thepennsylvaniastateuniversity thegraduateschool

The Pennsylvania State University

The Graduate School

P-WAVE FESHBACH RESONANCES IN QUASI-1D AND TWO

NOVEL LASER SOURCES FOR MANIPULATING LITHIUM

ATOMS

A Dissertation in

Physics

by

Francisco R. Fonta

© 2020 Francisco R. Fonta

Submitted in Partial Fulfillment

of the Requirements

for the Degree of

Doctor of Philosophy

December 2020

The dissertation of Francisco R. Fonta was reviewed and approved by the following:

Kenneth O’Hara

Adjunct Associate Professor of Physics

Kurt Gibble

Professor of Physics

Dissertation Advisor, Chair of Committee

Jainendra K. Jain

Evan Pugh University Professor & Erwin W. Mueller Professor of Physics

Kristen Fichthorn

Merrell Fenske Professor of Chemical Engineering and Professor of Physics

Richard Robinett

Professor of Physics

Associate Head for Undergraduate and Graduate Studies

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Abstract

In this thesis we present the development of two novel 671 nm laser sources forlaser cooling and manipulation of lithium atoms and a study of p-wave Fesh-bach resonances in quasi-1D with a focus on mechanisms for suppressing inelasticcollisions.

We begin with presenting two novel high-power all solid-state laser sources foruse in lithium atom experiments. Both lasers are capable of emitting light at 671nmwith output powers on the order of Watts. The lithium D-Line transitions areall near 671 nm, making high power laser sources at 671 nm crucial for ultracoldlithium atom experiments.

We further present our studies of p-wave Feshbach resonances in quasi-1D. In atheoretical study, we calculate the binding energies and closed channel fraction ofthe p-wave Feshbach molecules in quasi 1D for both 6Li and 40K. We show thatin the two body limit a p-wave halo dimer exists in quasi-1D and calculate theexperimental conditions required to reach the halo-dimer regime. The halo-dimerregime is of interest as it may allow the realization of long-lived samples of ultracoldfermions with p-wave interactions and ultimately yield unconventional superfluidityin a dilute Fermi gas. The expected stability of the gas is a direct result of thelarge spatial extent of the halo-dimer molecules as relaxation to more deeply boundmolecular states should be suppressed due to poor spatial overlap of the halo- andtightly-bound-molecular wavefunctions.

Finally we analyze the measured suppression of three body recombination totightly bound molecules in our quasi-1D 6Li atom experiments. We compare thethree body loss constant with predicted scaling laws and develop an intermedi-ate theory based on Breit-Wigner analysis that well explains the magnetic fielddependence of the loss constant as well as the energy dependence.

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Table of Contents

List of Figures vi

List of Tables xi

Acknowledgments xii

Chapter 1Introduction 11.1 Dissertation Outline . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Chapter 2Background Information 112.1 Lithuim Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2 Basic Laser Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2.1 Hermite-Gaussian beams and cavity resonance conditions . . 182.2.2 Output power . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3 Basic Scattering Theory . . . . . . . . . . . . . . . . . . . . . . . . 252.3.1 Partial wave scattering . . . . . . . . . . . . . . . . . . . . . 252.3.2 Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.3.3 Feshbach Resonaces . . . . . . . . . . . . . . . . . . . . . . . 32

Chapter 3High-power, frequency-doubled Nd:GdVO4 laser for use in lithium

cold atom experiments 373.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.2 Optical characterization of the Nd:GdVO4 crystal . . . . . . . . . . 423.3 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.4 Thermal lensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.5 Characterization of the fundamental and frequency-doubled laser . . 55

3.5.1 Single longitudinal mode operation and linewidth . . . . . . 55

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3.5.2 Mode quality of fundamental and second harmonic . . . . . 573.5.3 Wavelength tunability . . . . . . . . . . . . . . . . . . . . . 593.5.4 Long term stability and residual intensity noise . . . . . . . 61

3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

Chapter 4The Nd:YVO4 Raman Laser System 654.1 Atom-Light interactions/Raman Transitions . . . . . . . . . . . . . 65

4.1.1 Two level atom . . . . . . . . . . . . . . . . . . . . . . . . . 664.1.2 Electric and magnetic dipole transitions . . . . . . . . . . . 694.1.3 Raman Transitions . . . . . . . . . . . . . . . . . . . . . . . 73

4.2 Nd:YVO4 Laser Setup . . . . . . . . . . . . . . . . . . . . . . . . . 764.3 Making the Raman beams . . . . . . . . . . . . . . . . . . . . . . . 794.4 Notes on general design for solid state lasers . . . . . . . . . . . . . 81

Chapter 5Experimental Conditions for Obtaining Halo P -Wave Dimers

in Quasi-1D 905.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 915.2 Physical Significance of Poles . . . . . . . . . . . . . . . . . . . . . 955.3 Poles Analysis of P -Wave Resonances in 3D . . . . . . . . . . . . . 975.4 1D Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

Chapter 6Suppression of Three Body Loss Near P -Wave Resonances in

Quasi-1D 1106.1 The three body loss experiment . . . . . . . . . . . . . . . . . . . . 1116.2 The three body loss analysis . . . . . . . . . . . . . . . . . . . . . . 118

6.2.1 Three body recombination scaling laws . . . . . . . . . . . . 1186.2.2 Intermediate regime theory . . . . . . . . . . . . . . . . . . . 1206.2.3 Comparing theory to experiment . . . . . . . . . . . . . . . 123

Chapter 7Conclusions and Future Outlook 129

Bibliography 134

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List of Figures

2.1 Energy level diagram of lithium showing the fine and hyperfinestructure of the electronic ground and first excited state. . . . . . . 13

2.2 The Zeeman sublevels of the 2S1/2 hyperfine states. . . . . . . . . . 172.3 Poles of the S-matrix on the complex momentum and complex energy

planes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.4 Schematic picture of a Feshbach resonance. When the energy of

a coupled closed channel molecule approaches the energy of thescattering state, interactions are resonantly enhanced. The differ-ence between the energies of the closed channel molecule and thescattering state may be tuned with a magnetic field. . . . . . . . . 34

2.5 Diagram showing the scattering length (a) and binding energy of theFeshbach molecule (b) near a magnetically tuned s-wave Feshbachresonance. The inset in b) shows the universal region. The upper-half of c) shows the scattering cross section near the 6Li broads-wave Feshbach resonance. The lower-half of c) shows the bindingenergy of the Feshbach molecule (red line) and the binding energyof the closed channel molecule (doted black line). We see that whenthe Feshbach molecule approaches threshold, the scattering crosssection reaches its unitarity limited value. . . . . . . . . . . . . . . 35

3.1 (a) Energy level diagram of Nd+3 in a GdVO4 host material. Therelevant transitions for pumping, lasing, and excited state absorp-tion are labeled. (b) Absorption coefficient for a 0.5% at. dopedNd:GdVO4 crystal near the 880 nm and 888 nm pumping transitions.(c) Gain and excited state absorption spectrum. . . . . . . . . . . . 43

vi

3.2 A schematic of the laser setup. The laser is pumped by a fiber-coupled diode bar. Lenses L1 and L2 image the end of the pumpfiber onto the Nd:GdVO4 crystal. Mirrors M1 – M4, which are allflat, form a bow-tie cavity with a round trip path length of 450 mm.The physical distance between mirrors M1 and M2 is 45 mm andthat between M3 and M4 is 156 mm. M4 is the output coupler.Uni-directional operation is enforced by the combination of the λ/2waveplate and the TGG crystal placed in a high magnetic field toprovide Faraday rotation. The rotatable thin etalon is used to tunethe operating wavelength. The beam output from M4 is collimatedby L3 and passes through an optical isolator before being sent to acommercial build-up cavity for second harmonic generation. . . . . 48

3.3 Output power as a function of absorbed pump power. The line isPout = ηsl(Pabs − Pth) where the threshold power Pth = 13.1 W, theslope efficiency ηsl = 24% and Pabs is the absorbed pump power. . . 49

3.4 Dioptric power as a function of absorbed pump power. The curve isEqn. 3.1 with the fractional thermal heat load ηh given by Eqn. 3.5.These equations provide a model of the thermal lensing in our systemwith no free parameters. . . . . . . . . . . . . . . . . . . . . . . . . 51

3.5 (a) Fabry-Perot spectrum showing single-longitudinal mode opera-tions. (b) Beat note between the free-running Nd:GdVO4 and anextended cavity diode laser demonstrating an upper bound on thefundamental laser linewidth of 450 kHz. . . . . . . . . . . . . . . . . 56

3.6 Measurement of the laser caustic for the 1342 nm laser in the (a)vertical and (b) horizontal directions. (c) Measurement of the lasercaustic for the 671 nm laser beam. The inset shows a typical beamprofile of the 671 nm laser beam as recorded by the CCD camera.The solid lines are fits to Eqn. 3.8 used to determine the beamquality parameter M2. . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.7 (a) Fundamental and second harmonic output power versus wave-length. The power of the fundamental laser is measured after theoptical isolator in Fig. 3.2, just before entering the SHG cavity. (b)Spectrum of water absorption coefficient at 300K and 50% relativehumidity [1, 2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.8 (a) Fundamental and second harmonic output power over severalhours. The fundamental power is measured after the optical isolator,just before entering the SHG cavity. (b) One-sided power spectraldensity of the residual intensity noise of the frequency-doubled laseroutput. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

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4.1 Diagram representing the two level atom. ω0 represents the energydifference between the ground and excited states, ω represents thefrequency of the field coupling the two states and δ represents thedetuning. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.2 Diagram representing the two photon process involved in Ramantransitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.3 Schematic showing the setup of the Nd:YVO4 laser. . . . . . . . . . 764.4 Schematic showing the setup of the Raman laser system . . . . . . 814.5 Cavity stability(a) and laser mode radius inside the crystal (b) as

function of the crystal’s thermal lens. Vertical dashed red andgreen lines show the thermal lens associated with our Nd:YVO4 andNd:GdV04 lasers respectively.(a) We see that the cavity is stablefrom a thermal focal length of 11.25 cm up to very large focallengths.(b) The boundaries of optimal mode matching is shown bythe dotted orange lines, see text for details. We see that the optimalmode matching can be found for thermal focal lengths rangingfrom 12.5→ 110 cm. Both our Nd:YVO4 and Nd:GdVO4 laser fitcomfortably within these ranges. . . . . . . . . . . . . . . . . . . . . 82

4.6 The black lines show Mod[ n2πΨr, 1] or equivalently Mod[ vn−v0vFSR

, 1]plotted as a function of fth for n = 1 to 4. Degeneracy occurs whenthe curves cross 0 as explained in the text. We see the cavity isdegenerate with 3rd and 4th order modes at fth = 15 and 22.5 cmrespectively. This sets some limits to TEM00 operation of our laserdesign. The location on the Nd:YVO4 and Nd:GdVO4 laser systemsare shown with dashed red and green lines respectively. Both lasersoperate away from degeneracy. . . . . . . . . . . . . . . . . . . . . . 89

5.1 The colored lines show the poles of the S-matrix moving on thecomplex momentum plane in (a) 3D and (b) quasi-1D. The arrowsshow the direction the poles move as the magnetic field is tunedfrom the BEC side to the BCS side of the resonance. The stars showthe locations of the poles in the complex k plane at B = Bres. (a) In3D the pole corresponding to a bound state(blue) moves down thepositive imaginary axis becoming a resonance as soon as it crossesthreshold. (b) In quasi-1D the bound state pole moves down positiveimaginary axis and then continues along the negative imaginary axisas a “virtual state” until kpole = − i

r1Dand only then does it become

a resonance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 985.2 Closed channel amplitude in 3D for (a) 6Li and (b) 40K. The inset

shows the closed channel amplitude close to resonance. . . . . . . . 101

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5.3 3D p-wave scattering cross section and bound state energy for (a)6Li and (b) 40K. The bound state energy (solid line) tunes linearlyas a function of the magnetic field, directly becoming a resonance(dash-dotted line) at B = Bres, above which point the energy of thisquasi-stable molecular state tunes linearly through the continuum. . 102

5.4 Closed channel amplitude in quasi-1D for (a) 6Li and (b) 40K for avariety of transverse confinements. The resonance becomes signifi-cantly more open channel dominated as the confinement increases. . 104

5.5 Scattering cross section and bound state energy in quasi-1D for (a)6Li and (b) 40K. We assumed a transverse confinement of 3 MHz (500kHz) for 6Li (40K). The energy of the bound state (solid line) mergeswith the continuum at Bres and then continues on as a virtual state(dotted line) before eventually becoming a resonance (dash-dottedline). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.6 (a,b) Field stability and (c,d) temperature required for achievinghalo dimers as transverse confinement is increased. (a,c) Showthe conditions necessary for 6Li while (b,d) show the conditionsnecessary for 40K. . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

6.1 Schematic showing the setup of the three body loss experiment . . . 1126.2 Sample decay curves in 3D and quasi-1D both on resonance and far

from resonance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1146.3 Loss feature is 3D and quasi-1D showing the confinement induced

resonance shift. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1176.4 Magnetic field dependence of L3 in 3D and quasi-1D. For technical

reasons, the two 1D data sets were taken at different lattice depths,resulting in differing values of the confinement induced shift of theresonance. All the data sets are thus shifted so that the resonancelocations overlap at 0 δB marked with the solid gray vertical line.The colored vertical lines show the field below which L3 is expectedto be unitarity limited. The solid red curve is the intermediatetheory of Ref. [3] fit to our 3D data. The solid blue and green curvesare equation 8 fit to our quasi-1D data sets. The dashed curves showthe far from resonance 1D scaling laws. Data points are averages of3 to 5 individual measurements; error bars are the standard error ofthe mean. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

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6.5 Log-Log plot of L3 versus temperature. Solid red curve shows3D unitary limit. Solid orange line shows the on resonance L3 ∝constant scaling law. Dashed blue curve show the L3 ∝ T 3 scalinglaw. Solid blue curve is a fit to equation 8 assuming KAD ∝ T 3.Data points are averages of 3 to 5 individual measurements; errorbars are the standard error of the mean. . . . . . . . . . . . . . . . 125

6.6 Log-log plot of KAD vs Temperature. The blue line is KAD =T 3(14 ± 2)m3/sK3 with the shaded regions representing the errorbars. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

6.7 Log-log plot of on resonance L3 vs lattice depth. Solid line showsL3 ∝ U−1

0 scaling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

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List of Tables

5.1 p-wave scattering properties for 6Li and 40K [4–10]. . . . . . . . . . 92

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Acknowledgments

I would like to take this moment to thank all of those that have helped methroughout my time in graduate school. I would especially like to thank my entireLab Group:

• Andrew, for paving the wave in the three body loss experiment.

• Arif, for showing me the ropes of the lab and being a friend to me.

• Finally my advisor Ken, for his constant patience and optimism. And forsticking with me not just to the end, but far past it.

I would like to thank all of my State College friends, with a special thanks going to:

• Kelly and Mike, who made my transition to State College easy and fun.

• Pat, who always listened to me vent about "waiting for parts."

• Miguel, for dnd, rock band, and thoughtful talks.

• Jimmy, for being hilarious and someone I know I can count on.

• Colin, for Little Szechuan, Shenanigans, and Stability.

• Cody and Kasey, for being the glue.

I would like to thank my entire wonderful, loving, supportive family. Words arenot enough to express how you helped me but here are some anyways:

• My grandma Julie, for insisting that "physics is just like art" (the greatest ofcompliments in my family) and for always understanding me.

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• My parents Paco and Celia, for raising me to be the person I am, you are thebest parents anyone could ever wish for.

• My brother Adam, for first introducing me to physics, and continuing to bethe person I can talk to about it and everything else most deeply.

Finally I would like to thank my fiancée Jenna for sharing this crazy ride that isgrad school with me with all its highs and lows, I am so excited to share the rest ofmy life with you.

This material is based on work supported by the National Science Foundation(NSF) under Award No. 1607648 and the Air Force Office of Scientific Research(AFOSR) under Award No. FA9550-15-1-0239. Any opinions, findings, and conclu-sions or recommendations expressed in this publication are those of the author anddo no necessarily reflect the views of the NSF or the AFOSR.

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Chapter 1

Introduction

The experimental realization of ultracold atomic gases such as the Bose-Einstein

condensate (BEC) in 1995 [11–13], and the degenerate Fermi gas (DFG) in 1999 [14]

led to a revolution in atomic physics. The field of atomic physics, which had been

dominated by spectroscopic studies 1 of atoms, expanded to include the study of

new exotic phases of matter and later the study of quantum computing. A key

component of this massive growth was the development of two extremely powerful

experimental tools: the laser, and Feshbach resonances. Among other things, the

laser allowed for the cooling and trapping of an atomic ensemble to sufficiently

cold temperatures needed to reach quantum degeneracy. Feshbach resonances on

the other hand allowed for the control over the inter-atomic interaction strength

necessary to explore novel phases of matter.1After this revolution in atomic physics, spectroscopic studies continued and even improved

massively.

1

Ultracold atomic gases are interesting because they are quantum gases, that is

they are states of matter where the effects of quantum degeneracy and quantum

statistics become important. Quantum statistics become important when the

thermal de Broglie wavelength, λT =√

2π~2/2mkbT , is on the order of the inter-

particle spacing. The thermal de Broglie wavelength roughly corresponds to the

size of the particle’s wavefunction; intuitively this means the effects of quantum

mechanics become dominant when the particle wavefunctions are overlapping.

A gas of identical bosons follows Bose statistics and will tend to bunch in the

same quantum state. At a sufficiently low temperatures these statistics lead to a

macroscopic population of the ground state known as the BEC. Identical fermions

on the other hand cannot occupy the same quantum state due to Pauli’s exclusion

principal. At zero temperature a gas of fermions will occupy all the lowest energy

quantum states up until the Fermi energy. At sufficiently low temperatures Fermi

statistics will cause a gas of fermions to approach this zero temperature distribution.

When the temperature is less than the Fermi energy we consider the gas to be a

degenerate Fermi gas. While the BEC transition is a true phase transition, the DFG

transition is not, rather it is a gradual process by which the distribution of atoms

changes from that of the Boltzmann distribution to the Fermi-Dirac distribution.

Let’s take a moment to see how low the temperatures must be to reach quantum

degeneracy. Using the condition that the thermal de Broglie wavelength must be

2

on the order of the inter particle spacing2 we get

√2π~2

2mkbT≈ 1n1/3 . (1.1)

Clearly the temperature needed depends on the density. Typical experiments with

ultracold gasses operate at densities of between 1015 cm−3 and 1011 cm−3, using

the mass of lithium (10−26kg) as an example, this leads to a temperature window

of 50 nK to 25 µK. In other words very cold, some might say ultra cold.

These ultracold temperatures are the reason why even though the BEC was

predicted way back in 1925 by Albert Einstein, it took 70 years before its experi-

mental realization 3. Specifically it was the invention of the laser and the techniques

of laser cooling and trapping of atoms that paved the way for the experimental

realization of ultracold quantum gases.

That light could exert a mechanical force on atoms was known as early as

1929. However at the time there was not sufficiently intense sources of light to

make much use of this information. That is not until the invention of the laser in

1960 [16]. By 1970 Ashkin had used an argon laser to demonstrate the trapping of

micro-spheres by radiation pressure [17]; work that became the basis for optical2This condition is not the same as the actual transition temperature for a BEC or the Fermi

temperature both of which depend on the specifics of the situation. Still this condition serves toprovide a good order of magnitude estimate.

3Strictly speaking superfluid liquid helium, which was discovered in 1938, features Bosecondensation [15]. Liquid helium is a strongly interacting system and is far more complex thanthe BEC envisioned by Einstein. It was not until ultracold atomic gas experiments that a weaklyinteracting and even non-interacting BEC was formed.

3

tweezers, a technique that has greatly influenced the fields of not only physics but

also biology and chemistry [18–20]. As early as the first paper by Ashkin it is

clear that the ultimate goal was to apply these same ideas of optical trapping and

manipulation to atoms and molecules. Indeed, only a few years later the theory to

cool a gas of atoms by laser radiation was developed [21].

The idea was to take advantage of the light Doppler shift. Atoms irradiated

by light whose frequency is on resonance with an atomic transition will absorb

photons and thus get a momentum kick. If the light is red de-tuned (tuned to a

frequency below the atomic resonance) atoms moving in the opposite direction

of the photons will be Doppler shifted onto resonance and preferentially absorb

more photons. This results in a velocity dependent force that cools the atoms.

Atoms will then re-emit the photons due to spontaneous emission without any

preferred direction. Together the absorption and spontaneous emission will cool

a gas of atoms to what is called the Doppler limit, TDoppler = ~Γ/2kB, where Γ is

the linewidth of the atomic transition. Using 6 counter-propagating lasers, this

technique was first used to cool a gas of atoms in 1985 [22]. The atoms, while

not trapped, were significantly slowed as if moving through a super viscous fluid

thereby earning this technique the name optical molasses [23, 24]. To first trap the

atoms magnetic fields were necessary. [25]

The first magnetically trapped neutral atoms relied on the magnetic field

applying forces onto the magnetic dipole moment of the atoms [25]. It was a few

4

years later in 1987 that atoms were first optically trapped by a combination of

magnetic fields and lasers [26]. The idea was to take advantage of the Zeeman

shift of atomic energy levels. In a magnetic field, atomic energy sub-levels are split

based on their ms spin quantum numbers and for a inhomogeneous field of the form

B(z) = bz this split becomes spatially dependent. Adding counter-propagating

circularly polarized σ+ and σ− lasers slightly red detuned from the atomic resonance

completes the picture. Now, atoms spatially displaced from the center of the trap

preferentially absorb photons either from the σ+ or σ− lasers. This produces

the restoring force necessary for a true atomic trap. Furthermore the atoms still

experience the optical molasses effect described above. This technique, now known

as the Magneto Optical Trap (MOT)4 is now the starting point for most ultracold

atom experiments.

We now have atoms trapped and cooled by light to the Doppler limit. The

Doppler limit, while very cold, (for example the Doppler limit of Na is 240µK)

is still not cold enough to reach quantum degeneracy. Luckily, and much to

everyone’s surprise, repeating the optical molasses experiments with circularly

polarized light lead to significantly lower temperatures (on the order of 50µK) [27].

This phenomenon was theoretically explained a year later by Dalibard and Cohen-

Tannoudji as polarization gradient cooling [28]. This time the scheme took advantage4As pointed out in my atomic physics class by David Weiss, it is a tragedy of history that this

technique became known as the MOT. There are no magnetic forces involved in the trappingof the atoms. Rather it is the Zeeman effect which is exploited. The technique should really becalled the Zeeman Optical Trap (ZOT). Sadly once a name sticks, it sticks for good.

5

of light shifts of the atomic energy levels. In short, atoms would preferentially make

transitions to the excited state at the height of potential created by the laser fields

and would decay to the ground state at the bottom of the potential well. Between

transition events, the atoms would move up the potential hill losing energy. This

technique was dubbed Sisyphus cooling after the Greek myth of Sisyphus, who was

doomed to roll a boulder up a hill only to have it fall to the bottom once it reached

the top.

The temperatures achieved by Sisyphus cooling, while much cooler than the

Doppler limit, still were not enough to realize a BEC. The final hurdle to reach

cold enough temperature was overcome by evaporative cooling in 1995 [13,29–31].

Evaporative cooling is commonplace in nature. We see it in action when we sweat

and cool as the sweat evaporates, or when our hot steaming coffee cools if we don’t

drink it fast enough. In cold atom experiments evaporative cooling is achieved by

continuously removing atoms from the high energy tail of the thermal distribution

while also allowing the gas to rethermalize. Generally, atom-atom interactions are

required for the gas to rethermalize. High energy atoms are usually removed from

the trap by lowering the depth of the trapping potential.

The story of the experimental realization of the BEC is a story of advances in

techniques for manipulating atoms with laser light. And the story does not end

there. Lasers are now used to trap atoms in optical dipole traps, lattice traps,

lower dimensional traps. They are further used to probe the atoms; atoms are

6

probed by absorption imaging, phase contrast imaging, Raman spectroscopy, Bragg

spectroscopy, and quantum gas microscopes to name a few techniques involving

lasers tuned about an atomic resonance.

The BEC and DFG are phenomenon based on quantum statistics that occur

even in non-interacting gases. While it is useful to have a test bed for basic quantum

mechanics, it is only in an interacting system that novel wholly new phases of matter

may be studied 5. Fortunately ultracold atom experiments have a tool by which

the interactions can be controlled: magnetically tuned Feshbach resonances [32].

Near such a Feshbach resonance the scattering length, which characterizes the

strength of an interaction, can be tuned from zero to the highest values allowable by

nature simply by tuning a magnetic field. It is hard to overstate the importance of

Feshbach resonances, and describing all the different phenomenon observed through

their use is far beyond the scope of this introduction. Since our lab is interested in

Fermi gases, let me just give the example of fermionic superfluidity.

Superfluidity is a remarkable phenomenon marked by macroscopic phase co-

herence and zero viscosity. Superfluidity has been seen in liquid helium with

bosonic superfluidity occurring for 4He and fermionic superfluidity occurring in

3He. fermionic superfluidity is also seen in superconductors, which are superfluids

of charge carriers; instead of zero viscosity, superconductors feature zero resistance.

bosonic superfluidity can be described by an interacting Bose Einstein condensate.5I recall many journal club presentations in which non-interacting systems were discussed.

The response was almost always the same: while the research was well done, ultimately it isuninteresting because we already know the solution to the non-interacting problem.

7

fermionic superfluidity on the other hand can be described by pairs of fermions

forming composite bosons which then Bose condense. The traditional form of this

pairing is described by the BCS theory of superconductivity [33] in which fermions

of opposite momentum pair up through attractive interactions. The fermion pairs

may also form molecules, which then Bose condense into a BEC. BCS pairing occurs

on the attractive side of the fermionic Feshbach resonance, while BEC pairing

occurs on the repulsive side.

The critical temperature at which BCS superconductivity is predicted to occur

in a DFG with attractive s-wave interactions is TC ≈ 0.61TF e−π/2kF |a| where TF is

the Fermi temperature and a is the s-wave scattering length [34]. Normally the s-

wave scattering length is too small for the critical temperature to be experimentally

achievable. However Feshbach resonances allow us to tune the scattering length

and thus the critical temperature into an experimentally realizable regime [35,36].

Such a strongly interacting DFG near a s-wave resonance was first seen in 2002 [37].

Since then the field has exploded and the full s-wave BEC-BCS crossover has been

characterized. Further s-wave superfluidity continues to be studied in ultracold

gases in 3D, in lattice systems, and in reduced dimensions. Compared with 3He,

dilute ultracold Fermi gasses are relatively simple systems in that all the relevant

experimental parameters, such as the density, interaction strength, temperature,

and even geometry can be controlled, making them an ideal test bed for fermionic

superfluidity.

8

There are more complex superfluid pairings than just s-wave. In ultracold

atomic gasses p-wave pairings may be accessed through p-wave Feshbach resonance.

The theorized p-wave superfluids have a much richer phase diagram than their s-

wave counterparts. While the 3D s-wave BEC-BCS crossover is a smooth transition,

the p-wave BEC-BCS crossover features many separate classical, quantum, and

topological phase transitions [8]. In two dimensions the p-wave pairing should

result in a px +ipy topological superfluid featuring non-Abelian excitation. In

one dimension the p-wave pairing should allow for the realization of the classic

Kitaev chain model which features Majorana fermions localized at the ends of the

chain. There is however a problem with attempts to realize a p-wave-superfluid

with ultracold atomic gas systems: the p-wave Feshbach resonance is accompanied

by significant inelastic two body and three body loss. There is hope as it has been

theorized that this three body loss may be suppressed in quasi-1D traps [38].

1.1 Dissertation Outline

Our group works with degenerate Fermi gases of 6Li trapped in both 3D and in

quasi-1D via a 2D optical lattice. Since an up-to-date and detailed description of

our methods for cooling, trapping, and imaging 6Li can be found in the dissertations

of my previous lab members Andrew Marcum [39] and Arif Mawardi Ismail [40], I

will not be reproducing them here.

My thesis work may be understood in two parts as the study of the two key

9

experimental tools discussed above: lasers and Feshbach resonances. In the first

part (Chapters 3 and 4) we built and characterized two novel solid state lasers

for use in lithium atom experiments. In the second part (Chapters 5 and 6) we

examined p-wave Feshbach resonances in quasi-1D; we found the experimental

conditions in which the p-wave Feshbach molecules may be described as halo dimers

and we analyzed the measured suppression of three body loss in quasi-1D.

The dissertation is structured as follows. In Chapter 2, I provide all the pertinent

background information necessary for understanding the rest of the dissertation

including the properties of 6Li, basic laser physics, and basic scattering theory.

In Chapter 3, I present a novel Nd:GdVO4 laser system for use in lithium atom

experiments. In Chapter 4, I present a novel Nd:YVO4 laser system designed to

drive Raman transition as well as a discussion on a general design for solid state

lasers. In Chapter 5, I discuss p-wave Feshbach molecules in quasi-1D and show

that under the right experimental conditions these molecules assume the halo dimer

form. In Chapter 6, I present our experimental measurements of the three body loss

in 3D and quasi-1D near a p-wave Feschbach resonance; I further compare these

measurements to known scaling laws and develop a theory valid at intermediate

fields which explains the near resonate loss feature. Finally in Chapter 7, I give a

conclusion and discuss future directions for further research.

10

Chapter 2

Background Information

In this chapter I will provide the background information necessary to understand

my research that will be presented in the succeeding chapters. This chapter is

organized as follows:

• 2.1 In this section I will discuss the properties of lithium atoms in the context

of ultracold Fermi gas experiments. Experiments in our lab are conducted on

gasses of 6Li; thus, everything we do is done in the context of 6Li. When we

consider the nature of p-wave Feshbach molecules they are molecules of 6Li.

When we discuss advances in solid-sate lasers for ultracold atom experiments,

it is lasers whose properties are designed around 6Li.

• 2.2 In this section I will go over basic laser theory. A key part of my research

is the development of novel solid-sate lasers for lithium atom experiments, it

is important to understand the basics of how lasers function going forward.

11

• 2.3 In this section I will go over basic scattering theory leading up to an

explanation of Feshbach resonances. The second part of my research has to do

with p-wave Feshbach resonances in quasi-1D; thus, a general understanding

of Feshbach resonances is important.

2.1 Lithuim Atoms

For our ultracold atom experiments we have chosen to use 6Li. Lithium is an alkali

atom meaning it has a single unpaired valence electron, which determines most

of its spectroscopic properties. Alkali atoms have traditionally been the atoms of

choice for ultracold atom experiments because their simple hydrogen-like electronic

structure makes them easier to understand, manipulate, and cool. A very good

review of the properties of 6Li can be found in ref [41], below I will reproduce a

brief summary of lithium’s internal state structure.

Atoms are generally prepared in the electronic ground state and manipulated

through light coupling to the first excited state. Thus, understanding the energy

structure of the 6Li electronic ground 22S1/2 and first excited states 22P1/2 , 22P3/21

is of vital importance.

The energy level diagram for 6Li is shown in Figure 2.1 [41]. We see that

transitions between the electronic ground state and first excited state occur at1Here we are using atomic spectroscopic notation the general form of whih is N2s+1Lj , where

N is is the principle quantum number, s is the total spin quantum number, L is the orbitalangular momentum quantum number, and j is total (spin + orbital) angular momentum quantumnumber.

12

Figure 2.1: Energy level diagram of lithium showing the fine and hyperfine structureof the electronic ground and first excited state.

approximately 671nm, these are called the D-line transitions. The energy of the

first exited state is split in two by what is known as the fine structure, yielding two

transitions frequencies known as the D1 and D2 lines separated by approximately

10 GHz.

The fine structure comes from spin orbit coupling, that is to say an interaction

13

between the valence electron’s intrinsic spin angular momentum and the magnetic

field of the nucleus as seen from the electrons rest frame. The spin orbit coupling

Hamiltonian is given by

Hso = f(r)L · S (2.1)

where f(r) is a function related to the nuclear electrostatic potential. Under this

Hamiltonian the spin S and angular momentum L quantum numbers no longer

make a good basis, rather it is the basis of total angular momentum J = S +L

that diagonalizes the Hamiltonian. With the angular momentum addition rules

|L− S| ≤ J ≤ |L + S| we see that the ground state is un-split while the excited

state splits into two levels given by J = 1/2 and J = 3/2.

The energy levels are further split as shown in Figure 2.1 by what is known as

the hyperfine structure. The hyperfine structure comes from the magnetic dipole

and electric quadruple moment of the nucleus and is given by the Hamiltonian,

Hhf = AjI · J +Bj

[3 (I · J)2 + 3/2 (I · J)− I(I + 1)J(J + 1)

]2I(2I − 1)J(2J − 1) (2.2)

where Aj and Bj are the magnetic dipole and electric quadruple hyperfine constants

and I is the nuclear spin. For lithium the total nuclear spin is I = 1, and the

values of the hyperfine constants can be found in a table in ref [41]. Under this

Hamiltonian, J is no longer a good basis, rather it is the total spin quantum number

14

F = J + I that forms a good basis. As shown in Figure 2.1 the 6Li hyperfine

splitting ∆Ehf is very small.

In the above discussion of the internal structure of 6Li we have been considering

the atoms in free space, however our experiments all take place in a bias magnetic

field. The presence of this magnetic field further modifies the atomic energy levels

by what is known as the Zeeman effect. The Zeeman Hamiltonian is

HZE = −µB~∑x

gxXz ·B (2.3)

where µB is the Bohr magneton and the sum is over the good angular momentum

quantum numbers such that gx is the g factor and Xz is the projection of the angular

momentum corresponding to these good quantum numbers. Which quantum

numbers are good depends on the strength of the Zeeman shift ∆EZE compared to

the hyperfine and fine structure energy shifts.

When the magnetic field is small so that ∆EZE ∆Ehf , the Zeeman Hamilto-

nian can be treated as a perturbation on the hyperfine Hamiltonian. In this case F

is still a good quantum number and

∆EZE = −µB~gfmfB. (2.4)

When the magnetic field is larger so that ∆EZE ≈ ∆Ehf the problem can no longer

be considered perturbatively. Instead eigenstates of the full interacting Hamiltonian

15

Figure 2.2: The Zeeman sublevels of the 2S1/2 hyperfine states.

Tables with all these constants can be found in ref [41]. The magnetic field

dependence of these 6 lowest Zeeman sublevels is shown in Figure 2.2 [39].

Finally at high magnetic fields (B ≥ 100G) when ∆EZE ∆Ehf , or equiva-

lently µBBA2S1/2

1, the hyperfine interaction can be ignored and the eigenstates are

well approximated by pure |J,mj; I,mI〉 product states. In this regime the lowest

Zeeman sublevels are still given by the above six states but with sin Θ± ≈ 0. As

shown in Figure 2.2 the Zeeman sublevels split into triplets with three low field

seeking states and three high field seeking states.

17

Most of our experiments involve the lowest three Zeeman sublevels. However,

access to all six can prove useful, and indeed the laser system described in chapter

4 was built to drive transitions between states |2〉 and |5〉.

2.2 Basic Laser Physics

2.2.1 Hermite-Gaussian beams and cavity resonance condi-

tions

A laser’s operation is dependent on the cavity resonance condition. Simply put,

this is the requirement that upon a full round trip within the cavity the light field

of the laser reproduces itself.

The electric field E(r, t) of a laser can be described by Hermite-Gaussian beams,

which provide a complete basis of solutions to the paraxial Helmholtz equation.

For a laser propagating in the z direction, the paraxial Helmholtz equation is

( ∂2

∂x2 + ∂2

∂y2 + 2ik ∂∂z

)E(r, t) = 0 (2.15)

where k is the laser wavenumber defined as k = 2π/λ where λ is the wavelength of

light. Equation 2.15 is solved by

E(r, t) = εEE0ul(x, z)um(y, z)ei(kz−ωt) (2.16)

18

where εE is the polarization vector, E0 is the amplitude of the field and ul(x, z)

and um(y, z) are eigenfunctions describing the laser fields in the x and y directions

respectively. The functions ul(x, z) and um(y, z) are the Hermite-Gaussian modes

whose spatial distributions depends on the indices l and m; A beam in the l and

m mode is said to be in the TEMlm mode. In general any laser beam may be

described by a linear combination of TEMlm modes although usually we desire a

laser, which is purely Gaussian denoted by the TEM00 mode.

Now let’s look at the functional form of ul(x, z) in the Hermite Gaussian

representation2 [42]:

un(x, z) =( 2π

)1/4√√√√ei(2n+1)ψx(z)

2nn!wx(z) ×Hn

( √2x

wx(z)

)e−ik

x22q(z) . (2.17)

Here Hn

(√2x

w(z)

)is the Hermite polynomial of order n, ψx(z) is the Gouy phase,

qx(z) is the complex beam propagation parameter and w(z) is the characteristic

radius of the beam such that

qx(z) = z + izR,x (2.18)

and

w(z) = w0,x

√1 + z2

z2R

(2.19)

2There are other representations such as the Laguerre-Gaussian modes. It does not matterwhich representation you use as any laser can be described as a linear combination of the differentmodes, however some calculations may be easier in one representation over the other. We chooseHermite-Gaussian.

19

where w0,x is the minimum beam radius and zR is the Rayleigh length defined as

zR,x =π2w2

0,x

λ. (2.20)

Both the x and y directions have the same functional form however in general they

may be of a different order n and have different complex beam parameters and

Gouy phase shifts. This may seem like many parameters are needed to describe the

laser beam, however notice that all the parameters are interconnected such that

the entire spatial distribution in each direction is uniquely defined by the complex

beam parameters qx(z) and qy(z).

Now we can calculate the resonance conditions directly from equation 2.17. For

the phase of the field to reproduce itself, we need the total phase pickup over the

cavity to be a multiple of 2π. For a TEMlm mode this yields

φ,l,m,total = kLopt + (l + 12)∆ψl,x + (m+ 1

2)∆ψm,y = 2πb . (2.21)

Here, b is some integer, and Lopt is the total cavity optical path length. Using the

definition of the wavenumber k = 2πvl,mc

we can rewrite this condition in terms of

the resonance frequency as

vl,m = vFSR

[b−

(l + 12)∆ψl,x + (m+ 1

2)∆ψm,y2π

], (2.22)

20

where vFSR = cLopt

is the free spectral range. We see that the resonant modes are

spaced in frequency by the free spectral range.

To calculate the resonance condition for the complex beam parameter we can

use the beam propagation ABCD matrices. For a given propagation matrix the

complex beam parameter transforms as

A B

C D

q = Aq +B

Cq +D. (2.23)

Thus, all we need to do is calculate the round trip propagation matrix for the cavity

Mr and we can derive the cavity eigenmodes with the condition

qMr = q . (2.24)

In general the cavity propagation matrix is different for the x and y directions, in

that case it is easy enough to solve for qx and qy separately.

A note on spatial mode quality of a laser beam: As stated before the laser

mode can be described by a linear combination of Hermite Gaussian modes but

usually we want a pure TEM00 mode. Practically speaking it is much too labor

intensive to break the beam down into amplitudes of each TEMlm mode. Instead

we describe the quality of the spatial mode with a beam quality factor M2 such

21

that the measured beam waist as a function of distance is

w(z) = w0

√1 +

(M2 z − z0

zR

)2(2.25)

andM2 = 1 for a pure Gaussian TEM00 mode whileM2〉1 3 signifies some admixture

of higher order modes.

2.2.2 Output power

One of the most important characteristics of a laser along with its frequency is

its output power. For end pumped lasers the output power Pout may be expressed

as [43–46]

Pout = max[ηsl(Pabs − Pthr), 0] (2.26)

where ηsl is the laser slope efficiency, Pabs is the power absorbed from the pump

laser by the gain medium, and Pthr is the threshold power. Equation 2.26 tells us

that there is a minuim power the gain medium must absorb before lasing can begin

after which the output power scales as ηsl.

We can characterize the laser more fully by relating these parameters to proper-

ties of the gain medium and the laser cavity. The absorbed power is related to the3For some context on the spatial mode of high M2 beams consider that our pump laser, which

outputs a top-hat profile, has a M2 on the order of 100.

22

gain medium absorption coefficient α and the total pump power Pp by

Pabs = (1− e−αlgain)Pp (2.27)

where lgain is the length of the gain medium. The threshold power is given by

Pthr = ωpIsatωLlgain

L(∫

sl(x, y, z)rp(x, y, z)dV)−1

(2.28)

where ωp and ωL are the pump and laser frequencies respectively, L is the total

cavity round trip loss, the integral term is the effective mode volume(Veff ) where

sl(x, y, z) is the spatial distribution of the laser mode, rp(x, y, z) is the spatial

distribution of the pump beam, and Isat = ωL~/σeτrad is the saturation intensity

(where σe is the emission cross section and τrad is the radioactive lifetime of the

gain medium).

Finally the slope efficiency is given by

ηsl = TLωLωpη0 (2.29)

where T is the transmission of the output coupler, and η0 is the overlap integral

defined as

η0 = (∫sl(x, y, z)rp(x, y, z)dV )2∫s2l (x, y, z)rp(x, y, z)dV . (2.30)

If we assume the pump is collimated across the gain medium we can get a simple

23

form for both η0 and Veff [47]

η0 =ω2L(ω2

L + 2ω2p)

(ω2L + ω2

p)2 (2.31)

Vefflgain

= π

2 (w2l + 2ω2

p) (2.32)

Along with that, knowing the details of our laser cavity and gain medium we

can predict the laser output power. Things of course are never so simple, since

predicting the values of these cavity parameters before constructing the laser is

often difficult.

There is another more general expression for the laser output power given by

Pout = PsatT[

G

T + Lpass− 1

](2.33)

where Psat is the gain medium saturation power and Lpass is the passive round trip

loss in the laser cavity defined by L = T + Lpass, and G is the laser gain. For end

pumped lasers G and Psat can be expressed as

G = η0ωLωp

PabsPsat

(2.34)

Psat = η0IsatVefflgain

(2.35)

such that equation 2.33 transforms into equation 2.26. In general equation 2.26

is more useful when characterizing the laser output power while equation 2.33 is

24

more useful when characterizing the passive cavity loss Lpass and choosing the

transmission T of the output coupler.

2.3 Basic Scattering Theory

Interactions are vital to ultracold atom experiments. The interactions between

particles give rise to interesting many body effects and ultimately novel phases of

matter. Ultracold atomic gas experiments are unique in that they are a system

where the strength of the interaction can be tuned between the two extremes of

non-interacting and unitarity-limited interactions through a Feshbach resonance.

Since the ultracold gases are very dilute, the effects of interactions may be well

described by two particle scattering 4. Below I give an overview of basic scattering

theory with the goal of explaining Feshbach resonances. All of the scattering theory

presented here may be found in [32,48,49].

2.3.1 Partial wave scattering

Consider two neutral alkali atoms whose interaction potential V (r) only depends

on their separation r = |r1 − r2|, then the Hamiltonian of the system may be

written as

H = p2

2µ + V (r) (2.36)

4At the unitarity limit the interactions are strong enough that even for the dilute gases we aredealing with, many-body effects must be taken into account to describe the full interaction.

25

where µ = m1m2/(m1 + m2) is the reduced mass and the system is equivalent

to a single particle of reduced mass µ scattering off a potential V (r). Now, for

scattering, all we care about is what happens to the particles after they collide and

scatter asymptotically far from the interaction potential, that is to say in the limit

r →∞.

We assume that the incoming particle is well described by a plane wave with

wavenumber k traveling in the z direction 5. Then the asymptotic wavefunction is

limr→∞

Ψ(r) = eikz + f(k, θ)eikr

r, (2.37)

where the first term represents the incoming particle and the second represents the

outgoing scattered spherical wave with amplitude f(k, θ).

Since the potential V (r) is spherically symmetric we can solve the problem

with a partial wave expansion in which we break up the incoming wave and the

scattering amplitude into its separate angular momentum components:

eikz =∑l

(2l + 1)ilPl(cos θ)[eikr − e−ikr+lπ

2ikr

](2.38)

f(k, θ) =∑l

(2l + 1)flPl(cos θ) (2.39)

where Pl are the Legendre polynomials and fl are the partial wave scattering5Actual particles are not plane waves but rather a superposition of many plane waves known

as a wave-packet. That said, plane wave scattering still captures the essential details of thescattering process.

26

amplitudes. By plugging in equations 2.38 and 2.39 to the full wavefunction 2.37

we get the partial wave decomposition:

limr→∞

Ψ(r) =∑l

(2l + 1)Pl(cos θ)2ik

[Sl(k)e

ikr

r− e−ikr+lπ

r

](2.40)

where Sl(k) is the partial wave scattering S-matrix defined in terms of the scattering

amplitude as

Sl(k) = 1 + 2ikfl(k) . (2.41)

With equation 2.40 we have decomposed the scattering wave function into an

incoming spherical wave and an outgoing spherical wave modified by the S-matrix

Sl. For an elastic collision the probability of finding an incoming wave must be

equal to the probability of finding an outgoing wave, therefore the scattering matrix

must only impart a phase shift δ(k)

Sl(k) = e2iδ(k) . (2.42)

We may also rewrite the scattering amplitude in terms of the scattering phase shift

as

fl(k) = e2iδ(k) − 12ik . (2.43)

While the scattering amplitudes and the S-matrix are useful for theoretically

describing the scattering process, the experimentally useful (measurable) quantity

27

is the scattering cross section σ, which loosely speaking gives the effective area of

the particle. The partial wave differential cross section is directly related to the

scattering amplitude bydσldΩ = |fl(k)|2 . (2.44)

By plugging in equation 2.43 to equation 2.44 and integrating, we can calculate

the total cross section as

σl =∫|fl(k)|2dΩ = 4π

k2 (2l + 1) sin2 δl(k) . (2.45)

From the scattering cross section we see that maximal scattering occurs for a phase

shift of δl = π/2.

For the low temperature collisions in ultracold atomic gases the relative momen-

tum between particles is small. Thus it makes sense to consider scattering in the

low energy, k → 0 limit. First note, that in the low energy limit, the scattering can

usually6 be described by only considering the lowest angular momentum partial

wave, that is s-wave scattering. In the low energy limit we can Taylor expand the

phase shift in powers of k as

k2l+1 cot δl = − 1al

+ rl2 k

2 + ... (2.46)

6In the case where other partial waves are resonantly enhanced, s wave scattering no longerdominates.

28

where rl is the effective range, and al is a constant7. This approximation is known as

the effective range expansion. In terms of the scattering matrix, this approximation

yields:

Sl =− 1al

+ 12rlk

2 + ik2l+1

− 1al

+ 12rlk

2 − ik2l+1 , (2.47)

Finally, a note on the magnitude of the phase shift at low energies. From equation

2.46 we can see that

limk→0

tan δl(k) ≈ δl(k) ∝ (kR0)2l+1 (2.48)

where R0 is some characteristic length scale of the potential. Thus, generally the

collision phase shift at low energies is small, especially for large l.

2.3.2 Resonances

Even though δl(k) is generally small at low energies it can sometimes happen that

δl rises quickly from approximately 0 to π in a small region of k. This phenomenon

is a scattering resonances. In this small region near the resonant energy E0 (or

equivalently momentum k0), the scattering cross section takes on the Breit-Wigner

form

σl = 4πk2 (2l + 1) (Γ/2)2

(E − E0)2 + (Γ/2)2 (2.49)

7For s-wave scattering al is called the scattering length. For p-wave scattering it is called thescattering volume.

29

Figure 2.3: Poles of the S-matrix on the complex momentum and complex energyplanes.

where Γ is the width of the resonance. The maximum of this resonant cross section

is the unitarity limit.

Now let’s look at this resonance in terms of the S-Matrix. Near a resonance

Sl = E − E0 − iΓ/2E − E0 + iΓ/2 . (2.50)

Thus, we see that resonances are actually poles of the S-matrix on the complex

plane as shown in Figure 2.3 [48]. In terms of complex energy and momentum

these poles are

E = E0 + iΓ/2 , (2.51)

p = k0 + iκ , (2.52)

where E0 = ~2k20/2µ and Γ = 2~2κk0/µ. Already this allows us to use equation 2.47

to find the resonances in terms of the effective range parameters. But there is even

deeper knowledge to be gained: purely positive imaginary poles of the S-matrix

30

correspond to bound states of the scattering potential and resonances occur when

the energy of these bound states are brought close to that of the scattering particles.

To see how purely positive imaginary poles of the S-matrix correspond to bound

states, consider an eigenfunction of the scattering potential of the form

Ψkl(r) = Aeikr

r+B

e−ikr

r, (2.53)

such that the S-matrix is defined as

Sl(k) = A

B= outgoing wave amplitude

incoming wave amplitude . (2.54)

Now consider a positive imaginary pole of the S-matrix such that k = iκ and B = 0,

then the wavefunction 2.53 becomes

Ψkl(r) = Ae−κr

r, (2.55)

which is exponentially damped and thus a bound state. In scattering theory,

the wavefunction 2.53 is called the regular solution and the amplitudes A and B

are known as the Jost functions F ∗l (k) and Fl(k) respectively defined such that

Fl(k) = e−iδl(k) and Sl(k) = F ∗l (k)/Fl(k).

Finally a complex energy pole of the form, E = E0 + iΓ/2, corresponds to a

quasi-bound state. There are rigorous arguments for this statement based on a

31

full treatment of scattering of wave-packets; here I will just present a heuristic

argument. Consider that time dependence of a bound state with energy Eb is

e−iEbt/~; thus, the time dependence of resonance with complex energy E is

e−iEt/~ = e−iE0t/~e−Γt/2~ . (2.56)

Here equation 2.56 describes a bound sate with energy E0 that decays with a

half-life of t = ~/Γ, that is to say, a meta-stable bound state.

2.3.3 Feshbach Resonaces

Usually the resonances described above are fixed by the bound states of the

scattering potential. Feshbach resonances provide a way to control the resonance

by tuning the energy of a molecular bound state relative to the scattering states

via tuning a magnetic field.

To see how this works consider two atoms scattering off each other. Each atom

has its own internal state structure generally defined by its spin structure and

related quantum numbers. Following [32] lets label the internal states of the two

atoms, which are far apart, by q1 and q2 and define a scattering channel |α〉 by the

internal states of the two atoms and the partial wave l such that |α〉 = |q1q2〉|lml〉.

Most ultracold atoms experiments are conducted at high magnetic field making

the Zeeman sublevels a good basis for the scattering channel |α〉; When the atoms

are brought close together during a collision the scattering channel is no longer a

32

good diagonal basis and the scattering potential may couple this initial scattering

channel |α〉 with some other channel |β〉. The channel |β〉 may be described as

either open or closed; as the name suggests, a collision can produce particles in an

open channel but cannot produce particles in a closed one. Let the total energy of

the scattering state be Etotal = Eα + E, where Eα is the internal state energy of

channel |α〉, and E is the relative kinetic energy of the collision. Then it follows

from conservation of energy that any channel |β〉 with Eβ ≤ Etotal is open and any

channel with Eβ ≥ Etotal is closed. If a collision produces the particles in the same

channel as the initial one it is considered elastic, if it produces particles in some

other open channel the collision is considered inelastic.

Figure 2.4 [32] shows the toy picture of a Feshbach resonance. An entrance

scattering channel with energy E is coupled to a closed channel, which supports

a bound state with energy Ec8. If the closed channel and entrance channel have

different magnetic moments then the energy of the closed channel bound state

Ec = δµc(B − Bc) may be brought close to the energy of the entrance channel

scattering state thereby satisfying the conditions for resonant scattering discussed

in section 2.3.2.

The more nuanced description is that the when the energy of the closed channel

molecule is brought close to that of the scattering state the coupling between the8This closed channel is generally not a closed scattering channel. Rather it is a closed

molecule whose quantum numbers are best descried by those which diagonalize the interactingHamiltonian. For lithium, these are triplet and singlet spin combinations of the two atoms withtheir corresponding singlet and triplet potentials.

33

Figure 2.4: Schematic picture of a Feshbach resonance. When the energy of acoupled closed channel molecule approaches the energy of the scattering state,interactions are resonantly enhanced. The difference between the energies of theclosed channel molecule and the scattering state may be tuned with a magneticfield.

two channels, which is normally weak, becomes resonantly enhanced. Thus, the

true molecular bound state of the system must be described by a superposition of

the two channels:

|ψmol〉 = Z|ψclosed〉+ (1− Z)|ψopen〉 , (2.57)

where |ψmol〉 is the true molecular eigenstate, |ψclosed〉 is the “bare” closed molecule,

|ψopen〉 is the entrance channel, and |Z|2 is the fraction of eigenstate in the closed

34

Figure 2.5: Diagram showing the scattering length (a) and binding energy of theFeshbach molecule (b) near a magnetically tuned s-wave Feshbach resonance. Theinset in b) shows the universal region. The upper-half of c) shows the scatteringcross section near the 6Li broad s-wave Feshbach resonance. The lower-half ofc) shows the binding energy of the Feshbach molecule (red line) and the bindingenergy of the closed channel molecule (doted black line). We see that when theFeshbach molecule approaches threshold, the scattering cross section reaches itsunitarity limited value.

channel. It is then the energy of this bound state Eb = δµb(B − B0) that when

brought close to the energy of then scattering state leads to resonant scattering as

shown in Figure 2.5 [32]

Figure 2.5 c) shows the scattering cross section and resonant bound state binding

energy for the broad s-wave lithium Feshbach resonance. We see that as the bound

state approaches threshold, the cross section assumes it’s unitarity limited value

for a broad range of energies. Thus far we have not discussed the effects of the

resonance on scattering length (scattering volume for p-wave resonances); Figure 2.5

a)/b) show that when the bound state approaches threshold the scattering length

diverges. Indeed it is usually through the scattering length that the resonance is

35

described. For s-wave resonances the scattering length is given by

a = abg

(1− ∆

B −B0

)(2.58)

where abg is the background scattering length, ∆ is the resonance width, and B0 is

the location of the resonance (where the bound state crosses threshold). In s-wave

resonances the bound state molecule takes on a universal form of Eb = ~2/2µa2

near threshold as shown in the inset of Figure 2.5 b). This type of universal bound

molecule is often called a halo dimer since the spatial extent of the wavefunction

diverges as the scattering length a→∞. Similarly p-wave resonances are described

by

v = vbg

(1− ∆

B −B0

)(2.59)

where instead of the scattering length we have the scattering volume. However

unlike in s-wave resonances, there is no universal behavior of the molecular bound

state near threshold.

36

Chapter 3

High-power, frequency-doubled

Nd:GdVO4 laser for use in

lithium cold atom experiments1

We report on an 888 nm-pumped Nd:GdVO4 ring laser operational over a wavelength

range from 1340.3 nm to 1342.1 nm with a maximum output power of 7.4 W at

1341.2 nm and a beam quality parameter M2 < 1.1. To our knowledge this is the

highest single-longitudinal-mode power obtained with a Nd:GdVO4 crystal laser.

We use a commercial frequency-doubling cavity to achieve 1.2 W at 671.0 nm and

4.0 W at 670.6 nm for use in lithium cold atom experiments. Respectively, these

wavelengths are approximately resonant with and 250 GHz blue-detuned from the1This paper has been published in Optics Express as: Francisco R. Fonta, Andrew S. Marcum,

Arif Mawardi Ismail, and Kenneth M. O’Hara, "High-power, frequency-doubled Nd:GdVO4 laserfor use in lithium cold atom experiments," Opt. Express 27, 33144-33158 (2019)

37

lithium D-lines. Thus, this source provides ample power for laser cooling of lithium

atoms while also offering substantial power for experiments requiring light 10’s to

100’s of GHz blue-detuned from the primary lithium transitions.

3.1 Introduction

Laser sources near 671 nm are the workhorses of lithium atom experiments; they

are used for optical cooling and trapping [50], driving Raman transitions [51], Bragg

scattering in lithium atom interferometers [52, 53], and isotope separation [54].

Moreover, a multi-Watt source of light blue-detuned from the lithium D-lines has

the potential to form a pinning lattice for a lithium quantum gas microscope [55]

or provide a near resonant lattice in which to produce ultra-cold lithium atoms at

high phase space density [56] when the lattice is used in combination with gray

molasses [57,58] or Raman sideband cooling [51]. Further applications of high-power

671 nm light include its use as a low noise pump for Cr:LiSAF based lasers [59]

and the generation of entangled photon pairs in the O-band of commercial silica

fibers by optical parametric down conversion [60].

Traditionally, external cavity diode lasers followed by tapered amplifiers have

been used to produce several hundred milliWatts of continuous wave (CW) single

longitudinal mode (SLM) light at wavelengths near 671 nm. However, due to

poor spatial mode quality, deterioration of the tapered amplifiers over time, and

the limited attainable power, development of other laser sources is desirable,

38

and frequency doubled 1342 nm solid state lasers have emerged as a promising

alternative.

The two main choices of laser crystal for producing light near 1342 nm are

Nd:YVO4 and Nd:GdVO4. The emission cross section at 1342 nm is slightly larger

for Nd:YVO4 than for Nd:GdVO4, so most of the development of these lasers has

focused on Nd:YVO4. However, Nd:GdVO4 has a larger thermal conductivity and

thus may be a promising alternative at higher pump powers [61–63]. Furthermore,

the emission spectrum of Nd:GdVO4 is shifted in wavelength relative to that of

Nd:YVO4, making it useful for providing light at wavelengths not accessible with

Nd:YVO4.

Frequency-doubled, single-longitudinal mode Nd:YVO4 ring lasers have seen

significant progress in recent years. In 2010, Camargo et al. demonstrated the

first single-frequency operation of an 808 nm pumped Nd:YVO4 ring laser with

an output power of 1.55 W at 1342.5 nm. Through intra-cavity second harmonic

generation, they obtained 680 mW at 671.1 nm tunable with a thin etalon over a

wavelength range of 1.25 nm [64]. In 2012, using an 808 nm pumped Nd:YVO4

ring laser with comparable performance, Eismann et al. demonstrated the utility

of this laser in lithium cold atom experiments [47]. Later in 2013, they significantly

improved the power of their laser by pumping at 888 nm. A pump wavelength of

888 nm results in a lower quantum defect and a lower absorption coefficient; this

significantly reduces heating in the crystal and distributes the heat over the entire

39

crystal length allowing for higher pump powers before thermal lensing becomes

detrimental [65]. With the new setup, they reached powers of 2.5 W at 1342 nm

and 2.1 W at 671 nm [66]. In 2015 Koch et al. achieved even greater power by using

an injection locked Nd:YVO4 ring laser pumped at 888 nm. They demonstrated

17.2 W at 1342 nm, the highest yet reported, and, after frequency doubling, 5.7 W

at 671 nm [67]. Recently in 2018, Cui et al. improved upon the effectiveness of

second harmonic generation, reporting a conversion efficiency of 93% and 5.2 W at

671 nm [68].

Nd:GdVO4 lasers have also seen progress. In the 2000’s, multiple groups

demonstrated 1341 nm light using Nd:GdVO4 standing wave lasers [61, 69, 70].

However, due to reduced mode competition from spatial hole burning in standing

wave lasers, these lasers were not single longitudinal mode. In 2013, Wang et al.

demonstrated an 808 nm pumped CW single longitudinal mode Nd:GdVO4 ring

laser emitting 3.1 W at 1341 nm [71]. In 2014, they frequency doubled their laser to

achieve 1.3 W at 670 nm [72]. In 2015, that same group improved the power of their

1341 nm Nd:GdVO4 laser to 4.6 W by using a crystal with un-doped endcaps [73].

The undoped endcaps serve to reduce the effects of thermal lensing. In that work,

neither frequency doubling nor a value for the beam quality (M2) were reported.

In this article, we investigate power scaling of Nd:GdVO4 single-longitudinal

mode ring lasers by employing a pump laser with a wavelength of 888 nm. As with

Nd:YVO4 lasers, pumping at 888 nm reduces the quantum defect and distributes the

40

heat load over the length of the crystal, reducing the detrimental effects of thermal

lensing. We characterize the thermal lensing and describe its behavior with a simple

model. Management of thermal lensing allows us to demonstrate an SLM output

power of 7.4 W at 1341.2 nm. Further, we provide a high-resolution measurement

of the gain and excited state absorption spectrum near 1342 nm and demonstrate

the tunability of this laser with a thin etalon in order to evaluate the usefulness

of this laser for lithium cold atom experiments. We demonstrate tunability over

a wavelength range from 1340.3 nm to 1342.1 nm. The peak power is attained

at 1341.2 nm, which after frequency doubling, produces light approximately 250

GHz to the blue of the D-line transitions in lithium. A powerful laser source at

this wavelength is of value for lithium experiments requiring far-off-resonance light

in order to reduce photon scattering. For example, lithium atom interferometers

using off-resonant light for Bragg scattering would benefit from the high power at

a detuned wavelength offered by this laser [52,53], as would blue-detuned optical

lattices used to form a pinning lattice for a quantum gas microscope [55] or a

high-phase-space-density source of lithium atoms [56]. The Nd:GdVO4 laser is at

the same time also useful for providing near resonant laser cooling and trapping

light since it can achieve an output power of 2 Watts at 1342.0 nm, twice the

wavelength of the D-line transitions in lithium. Watt-level laser power at 671.0 nm

is sufficient even for demanding applications such as gray molasses cooling [57,58].

In addition to demonstrating its tunability, we also characterize the laser’s linewidth,

41

spatial mode quality, as well as long- and short-term power stability. Finally, we

demonstrate frequency doubling with an efficiency of 66% using a commercial

build-up cavity.

3.2 Optical characterization of the Nd:GdVO4 crys-

tal

Neodymium-doped vanadate crystals are amenable to diode-laser pumping with

808 nm light as they exhibit strong absorption features near this wavelength

corresponding to transitions from the ground Stark sublevel of the 4I9/2 manifold

in Nd3+ to the 4F5/2 manifold (see Fig. 3.1(a)). The peak absorption coefficient at

808.4 nm for Nd:GdVO4 is 57 cm−1 for 1 % at. Nd doping concentration and light

polarized along the c-axis of the crystal [74]. Unfortunately, the large quantum

defect for a laser emitting at 1342 nm combined with the small volume over which

pump light is absorbed poses challenges for management of the heat load in the

crystal when scaling lasers pumped with 808 nm light to high power. Heat deposited

in the crystal causes thermal lensing due to the temperature dependence of the

index of refraction, stress and strain in the crystal, and bulging of the end face that

can ultimately result in fracture.

Intra-band pumping directly into the laser emitting level with 880 nm light (see

Fig. 3.1(a)) is a promising alternative and has been used to demonstrate a 5.1 W

42

Gl

[]

(a) (b)

(c)

4I9/2

4I11/2

4I13/2

4I15/2

4F3/2

4F5/2

4G5/2

4G7/2

80

8 n

m

88

0 n

m

88

8 n

m

1.3

4 m

m

©

©

½

©1

.34

mm

Figure 3.1: (a) Energy level diagram of Nd+3 in a GdVO4 host material. Therelevant transitions for pumping, lasing, and excited state absorption are labeled.(b) Absorption coefficient for a 0.5% at. doped Nd:GdVO4 crystal near the 880 nmand 888 nm pumping transitions. (c) Gain and excited state absorption spectrum.

multi-longitudinal mode intra-cavity frequency-doubled laser operating at 670 nm

with an M2 < 2 [70]. Here, the quantum defect compared to 808 nm pumping is

significantly reduced. Further, the absorption coefficient is reduced compared to

808 nm light resulting in the absorbed power being spread over a somewhat larger

volume. Alternatively, 888-nm pumping from the thermally occupied second Stark

level in the 4I9/2 manifold should be possible (see Fig. 3.1(a)). Pumping with 888

nm light in Nd:YVO4 has been demonstrated to result in a significant reduction in

the absorption coefficient, allowing the pump light to be absorbed over the entire

43

length of a 30 mm long crystal [65,67]. Here, we demonstrate, for the first time, an

888-nm-pumped Nd:GdVO4 laser.

We begin by characterizing the optical properties of the Nd:GdVO4 crystal

starting with a measurement of its absorption coefficient for intra-band pumping

with 880 nm and 888 nm light. The measurement is made with a 0.5% at.-doped

Nd:GdVO4 crystal that is a-cut and 5 mm in length. The absorption coefficient

as a function of wavelength is shown in Fig. 3.1(b). The measurement is made

using a low power (10 mW) fiber-coupled diode laser (Thorlabs, L880P010) that is

temperature tuned in order to control its wavelength. The wavelength is measured

with a multi-wavelength meter (Agilent, 86120B). For a wavelength of 880 nm,

light polarized along the c-axis of the crystal is most strongly absorbed, with an

absorption coefficient of nearly 8 cm−1. Light at a wavelength of 888 nm, on the

other hand, is more strongly absorbed if it is polarized along the a-axis of the

crystal and the peak absorption coefficient of 0.9 cm−1 is reduced from that of 880

nm light by nearly an order of magnitude. Thus, as with Nd:YVO4, pumping an

Nd:GdVO4 crystal at 888 nm allows the absorbed pump power to be spread over

the length of a crystal several cm in extent. In fact, the ability to completely absorb

all of the pump light over the length of the crystal requires the use of exceptionally

long crystals or higher Nd doping concentrations than what is explored here.

We then measure the gain and excited state absorption coefficient at wavelengths

near 1342 nm for an Nd:GdVO4 crystal pumped with high power 888 nm light. In

44

this case we use the crystal that is ultimately used for constructing the laser, which

is a 4× 4× 25mm3 Nd:GdVO4 crystal that is a-cut and has 0.5% at. doping. The

crystal is wrapped in indium foil and mounted in a water cooled block of copper

which is maintained at a temperature of 17C. The high-power pump source

capable of 60 W output is a fiber-coupled diode bar (QPC Lasers, BrightLase

Ultra-100) with a spectral full-width at half maximum of 2.3 nm which is operated

at an output power of 50 W for this measurement. The output fiber core diameter

is 400µm with a 0.22 numerical aperture (NA). The output of the fiber is imaged

onto the crystal with a 75 mm and a 175 mm lens so that a top-hat pump beam

profile approximately 1 mm in diameter propagates through the gain medium. Of

the 50 W incident, nominally 40 W of 888 nm pump light is absorbed by the crystal.

The gain and excited state absorption is measured with light from a 1342 nm

extended cavity diode laser (ECDL) that is based on a fiber-coupled single-angled

facet gain chip (Thorlabs, SAF1174P). (The design of this laser is similar to that

presented in [75].) The gaussian beam from the ECDL is focused to a 400µm waist

(1/e2 intensity radius) at the location of the gain medium. Approximately 15 mW

of 1342 nm light is incident on the crystal.

The natural logarithm of the power gain, G(λ), as a function of wavelength is

shown in Fig. 3.1(c). This quantity, ln [G(λ)], is proportional to σe − σesa where σe

is the stimulated emission cross section and σesa is the excited state absorption cross

section. The gain features near 1341 nm are associated with stimulated emission

45

from the 4F3/2 manifold to the 4I13/2 manifold. The excited state absorption

features near 1337 nm are associated with transitions from the 4F3/2 manifold to

the 4G7/2 manifold (see Fig. 3.1(a) & 3.1(b)). Note that the peak of the gain is

approximately 0.8 nm (130 GHz) to the blue of 2× λLi where λLi is the wavelength

of the D-line resonances. Also, note that there is still significant gain at the

location of 2 × λLi. This can be contrasted with the gain observed in Nd:YVO4

lasers where the peak of the gain curve lies approximately 0.14 nm (25 GHz) to

the red of 2 × λLi [47, 66]. Thus, while Nd:YVO4 lasers can supply high-power

light near-resonant with 2× λLi, the Nd:GdVO4 laser can be used in applications

requiring high-power light far-detuned from the lithium resonances and moderate

power for light at resonance.

3.3 Experimental setup

A schematic of our laser setup is shown in Fig. 3.2. Our pump source is the same

fiber coupled diode bar described above which can provide up to 60 W at 888 nm in

a top hat profile. Lenses L1 (75 mm) and L2 (175 mm) focus the pump light onto

the laser crystal with a spot size radius of 467µm. The gain medium is the 0.5%

doped 4× 4× 25mm3 Nd:GdVO4 laser crystal described above. The Nd:GdVO4

crystal is anti-reflection coated for 888 nm, 1064 nm and 1342 nm light. Four

flat mirrors form the ring cavity. Our total cavity length is 450 mm giving a free

spectral range of 670 MHz. Because all the mirrors are flat we rely on the thermal

46

lens from the Nd:GdVO4 crystal to produce a stable cavity. At our maximum pump

power of 60 W, the crystal absorbs 46 W which produces a thermal lens with a

focal length of 19 cm (see Sect. 3.4 which describes thermal lensing in our gain

medium). Mirrors M1 – M3 are highly reflective for 1342 nm light and transmissive

for 888 nm light. Our output coupler (M4) has a reflectivity of 96% which is

close to optimal for our cavity losses. To ensure unidirectional operation we use a

home made optical diode consisting of an optical faraday rotator (OFR) (described

below) and zero order half-waveplate. For control of the wavelength we use a single

uncoated etalon made of undoped yttrium aluminum garnate (YAG) that is 250µm

thick. To reduce the effects of acoustic noise we house our fundamental laser in an

acrylic box.

We send the 1342 nm light from this fundamental laser to a commercial frequency-

doubling enhancement cavity (unmodified Toptica, SHG Pro) to attain light at 671

nm. Backscatter from the frequency doubler occasionally breaks the unidirectional

operation of our fundamental laser. To prevent loss of unidirectional operation we

insert an optical diode (which has approximately 10% insertion loss) between our

fundamental laser and the frequency doubler.

The Faraday rotator used to ensure uni-directional oscillation is home built

following the design outlined by Gauthier et al. in [76]. It consists of an assembly

of three right cylindrical neodymium ring magnets with an anti-reflection coated

terbium gallium garnet (TGG) crystal 5 mm in diameter and 7 mm in length placed

47

888nm

Pump Laser

SHG Cavity

TGG

L1 L2 M1 M2

M3M4 λ/2

YAG

Etalon

Op"cal Faraday

Rotator

1342nm

671nm

Op"cal

Diode

L3

Beam

dumpNd:GdV04

Figure 3.2: A schematic of the laser setup. The laser is pumped by a fiber-coupleddiode bar. Lenses L1 and L2 image the end of the pump fiber onto the Nd:GdVO4crystal. Mirrors M1 – M4, which are all flat, form a bow-tie cavity with a roundtrip path length of 450 mm. The physical distance between mirrors M1 and M2is 45 mm and that between M3 and M4 is 156 mm. M4 is the output coupler.Uni-directional operation is enforced by the combination of the λ/2 waveplate andthe TGG crystal placed in a high magnetic field to provide Faraday rotation. Therotatable thin etalon is used to tune the operating wavelength. The beam outputfrom M4 is collimated by L3 and passes through an optical isolator before beingsent to a commercial build-up cavity for second harmonic generation.

inside the bore. The central right cylindrical ring magnet has an outer diameter of

38.1 mm, an inner diameter of 6.35 mm, and a thickness of 19.05 mm. Two right

cylindrical magnets with their magnetization opposite that of the central magnet

are placed at either end. These outer right cylindrical magnets have the same outer

and inner diameter as that of the central magnet and each have a thickness of 12.7

mm. Each magnet is grade N38. An aluminum housing holds the assembly together.

The purpose of the two outer magnets is to increase the field at the center of the

central magnet by 54%. Along the axis of the TGG crystal, this magnet assembly

48

Figure 3.3: Output power as a function of absorbed pump power. The line isPout = ηsl(Pabs − Pth) where the threshold power Pth = 13.1 W, the slope efficiencyηsl = 24% and Pabs is the absorbed pump power.

provides an integrated magnetic field of IB =∫ `TGG0 B(z)dz = 6.8 T mm. For the

Verdet constant V = 20.3 rad T−1 m−1 measured at a wavelength of 1342 nm by

Eismann et al. in [47], the predicted rotation provided by our Faraday rotator is

φ = 7.9. The compact design of this Faraday rotator allows the total round trip

length of the ring resonator to be relatively small (450 mm).

Figure 3.3 shows the power output from the Nd:GdVO4 laser as a function

of absorbed pump power. Below an absorbed pump power of 19 W, the laser is

unstable presumably due to the focal length provided by the thermal lens being

too large. For absorbed pump powers between 19 W and 45 W, the output power

nominally follows a straight line from which we can determine a threshold power

of Pth = 13.1 W and a slope efficiency of ηsl = 24%. The maximum output power

is 7.4 W. The peak output powers near the maximum absorbed pump power may

49

show signs of saturation. Unfortunately, higher pump powers could not be provided

as the thermoelectric cooler used to regulate the temperature of the pump diode

bar could not maintain the temperature needed to operate at 888 nm at higher

diode current.

3.4 Thermal lensing

In end-pumped solid-state lasers, a fraction of the power from the pump beam is

deposited as heat in the laser crystal leading to thermal lensing, thermal induced

diffraction losses, and eventually to thermal fracture of the laser crystal. Accounting

for these thermal effects presents a significant challenge for the construction of solid-

state lasers; in particular, in 1342 nm solid-state lasers these thermal effects quickly

become severe due to the high quantum defect between the pump wavelength

and the laser wavelength and are often the limiting factor in attaining higher

power operation. In this section, we quantify the thermal effects in our system

by measuring the dioptric power as a function of the absorbed pump power and

compare our measured values to a theoretical model taking into account the

fractional thermal load due to both lasing and fluorescence.

Figure 3.4 shows our measured dioptric power. To determine the dioptric power,

we measured the output parameters of our 1342 nm laser with a beam propagation

profiler (Coherent ModeMaster PC). Complex paraxial resonator analysis of our

laser cavity [77] then gives the thermal lens necessary to produce the measured

50

Figure 3.4: Dioptric power as a function of absorbed pump power. The curve isEqn. 3.1 with the fractional thermal heat load ηh given by Eqn. 3.5. These equationsprovide a model of the thermal lensing in our system with no free parameters.

laser output mode. This method, while accurate and simple to implement, cannot

determine the thermal lens without laser action. Hence our data in Fig. 3.4 begins

at the onset of lasing. At our maximum output power, we measure a thermal lens

of 19 cm. The solid line in Fig. 3.4 is our theoretical model presented below.

The dioptric power for a laser crystal optically pumped by a top hat distribution

is given by [78]

D =ηh

dndtPabs

2πKcw2p

, (3.1)

where ηh is the fractional thermal load, dn/dt = 4.7× 10−6K−1 is the thermo optic

coefficient of Nd:GdVO4, Pabs is the absorbed pump power, Kc = 11.7 W/mK is

the thermal conductivity of Nd:GdVO4, and wp is the average pump radius inside

the laser crystal [62, 63]. The average pump beam radius is given by the average of

51

wp(z) weighted by the absorption in the crystal:

wp =

∫ L0 wp0

√1 + θ2

p

w2p0

(z − z0)2 exp(−αz)dz∫ L0 exp(−αz)dz

, (3.2)

where wp0 is the pump beam waist, θp is the far field half angle in the crystal, z0 is

the location of the focus, and α is the absorption coefficient. When aligning the

laser, we positioned the waist of the pump beam to be coincident with the center

of the crystal to the best of our ability. Thus, for our system, we take z0 to be the

center of the crystal. Further, we measured θp = 32.5 mrad, wp0 = 467µm, and

α = 0.59 cm−1. These parameters yield a waist of wp = 524µm.

Here we derive the fractional thermal load of the end-pumped Nd:GdVO4 laser

lasing at 1342 nm taking into account both laser action and fluorescence. Following

similar derivations [79, 80], we start with the rate equation for a 4-level system

given bydN

dt= αp

λphcIp −

N

tf− (σe + σesa) λl

hcIcircN. (3.3)

Here, N represents the population inversion density. The first term represents the

pumping rate, where αp is the pump absorption coefficient, Ip is the pump intensity,

h is Planck’s constant, c is the speed of light, and λp is the wavelength of pump

photons. The second term represents the fluorescent decay rate, where tf is the

fluorescence lifetime. The third term represents the stimulated emission rate, where

σe and σesa are the excited state emission and absorption cross sections respectively,

52

λl is the laser wavelength, and Icirc is the intensity of laser light in the cavity. The

steady state solution is given by

N = λphc

tfαpIp1 + Icirc/Isat

, (3.4)

where Isat = hc/(σe + σesa)tfλl is the laser saturation intensity.

Now we can consider the power densities associated with pumping, lasing,

fluorescence, and ultimately heating. Power densities can be derived by multiplying

the rate of the process by the associated photon energy. The power densities are

Qp = αpIp (from pumping), Qf = 1trad

hcλfN (from fluorescence) where trad is the

radiative life time and λf is the fluorescence wavelength, and Ql = σeλlhcIcircN (from

lasing). The power density associated with heating is the difference between that of

the pump power and that of the florescence and laser powers: Qh = Qp− (Qf +Ql).

Finally, the total fractional thermal load is the ratio between the heat power density

and the pump power density and is given by

ηh = Qh

Qp

= 1−

tftrad

λpλf

+ IcircIsat

(σe

σe+σesa

)λpλl

1 + Icirc/Isat

. (3.5)

The limits Icirc → 0 and Icirc/Isat 1 yield the often quoted fractional thermal

loads for non-lasing and lasing conditions respectively. These are [81,82]:

ηnon = 1− tftrad

λpλf, and ηlase = 1−

(σe

σe + σesa

)λpλl. (3.6)

53

The total fractional thermal load depends on the intensity of light circulating

in the cavity which can be approximated as

Icirc = Pout

(1−R)πw2l

' ηsl(Pabs − Pth)(1−R)πw2

p

, (3.7)

where Pout = ηsl(Pabs − Pth) is the output power that has been measured earlier

in this paper, R is the output coupler reflectivity, and wl is the radius of the laser

mode in the crystal which we approximate as the radius of the pump beam wp.

We now have the fractional thermal load expressed solely in terms of previously

measured quantities. We use the values for the fluorescence and radiative lifetime

found in [82], and the values for the excited state absorption and emission cross

sections found in [83]. Substituting the total fractional thermal load (Eqn. 3.5) into

the equation for dioptric power (Eqn. 3.1) completes the model for the thermal

lens. Figure 3.4 shows that there is good agreement between our measured values

and our model. The small remaining disagreement between the model and the

measurements may be due to thermal lensing in other elements of the cavity such

as the TGG crystal. However, since these elements are not exposed to the pump

light, we expect thermal lensing in them to be relatively small. For example, we

estimate that the thermal lens in TGG has a focal length of 1.2 m due to absorption

of 1341 nm light at maximum operating power.

At maximum pump power, the focal length of the thermal lens in the Nd:GdVO4

crystal is found to be 190 mm. The paraxial resonator analysis used to determine

54

this thermal lens from the measured output mode of our laser beam also predicts

the waist radius of the lasing mode at the center of the gain medium. This waist

radius is found to be 426µm. This radius is 81% of the average pump radius wp.

This is an appropriate ratio for the mode to pump radius as it is small enough to

avoid diffraction losses [43, 82] yet large enough to ensure suppression of transverse

modes.

3.5 Characterization of the fundamental and frequency-

doubled laser

3.5.1 Single longitudinal mode operation and linewidth

Single-longitudinal mode operation of the fundamental laser was first verified using

a scanning Fabry-Perot (FP) interferometer with a free spectral range of 300 MHz

(see Fig. 3.5(a)). Reliable single-mode operation could be achieved even in the

absence of an intra-cavity etalon, only the Faraday rotator and half-wave plate

are required. Whereas other groups have reported needing one or more etalons to

achieve single-mode operation, we ascribe the robustness of the single-frequency

behavior in our laser to the relatively short cavity length of the fundamental laser

which gives a free spectral range of 670 MHz. The linewidth of the 1342 nm

radiation measured with the FP interferometer is limited by its finesse.

To more accurately determine an upper limit on the linewidth of the free-running

55

(a) (b)

FWHM:

450 kHz

300 MHz

Figure 3.5: (a) Fabry-Perot spectrum showing single-longitudinal mode operations.(b) Beat note between the free-running Nd:GdVO4 and an extended cavity diodelaser demonstrating an upper bound on the fundamental laser linewidth of 450 kHz.

fundamental laser, we measure a beat note between the 1342 nm solid-state ring

laser and the extended cavity diode laser that is based on a fiber-coupled single-

angled facet gain chip which was described above. As the ECDL is acoustically

well isolated from the environment, we expect that its free-running linewidth is

quite narrow. The linewidth of an ECDL laser with a similar design has been

shown to be < 10 kHz [75]. Both lasers are made to illuminate an InGaAs amplified

detector with a 150 MHz bandwidth (Thorlabs PDB450C). The resulting beat note

between the two free running lasers as measured on a spectrum analyzer is shown

in Fig. 3.5(b). The resolution bandwidth of the spectrum analyzer is 300 kHz and

its video bandwidth is 10 kHz. The sweep time is set to 50 ms. The measured

beat note is fit to a gaussian and found to have a full-width at half maximum

(FWHM) of 450 kHz. This linewidth is sufficient for use of this laser in cooling and

trapping experiments with lithium as the natural linewidth of the D-line transitions

is 5.9MHz.

56

3.5.2 Mode quality of fundamental and second harmonic

To evaluate the mode quality of our fundamental laser beam, we use a commercial

beam propagation profiler (Coherent ModeMaster PC). This beam profiler utilizes

the knife edge method to measure the beam width along two orthogonal directions.

To measure the beam width at different positions along the propagation axis the

profiler moves a telescopic lens so that different planes along the propagation axis

are imaged onto the moving knife edge. After determining the widths at the location

of the knife edge, the widths of the propagating beam external to the profiler can

be determined from the known focal length and principal plane of the telescopic

lens. When measuring the beam propagation with the profiler we place a 1000

mm focal length lens following lens L3 in Fig. 3.2. The 1/e2 intensity radius as a

function of position external to the profiler is shown in Fig. 3.6(a) and 3.6(b). The

origin in these figures corresponds to the input bezel of the beam profiler. A fit to

the function

w(z) = w0

√√√√1 + (M2)2 (z − z0)2

z2R

(3.8)

is shown for each of the directions. Here, w0 is the beam waist, z0 is the location of

the beam waist, zR = πw20/λ is the Rayleigh length of a gaussian beam, λ = 1342 nm,

and M2 is the beam quality parameter. The fit parameters are z0, w0, and M2.

Propagation of the beam in the horizontal direction is consistent with that of an

ideal gaussian beam for which M2 = 1. The beam quality in the vertical direction

57

(a)

(b)

(c)

Figure 3.6: Measurement of the laser caustic for the 1342 nm laser in the (a) verticaland (b) horizontal directions. (c) Measurement of the laser caustic for the 671 nmlaser beam. The inset shows a typical beam profile of the 671 nm laser beam asrecorded by the CCD camera. The solid lines are fits to Eqn. 3.8 used to determinethe beam quality parameter M2.

is less than ideal but is still very good with M2 < 1.1. The laser beam is slightly

astigmatic with astigmatism given by (z0,horiz − z0,vert)/zR = 23% where zR is the

average Rayleigh length for the two directions. Also, the beam is close to circular

with a waist asymmetry given by w0,horiz/w0,vert = 0.94. The astigmatism and slight

ellipticity of the beam is due to elements that break the symmetry of the cavity

such as the orientation dependent thermal conductivity and natural birefringence

of Nd:GdVO4 or the non-zero angle of incidence of the cavity mode on the output

coupler. Still the beam quality is quite good. Attaining nearly ideal beam quality

is crucial for achieving good mode matching to the frequency-doubling cavity and

thereby high second harmonic generation efficiency.

The 671 nm radiation output from the second harmonic generation cavity should

58

be a nearly ideal gaussian beam. To verify this we also performed a measurement

of the 671 nm laser caustic. In this case, we focused the laser output from the

frequency-doubling cavity with a 200 mm focal length lens and measured the beam

profile at a number of positions along the beam path with a charge-coupled device

(CCD) camera (Thorlabs DCU223M). Fig. 3.6(c) shows the 1/e2 intensity radius

from gaussian fits to the laser profile as a function of position along the optical

axis. The solid curves show a fit to Eqn. 3.8 with λ = 671 nm. Both the horizontal

and vertical beam profiles are consistent with a value of M2 = 1 within their 95%

confidence interval. The fact that the caustics are fit to values of M2 slightly less

than one is presumably due to measurement error (e.g. small errors made in fitting

the width of the gaussian profile on the CCD camera).

3.5.3 Wavelength tunability

The wavelength of the fundamental laser can be tuned by rotating the YAG etalon

inside the laser cavity. The free spectral range of the 250µm thick etalon is 330

GHz. The finesse of the uncoated YAG etalon is 1.0. As shown in Fig. 3.7(a), the

wavelength of the fundamental laser can be tuned with this etalon from 1340.3

nm to 1342.1 nm, nominally over the 1.8 nm (300 GHz) width of the gain profile.

Mode hops of the thick etalon occur at either end of this nominally 300 GHz tuning

range. The power of the fundamental laser plotted in Fig. 3.7 is measured after the

optical isolator shown in Fig. 3.2 which results in a moderate (10%) loss of power.

59

(a)

(b)

Figure 3.7: (a) Fundamental and second harmonic output power versus wavelength.The power of the fundamental laser is measured after the optical isolator in Fig. 3.2,just before entering the SHG cavity. (b) Spectrum of water absorption coefficientat 300K and 50% relative humidity [1, 2].

Each data point is the average of three or more experimental runs to better account

for day to day fluctuations in performance. The error bars reflect the standard

deviation in the mean of these runs. Two prominent dips in power are observed at

wavelengths near 1340.5 nm and 1341.6 nm. These drops in power are associated

with strong water absorption lines [1, 2] as shown in Fig. 3.7(b).

Figure 3.7(a) also shows the output power of the second harmonic generation

cavity. At peak power, we achieve a second harmonic generation efficiency of 66%

determined by the ratio of the second harmonic power to the fundamental power

measured after the optical isolator. The maximum power we attain is 4 W at a

wavelength of 670.6 nm which is 0.36 nm (240 GHz) to the blue of the D2 line for

60

-80

-90

-100

-110

-120

-130

-140

dBc

(a) (b)

Pw

P2w

RIN

Detector photocur.

2w

Detector no photocur.

Figure 3.8: (a) Fundamental and second harmonic output power over several hours.The fundamental power is measured after the optical isolator, just before enteringthe SHG cavity. (b) One-sided power spectral density of the residual intensity noiseof the frequency-doubled laser output.

7Li and 0.38 nm (250 GHz) to the blue of the D2 line for 6Li. All four relevant

transitions for 7Li and 6Li are shown in Fig. 3.7(a). The 7Li D1 and D2 lines

respectively occur at 670.976 nm and 670.961 nm and the 6Li D1 and D2 lines

occur at 670.992 nm and 670.977 nm. The second harmonic power attained at the

D1 line of 7Li and the D2 line of 6Li is 1.2 W.

3.5.4 Long term stability and residual intensity noise

The long term power stability of both the fundamental and frequency-doubled

laser is ascertained by recording the power incident on a photodiode at half second

intervals over a period of several hours. The time traces for both the fundamental

laser and the frequency-doubled laser are shown in Fig. 3.8(a). For the fundamental

laser, the standard deviation of the laser power is σ = 0.7% over a five hour period.

For the second harmonic, σ = 0.8% over the same five hour period.

We measure the relative intensity noise of the frequency doubled laser by first

61

recording a time series of the instantaneous power in a nominally 50µW laser beam.

The power is measured using an amplified low-noise photodiode with a 50 MHz

bandwidth (Thorlabs PDA8A). The time series is recorded using a modular digital

oscilloscope with a 100 MHz bandwidth and a 16 Mpts memory depth (Agilent

U2701A) running at 500 MS/s. The digital oscilloscope is AC coupled so that it

records the fractional intensity fluctuations ε(t) = (I(t)− 〈I〉)/ 〈I〉 after the signal

is normalized by the average intensity 〈I〉 (here, 〈...〉 denotes a time average). The

one-sided power spectral density of the residual intensity noise is then given by:

SRIN = limT→∞

2T

⟨∣∣∣∣∣∫ T

0ε(t)e+i2πfdt

∣∣∣∣∣2⟩

(3.9)

which is computed for the data using a fast Fourier transform. The power spectral

density (PSD) we report is an average of 400 individual noise spectra.

The resulting one-sided PSD is shown in Fig. 3.8(b). The upper most power

spectral density in the plot, shown in orange, is the SRIN measured for the 671

nm laser. The lower blue curve is the electronic noise spectrum obtained when an

incoherent light source is used to produce the same photocurrent in the detector.

Finally, the lowest curve shown in green is the electronic noise spectrum obtained

when no light falls on the detector. The peak in the power spectral density for the

671 nm light centered at 230 kHz is due to relaxation oscillations in the solid-state

laser. The smaller peak at 50 kHz is also associated with noise in the solid-state

laser, rather than noise arising from the frequency doubling process, but its explicit

62

origin is not well understood. Above 1 MHz the SRIN of the 671 nm light falls

below the noise floor of the detection method. The narrow feature at 20 MHz is

due to phase modulation of the 1342 nm light used to lock the doubling cavity

to the fundamental laser frequency with the Pound-Drever-Hall technique [84].

The integral of the one-sided PSD from 500 Hz to 10 MHz yields an rms noise

εrms = 8.7 × 10−3. We verify that this rms noise is consistent with the directly

measured rms intensity fluctuations.

3.6 Conclusions

We have constructed a high-power, single-longitudinal mode Nd:GdVO4 ring laser

by intra-band pumping at 888 nm directly to the laser emitting level. A simple

model of the thermal load resulting from 888-nm pumping predicts the observed

thermal lensing in the gain medium. We show that this laser is suitable, after

frequency doubling, for cold atom experiments with lithium. In particular, we

show that more than 1 Watt can be obtained at a wavelength of 671.0 nm which is

resonant with the D-lines in lithium. Further, the laser system has a sufficiently

narrow linewidth, long-term stability, and small residual intensity noise for it to

be well suited for providing a reliable source of laser cooling and trapping light

at this wavelength. What is more, we have demonstrated that 4 Watts of power

can be attained at a laser frequency which is approximately 250 GHz detuned to

the blue of the D-line transitions in lithium. Thus, this laser source can provide

63

high-power light at a detuned wavelength which is desirable for applications that

require reduced spontaneous emission such as Bragg scattering beams in lithium

atom interferometers or blue-detuned optical lattices.

Higher output powers of the fundamental laser can be achieved by removing

the lossy Faraday rotator and half-wave plate from the cavity and achieving uni-

directional operation either by injection locking from a microchip laser or extended

cavity diode laser [67] or by self-injection locking where the lossy Faraday rotator

is placed in a weakly coupled external cavity [85]. Furthermore, second harmonic

generation efficiency as high as 93% has been demonstrated with periodically poled

potassium titanyl phosphate (ppKTP) in an external build-up cavity at a similar

wavelength [68]. Such improvements to the laser operation will only increase its

efficacy for potential applications, both in and outside of lithium atom experiments.

64

Chapter 4

The Nd:YVO4 Raman Laser

System

The high power Nd:GdVO4 laser presented in the previous chapter was built to

demonstrate the potential of using Nd:GdVO4 lasers systems broadly for lithium

atom experiments. Before building that novel laser I built a Nd:YVO4 laser using

the same basic design. The purpose of this Nd:YVO4 laser system was to drive

6Li Raman transitions. In this chapter I will discuss Raman transitions and the

design/construction of the Nd:YVO4 system.

4.1 Atom-Light interactions/Raman Transitions

In this section I will give a theoretical overview of atom light interactions with

the goal of explaining Raman transitions. Raman transitions are two photon

65

transitions between an atomic ground state energy level |g〉 and an excited state

energy level |e〉 using an intermediate state |i〉. I will start with the simplest case

of a two level atom under the effects of an oscillating electromagnetic field 4.1.1.

Then I will discuss electric dipole and magnetic dipole transitions along with their

corresponding selection rules 4.1.2. Finally I will discuss Raman transitions 4.1.3.

4.1.1 Two level atom

Consider a two level atom1 interacting with an oscillating electromagnetic(EM)

field [86,87] as shown in figure 4.1. Let |g〉 represent the ground state and |e〉 the

excited state. We will treat the electromagnetic radiation semi-classically; thus, the

Hamiltonian can be written

H(t) = H0 +HInt cos(ωt), (4.1)

where H0 is the Hamiltonian unperturbed by the EM field such that

H0|g〉 = 0|g〉 (4.2)

H0|e〉 = ~ω0|e〉 (4.3)

1A real atom of course has many energy levels, however the two level atom serves as a goodapproximation when the electromagnetic fields couple the two energy levels much more stronglythan any of the other atomic energy levels as is the case for fields whose frequency is resonantwith the atomic transition.

66

Figure 4.1: Diagram representing the two level atom. ω0 represents the energydifference between the ground and excited states, ω represents the frequency of thefield coupling the two states and δ represents the detuning.

and Hint is the interaction between the EM field and the atoms. Usually this

interaction is either through the electric dipole moment

HintE = −µe · E = er · εeE0 (4.4)

or the magnetic dipole moment

HintB = −µ ·B = µB(L+ gsS + gII) ·B. (4.5)

67

For now we will take the interaction to be completely general. In 4.1.2 I will

discuss the implications of these different dipole interactions. In all generality a

wavefunction describing this system can be expressed as:

ψ(t) = cg(t)|g〉+ ce(t)|e〉e−iω0t, (4.6)

where |cg|2 and |cg|2 give the ground and excited state populations respectively.

Solving Schrodinger’s equation gives the time dependence of the ground and excited

state amplitudes as:

icg = Ω2 [ei(ω−ω0)t + e−i(ω+ω0)t]ce (4.7)

ice = Ω∗2 [ei(ω+ω0)t + e−i(ω−ω0)t]cg (4.8)

Where Ω is the Rabi frequency defined as

Ω = 〈g|Hint|e〉~

. (4.9)

To solve the equation further we need to make the rotating wave approximation.

In the rotating wave approximation, we assume that the detuning is small so that

δ = ω − ω0 << ω0 and consequently neglect the ω + ω0 terms which oscillate on

a much quicker time scale and average to zero. Under this approximation, the

coupled equations 4.7 and 4.8 can be solved analytically. Specificity under the

68

initial condition that the entire population is in the ground state, the excited state

population evolves as:

|ce|2 = Ω2

Ω′2 sin2(Ω′t

2 ), (4.10)

where Ω′ =√

Ω2 + δ2 is the generalized Rabi frequency.

Now let’s interpret this result. Equation 4.10 shows that under the effects of

an oscillating field an atomic population will oscillate between the ground state

and excited state. If the field is on resonance (δ = 0) then it is possible to take

the entire population from the ground state to the excited state. In experiments

this feature is exploited by what is called a π pulse. If a resonant field is pulsed on

with a pulse duration t such that Ωt = π then the entire ground population will

be brought into the excited state. This is one of the main ways in which we drive

transitions between atomic states.

The above discussion neglects spontaneous emission from the excited state.

However, as long as the Rabi frequency is much larger than the spontaneous

emission rate, atoms may still be coherently transferred to the excited state. [41]

4.1.2 Electric and magnetic dipole transitions

To see how these transitions play out in actual atoms, specifically 6Li, we need to

consider the dipole interactions in detail.

First let’s calculate the coupling the between an initial state |L, S, J,mj, I,mi〉

and a final state |L′, S

′, J

′,m

′j, I

′,m

′i〉 due to the electric dipole interaction as the

69

electric dipole coupling is much stronger than its magnetic counterpart. That is we

need to calculate the electric-dipole transition matrix elements

〈L′, S

′, J

′,m

′

j, I′,m

′

i|µe|L, S, J,mj, I,mi〉 (4.11)

where µe = r · εe is the electric dipole operator. Following the derivation in [41] we

can break down this complex matrix element into a much reduced matrix element.

First we recognize that µe is a spherically irreducible tensor that may be written in

the form µ(1, q) where q= -1,0,1 represents σ−, π, and σ+ radiation respectively.

Next we make extensive use of the Wigner-Eckart theorem to obtain:

〈L′, S

′, J

′,m

′

j, I′,m

′

i|µ(1, q)|L, S, J,mj, I,mi〉 = δI,I′ (−1)F′+F+J ′+I−m′

F+1

√(2F ′ + 1)(2F = 1)

J

′I F

′

F 1 J

J

′ 1 J

−mF ′ q mF

〈J ′|µ(1)|J〉. (4.12)

Here the large curly bracket is the Wigner 6-j symbol and the lage parenthesis is the

Wigner 3-J symbol. This may seem like a monster equation but it is actually quite

beautiful. First off the Wigner symbols’ values can easily be looked up in tables.

Secondly the Wigner 3-J symbol immediately gives us the transition selection rules

of ∆F = ±1 and ∆mF = q. Knowing that the electric dipole interaction cannot

change I or S further restricts the selection rules to ∆L = ±1. Thirdly the entire

calculation of the electric-dipole transition matrix elements now boils down to

70

calculating 〈J ′|µ(1)|J〉, which for 6Li ends up having only two possible values, one

for the D1 line and one for the D2 line. These values have already been calculated

and are in a table in ref [41].

Now that we have the electric-dipole transition matrix elements we can calculate

the Rabi frequency as

Ω = 〈g| − µe ·E|e〉~

= µgeE0

~(4.13)

where µge is the electric-dipole transition matrix element between the ground and

excited state and E0 is the strength of the electric field.

All is not good news in our calculations of the electric-dipole transition matrix

elements. The selection rule of ∆L = ±1 prevents electric dipole transitions from

driving transitions between the different magnetic sublevels of the ground state

Zeeman manifold. This is a big problem since the majority of the experiments

we would like to conduct involve manipulation of atoms within the ground state

Zeeman manifold!

One solution is to try to make use of the magnetic dipole transitions. Examining

equation 4.5 it is clear that we can deal with the contributions from the orbital,

electron spin, and nuclear spin separately. Thus, to calculate the transition matrix

elements we only need to know

〈L′,m

′

L|L ·B|L,mL〉 , (4.14)

〈S ′,m

′

S|S ·B|S,mS〉 , (4.15)

71

〈I ′,m

′

I |I ·B|I,mI〉 . (4.16)

Using angular momentum ladder operators as described in most quantum mechanics

textbooks [49] the selection rules can be shown to be

∆L,∆S,∆I = 0 (4.17)

∆mL,∆mS,∆mI = 0,±1. (4.18)

These selection rules, specifically ∆L = 0, allow for transitions between states

within the ground state manifold.

In our experiments we use radio-frequency pulses from an antenna located within

the vacuum chamber to drive magnetic dipole transitions between the low-field

seeking ground hyperfine levels of 6Li |1〉,|2〉,|3〉. For details on the setup for our

RF transitions see J.R Williams’ thesis [88]. There are limitations to this setup.

The antenna has to be put inside the vacuum chamber to reach sufficient power to

drive the transition due to the small magnetic dipole transition matrix elements.

This makes it very difficult to change or repair the system as doing so would require

opening the vacuum system to atmosphere. Furthermore the low powers required

for the RF pulses means the pulse duration required for a π pulse is quite long,

between 50 and 100 µs. Finally due to the necessity of impedance matching between

the antenna and the transmission lines the bandwidth of the antenna is limited

72

Figure 4.2: Diagram representing the two photon process involved in Ramantransitions.

to several hundred MHz. This limits the transitions that can be made to those

between the three lowest states of the ground hyperfine manifold. If we want quick

transitions and transitions between low field and high field seeking states of the

ground hyperfine manifold, such as |2〉 → |5〉 transitions, some other technique is

needed; Raman transitions are needed.

4.1.3 Raman Transitions

Raman transitions are two photon transitions between a ground and excited state

involving a third intermediate state as show in figure 4.2. Assume that one laser

beam with frequency ωL1 couples a state |1〉 to an intermediate state |i〉 with a

detuning ∆ = ωL1 − (ωi − ω1) and another laser with frequency ωL2 couples a

different state |2〉 to the intermediate state |i〉 with a total two photon detuning

73

δ = (ωL1− ωL2)− (ω2− ω1). As the couplings to the intermediate state are electric

dipole transition matrix elements with selection rules ∆L = ±1 the total two

photon selection rules allow for ∆L = 0 transitions and thus Raman transition can

drive coupling between the entire ground state hyperfine manifold.

Raman transitions are different than two single photons. Atoms are not brought

from state |1〉 to state |i〉 and then from state |i〉 to state |2〉. Rather they are a

coherent two photon process where atoms are brought directly from sate |1〉 to

state |2〉, with virtual transitions to state |i〉 acting as an intermediary. Indeed

when the single photon detuning ∆ is much larger than the Rabi frequencies Ω1i,

Ω2i coupling states |1〉 and |2〉 to state |i〉 and much larger than the two photon

detuning δ, the system may be considered an effective two level system [87] with

an effective Rabi frequency,

Ωeff = Ω1iΩ2i

2∆ . (4.19)

In this effective two level system it is the two photon detuning that determines if

the transition is on resonance.

As with the RF magnetic dipole transitions Raman transitions have their own

difficulties. In a real multilevel atom there are more than 3 energy levels, which

makes it impossible to make the single photon detuning arbitrarily large. Thus, the

Raman transition can never be a truly coherent process, there will always be some

chance of spontaneous emission from the intermediate state leading to heating in

the gas. This heating is mitigated by having very intense laser beams yielding

74

a large Ωeff and thus allowing short duration π pulses. The next difficultly is a

practical one: It is difficult to get two phase coherent laser beams separated enough

in frequency to address to ground hyperfine manifold.

For 6Li, two phase coherent lasers with a frequency separation of approximately

1 GHz is needed. One method of generating the lasers is to use two separate diode

laser whose beat note signal is phase locked to a microwave source at the transition

frequency. Before the development of our Nd:YVO4 laser system descried below,

this phase locking system was implemented in our lab. Details of this system may

be found in a Yi Zhang’s thesis [89]. There were two problems with this setup: (1)

creating a phase lock over GHz separation is very difficult and the lock was never

quite robust enough; (2) diode lasers at 671nm output relatively little power and

after implementing the phase lock each beam ended with 2-4 mW of power which,

while sufficient, is not great.

Another way to get two phase coherent beams separated by GHz in frequency is

to derive both beams from the same laser source. The idea is to split the laser into

two beams with a beam splitter and then shift the frequencies of the beams with a

cascade of AOMs (Acousto-Optic Modulators). A great advantage of this technique

is that because the beams are derived from the same source they are guaranteed to

be phase coherent. The disadvantage is that on each pass through an AOM the

laser loses a lot of power. Diode lasers do not emit enough power at 671 nm for

this scheme to be feasible. Frequency doubled solid-state lasers on the other hand

75

Figure 4.3: Schematic showing the setup of the Nd:YVO4 laser.

have been shown to reach powers on the order of watts [47,64,66,67]. Motivated

by this work we decided to build a high power frequency doubled Nd:YVO4 laser

with the idea of achieving enough power to drive fast Raman transition.

Our solid-state Nd:YV04 laser system is capable of delivering up to 4W of power

at 671 nm. Our AOM system has an efficiency of approximately 20%, leaving us

with up to 400 mW in each beam to drive Raman transitions. The power does drift

from day to day, however our Rabi frequencies are regularly high enough to drive π

pulses in less than 500 ns.

4.2 Nd:YVO4 Laser Setup

The setup for our Nd:YVO4 laser system is shown in figure 4.3. A commercial

fiber-coupled diode stack (QPC Lasers PR-6008-0008) capable of outputting 50W

76

of light at a wavelength of 888nm with a top hat spatial profile acts as our pump

source. Lenses L1 and L2 image the pump light to a spot size radius of 467 µm onto

the Nd:YVO4 gain medium. The gain medium is a 4× 4× 25mm3, a-cut, 0.5% at.

doped Nd:YVO4 crystal housed in a home built water cooled mount. The Nd:YVO4

gain medium absorbs 40W of the pump light leaving 10W of power that must

be dumped2. The cavity is formed with 4 flat mirrors in a bow-tie configuration.

Mirror M4 is the output coupler with a reflectivity of 90.2% at 1342nm. The cavity

has a total length of 45cm resulting in a free spectral range of vFSR = 670MHz. The

large free spectral range allows us to achieve single longitudinal mode operation and

control the wavelength with a single etalon made of undoped YAG that is 250µm

thick. Since there are no curved mirrors in the cavity the cavity stability relies on

the thermal lens of the Nd:YVO4 gain medium. At maximum absorbed power the

Nd:YVO4 crystal has a thermal lens with a focal length of 16cm. Unidirectional

operation is ensured with a home built optical Faraday rotator and a zero order

half wave plate 3. The Faraday rotator is the same as that described in section

3.3 only instead of a TGG crystal an un-doped YAG crystal 5mm in diameter and

18mm in length is placed within the magnet assembly.

The 1342 nm output from the Nd:YVO4 laser is then sent to a commercial2It is important to dump the excess pump laser power far from the laser cavity. If the beam

dump is located too close to the laser cavity the resulting thermal gradients can cause noise in thelaser. The resulting noise can be strong enough to break single mode longitudinal mode operationand even unidirectional operation.

3Due to the birefringence of the Nd:YVO4 gain medium a direction dependent rotation of thepolarization is enough to break the symmetry and ensure unidirectional operation.

77

frequency doubler (Toptica, SHG Pro) to attain light at 671nm. An optical diode

is placed between the fundamental laser and the frequency doubler to prevent

back-scatter from disrupting unidirectional operation. Under optimal conditions4

the fundamental laser outputs 6W at 2λLi (twice the lithuim D-Line transition

wavelength) with a spatial mode M2 = 1; After the frequency doubler the sytem

outputs 4W at λLi with M2 = 1.

As you may have noticed, the setup for this Nd:YVO4 laser is the same as

that for our Nd:GdVO4 laser system (described in section 3.3) with a few notable

exceptions. First, the gain medium is Nd:YVO4 instead of Nd:GdVO4. We choose

Nd:YVO4 as our gain medium because the peak of its gain is closer to the 6Li

D-line transitions. Moreover as the feasibility of the crystal for use in lithium atom

experiments had already been demonstrated [47, 66] it was a easy choice for the

construction of our Raman laser system. Second, the pump source is QPC Lasers

PR-6008-0008 instead of QPC Lasers, BrightLase Ultra-100. Simply put, this is

just an earlier model from the same company of a fiber coupled diode stack capable

of emitting high power light at 888nm. Finally instead of the TGG crystal used in

the OFR of our Nd:GdVO4 laser, here we have used undoped YAG that has a 5 mm

diameter and is 18 mm long. YAG has a Verdet constant of V = 1.38 rad T−1 m−1 at4It is very difficult to align the ring cavity and frequency doubler perfectly to achieve these

optimal conditions. The laser system’s output power drifts from day to day due to changes inhumidity and temperature affecting cavity alignment and so every now and then we must realignthe laser system. It is experimentally unfeasible to spend the hours necessary to achieve fullyoptimal operation. Thus, on a day to day basis it is common to operate with approximately 4Win the fundamental laser and 3W after frequency doubling which is easy to achieve after a quickoptimization.

78

1342nm [90]; our magnetic assembly provides IB =∫ `TGG

0 B(z)dz = 6.8 T mm; thus,

with YAG our optical Faraday rotator provides a rotation of φ = 1.4 compared to

the rotation of φ = 7.9 achieved with TGG. The φ = 1.4 rotation provided by

YAG was enough to achieve robust unidirectional operation in our Nd:YVO4 laser

but proved unstable in our Nd:GdVO4 laser at which point we opted to upgrade to

TGG.

It is not mere coincidence that the setup of the two lasers is so similar. Indeed

the design is quite general as discussed in section 4.4 and can be used for any solid

state laser whose thermal lens is in the appropriate range and whose gain medium

is birefringent.

4.3 Making the Raman beams

Figure 4.4 [40] shows the setup for creating two Raman beams separated in frequency

by ∆ω ≈ 1.8 GHz. We start with up to 6W of light at a wavelength of 2λLi nm

from the fundamental Nd:YVO4 laser. This light is sent to the Toptica SHG Pro

frequency doubler, which outputs up to 4W of light at the lithium D-line transition

wavelength λLi. This light is then split into two beams by a half-wave plate and

polarizing beam splitter. The beams are then double passed5 through acousto-

optic modulators(AOMs) to shift their frequencies. In total we use three AOMs

labeled AO1, AO2 and AO3, all of which are driven by the same Direct Digital5After a single pass through an AOM a laser beam has its frequency shifted by the AOM’s

drive frequency. After a double pass the beams are shifted by twice the drive frequency

79

Synthesizer(DDS) with a frequency of ωDDS ≈ 300 MHz. The first beam is double

passed through AO1, which downshifts its frequency by 2 × ωDDS. The second

beam is double passed though AO2 and AO3, which both upshift its frequency by

2 × ωDDS for a total frequency shift of 4 × ωDDS. After this series of AOMs the

frequency difference between the two beams is

∆ω = 6ωDDS ≈ 1.8 GHz, (4.20)

The frequency difference between the two beams is tuned by tuning the drive

frequency of the DDS. The two beams are then recombined with a PBS and sent

though a fourth AOM (AO4) after which they are coupled to an optical fiber and

sent to the experiment. The fourth AOM acts as a switch allowing us to keep the

laser system on while pulsing the Raman beams off and on within the experiment.

The cascade of AOMs is pretty inefficient and we are left with approximately just

20% of the starting 4W at the output of the fiber to the experiment. However since

we started with such high powers, a 20% efficiency still leaves us with a whopping

800 mW of power in our Raman beams! With this Raman system we can drive

state |2〉 → |5〉 transition with π pulses as short as 350 ns!

80

Figure 4.4: Schematic showing the setup of the Raman laser system

4.4 Notes on general design for solid state lasers

One of the most difficult things when designing a cavity for end pumped solid state

lasers is dealing with the thermal lens induced in the gain medium by the pump

laser. In particular this thermal lens can affect the overall stability of the cavity

as well as the cavity eigenmode. The problem is it is usually difficult to predict

the exact thermal lens that will be induced by the pump laser 6. Furthermore the

cavity must be stable for both the thermal focal length while the system is lasing6It is my hope that the theory developed in section 3.4 will help in achieving more accurate

predictions of the thermal lens.

81

(a)

(b)

Figure 4.5: Cavity stability(a) and laser mode radius inside the crystal (b) asfunction of the crystal’s thermal lens. Vertical dashed red and green lines show thethermal lens associated with our Nd:YVO4 and Nd:GdV04 lasers respectively.(a)We see that the cavity is stable from a thermal focal length of 11.25 cm up to verylarge focal lengths.(b) The boundaries of optimal mode matching is shown by thedotted orange lines, see text for details. We see that the optimal mode matchingcan be found for thermal focal lengths ranging from 12.5 → 110 cm. Both ourNd:YVO4 and Nd:GdVO4 laser fit comfortably within these ranges.

and non-lasing. Thus, the cavity must be designed to function across a range of

induced thermal focal lengths.

The cavity may be described using ABCD matrix propagation by the resonator

82

matrix [42]

Mr =

Ar Br

Cr Dr

. (4.21)

This resonator matrix is constructed using matrix multiplication to calculate the

matrix describing the full round-trip propagation though the cavity. We may now

define a cavity stability parameter as

pstab = Ar +Dr

2 , (4.22)

such that cavity is said to be stable if

− 1 < pstab < 1. (4.23)

Furthermore we may use the resonator matrix to calculate the cavity eignmode

qr and consequently the radius of the beam within the cavity wr(z). The cavity

eigenmode is that which returns to its original value after a full round trip. That is

to say it is the solution to the equation:

Mr × qr = qr. (4.24)

The cavity eigenmode is the complex beam parameter describing the Hermite-

Gaussian beam in the cavity, specifically qr = z + izR,r. We may thus use it to

83

calculate the radius of the beam within the cavity as

wr(z) = w0

√√√√1 + z2

z2R,r

, (4.25)

where w0 =√λzR,r/π. In the above we have assumed a circular beam such that

qx = qy = qr.

While cavity stability is self explanatory, the importance of a good cavity

eigenmode is a bit more subtle. The overlap between the pump laser and the

cavity eigenmode within the crystal heavily influences the output power of the laser.

Moreover thermal effects due to the pump laser affect the optimal mode-to-pump

ratio, which has been found to be [43]

wlwpa≈ .8, (4.26)

where wl is the radius of the cavity eigenmode within the laser crystal, and wpa is

the average radius of the pump beam within the laser crystal as defined in equation

3.2. It is important that the cavity be designed to achieve this optimal overlap.

For our cavity design in which the total cavity length is 45cm and the only focal

element is the thermal lens fth it is easy to calculate the resonator matrix as

Mr =

1 0

− 1fth

1

×1 45cm

0 1

=

1 45cm

− 1fth

1− 45cmfth

. (4.27)

84

From here it is easy to calculate the cavity stability parameter and the eigenmode

radius at the crystal as a function of the thermal focal length as shown in figure

4.5. We see in figure 4.5 (a) that our cavity design is stable over a large range of

thermal lenses, from fth = 11.25cm on to very large values. What this tells us is we

do not have to worry about the thermal effects with regard to cavity stability as

long as they are not strong enough to produce fth = 11.25cm. However the effects

of the thermal lens on the mode radius produces more stringent limits.

For our laser system to be efficient and high power we need the mode-to-pump

ratio to be close to 80%. Our pump beam is imaged to a spot size of 467 µm

by lens L1 and L2 (see figures 3.2, 4.2), the location of which can be shifted by

moving L2. If we limit ourselves to the situation where the pump beam waist is

located somewhere within the laser crystal7 we may obtain an average pump beam

radius range of wpa = 515 to 706 µm. The doted orange lines in figure 4.5 (b) show

wl,min = .8× 515µm and wl,max = .8× 706µm, and enclose the region in which we

can obtain optimal mode-pump ratio by moving lens L2. Thus, with our cavity

design, optimal operation can be found between fth = 12.5 and 110 cm. This is

quite a large range and our Nd:YVO4 and Nd:GdVO4 laser both fall comfortably

within it as shown by the dashed red and green lines.

Also part of our design criterion is the requirement that the spatial mode be

pure TEM0,0, that is to say we want a pure Gaussian beam. In general, due to7We want the pump beam to be roughly collimated throughout the length of the crystal. The

least we can do is to have the pump beam waist reside within the laser crystal.

85

the Gouy phase shift, the laser cavity is resonant with only one spatial mode at a

time8, and consequently only one mode is resonantly enhanced and dominates the

cavity gain. For end-pumped lasers the pumped gain medium acts as a spatial filter

helping ensure the resonant mode is preferentially the TEM0,0 mode. However care

must still be taken in calculating the cavity resonance conditions to ensure no low

lying higher order TEMl,m modes are degenerate with the TEM0,0 mode. The gain

of a degenerate higher order mode would be resonantly enhanced disrupting the

good spatial mode of the laser resulting in an unwanted M2 > 1 and potentially

even disrupting single longitudinal mode operation.

Let’s calculate the resonance condition for our cavity, taking care to include the

Gouy phase shift. On resonance the laser must reproduce itself after a full round

trip in the cavity. In terms of the phase, after a full round trip the total phase

pickup φtotal must be multiple of 2π:

φ,l,m,total = 2πb (4.28)

where b is an integer, l,m represents the TEMl,m Hermite-Gaussian mode and,

φtotal = 2πvl,mc

Lopt + Φl,m. (4.29)

8This statement isn’t strictly true, since there are infinite higher order modes some very highorder modes are bound to be degenerate with the TEM0,0 in the cavity. However there is a bigdifference between a resonantly enhanced TEM1,1 compared to a TEM100,100 mode. The spatialextent of the very high order modes ensure that they are suppressed even if they are degenerate.It is really the low lying higher order modes that we have to worry about.

86

Here vl,m is the frequency of the TEMl,m mode, Lopt is the total optical path length

and Φl,m is the total Gouy phase shift defined as

Φl,m = (l + 12)Ψx + (m+ 1

2)Ψy (4.30)

where Ψx and Ψx are the Gouy phases associated with Horizontal(x) and Vertical(y)

directions respectively. To make the calculation simpler we assume the cavity

eigenmode is the same for both the x ad y directions9, giving Ψx = Ψy =Ψr. In

this approximation

Φl,m = (l +m+ 1)Ψr = (n+ 1)Ψr = Φn, (4.31)

where we have have defined n = l +m as the total Hermite-Gaussian order. The

total cavity round-trip Gouy phase may be calculated from the resonator matrix [42]

as

Ψr = Arg[Ar + Br

qr]. (4.32)

Combining equations 4.28, 4.29, 4.31 and 4.32 we may now write the resonance

condition in terms of the laser frequency as

vn = vFSR(b− n+ 12π Arg[Ar + Br

qr]), (4.33)

9This is the same as neglecting the ellipticity of the laser beam. From 3.5.2 we havew0,horiz/w0,vert = 0.94 showing that this is a good approximation in our laser system.

87

where vFSR = c/Lopt is the free spectral range of the cavity. The cavity is degenerate

with the nth order Hermite-Gaussian mode if the equation

vn − v0

vFSR= ∆vnvFSR

= (∆b− n

2πArg[Ar + Br

qr]) = 0 (4.34)

is satisfied or equivalently since ∆b is an integer the equation

Mod[ n2πΨr, 1] = 0. (4.35)

is satisfied.

Since the Gouy phase depends on the cavity parameters including the thermal

lens it is possible that over the range in which our cavity is stable there exist

some thermal focal length in which equation 4.34 is satisfied. This would show us

regions, which our cavity design would struggle to achieve a spatial mode quality

of M2 = 1. In Figure 4.6 we have plotted Mod[ n2πΨr, 1] as a function of fth for

our cavity design for the first 4 Hermite-Gaussian orders. The location on the

Nd:YVO4 and Nd:GdVO4 laser systems are shown with dashed red and green lines

respectively. Degeneracy with higher order modes occurs when the black curves

cross 0, this occurs for 3rd order modes at a focal length of 15 cm 10 and for 4th

order modes at a focal length of 22.5 cm. In conclusion, while degeneracy with

higher order modes excludes operation near fth = 15 and 22.5 cm, the range of10In many ways we narrowly dodged a bullet with our Nd:YVO4 laser, which has a thermal

focal length of 16 cm.

88

Figure 4.6: The black lines showMod[ n2πΨr, 1] or equivalentlyMod[ vn−v0vFSR

, 1] plottedas a function of fth for n = 1 to 4. Degeneracy occurs when the curves cross 0 asexplained in the text. We see the cavity is degenerate with 3rd and 4th order modesat fth = 15 and 22.5 cm respectively. This sets some limits to TEM00 operation ofour laser design. The location on the Nd:YVO4 and Nd:GdVO4 laser systems areshown with dashed red and green lines respectively. Both lasers operate away fromdegeneracy.

operation remains large for our cavity design. Importantly neither our Nd:YVO4

or Nd:GdVO4 implementations are degenerate with higher order modes.

89

Chapter 5

Experimental Conditions for

Obtaining Halo P -Wave Dimers

in Quasi-1D1

We calculate the binding energy and closed channel fraction of p-wave Feshbach

molecules in quasi-1D by examining the poles of the p-wave S-matrix. We show

that under the right experimental conditions, the quasi-1D p-wave molecule behaves

like a halo dimer with a closed channel fraction approaching zero at resonance and

a binding energy following the universal relation Eb ∼ 1/a21D, where a1D is the 1D

scattering length. We calculate these experimental conditions for both 6Li and 40K

over a range of transverse confinements. We expect that in this halo dimer regime1This paper has been submitted to Physical Review A as, Francisco R. Fonta and Kenneth M.

O’Hara, "Experimental Conditions for Obtaining Halo P -Wave Dimers in Quasi-1D"

90

the three body loss associated with the p-wave Feshbach resonance will be greatly

suppressed, potentially allowing for a stable p-wave superfluid to be created. For

an easy comparison between the 3D and quasi-1D cases, we provide the same poles

analysis of the Feshbach molecules applied to the 3D p-wave resonance and show

there is a qualitative difference between the two.

5.1 Introduction

Ultracold dilute Fermi gases near p-wave Feshbach resonances are of great interest

due to the rich phases of matter associated with p-wave pairing. P -wave pairing

is characterized by a more elaborate order parameter than that of s-wave pairing

due to the different projections of the non-zero (l = 1) angular momentum. The

distinct symmetries of the different angular momentum projections allow for sharp

phase transitions between qualitatively different ground states as one tunes across

the p-wave Feshbach resonance from the BEC side to the BCS side [8, 91–94].

The s-wave BEC-BCS crossover in contrast features a smooth transition with no

qualitative differences on either side of the resonance.

Experimentally, a dilute Fermi gas in three-dimensions (3D) with controlled,

resonant, p-wave interactions can in principle be realized with an optically trapped

gas of fermionic alkali atoms (e.g. 6Li or 40K) magnetically tuned near a p-wave

Feshbach resonance. Such a gas would, at zero temperature, feature the sharp phase

transitions mentioned above. Furthermore, by changing the trap configuration, p-

91

wave pairing may be explored in reduced dimensions. In two dimensions, a px + ipy

topological superfluid is expected [8, 95–97]; and in one dimension a fermionic

Tonks Girardeau gas is predicted [98–103]. Furthermore, p-wave interactions on

a one-dimensional lattice should reproduce the classic Kitaev chain model, which

predicts the long sought after Majorana fermions [104]. However, despite significant

experimental advancements in characterizing and manipulating p-wave Feshbach

resonances [4, 7, 105–107], little progress has been made in stabilizing any of these

novel phases. This is because unlike s-wave Feshbach resonances, p-wave Feshbach

resonances are accompanied by significant three-body and two-body loss [6,108–110].

While two-body loss can be mitigated in cases where the p-wave resonance occurs

for atoms in their lowest hyperfine state, three-body loss is unavoidable. Three-body

loss occurs when three particles are involved in a collision and subsequently two of

the particles form a deeply bound dimer molecule, while the third particle allows

for the conservation of energy and momentum in the exothermic reaction [111].

This loss mechanism can be enhanced by a Feshbach resonance through a process

where a Feshbach molecule resonantly formed in the continuum collides with a

third particle and subsequently decays to a more deeply bound molecular state. It

is therefore important to understand the nature of Feshbach resonances and their

Vbg (a30) ∆B (G) re (a−1

0 ) Bres (G) δµc (µK/G)6Li -70 x 103 -40 -0.182 159.1 14240K -10.49 x 105 -21.95 -0.0416 198.9 11.7

Table 5.1: p-wave scattering properties for 6Li and 40K [4–10].

92

underlying Feshbach molecules [32,112,113].

Resonant scattering occurs when a molecular bound state of the interacting

particles is brought close to the continuum of free particle states (e.g. by application

of a magnetic field). This connection between bound states and resonant interactions

is seen most readily in the scattering S-matrix where the molecular bound states

exist as poles of the S-matrix [48]. For s-wave collisions near threshold and tuned

close to resonance, the bound state associated with the Feshbach resonance is

a halo dimer with a binding energy Eb = ~2/(ma2), which only depends on the

scattering length a. This is the so called universal regime where regardless of the

specific atomic species the only length scale governing the molecular state is the

scattering length. The spatial wave function of this molecule is proportional to

ψl(r) ∼ e−r/a indicating that the molecular state becomes extremely delocalized as

the scattering length diverges. This extremely delocalized molecule has virtually

no wave function overlap with more deeply bound molecular states thus effectively

suppressing three body loss. Conversely, due to the centrifugal barrier, p-wave

resonances in 3D feature no such universal regime [107]. The underlying p-wave

Feshbach molecular state is well localized with significant overlap with the more

deeply bound molecular states resulting in a very high likelihood of three body loss.

In the coupled channel picture, the Feshbach molecule is a dressed state

|ψmol〉 = Z|ψclosed〉+ (1− Z)|ψopen〉, (5.1)

93

which is a superposition of a free particle scattering state ψopen and a bare molecular

state ψclosed. It is largely the closed channel fraction |Z|2 that leads to the three-

body loss as the closed channel wave function has significant overlap with more

deeply bound molecular states. In the two-body limit in 3D, the closed channel

fraction, |Z|2, tends towards zero as one approaches an s-wave Feshbach resonance.

In contrast, as one approaches a p-wave Feshbach resonance, the closed channel

fraction, |Z|2, remains significant and stays approximately constant. In the actual

BEC-BCS crossover of an ultracold Fermi gas, many-body effects modify the

Feshbach resonance [114–117]; in particular even for s-wave resonances the closed

channel fraction has been shown to be non-zero, albeit very small and density

dependent, at unitarity on through to the BCS side [118,119].

Recently, several studies have investigated ways to suppress three body loss

associated with p-wave resonances by considering scattering in lower dimensions [38,

120–122]. L. Zhou and X. Cui, for example, have shown that in quasi-1D the p-wave

molecular wave function is significantly more delocalized than in 3D suggesting that

quasi-1D is a promising method for suppressing three-body loss [38]. Motivated by

this work, our group as well as others has begun to study p-wave Fermi gasses in

quasi-1D. In a previous work we have measured the three-body loss in lithium in

quasi-1D [123]; while a significant suppression has been observed, it is not clear

that it is sufficient to stabilize the gas for adequate time to reach equilibrium.

Here we expand on the results from L. Zhou and X. Cui. By examining the

94

poles of the S-matrix we show that in quasi-1D a p-wave halo dimer exists. in

the two-body limit We see that p-wave resonances in quasi-1D behave similarly to

narrow s-wave resonances. We go on to characterize the 1D resonance for 40K and

6Li to determine the temperature, field stability and transverse confinement needed

to reach the halo dimer regime. Further we determine the closed channel fraction

of the Feshbach-dressed molecule in quasi-1D and compare it to that in 3D. All

of our calculations are in the two-body limit; while it is known that many-body

effects will change the details of the resonance for an ultracold Fermi gas, it is our

hope that the two body physics presented here captures enough of the picture to

serve as an effective guide for future experiments.

For the calculations in this paper related to 6Li we consider the p-wave resonance

between atoms in the |F = 12 ,mf = +1

2

⟩state. For calculations related to 40K we

consider the p-wave resonance between atoms in the |F = 92 ,mf = +7

2

⟩state with

orbital angular momentum projected onto ml = 0. The resonance parameters we

use are reported in Table 5.1.

5.2 Physical Significance of Poles

Here we present a brief discussion relating the poles of the S-matrix to bound states

of the molecular potential [48]. Consider a partial wave scattering state ψl(k, r)

that is the solution to a radial interacting potential. The asymptotic behavior of

95

ψl(k, r) is

ψl(k, r)→i

2r [e−i(kr−lπ2 ) + Sl(k)ei(kr−lπ2 )]. (5.2)

For any such state, there exists a corresponding regular solution given by ϕl(k, r) =

Fl(k)ψl(k, r) behaving as

ϕl(k, r)→i

2r [Fl(k)e−i(kr−lπ2 ) + Fl(−k)ei(kr−lπ2 )], (5.3)

where Fl(k)is the Jost function and is related to the S-matrix by Sl(k) = Fl(−k)/Fl(k).

It is clear that zeros of the Jost function are poles of the S-matrix for which we

consider solutions extended into the complex momentum plane. Consider a pole of

the S-matrix (corresponding to a zero of Fl(k)) where the pole is purely positive

imaginary, k = i~

√m|Eb|. Then the regular solution, ϕl(k, r) ∼ e−|k|r, is a true

bound state solution of the Schrödinger equation with energy Eb. Conversely, a

purely negative imaginary pole, k = −i~

√m|E|, results in ϕl(k, r) ∼ e|k|r; a solution

that cannot be normalized which we call a virtual state. This state, while unphysi-

cal, still affects the underlying scattering process. Finally, a complex pole results

in a state with complex energy, Epole = Er − iΓ/2. This corresponds to what we

call a resonance, a quasi-stable state with energy Er and lifetime τ = 1Γ [8, 9]. It is

this quasi-stable state embedded in the continuum which is thought to resonantly

enhance three body loss.

The poles of the scattering matrix thus give us direct access to the energy of

96

the molecular bound state and the resonant state that is involved in three body

loss. Furthermore, we may use the dressed state energy, Eb, to calculate the closed

channel amplitude as Z = ∂(−Eb)/∂(Ec) where Ec is the energy of the closed

channel molecular state [10,32,113].

5.3 Poles Analysis of P -Wave Resonances in 3D

To elucidate the problems leading to three body loss near p-wave resonances

we begin by examining the p-wave resonance in three dimensions. In all of the

following we make the usual assumptions that the interatomic forces are short range

and isotropic. We also assume that we are within the neighborhood of a p-wave

resonance. The scattering process is then well described by an l = 1 partial wave

S-matrix [8]

S =− 1w

+ 12rek

2 + ik3

− 1w

+ 12rek

2 − ik3 , (5.4)

where k is the relative momentum of the two atoms, w is the scattering volume and

re is the effective range, which for 3D p-wave resonances has units of inverse length..

It is important to note that for the magnetically tuned Feshbach resonances we are

interested in, the scattering volume w is a function of magnetic field

w(B) = wbg

(1− ∆B

B −B0

), (5.5)

97

Figure 5.1: The colored lines show the poles of the S-matrix moving on the complexmomentum plane in (a) 3D and (b) quasi-1D. The arrows show the direction thepoles move as the magnetic field is tuned from the BEC side to the BCS side ofthe resonance. The stars show the locations of the poles in the complex k plane atB = Bres. (a) In 3D the pole corresponding to a bound state(blue) moves downthe positive imaginary axis becoming a resonance as soon as it crosses threshold.(b) In quasi-1D the bound state pole moves down positive imaginary axis and thencontinues along the negative imaginary axis as a “virtual state” until kpole = − i

r1Dand only then does it become a resonance.

where wbg is the background scattering volume, B0 is the bare resonance position

and, ∆B is the resonance width.

The roots of − 1w

+rek2/2− ik3 with respect to k give the 3 poles of the S-matrix.

98

As we vary the scattering volume (by varying the magnetic field) the poles move on

the complex plane, see Fig. 5.1(a). Importantly, one of these poles moves along the

positive imaginary axis to cross threshold and become a resonance. It is this pole

corresponding to a true molecular bound state which then becomes a meta-stable

state in the continuum that would potentially decay into a deeper molecular state

upon collision with a third atom. The wavenumber of this state is

kpole = −(ire6 + (i+√

3)r2e

12(−r3e − 108α + 6

√6√α(r3

e + 54α)) 13

+ i−√

312 (6

√6√α(r3

e + 54α)− r3e − 108α) 1

3 ), (5.6)

where α is 1w. As we approach resonance, the leading term in an expansion with

respect to α gives kpole →√

2wre

. Thus, as we approach resonance the bound

state energy scales as Epole = ~2k2pole/m → 2~2/(mwre). There are two striking

differences from the classic unitarity limited bound state energy in s-wave resonances.

First, the effective range is included in the energy implying that the behavior is

not universal across atomic species. Second, the binding energy scales as 1/w in

contrast with the 1/a2 scaling of an s-wave Feshbach resonance in 3D. Because of

this the p-wave binding energy approaches threshold as (B −Bres) instead of the

typical s-wave behavior of (B −Bres)2.

To calculate the closed channel amplitude, we note that Ec = δµc(B −B0) is

99

the closed channel energy. We may then rewrite the scattering volume as

w = wbg

(1− δµc∆B

Ec

). (5.7)

Thus, we may rewrite kpole(w, re)→ kpole(Ec, re) and consequently we may rewrite

the binding energy Eb in terms of the closed channel energy Ec. Simple differentia-

tion, Z = ∂(−Eb)/∂(Ec), yields the closed channel amplitude.

Figure 5.2 shows the closed channel amplitude calculated for both 6Li and 40K

close to their respective 3D p-wave resonances. Both closed channel fractions remain

approximately constant as they approach resonance. We calculate Z = 0.8 and

Z = 0.76 for 6Li and 40K respectively which are consistent with the measurements

by J. Fuchs et al. [9] for 6Li and J. Gaebler et al. [10, 124] for 40K . The closed

channel fraction is thus large over the entire resonance for both atomic species.

This is in stark contrast to s-wave resonances where even for narrow resonances

the closed channel fraction approaches zero as we approach resonance.

Figure 5.3(a) and (b) shows the 3D p-wave scattering cross section as well as

the Feshbach bound state energy, Eb, for 6Li and 40K respectively. Note that both

resonances are extremely narrow and that the bound state (solid line) tunes directly

through the continuum to form a resonant state (dash-dotted line). The collision

energy associated with maximal scattering cross section directly follows the energy

of the resonant state.

100

Figure 5.2: Closed channel amplitude in 3D for (a) 6Li and (b) 40K. The insetshows the closed channel amplitude close to resonance.

5.4 1D Analysis

Now we extend the poles analysis to one dimension. In quasi-1D, L. Zhou and X.

Cui found that you may write a new effective 1D S-matrix [38]

S1D =− 1a1D

+ 12r1Dk

2 + ik

− 1a1D

+ 12r1Dk2 − ik

, (5.8)

101

Figure 5.3: 3D p-wave scattering cross section and bound state energy for (a) 6Liand (b) 40K. The bound state energy (solid line) tunes linearly as a function ofthe magnetic field, directly becoming a resonance (dash-dotted line) at B = Bres,above which point the energy of this quasi-stable molecular state tunes linearlythrough the continuum.

where a1D is the effective 1D scattering length and r1D is the 1D effective range.

The parameters are related to the 3D scattering volume and 3D effective range by

1a1D

= a2⊥3

( 1w− 1

2rea⊥2

)− 1a2⊥ζ(−1

2 , 1), (5.9)

102

r1D = a2⊥re3 − a⊥√

2ζ(1

2 , 1). (5.10)

Here a⊥ is the transverse confinement length given by a⊥ =√

~m 2πf⊥

. For the

remainder of the paper we will quantify the confinement by the transverse trapping

frequency f⊥.

Figure 5.1 shows how the poles of the 1D S-matrix move on the complex

momentum plane. In contrast to the 3D case, there are only 2 poles. More

importantly the pole corresponding to the true bound state crosses threshold and

then remains on the pure imaginary axis briefly before picking up a real part.

In physical language the bound state becomes a virtual state and then becomes

a resonance. It is this threshold behavior (bound state → virtual state) that

encapsulates the universal regime. Explicitly solving for the bound state pole yields

kpole1D =i−

√−1 + 2r1D

a1D

r1D. (5.11)

Taking the limit as we approach resonance ( 1a1D→ 0), kpole1D → 1

a1Dreproducing the

universal limit Epole → ~2/(ma21D). This suggests that as we approach resonance

the underlying molecular state is a halo dimer. To quantify this molecular state

more fully, we will calculate the closed channel fraction.

We calculate the closed channel amplitude the same way as with the 3D case,

that is to say owing to the poles we now have an expression for the 1D bound

state energy Eb,1D in terms of the closed channel energy. Figure 5.4 (a) ( (b)

103

Figure 5.4: Closed channel amplitude in quasi-1D for (a) 6Li and (b) 40K for avariety of transverse confinements. The resonance becomes significantly more openchannel dominated as the confinement increases.

) shows the closed channel amplitude calculated for 6Li (40K) for a variety of

transverse confinements. (Note that the confinement shifts the resonance position;

we have shifted the origins to fit all the curves onto one plot.) The 1D p-wave

closed channel amplitude resembles the closed channel amplitude of narrow s-wave

resonances. Furthermore, as the confinement is increased the resonances become

104

more and more open channel dominated.While for our two-body calculations the

closed channel amplitude goes to zero on resonance we expect many-body effects to

keep Z non-zero throughout the resonance as they do in the s-wave case [118,119].

However, that Z goes to zero in the two-body limit should imply that the closed

channel fraction becomes extremely small in the full many-body limit. It should

be noted here that many-body effects also limit the universality of narrow s-wave

resonances; when many-body effects are taken into account it has been shown

that only resonances which are broad compared to the Fermi energy are truly

universal [114–118]. This may be an advantage for 1D p-wave resonances over their

narrow s-wave counterparts as the 1D p-wave resonances can be broadened by

increasing the transverse confinement; however, a full many-body treatment of the

problem is beyond the scope of this paper.

Next we consider the scattering cross section itself. Figure 5.5(a) ( (b) )

shows the scattering cross section as well as the real part of the bound state pole,

Re[Epole], for 6Li (40K). For an experimentally realizable trap geometry which

can provide extremely tight confinement in two-dimensions, we consider a square

two-dimensional standing-wave lattice made from retro-reflected 532 nm light with

a depth of 200 ER (where ER is the recoil energy for a 532 nm photon). This would

correspond to a transverse confinement frequency of 3 MHz for 6Li and 500 kHz

for 40K. The solid line in energy represents when the pole is a true bound state,

the dashed line when it is a virtual state, and the dash dotted line when the pole is

105

Figure 5.5: Scattering cross section and bound state energy in quasi-1D for (a) 6Liand (b) 40K. We assumed a transverse confinement of 3 MHz (500 kHz) for 6Li(40K). The energy of the bound state (solid line) merges with the continuum at Bresand then continues on as a virtual state (dotted line) before eventually becoming aresonance (dash-dotted line).

a resonance. Similar to narrow s-wave resonances, we see that the energy of the

two-body state varies quadratically with magnetic field when the state is a true

bound state and a virtual state near resonance. Once the two-body state becomes

a quasi-bound state it starts to vary linearly with field.

106

We want to estimate the experimental conditions necessary to access the uni-

versal regime, kpole ∼ 1a1D

. It is clear from Fig. 5.5 that once the pole becomes a

resonance there is a sharp change in behavior after which the energy of the pole scales

linearly with the magnetic field. This transition occurs at a1D[f⊥, B] = 2r1D[f⊥]

and thus the unwanted resonance regime is avoided for 1a1D

< 12r1D

. Next we want

to ensure kpole is well approximated by a first order expansion. Expanding kpole to

second order

kpole → i( 1a1D

+ r1D

21a2

1D). (5.12)

we obtain the condition 1a1D

<< 2r1D

. We will take the requirement that no

resonant states are formed to be sufficient as it is a factor of 4 more stringent

than 1a1D

< 2r1D

; we believe under these conditions the first order approximation

is adequately satisfied, however there may still be some small deviation from the

halo dimer form. To maintain this condition, very strict control of the magnetic

field stability is necessary. For our suggested trap configuration, a field stability of

δB < 3.6mG for 6Li, and δB < 187mG for 40K is required. Furthermore, looking

at the collisional energy where the scattering cross section is resonant, we may

estimate the temperatures needed to reach this 1D p-wave halo-dimer regime. For

the trap configuration in Fig. 5.5 this corresponds to a temperature of T < 0.21µK

for 6Li and T < 1µK for 40K. Note that the conditions for potassium are less

stringent than that for lithium and thus it may be easier to suppress three body

loss in potassium. However, unlike lithium, potassium not only suffers from three

107

Figure 5.6: (a,b) Field stability and (c,d) temperature required for achieving halodimers as transverse confinement is increased. (a,c) Show the conditions necessaryfor 6Li while (b,d) show the conditions necessary for 40K.

body loss but also suffers from two body loss due to dipolar relaxation [108].

The confinement clearly plays a crucial role in achieving p-wave halo dimers.

To identify requirements for future experiments aimed at realizing long-lived halo

p-wave molecules, we have plotted in Fig. 5.6 the magnetic field stability and

temperature necessary to access the universal regime for 6Li and 40K as a function

of transverse confinement. We use the conditions a1D[f⊥, Bhalo] < 2r1D[f⊥] and

Thalo < ~2/(kBmr1D[f⊥])2) to estimate the magnetic field stability and temperatures

for which we expect halo dimers. Note that both δBhalo and Thalo increase very

rapidly as the transverse confinement increases making increasing the confinement

108

a promising avenue for attaining the halo dimer region.

Thus far we have discussed the conditions for accessing the universal regime

(achieving halo dimers) in quasi-1D p-wave Fermi gases. However, three body loss

may be suppressed even beyond the halo dimer regime. The energy of the pole

in quasi-1D as a function of magnetic field (see Fig. 5.5) shows that there is a

sizeable region in which there is no resonant state embedded in the continuum even

though the scattering p-wave cross-section is still unitarity limited. This region is

roughly twice as large as the halo dimer regime and consists of where the pole is a

true bound state, a virtual sate, and a resonant state below threshold. Without

a quasi-stable bound state embedded in the continuum, three body loss would

have to occur between three separate atoms rather than between one atom and

one quasi-stable molecule. Thus, we expect three body loss to be significantly

suppressed within this entire region.

109

Chapter 6

Suppression of Three Body Loss

Near P -Wave Resonances in

Quasi-1D

As discussed in section 5.1, p-wave Feshbach resonances in ultracold atomic gasses

have the potential to be used to study exotic p-wave superfluids. However, the

significant three body loss near p-wave Feshbach resonances presents a large barrier

to realizing a p-wave superfluid. There is hope, as discussed in Chapter 5, that

three body loss may be strongly suppressed in quasi-1D.

In this chapter we analyze the three body loss rate constant measured in our

lab. Our experiments were conducted far from the experimental conditions put

forward in 5 necessary to reach the halo dimer regime. However, while three body

110

loss remains resonantly enhanced, we still observe a factor of 20 suppression of the

three body loss rate constant in quasi-1D as compared to 3D as a result of the

different three body recombination scaling laws.

6.1 The three body loss experiment

A detailed description of the experimental setup and procedure for measuring three

body loss near the |1〉 - |1〉 p-wave Feshbach resonance is given in Andrew Marcum’s

thesis [39]. In this section I will only give a brief summary of the experiment.

We start by capturing and cooling the atoms in a D2-MOT. Because the

hyperfine levels of lithium are unresolved the D2-MOT cooling is limited by the

Doppler limit; after the MOT stage we have approximately 2.3× 109 atoms at a

temperature of 350 µK and a phase space density of ρ = 0.29× 10−5. In the next

stage the atoms are further cooled by a D1-Line gray optical molasses [125, 126]

after which we can achieve up to 1.2× 109 atoms at a temperature of 45 µK and

a phase space density of ρ = 2.9 × 10−5. We load an equal mixture of states |1〉

and |2〉 from this gray optical molasses into a crossed optical dipole trap formed

by one 1060m beam and another 1070nm beam intersecting at an angle of 12

with a waist radius of 30 µm and total output power of up to 80W. The atoms are

then evaporatively cooled by exponentially decreasing the power in the beams to

1W/beam. Before evaporative cooling the magnetic field is brought to 320.4 G to

achieve a local maximum in the |1〉 − |2〉 s-wave scattering length; this ensures that

111

!"#

λ$%

!"#

λ$%

!"

!#

Figure 6.1: Schematic showing the setup of the three body loss experiment

the gas can rethermalize during the evaporation process. From this starting point

the procedure changes depending on if we are measuring the three body loss in 3D

or quasi-1D.

To measure the the three body loss in quasi 1D we load the atoms into a 2D

optical lattice as shown in Figure 6.1 [123]. The optical lattice is formed by two

orthogonal retro-reflected 1064 nm beams focused to horizontal waist of 33µm and

vertical waist of 300µm. The resulting interference pattern forms a 2D optical

lattice; Since the interference pattern is only in two dimensions the result is a lattice

of elongated 1D tubes, which serve as the quasi-1D traps. This setup simultaneously

creates an array of quasi-1D traps. At full power in the lattice beams, the transverse

and longitudinal trapping frequencies were measured to be ω⊥ = 2π × 281 kHz and

ω‖ = 2π×200 Hz. The transverse trapping frequency corresponds to a lattice depth

112

of 23 Er, where Er is the recoils energy of 6Li in the lattice. The time tunneling

time between tubes (lattice sites) is τ = 4.9ms.

To ensure that we are in the quasi-1D limit we need to ensure that only the

ground band of the lattice is populated. We verify this by adiabatically ramping

down the lattice depth and then imaging the cloud after a time of flight. The

adiabatic ramp preserves the band structure; thus, if the time of flight only shows

the first Brillouin zone we know the atoms are fully in the lowest Bloch band. This

technique is known as band mapping the results of which are shown in Figure 6.1.

To load the atoms into the optical lattice we use the technique presented in [127].

As shown in Figure 6.1 each retro-reflected arm of the lattice beams contain a

liquid crystal retarder (LCR) and quarter-wave plate (λ/4); together these allow

us to dynamically adjust the polarization of the retro-reflected beams. When

the polarizations are orthogonal, no interference occurs and therefore no lattice

is formed (this is the 3D configuration); when the polarizations are the same,

interference occurs and the 2D lattice is formed (this is the lattice configuration).

We first load the atoms from the cross optical dipole trap into the "lattice"1 beams

in the 3D configuration. Then we load the lattice by adiabaticly ramping the

polarizations of the retro-reflected beams to the lattice configuration.

The atoms start in an equal mixture of states |1〉 and |2〉. To prepare the atoms

for the three body loss measurement, we clear the entire state |1〉 population while1From here on I will be putting the "lattice" beams in quotes as they are used to form a lattice

trap as well as a 3D trap.

113

Figure 6.2: Sample decay curves in 3D and quasi-1D both on resonance and farfrom resonance.

in the 3D configuration of the lattice beams by ramping the magnetic field to the

location of the |1〉-|1〉 3D p-wave resonance. After the state |1〉 population decays

via three body loss we are left with a spin polarized gas of atoms in state |2〉. The

atoms are then loaded into the lattice configuration and are ready for the loss

measurement.

To perform the loss measurement we use a double radio-frequency(RF) pulse

technique. First we ramp the magnetic field to the field of interest, near the |1〉−|1〉

114

resonance, then we use a RF pulse to transfer state |2〉 to state |1〉, after a variable

wait time we transfer the atoms back to state |2〉 with a second RF pulse. Finally,

phase contrast imaging is used to extract the number of atoms remaining. the

double RF pulse technique is required to avoid decay during the field ramps. This

technique is used to measure the decay curves in both 3D and in quasi-1D. Sample

decay curves are shown in Figure 6.2.

To perform the three body loss measurement in 3D the atoms are still loaded

from the crossed optical dipole trap into the "lattice beams", only now with the

retro-reflection completely blocked. In this blocked configuration of the lattice

beams the same procedure is performed as for the quasi-1D measurements. Namely

a spin polarized gas of state |2〉 atoms is made by clearing state |1〉 via three body

loss, and the decay curves are measured with the double RF pulse technique.

To extract the three body loss constant L3 from the decay curves, the curves

must fit the solution to the differential equation

N

N= L3〈n2〉, (6.1)

where N is the atom number and, 〈n2〉 is the mean squared density. In 3D the

above differential equation is solved by

N(t) = (2γt+ 1/N20 )−1/2 (6.2)

115

where N0 is the initial atom number and

γ = L3(T,B)( m

2√

3πkBT)3ω2/3 (6.3)

where ω is the mean trap frequency. Because our quasi-1D measurements come from

an array of tubes each of which independently follows equation 6.1, the quasi-1D

decay curves follow the form

N(t) =100∑

i,j=−100(2γ1Dt+ 1/NT

0 (i, j)2)−1/2 (6.4)

where the sum is over all of the individual tubes, NT0 (i, j) is the initial atom number

in a given tube, which varies according to the 3D envelope of the cloud, and

γ1D = L3(T,B)( m3

2√

27π3~2kBT)3ω2⊥ω

2‖. (6.5)

The extracted three body loss constants L3(T,B) are plotted as a function of mag-

netic field, temperature and, lattice depth in Figures 6.4, 6.5, and 6.7 respectively.

Lastly, a note on the location of the p-wave Feshbach resonances. In quasi-1D

the location of the resonance is shifted such that the 1D scattering length diverges

when

vp = − a3⊥

a⊥kp + 3√

2|ζ(−1/2)|(6.6)

where vp is the 3D scattering volume, a⊥ is the confinement length, kp is the

116

Figure 6.3: Loss feature is 3D and quasi-1D showing the confinement inducedresonance shift.

3D effective range, and ζ(−1/2) is the Zeta function. This shift is known as the

confinement induced resonance. Figure 6.3 shows the loss feature in 3D an quasi-1D

as well as the predicted confinement induced resonance shift. In Figure 6.4 we have

shifted all the curves so that the resonance positions all fall at the same point. This

is done to make comparisons between the 3D and quasi-1D case easier.

117

6.2 The three body loss analysis

6.2.1 Three body recombination scaling laws

Here we will be comparing the 1D and 3D three body recombination rate scaling

laws.

In 2007, Mehta, Esry, and Green found the threshold scaling laws for three body

recombination is one dimension [128]. Using an adiabatic hyperspherical approach

they found that in 1D, at threshold, the three body recombination rate constant

must scale as

K3,1D ∝ (ka)2κmin (6.7)

where k is the relative momentum of the scattering states, a is the scattering length

and κmin is a constant associated with the lowest continuum channel adiabatic

potential. For p-wave fermions far from resonance they found that κmin = 3,

yielding the far from resonance scaling law

limk→0

K3,1D ∝ (ka)6 ∝ T 3, (6.8)

where T is the temperature of the interacting gas. For p-wave fermions on resonance

they found that κmin = o, yielding the on resonance scaling law

lima→∞

K3,1D ∝ (ka)0 = const. (6.9)

118

Thus, the unitarity limited recombination rate is a constant not dependent on

either magnetic field or temperature.

Now let’s look at the 3D scaling laws that were also found by Esry, and Green

as well as Suno in 2003 [111,129]. Far from resonance in 3D they found that

limk→0

K3,3D ∝ (k3Vp)8/3 ∝ T 4, (6.10)

where Vp is the scattering volume. The unitarity limited three body recombination

rate in 3D has further been shown to be

Kmax3,3D = λ

36√

3π2~5

m3(kB)2 ∝ T−2. (6.11)

where λ is some atomic species dependent constant.

From equations 6.8, 6.8, 6.10,6.11 it is easy to compare the 1D and 3D recombi-

nation rate constants. On resonance we have

Kmax3,1D

Kmax3,3D∝ T 2 (6.12)

and off resonance we haveK3,1D

K3,3D∝ T. (6.13)

Thus, at low temperatures the three body recombination rate constant should be

significantly suppressed in 1D compared to its 3D counterpart. This suppression

119

should be especially strong on resonance where the ratio between the two scales as

T 2.

6.2.2 Intermediate regime theory

The scaling laws tell us about the far from resonance behavior and the on resonance

behavior of the three body recombination rate constant, but what about all the

fields in between? In 3D some intermediate field theories have already been

developed [3, 109]. For example Waseem et al. used rate equations to find

L3 ≈ 9KAD(6π/k2T )3/2e−k

2r/k

2T , (6.14)

where L3 is the thermally averaged three body recombination rate constant, KAD

is the atom dimer relaxation coefficient, kT = (3mkBT/2~2)1/2 is the thermal

momentum and kres = (|vp|kp)−1/2 is a momentum scale defined by the scattering

parameters. Below we develop the first intermediate theory for the recombination

rate constant in quasi-1D.

We base our intermediate theory on Breit-Wigner Scattering theory [48,109,130–

134], in which we assume that the dominant loss mechanism stems from a resonantly

formed quasi-bound molecule decaying to a deeper molecular state upon collision

with a third atom. Under these circumstances the atom loss can be expressed as

n = −36

2~km

σinp1dn2 = −K3n

3, (6.15)

120

where σinp1dis the inelastic scattering cross section, and the factor of 3/6 is added

since every inelastic collision event results in 3 lost atoms and there are N3/6

triplets per unit volume. the inelastic cross section takes the Breit-Wigner form

σinp1d= 3πk2

Γe1DΓ0

(E − Eres)2 + (Γe1D+Γ0)2

4

, (6.16)

where Eres is the binding energy of the quasi-bound molecule, Γe1D is the resonant

energy width of the quasi-bound molecule, and Γ0/~ is the inelastic atom-dimer

relaxation rate. The inelastic atom-dimer energy width may be rephrased in terms

of the density n as Γ0 = ~KADn where KAD is the atom-dimer relaxation coefficient.

Comparing equation 6.16 and 6.15 we see that the three body recombination rate

constant can be expressed as

K3 = 3 π~mk

Γe1DKAD

(E − Eres)2 + (Γe1D )2

4

. (6.17)

Here we have assumed Γe >> Γ0, such that Γ0 may be dropped from the denomi-

nator of equation 6.16.

Thus far our analysis has been completely general. To apply it to quasi-1D

p-wave resonances we note that the important quantities such as Eres and Γe are

derived from the scattering S-matrix. As show in chapter 5 the k dependence of the

quasi-1D p-wave S-matrix (see equation 5.8 ) is the same as the 3D s-wave S-matrix.

Therefore, instead of of the usual p-wave resonant energy width Γe3D ∝ E3/2, we

121

have Γe1D =√

4~2E/(mr21D) and instead of the usual p-wave binding energy we

have Eres = 2~2/(ma1Dr1D) as in the s-wave case.

To actually obtain our measure three body recombination rate we must take the

thermal average. Noting that in quasi-1d the gas is only thermal in one dimension

we take the thermal average in only one dimension yielding:

L3 = 1√πkBT

∫ ∞0

K3√Ee− EkBT dE. (6.18)

We can obtain an analytic solution to the integral by preforming integration by

parts to separate out the contributions due to the diverging density of states in

one dimension and the contributions due to the quasi bound molecule embedded in

the continuum. To do so we assume that Γe1D << kBT2 yielding:

L3 = 3π~3

m3/2KAD

√

4~2

mr21D

E2res

+ 2πe−EreskBT

Eres√πkBT

. (6.20)

While equation 6.18 can just as well be evaluated numerically, the great advantage

of equation 6.20 is we can explicitly identify two separate components resulting

in three body loss. The first term in the brackets is derived from the low energy

density of sates and dominates far from resonance, while the second term in the2The approximation Γe1D

<< kBT holds for all of our measured temperatures. Taking itallows us to use the identity limε→0 ε/(x2 + ε2) = πδ(x) to make the approximation

Γe1D

(E − Eres)2 + Γ2e1D

4

≈ 2πδ(E − Eres). (6.19)

122

Figure 6.4: Magnetic field dependence of L3 in 3D and quasi-1D. For technicalreasons, the two 1D data sets were taken at different lattice depths, resulting indiffering values of the confinement induced shift of the resonance. All the data setsare thus shifted so that the resonance locations overlap at 0 δB marked with thesolid gray vertical line. The colored vertical lines show the field below which L3 isexpected to be unitarity limited. The solid red curve is the intermediate theory ofRef. [3] fit to our 3D data. The solid blue and green curves are equation 8 fit toour quasi-1D data sets. The dashed curves show the far from resonance 1D scalinglaws. Data points are averages of 3 to 5 individual measurements; error bars arethe standard error of the mean.

brackets is due to the quasi-bound state embedded in the continuum and dominates

closer to resonance.

6.2.3 Comparing theory to experiment

Figure 6.4 [123] shows our magnetic field dependence of the thermally averaged

three body recombination constant L3. The suppression of three body loss is clearly

123

seen at our operational temperatures of approximately 2 µK. At these temperatures

and at our trap depths we see a 29 fold suppression of three body loss.

The colored vertical lines show the beginning of the unitarity limited regime

marked by the maximum of the three body loss. The intermediate field theory

should only fit outside of this unitarity limited regime.

The red line is a fit to equation 6.14 with KAD as the only free parameter. From

this fit we obtain KAD,3D = 6.5(1.0)× 10−17m3/s. Using the same theory, Waseem

et al. measured a value of KAD,3D = 1.3(5) × 10−15m3/s ; we attribute the large

difference in these measured values to the differences in temperature. Indeed KAD

has already been shown to be temperature dependent in s-wave resonances [135]

and we expect the same to be true for p-wave resonances.

The blue and green lines show the fits to the quasi-1D intermediate theory (see

equation 6.20) with KAD as the only free parameter. We see that our intermediate

theory fits the data quite well. From our 1D measurements we obtain KAD,1D =

1.4(3) × 10−16m3/s at 2 µK and KAD,1D = 3.6(1.6) × 10−17m3/s at 0.78 µ K

respectively.

Finally the dashed blue and green lines show the far from resonance scaling

laws, equation 6.8. We see fairly good agreement with the scaling law in the 2 µ K

data, but don’t see good agreement in the .78 µK data. Furthermore, in both cases

the intermediate theory fits the data better than the scaling law. We believe we

are simply not far enough from resonance to confirm the scaling law. Moreover,

124

Figure 6.5: Log-Log plot of L3 versus temperature. Solid red curve shows 3Dunitary limit. Solid orange line shows the on resonance L3 ∝ constant scalinglaw. Dashed blue curve show the L3 ∝ T 3 scaling law. Solid blue curve is a fitto equation 8 assuming KAD ∝ T 3. Data points are averages of 3 to 5 individualmeasurements; error bars are the standard error of the mean.

measuring L3 further from resonance is not possible in our current experiment

since we observe heating in the lattice at the longer decay times needed to take the

measurement further from resonance.

Figure 6.5 shows the temperature dependence of the three body recombination

rate constant. We see that L3 in quasi-1D and on resonance is independent of

temperature, confirming the on resonance scaling law. The stark contrast between

the on resonance scaling law in quasi-1D (orange line) and 3D (red line) is evident

and shows the origin of the relative suppression of three body loss in quasi-1D at

125

low temperatures. At our lowest temperatures we see a factor of 79 suppression

of the three body loss coefficient in quasi-1D relative to 3D. Once again the far

from resonance scaling law (Dashed blue line) only roughly fits the data. As stated

above we believe we are on the cusp of intermediate regime but for technical reasons

cannot quite take data further from resonance. As such we would like to fit the

temperature dependence of L3 with our intermediate theory.

To do so we need to, in some way, account for the temperature dependence

of KAD. To our knowledge there is no independent theory or measurement of

KAD’s temperature dependence for p-wave resonances. However the form of our

Breit-Wigner theory (see equation 6.20) may give us a clue as to this temperature

dependence. Notice that the first term in the brackets of equation 6.20, which

stems from the low energy contribution and dominates far for resonance, has

no explicit temperature dependence. If our intermediate theory is to have any

hope of capturing the physics of the far from resonance scaling law, KAD must be

proportional to T 3.

We have constructed a modified intermediate theory assuming KAD = AT 3 and

leaving A as the only free parameter in equation 6.5 fit the off resonant quasi-1D

data. Our modified intermediate theory (Blue line) once again fits the data well

with KAD = T 3(14± 2)m3/sK3. As the theory was constructed from the far from

resonance scaling law we take this as more partial confirmation of the far from

resonance scaling law.

126

Figure 6.6: Log-log plot of KAD vs Temperature. The blue line is KAD = T 3(14±2)m3/sK3 with the shaded regions representing the error bars.

Furthermore KAD = T 3(14± 2)m3/sK3 is mostly consistent with our previously

reported values of KAD as well as that measured by Waseem et al. [3] as shown

in Figure 6.6. This is striking as the form of KAD was found from temperature

defendant data at a single magnetic field by phenomenologically connecting an

intermediate theory with far from resonate scaling laws. Based on this, an inter-

esting avenue of future research could be fully measuring and characterizing the

temperature dependence of KAD in p-wave Feshbach resonances3 as was done by

Li et al. in narrow s-wave resonances [135] .

We have shown that on resonance scaling laws in quasi-1D imply a strong

relative suppression of three body loss compared to the 3D case at low temperatures.3Either confirming or denying KAD ∝ T 3

127

Figure 6.7: Log-log plot of on resonance L3 vs lattice depth. Solid line showsL3 ∝ U−1

0 scaling.

However this relative suppression really comes from the 3D loss rate increasing while

the quasi-1D loss rate remains the same. If we are to have any hope of realizing

p-wave superfluids in ultracold atom experiments a more absolute suppression is

needed. To this end we measure the scaling of three body loss rate in quasi-1D

with lattice depth U0 as shown in Figure 6.7. We observe a L3 ∝ U−10 scaling of the

loss rate constant with lattice depth. In terms of the transverse confinement length

this is a L3 ∝ a−4⊥ scaling, showing that increasing the transverse confinement is

a promising route to achieving enough absolute suppression of three body loss to

realize a stable p-wave superfluid.

128

Chapter 7

Conclusions and Future Outlook

The work presented in this thesis may be thought of in two parts: first the

development of two novel all solid-state laser sources for use in lithium atom

experiments, and second the analysis of p-wave Feshbach resonances in quasi-1D

with a focus on mechanisms to reduce three body loss. These two disparate parts

are unified in that together they advance two of the most essential tools for ultracold

atomic gas experiments. Below I will give a summary of our main results with an

eye towards the future.

First we developed two frequency-doubled solid-state lasers capable of emitting

light at 671nm (the lithium D-line transition) with powers on the order Watts.

Laser light near 671 nm is the workhorse of lithium atom experiments; it is used to

produce the MOT and the gray molasses in order to trap and cool lithium atoms,

further it is used to image the atomic cloud. Both lasers are end pumped by 888nm

129

light, one features Nd:YVO4 as the gain crystal and the other Nd:GdVO4. Both

laser cavities are based on the same design, which has been shown to be generally

applicable to end pumped solid-state lasers with significant thermal lensing in

the gain crystal. Importantly the frequency doubled Nd:YVO4 laser is capable

of emitting 4 W of light at 671 nm, which is ample power to derive all the near

resonant light needed for lithium atom experiments. In our experiments the 4

W of power was used in conjunction with a cascade of AOMs to produces laser

beams capable of driving two photon Raman transitions. The frequency doubled

Nd:GdVO4 laser on the other hand is capable of emitting 1.2 W of light at 671

nm, which is still sufficient power to derive all the near resonant light needed for

lithium atom experiments. Furthermore it is capable of outputting 4 W 250 GHz

blue detuned from the lithium transition frequency. This makes Nd:GdVO4 lasers

ideal for applications that require blue detuned light; for example we believe this

blue detuned light may be used to form a pinning lattice as part of a lithium atom

quantum gas microscope.

Either laser is capable of fully replacing the current standard used to produce

the necessary 671 nm light: external cavity diode lasers (ECDL) followed by tapered

amplifiers. It is hard to overstate the advantage of these novel solid-state lasers

over their ECDL counterparts1. The first advantage is the most straightforward,

ECDLs even after tapered amplifiers cannot produce anywhere near the powers we1The comparison is at 671 nm for use in lithium atom experiments. I believe external cavity

diode lasers may be more effective at other wavelengths for use in different atomic experiments. Iwant to emphasize that the focus here is lasers at 671 nm.

130

have achieved with our solid state lasers. Most of the problems with ECDL systems

stem from the tapered amplifiers, which yield a poor spatial mode quality and have

a finite lifetime after which they become noisy and then stop working all together.

The short lifetime of the taper amplifiers means lithium groups, which rely on them,

must keep buying new ones. To add insult to injury, some companies, which sell

the tapered amplifiers, are often out of stock, while other companies have been

known to sell faulty chips that are dead on arrival. Our solid-state lasers, aside from

providing significantly more power, should be very long lived with robust operation

only requiring the occasional realignment to achieve ideal operating conditions.

It is my hope that these solid-state lasers and others like them will eventually

replace all the ECDL and tampered amplifier systems currently used in lithium

atom experiments.

Second we investigated p-wave Feshbach resonances in quasi-1D. P -wave Fes-

hbach resonances hold the promise of being able to realize p-wave superfluids in

ultracold atomic gas experiments. P -wave superfluids feature a far richer phase

diagram than their s-wave counterparts. In three dimensions there should be clas-

sical, quantum, and topological phase transitions across the BEC-BCS crossover.

In two dimensions a px + ipy superfluid with non-abelian excitations is expected.

In one dimension a fermionic Tonks-Girardeau gas could be realized as well as the

classic Kitaev chain model featuring Majorana edge states. The Majorana edge

states in 1D and the non-abelian excitations in 2D should allow for topologically

131

protected states, which can be useful in quantum computing. Unfortunately so far

inelastic two-body and three-body loss have prevented stabilization of the atomic

gas near p-wave Feshbach resonances. Two-body loss can be avoided for atoms in

the lowest hyperfine states. Three-body loss on the other hand in unavoidable. In

this thesis we have presented on mechanisms and efforts to suppress three body

loss in quasi-1D near p-wave Feshbach resonances.

We have shown that in quasi-1D the p-wave Feshbach molecule is a halo dimer

in the two body limit. Halo dimers are hugely delocalized molecules, which have

virtually no wavefunction overlap with deeper molecular wavefunctions. As a result

the probability of decay from a halo dimer to a deeper molecular state is extremely

low. Halo dimers are featured in s-wave resonances and are a large part of the

reason why three body loss is manageable near s-wave resonances. We hope that

by localizing the quasi-1D Fermi gas to the regime in which p-wave halo-dimers are

formed, three body loss may be sufficiently suppressed to allow for the stabilization

of a p-wave superfluid. We have found the experimental conditions necessary to

reach this halo dimer regime, which are presented in Chapter 5.

We have further performed experiments measuring the three body loss coefficient

in quasi-1D and 3D. We have seen up to a factor of 79 suppression of the three body

loss coefficient in quasi-1D at our lowest attainable temperatures. We have compared

our measurements to the predicted scaling laws and confirmed that on resonance

the quasi-1D three body recombination rate is independent of temperature. Due

132

to heating in the lattice we were not able to take measurements far enough from

resonance to confirm the far from resonance scaling law. We further developed a

theory for intermediate fields that explains the observed loss feature. Finally we

observed that the on resonance loss rate coefficient scales inversely with lattice

depth. Therefore, increasing the transverse confinement (by increasing the lattice

depth) is a promising route to achieving more suppression of three body loss and

maybe one day realizing a p-wave superfluid in quasi-1D.

As an avenue for future research we propose using retro-reflected 532 nm light

with a depth of 200 Er to provide tight enough transverse confinement to potentially

access the halo dimer regime and achieve even greater suppression of three body

loss. Comparing our intermediate theory for the three body loss coefficient in quasi-

1D to the 1D scaling laws we conjectured that the p-wave atom-dimer relaxation

coefficient was proportional to T 3. Another interesting avenue of future research

would be to measure the atom-dimer relaxation coefficient in 3D over a large range

of temperatures to fully characterize its temperature dependence. Finally yet

another direction for future research would be to add a lattice along the direction

of the quasi-1D tubes. This should allow the observation of a dissipation induced

insulator in which the large decay rate near a p-wave resonance prevents hoping

between lattice sites due to the quantum Zeno effect [136]. This dissipation induced

blockade mechanism may be another route to stabilizing a p-wave superfluid. In

short, there is hope yet for stabilizing a p-wave superfluid.

133

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VitaFrancisco R. Fonta

EducationPhD Pennsylvania State University, University Park, PA Fall 2020B.S. University of Chicago, Chicago, IL Spring 2013

Selected Publications

Francisco R. Fonta, Andrew S. Marcum, Arif Mawardi Ismail, and Kenneth M.O’Hara, "High-power, frequency-doubled Nd:GdVO4 laser for use in lithium coldatom experiments," Opt. Express 27, 33144-33158 (2019)

Andrew S. Marcum, Francisco R. Fonta, Arif Mawardi Ismail, and KennethM. O’Hara, "Suppression of Three-Body Loss Near a p-Wave Resonance Due toQuasi-1D Confinement" , In preparation

Francisco R. Fonta and Kenneth M. O’Hara, "Experimental Conditions forObtaining Halo P -Wave Dimers in Quasi-1D" , Submitted to Physical Review A

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