 1,mF = 0〉 −√

1/4√

1/12

|F = 1,mF = 1〉 −√

1/12√

1/12

Table A.9: Clebsch-Gordan coefficients for the D1-line transition with σ−-polarized lightsuch that m′F = mF − 1.


|F = 2,mF = −2〉 −√

1/3

|F = 2,mF = −1〉 −√

1/12√

1/4

|F = 2,mF = 0〉 0√

1/3

|F = 2,mF = 1〉√

1/12√

1/4

|F = 2,mF = 2〉√

1/3

|F = 1,mF = −1〉√

1/4√

5/12

|F = 1,mF = 0〉√

1/3 0

|F = 1,mF = 1〉√

1/4 −√

1/12

Table A.10: Clebsch-Gordan coefficients for the D1-line transition with π-polarized light suchthat m′F = mF .

136

Appendix B

Two-Level System

When the atom is in the presence of a weak external field, nearly on resonance with asingle atomic transition, the physics simplifies to a two-level system. In this simplification,transitions to other states are negligible. Only a ‘ground’ |g〉 state and ‘excited’ |e〉 state areconnected by absorption and perhaps stimulated emission of the field. The field given by

E(r, t) = εE0e+i(k·r−ωt) + εE∗0e

−i(k·r−ωt) (B.1)

is monochromatic with angular frequency ω with ε denoting the unit polarization vector.The lowest order contribution in the multipole expansion of the atom-field interaction,

the dipole approximation, assumes that the wavelength of the field is much longer than thespatial extent of the atom. Any variations of the field over the atom system are ignored. Thisis generally true for cold atomic systems since atomic dimensions are of order an Angstromand optical transitions are hundreds of -nms or thousands of Angstroms.

The total Hamiltonian of this system is the sum of the free atomic Hamiltonian HA

HA = ~ω0|e〉〈e| (B.2)

(in which we have set the ground-state energy to zero) and the atom-field interaction Hamil-tonian HAF given by

HAF = −d · E . (B.3)

The atomic dipole operator d is defined in terms of the electron position re as

d = −qere (B.4)

The dipole operator can be decomposed with

I = |e〉〈e|+ |g〉〈g| (B.5)

and simplified assuming that〈g|d|g〉 = 〈e|d|e〉 = 0, (B.6)

APPENDIX B. TWO-LEVEL SYSTEM 137

as

d = 〈g|d|e〉|g〉〈e|+ 〈e|d|g〉|e〉〈g|= µeg|g〉〈e|+ µge|e〉〈g|= 〈g|d|e〉(σ + σ†)

(B.7)

The dipole matrix elements are defined as µab = 〈a|d|b〉.The total Hamiltonian is given by the following expression

H = HA +HAF = ~ω|e〉〈e| −(µge|g〉〈e|+ µeg|e〉〈g|

)·(E0e

−iωt + E∗0e+iωt). (B.8)

The state vector for the system is a linear combination of the eigenstates of the freeatomic Hamiltonian

|ψ〉 = cg(t)|g〉+ ce(t)|e〉.

The time-dependent Schrodinger equation

i~d

dt|ψ〉 = H|ψ〉,

produces a pair of coupled differential equations for the probability amplitudes cg(t) andce(t):

i~ce(t) = ~ωce(t)− µeg

(E0e

−iωt + E∗0e+iωt)cg(t) (B.9)

i~cg(t) = −µge

(E0e

−iωt + E∗0e+iωt)ce(t). (B.10)

In the rotating frame as defined by

ce(t) = ce(t)eiωt (B.11)

yields the following for these coupled equations

i~ce(t) = −~∆ce(t)− µeg

(E0 + E∗0e

+2iωt)cg(t) (B.12)

i~cg(t) = −µ∗eg

(E0e

−2iωt + E∗0)ce(t) (B.13)

with detuning given by ∆ = ω − ω0.In the above expression, the terms E0e

±2iωt will oscillate rapidly as e±i(ω+ω0)t comparedto the others oscillating as e±i∆t. The slowly oscillating terms will dominate the dynamics ofthe system. In the rotating wave approximation, we chose to focus on the slow dynamics oralternatively renormalize to a coarse grained fs-time scale, assuming that |ω−ω0| << ω+ω0.Terms oscillating at optical frequencies are replaced by their zero average value. This isa reasonable approximation to make given the physical response time of modern opticaldetectors.


With the rotating wave approximation implemented, the equations for the probabilityamplitudes become

˙ce(t) = i∆ce(t)− iΩR

2cg(t) (B.14)

˙cg(t) = −iΩ∗R2ce(t) (B.15)

where the Rabi frequency Ω has been defined as

ΩR := −2E0µ

~= −2E0〈e|d|g〉

~. (B.16)

For a Gaussian beam of beam waist w0, the intensity at the beam’s center is given by

I =P

πw20

where P is the total beam power. Rewriting this in terms of the electric field amplitude E0

gives

I =ε0cE2

0

2

Rewriting the field amplitude in terms of the total power yields the following expression forthe Rabi frequency:

Ω =µge

~w0

√2P

πε0c. (B.17)

The same above equations are alsp generated by an effective Hamiltonian for the atomin the rotating-frame:

HA = ~∆|e〉〈e|.

The effective interaction Hamiltonian is then

HAF = −~2

(Ω∗R|g〉〈e|+ ΩR|e〉〈g|

)(B.18)

B.1 Flip-flop

B.1.1 On resonance

Solving the coupled equations for the amplitudes of the atomic states in the rotating frameyields the driven dynamics of this two-level system.

When the light is exactly on resonance with the splitting of the atomic states (∆ = 0),the coupled equations for the probability amplitudes are

cg = −iΩ2ce


˙ce = −iΩ2cg.

(B.19)

Decoupled these reveal the final solution for the amplitudes in terms of the initial condi-tions for a field on resonance with the energy splitting of the atomic states (∆ = 0),

cg(t) = cg(0) cos

(Ωt

2

)− ice(0) sin

(Ωt

2

)ce(t) = ce(0) cos(

Ωt

2)− icg(0) sin(

Ωt

2). (B.20)

The square of the amplitudes gives the probability of finding the atom in either state asa function of time. If the atom is prepared so as to be in the ground state at t = 0 then wehave the following ground- and excited-state populations as function of time

Pg(t) = |cg(t)|2 = cos2

(Ωt

2

)=

1

2(1 + cos Ωt)

Pe(t) = |ce(t)|2 = sin2

(Ωt

2

)=

1

2(1− cos Ωt)

where the probability for either state oscillates as a function of interaction time.For an atom beginning in the ground state, if the field is pulsed on for a duration of

t = π/Ω, then the atom will have unit probability to transition to the excited state. Thistype of pulse is referred to as a π-pulse and acts as the ‘mirror’ in our atom interferometer.Similarly, a beamsplitter pulse or π/2-pulse is realized when the field is pulsed on for aduration (t = π/2Ω) = such that the atom has a 50% probability of being transferred to theexcited state.

B.1.2 Almost on resonance

For a nonzero detuning, ∆ 6= 0, we begin again with the coupled differential equations forthe probability amplitudes

cg = −iΩ2ce

˙ce = i∆ce − iΩ

2cg.

Decoupling, solving and factoring produces(∂t − i

∆

2+ i

Ω

2

)(∂t − i

∆

2− iΩ

2

)cg = 0(

∂t − i∆

2+ i

Ω

2

)(∂t − i

∆

2− iΩ

2

)ce = 0


where now we can define the generalized Rabi frequency as

Ω :=√

Ω2 + ∆2. (B.21)

General solutions to the above equations can be found in terms of the initial probabilityamplitudes at time t = 0

cg(t) = ei∆t/2[cg(0) cos

(Ωt

2

)− i

Ω[∆cg(0) + Ωce(0)] sin

()Ωt

2

]ce(t) = ei∆t/2

[ce(0) cos

(Ωt

2

)+i

Ω[∆ce(0)− Ωcg(0)] sin

(Ωt

2

)](B.22)

Therefore, for an atom initially in the ground state, the probability of excitation to thehigher energy state is

Pe(t) =Ω2

Ω2sin2

(Ωt

2

)=

Ω2

Ω2

(1

2− 1

2 cos Ωt

)(B.23)

At nonzero detuning, we observe both a reduction in amplitude and an increase to thegeneralized Rabi frequency.

141

Appendix C

Bloch sphere

The Bloch sphere is a geometric interpretation of the two-level atomic system, interactingwith an external optical field with Hamiltonian given by a sum of the free-atomic Hamiltonianand the interaction Hamiltonian

H = HA + Hint. (C.1)

Assuming that the driving field’s frequency is close to the splitting between states, atany instant the wave function for the atom must be a linear combination of the two energyeigenstates

|ψ〉 = c1|1〉+ c2|2〉or written in the configuration basis the wave function for the system is the following:

Ψ(r, t) = c1(t)ψ1(r, t) + c2(t)ψ2(r, t).

Substitution of the above general wave function into the time-dependent Schrodingerequation leads to the following expression for the coefficients c1 and c2,

i~dc1

dt= 〈1|Hint|1〉c1 + exp(−iω0t)〈1|Hint|2〉c2 (C.2)

i~dc2

dt= 〈2|Hint|1〉c1 + exp(iω0t)〈2|Hint|1〉c1 (C.3)

This two-dimensional state space warrants a description in terms of the Pauli matrices.Writing down the density matrix ρ which describes the quantum system as a statisticalensemble generically given by

ρ =∑n,m

cnc∗m|n〉〈m| (C.4)

for pure state |n,m〉 with amplitude cn,m.For the above two-level system, the density matrix is

ρ = |Ψ〉〈Ψ| =

[|c1|2 c1c

∗2

c2c∗1 |c2|2

]=

[ρ11 ρ12

ρ21 ρ22

](C.5)

APPENDIX C. BLOCH SPHERE 142

given in terms of the off-diagonal ‘coherences’, ρ12 and ρ21, which describe the couplingbetween levels due to the interaction with the field and the diagonal ‘populations’, ρ11 andρ22, which detail the probabilities for the atom to be found in the associated pure state.

The dynamics of the density matrix is described by the (Liouville variant form) time-dependent Schrodinger equation

i~∂ρ

∂t= [H, ρ] (C.6)

with the following two-level Hamiltonian

H = −~

[ΩAC

1ΩR2ei(δt−φγ)

Ω∗R2e−i(δt−φγ) ΩAC

2

]. (C.7)

In the rotating frame, defined by

c1 = c1e−iδt/2 (C.8)

c2 = c2eiδt/2 (C.9)

the equations of motion for the populations and coherences are as follows

i~∂ρ11

∂t= −~ΩR

2e−iφγ ρ21 +

~Ω∗R2eiφγ ρ12 (C.10)

i~∂ρ22

∂t= −~Ω∗R

2eiφγ ρ12 +

~ΩR

2e−iφγ ρ21 (C.11)

i~∂ρ12

∂t= ~(δ − δAC)ρ12 +

~ΩR

2e−iφγ (ρ11 − ρ22) (C.12)

i~∂ρ21

∂t= −~(δ − δAC)ρ21 −

~Ω∗R2eiφγ (ρ11 − ρ22). (C.13)

The density operator can be expanded in the complete basis of Pauli matrices (σx, σy, σz)as

ρ =1

2

(1 + (ρ22 − ρ11)σz + (ρ21 + ρ12)σx + (ρ21 − ρ12)σy

). (C.14)

where σx, σy, and σz are defined as

σx =

(0 11 0

)σy =

(0 −ii 0

)σz =

(1 00 -1

)(C.15)

The expectation value of an arbitrary operator A can be computed using the trace formula

〈A〉 = Tr[ρA] = 〈ψ|A|ψ〉.


The Bloch vector is defined as

R = ux + vy + wz (C.16)

for which the expectation values of the Pauli spin matrices define the amplitudes along theaxes:

u = Tr(σxρ) = ρ12 + ρ21

v = Tr(σyρ) = −i(ρ21 − ρ12)

w = Tr(σzρ) = ρ11 − ρ22. (C.17)

The dynamics of the expectation values are described in the absence of spontaneousemission by the following coupled differential equations:

∂u

∂t= −(δ − δAC)v − i

2

(ΩRe

−iφγ − Ω∗Reiφγ)w (C.18)

∂v

∂t= (δ − δAC)u− 1

2

(ΩRe

−iφγ + Ω∗Reiφγ)w (C.19)

∂w

∂t= −i

(Ω∗Re

iφγ ρ12 − ΩRe−iφγ ρ21

). (C.20)

For a real Rabi frequency, ΩR = Ω∗R, the equations become

∂u

∂t= −(δ − δAC)v − ΩR sinφγw (C.21)

∂v

∂t= (δ − δAC)u+ ΩR cosφγw (C.22)

∂w

∂t= −ΩR cosφγv + ΩR sinφγu. (C.23)

The time-evolution of R can be represented as torque produced by the field Ω written as

dR

dt= R×Ω (C.24)

with the field vector defined as follows:

Ω = Ωr cosφγx + ΩR sinφγy − (δ − δAC)z. (C.25)

The rate at which the Bloch vector R rotates around Ω is given by

|Ω| =√

Ω2R + (δ − δAC)2, (C.26)

which is equal to the generalized two-photon Rabi frequency. Incorporating spontaneousemission would add a time-dependence to the magnitude of the vector.


C.1 Simulations of interferometry

The Bloch sphere provides a nice visualization in the context of atom interferometry, par-ticularly when considering light-pulses that may or may not induce the expected rotation.

C.1.1 Mathematica code

ketΨ[ν , ϕ ]:=Cos[ν/2],Exp[I ∗ ϕ]Sin[ν/2]ketΨ[ν , ϕ ]:=Cos[ν/2],Exp[I ∗ ϕ]Sin[ν/2]ketΨ[ν , ϕ ]:=Cos[ν/2],Exp[I ∗ ϕ]Sin[ν/2]Ψ00 = ketΨ[0, 0]; (*Initiallyin(1, 0)state*)Ψ00 = ketΨ[0, 0]; (*Initiallyin(1, 0)state*)Ψ00 = ketΨ[0, 0]; (*Initiallyin(1, 0)state*)

(*timeevolutionoperatorwhereθ = Ωr ∗ t,(*timeevolutionoperatorwhereθ = Ωr ∗ t,(*timeevolutionoperatorwhereθ = Ωr ∗ t,φisthelaserphaseandαknowsaboutthe2− photondetuning*)φisthelaserphaseandαknowsaboutthe2− photondetuning*)φisthelaserphaseandαknowsaboutthe2− photondetuning*)U [θ , φ , α ]:=Cos[θ/2] ∗ (1, 0, 0, 1)−U [θ , φ , α ]:=Cos[θ/2] ∗ (1, 0, 0, 1)−U [θ , φ , α ]:=Cos[θ/2] ∗ (1, 0, 0, 1)−I ∗ Sin[θ/2] ∗ (PauliMatrix[1] ∗ Cos[φ] ∗ Cos[α] + PauliMatrix[2] ∗ Sin[φ] ∗ Cos[α]+I ∗ Sin[θ/2] ∗ (PauliMatrix[1] ∗ Cos[φ] ∗ Cos[α] + PauliMatrix[2] ∗ Sin[φ] ∗ Cos[α]+I ∗ Sin[θ/2] ∗ (PauliMatrix[1] ∗ Cos[φ] ∗ Cos[α] + PauliMatrix[2] ∗ Sin[φ] ∗ Cos[α]+PauliMatrix[3] ∗ Sin[α])PauliMatrix[3] ∗ Sin[α])PauliMatrix[3] ∗ Sin[α])

Utilde[OMr ,OM , t , φ ]:=Utilde[OMr ,OM , t , φ ]:=Utilde[OMr ,OM , t , φ ]:=Cos[OMr ∗ t/2] ∗ (1, 0, 0, 1)−Cos[OMr ∗ t/2] ∗ (1, 0, 0, 1)−Cos[OMr ∗ t/2] ∗ (1, 0, 0, 1)−I ∗ Sin[OMr ∗ t/2]∗I ∗ Sin[OMr ∗ t/2]∗I ∗ Sin[OMr ∗ t/2]∗(PauliMatrix[1] ∗ Cos[φ] ∗OM/OMr + PauliMatrix[2] ∗ Sin[φ] ∗OM/OMr+(PauliMatrix[1] ∗ Cos[φ] ∗OM/OMr + PauliMatrix[2] ∗ Sin[φ] ∗OM/OMr+(PauliMatrix[1] ∗ Cos[φ] ∗OM/OMr + PauliMatrix[2] ∗ Sin[φ] ∗OM/OMr+PauliMatrix[3] ∗ Sqrt[OMr∧2−OM∧2]/OMr)PauliMatrix[3] ∗ Sqrt[OMr∧2−OM∧2]/OMr)PauliMatrix[3] ∗ Sqrt[OMr∧2−OM∧2]/OMr)

UUtilde[θ , φ , δ ,OM ,OMr ]:=UUtilde[θ , φ , δ ,OM ,OMr ]:=UUtilde[θ , φ , δ ,OM ,OMr ]:=Cos[θ/2] ∗ (1, 0, 0, 1)−Cos[θ/2] ∗ (1, 0, 0, 1)−Cos[θ/2] ∗ (1, 0, 0, 1)−I ∗ Sin[θ/2]∗I ∗ Sin[θ/2]∗I ∗ Sin[θ/2]∗(PauliMatrix[1] ∗ Cos[φ] ∗OM/OMr + PauliMatrix[2] ∗ Sin[φ] ∗OM/OMr−(PauliMatrix[1] ∗ Cos[φ] ∗OM/OMr + PauliMatrix[2] ∗ Sin[φ] ∗OM/OMr−(PauliMatrix[1] ∗ Cos[φ] ∗OM/OMr + PauliMatrix[2] ∗ Sin[φ] ∗OM/OMr−PauliMatrix[3] ∗ δ/OMr)PauliMatrix[3] ∗ δ/OMr)PauliMatrix[3] ∗ δ/OMr)(*Hopffibration; mappingSU(2)toSO(3)*)(*Hopffibration; mappingSU(2)toSO(3)*)(*Hopffibration; mappingSU(2)toSO(3)*)SPINORmapR3[spinor ]:=Module[new, x1, x2, x3, x4, z1, z2, z3,SPINORmapR3[spinor ]:=Module[new, x1, x2, x3, x4, z1, z2, z3,SPINORmapR3[spinor ]:=Module[new, x1, x2, x3, x4, z1, z2, z3,new = ;new = ;new = ;x1 = Re[spinor[[1]]];x1 = Re[spinor[[1]]];x1 = Re[spinor[[1]]];x2 = Im[spinor[[1]]];x2 = Im[spinor[[1]]];x2 = Im[spinor[[1]]];x3 = Re[spinor[[2]]];x3 = Re[spinor[[2]]];x3 = Re[spinor[[2]]];x4 = Im[spinor[[2]]];x4 = Im[spinor[[2]]];x4 = Im[spinor[[2]]];

z1 = 2 ∗ (x1 ∗ x3 + x2 ∗ x4);z1 = 2 ∗ (x1 ∗ x3 + x2 ∗ x4);z1 = 2 ∗ (x1 ∗ x3 + x2 ∗ x4);z2 = 2 ∗ (x2 ∗ x3− x1 ∗ x4);z2 = 2 ∗ (x2 ∗ x3− x1 ∗ x4);z2 = 2 ∗ (x2 ∗ x3− x1 ∗ x4);z3 = (x1)∧2 + (x2)∧2− (x3)∧2− (x4)∧2;z3 = (x1)∧2 + (x2)∧2− (x3)∧2− (x4)∧2;z3 = (x1)∧2 + (x2)∧2− (x3)∧2− (x4)∧2;new = z1, z2, z3;new = z1, z2, z3;new = z1, z2, z3;out = newout = newout = new]]]


points1 = Table[SPINORmapR3[Utilde[Pi/2 ∗OM,Pi/2, 1, 0].Ψ00], OM, 1, 10, 0.1];points1 = Table[SPINORmapR3[Utilde[Pi/2 ∗OM,Pi/2, 1, 0].Ψ00], OM, 1, 10, 0.1];points1 = Table[SPINORmapR3[Utilde[Pi/2 ∗OM,Pi/2, 1, 0].Ψ00], OM, 1, 10, 0.1];points1v2 = Table[SPINORmapR3[Utilde[Pi/2 ∗ (1 + ε) ∗OM,Pi/2 ∗ (1 + ε), 1, 0].Ψ00],points1v2 = Table[SPINORmapR3[Utilde[Pi/2 ∗ (1 + ε) ∗OM,Pi/2 ∗ (1 + ε), 1, 0].Ψ00],points1v2 = Table[SPINORmapR3[Utilde[Pi/2 ∗ (1 + ε) ∗OM,Pi/2 ∗ (1 + ε), 1, 0].Ψ00],ε, 0, 0.25, 0.01, OM, 1, 2, 0.1];ε, 0, 0.25, 0.01, OM, 1, 2, 0.1];ε, 0, 0.25, 0.01, OM, 1, 2, 0.1];points2 = Table[SPINORmapR3[Utilde[Pi/2 ∗OM,Pi/2, 1, 0].Ψ00], OM, 1, 2, 0.05];points2 = Table[SPINORmapR3[Utilde[Pi/2 ∗OM,Pi/2, 1, 0].Ψ00], OM, 1, 2, 0.05];points2 = Table[SPINORmapR3[Utilde[Pi/2 ∗OM,Pi/2, 1, 0].Ψ00], OM, 1, 2, 0.05];points3 = Table[SPINORmapR3[Utilde[Pi ∗OM,Pi, 1, 0].Ψ00], OM, 1, 10, 0.1];points3 = Table[SPINORmapR3[Utilde[Pi ∗OM,Pi, 1, 0].Ψ00], OM, 1, 10, 0.1];points3 = Table[SPINORmapR3[Utilde[Pi ∗OM,Pi, 1, 0].Ψ00], OM, 1, 10, 0.1];points4 = Table[SPINORmapR3[Utilde[Pi ∗OM,Pi, 1, 0].Ψ00], OM, 1, 2, 0.05];points4 = Table[SPINORmapR3[Utilde[Pi ∗OM,Pi, 1, 0].Ψ00], OM, 1, 2, 0.05];points4 = Table[SPINORmapR3[Utilde[Pi ∗OM,Pi, 1, 0].Ψ00], OM, 1, 2, 0.05];

Graphics3D[Opacity[0.1], Sphere[], Dashed,Line[−1.25, 0, 0, 1.25, 0, 0],Graphics3D[Opacity[0.1], Sphere[], Dashed,Line[−1.25, 0, 0, 1.25, 0, 0],Graphics3D[Opacity[0.1], Sphere[], Dashed,Line[−1.25, 0, 0, 1.25, 0, 0],Dashed,Line[0,−1.25, 0, 0, 1.25, 0],Dashed,Line[0,−1.25, 0, 0, 1.25, 0],Dashed,Line[0,−1.25, 0, 0, 1.25, 0],Dashed,Line[0, 0,−1.25, 0, 0, 1.25],Dashed,Line[0, 0,−1.25, 0, 0, 1.25],Dashed,Line[0, 0,−1.25, 0, 0, 1.25],Arrowheads[Medium],Arrow[1.25, 0, 0, 1.35, 0, 0],Arrowheads[Medium],Arrow[1.25, 0, 0, 1.35, 0, 0],Arrowheads[Medium],Arrow[1.25, 0, 0, 1.35, 0, 0],Arrowheads[Medium],Arrow[0, 1.25, 0, 0, 1.35, 0],Arrowheads[Medium],Arrow[0, 1.25, 0, 0, 1.35, 0],Arrowheads[Medium],Arrow[0, 1.25, 0, 0, 1.35, 0],Arrowheads[Medium],Arrow[0, 0, 1.23, 0, 0, 1.35],Arrowheads[Medium],Arrow[0, 0, 1.23, 0, 0, 1.35],Arrowheads[Medium],Arrow[0, 0, 1.23, 0, 0, 1.35],Text[Style[x, 14], 1.45, 0, 0],Text[Style[y, 14], 0, 1.45, 0],Text[Style[x, 14], 1.45, 0, 0],Text[Style[y, 14], 0, 1.45, 0],Text[Style[x, 14], 1.45, 0, 0],Text[Style[y, 14], 0, 1.45, 0],Text[Style[z, 14], 0, 0, 1.45], PointSize[Medium],Pink,Point[points1],Text[Style[z, 14], 0, 0, 1.45], PointSize[Medium],Pink,Point[points1],Text[Style[z, 14], 0, 0, 1.45], PointSize[Medium],Pink,Point[points1],PointSize[Medium],Blue,Point[points2],Boxed→ False]PointSize[Medium],Blue,Point[points2],Boxed→ False]PointSize[Medium],Blue,Point[points2],Boxed→ False]Graphics3D[Opacity[0.1], Sphere[], Dashed,Line[−1.25, 0, 0, 1.25, 0, 0],Graphics3D[Opacity[0.1], Sphere[], Dashed,Line[−1.25, 0, 0, 1.25, 0, 0],Graphics3D[Opacity[0.1], Sphere[], Dashed,Line[−1.25, 0, 0, 1.25, 0, 0],Dashed,Line[0,−1.25, 0, 0, 1.25, 0],Dashed,Line[0,−1.25, 0, 0, 1.25, 0],Dashed,Line[0,−1.25, 0, 0, 1.25, 0],Dashed,Line[0, 0,−1.25, 0, 0, 1.25],Dashed,Line[0, 0,−1.25, 0, 0, 1.25],Dashed,Line[0, 0,−1.25, 0, 0, 1.25],Arrowheads[Medium],Arrow[1.25, 0, 0, 1.35, 0, 0],Arrowheads[Medium],Arrow[1.25, 0, 0, 1.35, 0, 0],Arrowheads[Medium],Arrow[1.25, 0, 0, 1.35, 0, 0],Arrowheads[Medium],Arrow[0, 1.25, 0, 0, 1.35, 0],Arrowheads[Medium],Arrow[0, 1.25, 0, 0, 1.35, 0],Arrowheads[Medium],Arrow[0, 1.25, 0, 0, 1.35, 0],Arrowheads[Medium],Arrow[0, 0, 1.23, 0, 0, 1.35],Arrowheads[Medium],Arrow[0, 0, 1.23, 0, 0, 1.35],Arrowheads[Medium],Arrow[0, 0, 1.23, 0, 0, 1.35],Text[Style[x, 14], 1.45, 0, 0],Text[Style[y, 14], 0, 1.45, 0],Text[Style[x, 14], 1.45, 0, 0],Text[Style[y, 14], 0, 1.45, 0],Text[Style[x, 14], 1.45, 0, 0],Text[Style[y, 14], 0, 1.45, 0],Text[Style[z, 14], 0, 0, 1.45],Text[Style[z, 14], 0, 0, 1.45],Text[Style[z, 14], 0, 0, 1.45],PointSize[Medium],Black,Point[points3], PointSize[Medium],Red,Point[points4],PointSize[Medium],Black,Point[points3], PointSize[Medium],Red,Point[points4],PointSize[Medium],Black,Point[points3], PointSize[Medium],Red,Point[points4],Boxed→ False]Boxed→ False]Boxed→ False]Graphics3D[Opacity[0.1], Sphere[], Dashed,Line[−1.25, 0, 0, 1.25, 0, 0],Graphics3D[Opacity[0.1], Sphere[], Dashed,Line[−1.25, 0, 0, 1.25, 0, 0],Graphics3D[Opacity[0.1], Sphere[], Dashed,Line[−1.25, 0, 0, 1.25, 0, 0],Dashed,Line[0,−1.25, 0, 0, 1.25, 0],Dashed,Line[0,−1.25, 0, 0, 1.25, 0],Dashed,Line[0,−1.25, 0, 0, 1.25, 0],Dashed,Line[0, 0,−1.25, 0, 0, 1.25],Dashed,Line[0, 0,−1.25, 0, 0, 1.25],Dashed,Line[0, 0,−1.25, 0, 0, 1.25],Arrowheads[Medium],Arrow[1.25, 0, 0, 1.35, 0, 0],Arrowheads[Medium],Arrow[1.25, 0, 0, 1.35, 0, 0],Arrowheads[Medium],Arrow[1.25, 0, 0, 1.35, 0, 0],Arrowheads[Medium],Arrow[0, 1.25, 0, 0, 1.35, 0],Arrowheads[Medium],Arrow[0, 1.25, 0, 0, 1.35, 0],Arrowheads[Medium],Arrow[0, 1.25, 0, 0, 1.35, 0],Arrowheads[Medium],Arrow[0, 0, 1.23, 0, 0, 1.35],Arrowheads[Medium],Arrow[0, 0, 1.23, 0, 0, 1.35],Arrowheads[Medium],Arrow[0, 0, 1.23, 0, 0, 1.35],Text[Style[x, 14], 1.45, 0, 0],Text[Style[y, 14], 0, 1.45, 0],Text[Style[x, 14], 1.45, 0, 0],Text[Style[y, 14], 0, 1.45, 0],Text[Style[x, 14], 1.45, 0, 0],Text[Style[y, 14], 0, 1.45, 0],Text[Style[z, 14], 0, 0, 1.45],Text[Style[z, 14], 0, 0, 1.45],Text[Style[z, 14], 0, 0, 1.45],PointSize[Small],Red,Point[points1v2[[1,All]]],PointSize[Small],Red,Point[points1v2[[1,All]]],PointSize[Small],Red,Point[points1v2[[1,All]]],PointSize[Small],Black,Point[points1v2[[2,All]]],PointSize[Small],Black,Point[points1v2[[2,All]]],PointSize[Small],Black,Point[points1v2[[2,All]]],PointSize[Small],Black,Point[points1v2[[4,All]]],PointSize[Small],Black,Point[points1v2[[4,All]]],PointSize[Small],Black,Point[points1v2[[4,All]]],PointSize[Small],Black,Point[points1v2[[5,All]]],PointSize[Small],Black,Point[points1v2[[5,All]]],PointSize[Small],Black,Point[points1v2[[5,All]]],PointSize[Small],Black,Point[points1v2[[6,All]]],PointSize[Small],Black,Point[points1v2[[6,All]]],PointSize[Small],Black,Point[points1v2[[6,All]]],PointSize[Small],Black,Point[points1v2[[7,All]]],PointSize[Small],Black,Point[points1v2[[7,All]]],PointSize[Small],Black,Point[points1v2[[7,All]]],PointSize[Small],Black,Point[points1v2[[8,All]]],PointSize[Small],Black,Point[points1v2[[8,All]]],PointSize[Small],Black,Point[points1v2[[8,All]]],


Trajectory of probability amplitude on the Bloch sphere

Figure C.1: Model of trajectory of state on the Bloch sphere.

PointSize[Small],Black,Point[points1v2[[9,All]]],PointSize[Small],Black,Point[points1v2[[9,All]]],PointSize[Small],Black,Point[points1v2[[9,All]]],PointSize[Small],Black,Point[points1v2[[10,All]]],PointSize[Small],Black,Point[points1v2[[10,All]]],PointSize[Small],Black,Point[points1v2[[10,All]]],PointSize[Small],Black,Point[points1v2[[3,All]]],Boxed→ False]PointSize[Small],Black,Point[points1v2[[3,All]]],Boxed→ False]PointSize[Small],Black,Point[points1v2[[3,All]]],Boxed→ False]ListPlot[0.5 ∗ (1− points1[[1;;20, 3]]), 0.5 ∗ (1− points3[[1;;20, 3]]),PlotRange→ Full]ListPlot[0.5 ∗ (1− points1[[1;;20, 3]]), 0.5 ∗ (1− points3[[1;;20, 3]]),PlotRange→ Full]ListPlot[0.5 ∗ (1− points1[[1;;20, 3]]), 0.5 ∗ (1− points3[[1;;20, 3]]),PlotRange→ Full]Plot[(Pi/2) ∗ Sqrt[1 + δ∧2], (Pi) ∗ Sqrt[1 + δ∧2], δ, 0, 10]Plot[(Pi/2) ∗ Sqrt[1 + δ∧2], (Pi) ∗ Sqrt[1 + δ∧2], δ, 0, 10]Plot[(Pi/2) ∗ Sqrt[1 + δ∧2], (Pi) ∗ Sqrt[1 + δ∧2], δ, 0, 10]

147

Appendix D

Magneto-optical traps

When an atom scatters a photon, a defined momentum with a particular direction andmagnitude is imparted to the atom as it absorbs the photon. However, if spontaneousemission then follows, repeating this process will ultimately result in a net force or kick ofthe atom in the direction of the light. Momentum conservation is not the only principle atplay in laser cooling atoms but it is the foundation upon which the rest of the subtletiesreside. By carefully choosing the attributes of the laser light, optical molasses results sonamed because of the vicious damping force imposed by it on the atoms.

D.0.1 Optical molasses

Consider an atom with nonzero velocity given by v irradiated by a collimated monochromaticlaser beam. If the laser is tuned to be near resonant with an atomic transition, the atomwill absorb photons from the beam and will scatter light from the with a rate given by

Rs =Γ

2

Ω2/2

δ2 + Ω2/2 + Γ2/4=

Γ

2

I/Isat

1 + I/Isat + 4δ2/Γ2. (D.1)

Here, δ = ω − ω0 + k · v is the difference between the laser frequency ω and Doppler-shifted atomic resonance ω0 + k · v. The isotropicity of emission results in the atom beingpreferentially ‘kicked’ in the direction of the absorbed photon’s momentum. The magnitudeof the resulting force upon the atom is as follows

Fs = ~kΓ

2

I/Isat

1 + I/Isat + 4δ2/Γ2. (D.2)

The Rabi frequency Ω and saturation intensity Isat are related by the natural line width ofthe transition as

I

Isat=

2Ω2

Γ2. (D.3)

In the limit of infinite intensity I →∞, the maximum scattering force is realized as

Fmax =~kΓ

2(D.4)

APPENDIX D. MAGNETO-OPTICAL TRAPS 148

and the maximum acceleration of an atom with mass M is

amax =~kΓ

2M=vr2τ

(D.5)

where τ is the lifetime of the excited state and vr = ~k/m is the velocity at which the atomrecoils after absorbing and emitting a photon with wavenumber k.

Optical molasses is a Doppler-cooling technique which exploits the velocity-dependenceof the radiation scattering force on atoms. Optical molasses consists of orthogonal pairsof counter-propagating laser beams with frequency ω, that are red-detuned or tuned belowatomic resonance of a cycling transition, ω < ω0. As an atom moves towards a beam, it seesthe beam’s frequency Doppler shifted, closer to resonance with the transitions δ± = δ ± kv.

The force due to radiation pressure from these counter-propagating beams is given by

Frad = ~k(Γ/2)3

(1

(δ − kv)2 + (Γ/2)2− 1

(δ + kv)2 + (Γ/2)2

)I

Isat. (D.6)

For small velocities, the force becomes viscous and ‘molasses-like’; it is proportional to v,

Frad =~k2γ3

2

δ

(δ2 + (γ/2)2)2

I

Isatv, (D.7)

with a velocity capture range of

±|∆|k

γ

2k= ±γλ

4π. (D.8)

The lower limit to cooling in this picture is referred to as the Doppler temperature,

TD =~Γ

2kB(D.9)

and is in principle limited by the line-width of the transition.

D.0.2 Magnetic trapping

The velocity-dependent force of optical molasses confines the atoms in momentum spacebut not position. While the atoms are sludging through the optical molasses, the preferreddirection for absorption combined with isotropic emission of momentum will cause the atomto eventually diffuse outward. To remedy this, a position-dependent force is created bycoupling a magnetic quadrupole field to the aforementioned optical molasses in a hybridtrap, referred to as a magneto-optical trap (MOT).

The quadrupole field imposes a quantization axis and creates spatially varying Zeemanshifts of the atomic energy levels,

∆EF,mF = µBgFmFB (D.10)


+1

0

mF

0

-1

-1

0

+1

E

B(z)

z

σ+ σ-

ω0

ω

Δ

Figure D.1: The combined effect of quadrupole field and optical molasses, here orthogonallypolarized, confines the atoms spatially in addition to cooling them to lower temperature.The hybrid trap is called a magneto-optical trap.

where gF is the Lande g-factor, mF is the magnetic quantum number, B is the field magnitudeand µB is the Bohr magnetron. The energy shift changes sign through the trap center atwhich |B| = 0.

The magneto-optical trap is created by adding counter-propagating laser beams, of op-posite circular polarization and red-detuned from resonance to the quadrupole field. At lowintensity, the total force on the atoms becomes

F± = ±~kγ

2

s0

1 + s0 + (2δ±/γ)2. (D.11)

The detuning δ± now has a contribution from the Zeeman shift given by

δ± = δ ∓ k · v ± µ′B/~, (D.12)

where µ′ ≡ (geFmeF − g

gFm

gF )µB is the effective magnetic moment of the transition. For an

atom moving outward from the trap center, the magnetic field will tune either ∆mF = 1 or


∆mF = −1 closer to resonance, depending upon the sign gg,eF . Fortunately, the transitionnot tuned closer will shift further from resonance, with the roles being reversed on the otherside of the trap’s zero. An appropriate choice of beam polarization, will push the atoms backtowards the center of the trap. For example, σ− polarization incident on an atom movingrightward at z′ > 0, for which the transition ∆mF = −1 is tuned closer, will push the atomsback to the center. The density of trapped atoms is limited by outward radiation pressureor the re-absorption of scattered photons.

151

Appendix E

α0, the static polarizability

Lithium’s atomic properties can be computed with high accuracy, only utilizing ab initiowave functions. This is a result of lithium’s simple electron structure. The Stark shiftmeasurements [74] are currently the most stringent test of the polarizability calculations forlithium. This measurement gave a polarizability difference of -37.14(2) a.u. for the 2s−2p1/2

transition and is in excellent agreement with the 7Li Hylleraas difference computed to be-37.14(4) a.u.

E.1 Nonrelativistic α(0)

Beginning with a nonrelativistic Hamiltonian for an infinitely heavy lithium atom in thepresence of an external electric field E

H =∑a

pa2−∑a

Z

ra+∑a>b

1

rab−∑a

E iria (E.1)

Transition Value Method2s− 2p1/2

Th. Hylleraas 37.14(3)Th. CICP 37.26

Th. RLCCSDT 37.104Expt. 37.146(17)Expt. 37.11(33)

2s− 2p3/2

Th. RLCCSDT 37.089 MBPTExpt. 37.30(42)

Table E.1: Scalar polarizability differences α0(nPJ)− α0(nS) in a.u. for 7Li.

APPENDIX E. α0, THE STATIC POLARIZABILITY 152

the nonrelativistic electric dipole polarizability α0 can be evaluated with second-order per-turbation theory as

α0 = −2

3

∑a,b

〈φ0|ria1

E0 −H0

rib|φ0〉 (E.2)

where φ0 and E0 are the ground-state wave function and nonrelativistic energy, respectively.Here, H0 represents the Hamiltonian in the absence of the external field. Modifying H0 byδH to include corrections from the finite mass of the nucleus, relativity and QED given by

δH = λHMP + α2Hrel + α3HQED (E.3)

where λ = −µ/M is the ratio of the reduced electron mass to the nucleus mass and MPdenotes the mass polarization correction.

With δH, a perturbative formula for the electric dipole polarizability yields

αE = α0 + δαE (E.4)

with

−3

2δαE =

∑a,b

[2〈φ0|δH

1

(E0 −H0)ria

1

E0 −H0

rib|φ0〉

+〈φ0|ria1

E0 −H0

(δH − 〈δH〉) 1

E0 −H0

rib|φ0〉]. (E.5)

The components of δH are given by [102]

HMP = −∑a<b

piapib, (E.6)

Hrel =∑a

[− p4

a

8+πZ

2δ3(ra)

]+∑a<b

[πδ3(rab)−

1

2pia

(δij

rab+riabr

jab

r3ab

)pjb

](E.7)

HQED =4Z

3

[19

30+ ln(α−2)− ln k0

]∑a

δ3(ra)

+

[164

15+

14

3lnα

]∑a<b

δ3(rab)−7

6π

∑a<b

P

(1

r3ab

). (E.8)

In the above expression, the Bethe logarithm is defined as follows

ln k0 ≡〈∑

a pia(H0 − E0) ln[2(H0 − E0)]

∑b p

ib〉

2πZ∑

c〈δ3(rc)〉(E.9)

APPENDIX E. α0, THE STATIC POLARIZABILITY 153

and P (1/r3ab) is given by

〈φ|P(

1

r3

)|ψ〉 = lim

a→0

∫d3rφ∗(r)

[1

r3Θ(r − a)

+4πδ3(r)(γ + ln a)

]ψ(r). (E.10)

154

Appendix F

Hyperpolarizability

The energy shift that results due to the presence of an electric field of strength E can becomputed perturbatively

∆E = ∆E2 + ∆E4. (F.1)

The first and third order corrections are zero because of parity selection rules.Considering a static field, the second order energy is given by

∆E2 = −E2

2[α(0) + α(2) 3M2 − L(L+ 1)

L(2L− 1)] (F.2)

for L 6= 0, 12, where α(0) and α(2) are the scalar and tensor dipole polarizabilities.

The fourth-order correction

∆E4 = −E4

24[γ0 + γ2g2(L,M) + γ4g4(L,M)] (F.3)

is given in terms of γ0, the scalar hyperpolarizability, and γ2 and γ4, the tensor hyperpolar-izabilities which is defined as [82]

γ0 = (−1)2L128π2

3

1√2L+ 1

∑LaLbLc

G0(L,La, Lb, Lc)T (La, Lb, Lc) (F.4)

γ2 = (−1)2L128π2

3

√L(2L− 1)

(2L+ 3)(L+ 1)(2L+ 1)

∑LaLbLc

G2(L,La, Lb, Lc)T (La, Lb, Lc) (F.5)

γ4 = (−1)2L128π2

3

√L(2L− 1)(L− 1)(2L− 3)

(2L+ 5)(L+ 2)(2L+ 3)(L+ 1)(2L+ 1)

∑LaLbLc

G4(L,La, Lb, Lc)T (La, Lb, Lc)

(F.6)

Here, the functions T (La, Lb, Lc) and GΛ(L,La, Lb, Lc) are given by the following expres-sions and parameterize the sum over intermediate states and the coupling as defined byClebsch-Gordarn coefficients, respectively.

APPENDIX F. HYPERPOLARIZABILITY 155

T (La, Lb, Lc) =∑kmn

〈n0L‖T1‖mLa〉〈mLa‖T1‖nLb〉〈nLb‖T1‖kLc〉〈kLc‖T1‖n0L〉[Ek(Lc)− E0(L)][Em(La)− E0(L)][En(Lb)− E0(L)]

−δ(Lb, L)(−1)2L−La−Lc∑m

|〈n0L‖T1‖mLa〉|2

[Em(La)− E0(L)]

∑k

|〈n0L‖T1‖kLc〉|2

[Ek(Lc)− E0(L)]

(F.7)

GΛ(L,La, Lb, Lc) =∑K1K2

(Λ, K1, K2)

(1 1 K1

0 0 0

)(1 1 K2

0 0 0

)(K1 K2 Λ0 0 0

)×

1 1 K1

L Lb La

1 1 K2

L Lb Lc

K2 K1 ΛL L Lb

(F.8)

In the case that L = 0, one only needs to consider to terms

γ0 =128π2

3

[1

9T (1, 0, 1) +

2

45T (1, 2, 1)

]. (F.9)

Explicitly these are defined as

T (1, 0, 1) =∑kmn

〈2S‖T1‖mP 〉〈mP‖T1‖nS〉〈nS‖T1‖kP 〉〈kP‖T1‖2S〉[Ek − E0][Em − E0][En − E0]

−∑m

|〈2S‖T1‖mP 〉|2

[Em − E0]

∑k

|〈2S‖T1‖kP 〉|2

[Ek − E0]

(F.10)

and

T (1, 2, 1) =∑kmn

〈2S‖T1‖mP 〉〈mP‖T1‖nD〉〈nD‖T1‖kP 〉〈kP‖T1‖2S〉[Ek − E0][Em − E0][En − E0]

.

(F.11)

However, the previous expressions do not take into account any frequency dependence ofthe hyperpolarizability. Following the methodology outline in Ref. [159] and Ref. [160] thefrequency dependence of the hyperpolarizability can be determined.

The polarization response of an atom in an applied oscillating electric field E(t) is charac-terized by the polarizability (linear) and hyperpolarizability (nonlinear) seen in an expansionof the induced dipole moment given by the following

µα(ωσ) = ααβ(−ωσ;ω1)Eβ(ω1)


+1

2βαβγ(−ωσ;ω1, ω2)Eβ(ω1)Eγ(ω2)

+1

6γαβγδ(−ωσ;ω1, ω2, ω3)Eβ(ω1)Eγ(ω2)Eδ(ω3)

(F.12)

where each term above representative of an induced dipole at a frequency for a particularset of field components

ωσ =∑i

ωi. (F.13)

Each combination of polarization (subscripts) and frequencies (arguments) for the appliedfield components corresponds to a particular nonlinear process. For the instance that theexternal field is optical, then the second term in the above expansion is zero due to parityselection rules.

Therefore, the dynamic (second) hyperpolarizability is given by

γαβγδ(−ωσ;ω1, ω2, ω3) = (F.14)

F.1 Positive and negative frequency components

For an atom in state |g〉 in the presence of an external optical field, the ac-Stark shift isgiven by

∆Eg =~|Ω(r)|2

4∆=|〈g|ε · d|e〉|2|E0(r)|2

~(ω − ω0)(F.15)

for a monochromatic field of the form

E(r) = εE (+)0 (r)e−iωt + c.c. (F.16)

Adding in the energy shift due to the counter-rotating term, undoing the rotation wayapproximation, leads to the following expression

∆Eg =|〈g|ε · d|e〉|2|E0(r)|2

~(ω − ω0)− |〈g|ε · d|e〉|

2|E0(r)|2

~(ω + ω0)=

2ω0|〈g|ε · d|e〉|2|E0(r)|2

~(ω20 − ω2)

, (F.17)

which produces the scaling with ω0 from previous expressions.


Previous formalism takes this into account explicitly by considering both the total re-sponse tensor, a sum of the ‘positive’ (nonsecular) and ‘negative’ (secular) frequency tensorsgiven by

Xn = Xn(+) +Xn(−) (F.18)

where each is defined as

Xn(+) = ~−n∑a1

∑a2

...∑an

〈0|z|a1〉

×〈a1|z|a2〉...〈an|z|0〉∑P

[(ωa1 − ωσ)

×(ωa2 − ωσ + ω1)...(ωan − ωn)]−1 (F.19)

and the other component for the second hyperpolarizability is

X3(−) = −~−3∑a1

∑a3

|〈0|z|a1〉|2|〈0|z|a3〉|2

×∑P

[(ωa1 − ωσ)(ωa3 − ω2)(ωa3 + ω3)]−1 (F.20)

The polarizability α(ω) can similarly be expressed in terms of an Ln dependence as

α(La) =8π

9(2L+ 1)

∑m

|〈n0L‖T1‖mLa〉|2

[Em(La)− E0(L)](F.21)

which can be revised to maintain the frequency dependence of α as

α1(La, ω) =8π

9(2L+ 1)

∑m

(Em(La)− E0(L))|〈n0L‖T1‖mLa〉|2

[(Em(La)− E0(L))2 − ω2]. (F.22)

At the tune-out frequency ω = ωto, this term vanishes and reinterpreting the contribu-tions to the hyperpolarizability in this approximation (at the tune-out) allows for a simpli-fication in that the T (1, 0, 1) term will transform into a sum that looks very much like theT (1, 2, 1) term, except with (n, Lb) = (n, S) instead of (n,D).

158

Appendix G

Matlab simulation of thermal cloud

At a temperature of 300 µK, the velocity width of the cloud prevents a closed functionalform for the anticipated contrast of the interference fringes. A Monte Carlo simulation isused to infer the effect of the such a warm cloud.

G.0.1 Code

The following script sets the parameters for the ‘precompute’ function given an intensity andpulse time pair.

BiCubicFUNisHAPPENing = 0; %do we interpolate?

HOTcloud = 0; %are we using atoms >0 K

if HOTcloud

HOTtau = true;

else

HOTtau = false;

end

flipFLOP = 1; %if true, determine pi/2 via plot&fit

PROject = 0; %true to QPN, false to not

STARKshift = 1; %ac Stark shifts

SUM = 1; %sum all atoms in state

CONJUgate = 0;

PROcess = 0;

nSims = 100;

w0 = 2.1;

sigr= 0.5;

TEMP = 300; %in K

APPENDIX G. MATLAB SIMULATION OF THERMAL CLOUD 159

sV =@(TEMP) sV(TEMP)/1000; %in mm/us

dL = -4.00e-3; %MHz

pow1= 0.013; pow2= 0.030; %power in Raman beams

Delta0 = -200; %MHz

sigD = 1.0; %line width

% Define pulse shape

pS = @(int)square(int);

% T-Values

TValues = 160:0.30:230;

if HOTtau

tauROOT= findtau(pow1,pow2,sV(300),sigr,

w0, Delta0, dL, 100, STARKshift);

else

tauROOT= findtau(pow1, pow2, 0, 0, w0,

Delta0, dL, 100, STARKshift);

end

%% determine pi/2 pulse length w/ plotting && fitting %%

if flipFLOP

TAUs= 0:0.002:0.500;

out= zeros(5,length(TAUs));

ii=1;

if HOTtau

for ii=1:5

out(ii,:)= arrayfun(@(TAU)

rabiFLOP(TAU,pow1,pow2,sV(300),...

sigr,w0,Delta0,dL,100,STARKshift),TAUs);

end

else

for ii=1:5

out(ii,:)= arrayfun(@(TAU)

rabiFLOP(TAU,pow1,pow2,0,0,...

w0,Delta0,dL,100,STARKshift),TAUs);

end

end

tauFIT = fitTAU(out,TAUs,pow1,pow2);

else

fprintf(’\n NO FLIP--FLOP, tau is %f\n’,tauROOT);


end

% Format file name to save data to

fileName = formattedFileName(tauROOT,dL,sigD,PROject,HOTcloud,SUM);

if BiCubicFUNisHAPPENing

int1= 2*pow1/pi/w0/w0*10^(-6); %J/mm/us

int2= 2*pow2/pi/w0/w0*10^(-6); %J/mm/us

intR = int2/int1;

diary(’precompute/preprecomputeDiary.txt’);

fprintf(’*** prePrecomputes Started %s ***\n’,

datestr(now,’yyyy-mm-dd HH:MM:SS.FFF’));

startTime = clock;

preprecompute(int1,intR,Delta0,sigD,dL,tau,fileName)

fprintf(’\n*** precompute Complete %s ***\n’,


fprintf(’Time elapsed is %f minutes\n’,etime(clock,startTime)/60);

diary(’off’);

end

% Format file name to save data to

fileName = formattedFileName(tauFIT,dL,sigD,PROject,HOTcloud,SUM);

jj=1; startTime0 = clock; output= zeros(5,length(TValues));

for jj=1:5

diary(’simulationLogFile.txt’);

fprintf(’*** Simulations Started %s ***\n’,


startTime = clock;

% Simulate over T values

output(jj,:)= arrayfun(@(T)

simulatef0is1(fileName,tauFIT,pow1,pow2,T,w0,...

sV(300),sigr,Delta0,sigD,dL,...

nSims,PROject,HOTcloud,SUM,CONJUgate),TValues);

fprintf(’*** Simulations Complete %s ***\n’,



fprintf(’Time elapsed is %f minutes \n’,

etime(clock,startTime0)/60);

diary(’off’);

pause(0.1);

end

G.0.2 Matlab functions

G.0.2.1 Preprecompute.m

This function precomputes the simultaneous conjugate interferometers for a particular choiceof parameters.

function preprecompute(int0,intR,Delta0,sigD, ...

delta,tau,pS,fileName)

if (sigD>0)

dDelta = (-4*sigD:sigD/10:4*sigD)+Delta0;

else

dDelta = (-2:0.1:2)+Delta0;

end

int = (0.00:0.05:2.00)*int0;

ndD = numel(dDelta);

nInt = numel(int);

fprintf(’\nStarted precomputing %s.mat at %s\n’,...

fileName,datestr(now,’yyyy-mm-dd HH:MM:SS.FFF’));

%precompute

map = zeros(ndD,nInt,10);

for i= 1:nInt

redmap= zeros(ndD,10);

for j= 1:ndD

redmap(j,:)=

SCIk(int(i),intR,dDelta(j),delta,tau,pS);

end

map(:,i,:)=redmap;

end

%test precompute


randDelta = normrnd(Delta0,sigD);

I0 = int0*exp(-2*4);

val = abs(raman(I0,intR,randDelta,delta,tau,0,0,2,2,pS)^2 ...

-bicubic(dDelta,int,map(:,:,8),randDelta,I0)^2);

if ~exist(’precompute’,’dir’)

mkdir(’precompute’);

end

% Save data to .mat file

save([’precompute/’,fileName],’map’,’dDelta’,’int’,’int0’);

% Calculate precompute time

precompTime = toc; endTime = now;

endDateFormatted = datestr(endTime,’yyyy-mm-dd’);

endTimeFormatted = datestr(endTime,’HH:MM:SS.FFF’);

% Output to command line

fprintf(’\nSaved %s.mat with

[map]= %sx%sx10.\n’,fileName,num2str(ndD), num2str(nInt));

% Open log file to track and record progress

logFileName =

sprintf(’precompute/precomputeLogFile%s.csv’,endDateFormatted);

% Create header of log file if new file

if ~exist(logFileName)

logFile = fopen(logFileName,’w’);

fprintf(logFile,’tau,precompTime,endTime,fileName,endTimeFormatted\n’);

else

logFile = fopen(logFileName,’a’);

end

% Write to log file

fprintf(logFile,’%d,%d,%f,%f,%f,%f,%s,%s\n’,...

tau,precompTime,endTime,fileName,endTimeFormatted);

fclose(logFile);

G.0.2.2 simulatef0is1.m

function out=


simulatef0is1(fileName,tau,pow1,pow2,T,waist,sigV,sigr,...

Delta0,sigD,dL,nSims,project,HOTcloud,SUM,CONJugate)

omrec = 63.13e-3; %MHz

vr= 0.85e-4; %mm/us

k = 9364.23;

int1= 2*pow1/(pi*waist^2)*10^(-6); %J/mm/us

int2= 2*pow2/(pi*waist^2)*10^(-6); %J/mm/us

exSUMMed = complex(zeros(1,nSims));

exCONJugate = complex(zeros(1,nSims));

exNORMal = complex(zeros(1,nSims));

exALL = complex(zeros(1,nSims));

exOUT = complex(zeros(1,nSims));

D = Delta0+randn(1,4)*sigD;

Tp=10; rMax= 2.0; x0= 0; y0= 0;

cup = @(T) exp(1i*commPhi(T,dL) + 1i*diffPhi(T,omrec));

cdwn = @(T) exp(1i*commPhi(T,dL) - 1i*diffPhi(T,omrec));

tic

for j=1:nSims

detectable= false;

while ~detectable

if HOTcloud

vx= randn()*sigV;

vy= randn()*sigV;

vz= randn()*sigV/vr;

% initial position is offset so that first pulse is centered

x0= randn()*sigr;

y0= randn()*sigr;

z0= randn()*sigr;

else

vx= 0;vy= 0;vz= 0; %mm/us

x0= 0;y0= 0;

end

xf(j)= x0+(2*T+Tp)*vx;yf(j)= y0+(2*T+Tp)*vy;

if xf(j)^2+yf(j)^2 < rMax^2

detectable= true;


end

end

% first pulse

x1 = x0; y1 = y0;

a = exp(-2*(x1^2+y1^2)/waist^2);

C1a= DOPPraman(k,vz,a*int1,a*int2,D(1),dL,tau,0,0,1,1);

C2a= DOPPraman(k,vz,a*int1,a*int2,D(1),dL,tau,0,2,1,2);

% second pulse

x2 = x0+vx*T; y2 = y0+vy*T;

a = exp(-2*(x2^2+y2^2)/waist^2);

C1b= DOPPraman(k,vz,a*int1,a*int2,D(2),dL,tau,0,0,1,1);




% third pulse

x3 = x0+vx*(T+Tp); y3 = y0+vy*(T+Tp);

a = exp(-2*(x3^2+y3^2)/waist^2);

C1c= DOPPraman(-k,vz,a*int1,a*int2,D(3),dL,tau,0,0,1,1);


C5c= DOPPraman(-k,vz,a*int1,a*int2,D(3),dL,tau,0,-2,1,2);


% fourth pulse

x4 = x0+vx*(2*T+Tp); y4 = y0+vy*(2*T+Tp);

a = exp(-2*(x4^2+y4^2)/waist^2);

%C1d= DOPPraman(-k,vz,a*int1,a*int2,D(4),dL,tau,0,0,1,1);

C4d= DOPPraman(-k,vz,a*int1,a*int2,D(4),dL,tau,2,2,2,2);

C5d= DOPPraman(-k,vz,a*int1,a*int2,D(4),dL,tau,0,-2,1,2);


C7d= DOPPraman(-k,vz,a*int1,a*int2,D(4),dL,tau,4,2,1,2);


C9d= DOPPraman(-k,vz,a*int1,a*int2,D(4),dL,tau,-2,-2,2,2);

%C10d= DOPPraman(-k,vz,a*int1,a*int2,D(4),dL,tau,-2,0,2,1);

%amplitudes for each path (interfering)

exSUMMed(j)= interFEAR(C2a*C4b*C4c*C4d,C1a*C2b*C6c*C7d,cup(T))...

+interFEAR(C1a*C1b*C1c*C5d,C2a*C3b*C5c*C9d,cdwn(T));

exCONJugate(j)= interFEAR(C2a*C4b*C4c*C4d,C1a*C2b*C6c*C7d,cup(T))...


+conj(C1a*C1b*C1c*C5d)*(C1a*C1b*C1c*C5d)...

+conj(C2a*C3b*C5c*C9d)*(C2a*C3b*C5c*C9d);

exNORMal(j)= interFEAR(C1a*C1b*C1c*C5d,C2a*C3b*C5c*C9d,cdwn(T))...



%amplitudes for each path (non-interfering)

exBACK(j)= conj(C2a*C3b*C1c*C5d)*(C2a*C3b*C1c*C5d)...




if SUM

exALL(j)= exSUMMed(j)+exBACK(j);

elseif (~SUM) && CONJugate

exALL(j)= exCONJugate(j)+exBACK(j);

elseif (~SUM) && (~CONJugate)

exALL(j)= exNORMal(j)+exBACK(j);

end

if project

pp= rand;

if (pp>=0) && (pp<exALL(j))

exOUT(j)= 1;

elseif (pp>=exALL(j))

exOUT(j)=0;

end

else

exOUT(j)= exALL(j);

end

end

OUTput = [T, sum(exOUT)];

out = sum(exOUT);

fileName1 = [fileName,’_’,num2str(nSimulations)];

save_data(’results/’,fileName1,OUTput);

elapsed= toc;

disp([’T = ’,num2str(T),’ in ’,num2str(elapsed),’ seconds’]);

end

166

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