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
ii
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
iv
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
v
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.
xii
• 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.
xiii
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
must be found as linear combinations of the product states |J,mj; I,mI〉. The full
Hamiltonian reads
Hint = Hhf +HZE = Hhf − µB(gjJ + gII) ·B. (2.5)
Usually the eigenstates are solved for numerically however for the 2S1/2 state the
solution can be found analytically. In terms of the |ms;mI〉 product states these
lowest Zeeman energy sublevels are given by
|1〉 = sin Θ+|12 , 0〉 − cos Θ+| −
12 , 1〉 (2.6)
|2〉 = sin Θ−|12 ,−1〉 − cos Θ−| −
12 , 0〉 (2.7)
|3〉 = | − 12 ,−1〉 (2.8)
|4〉 = cos Θ−|12 ,−1〉+ sin Θ−| −
12 , 0〉 (2.9)
|5〉 = cos Θ+|12 , 0〉+ cos Θ+| −
12 , 0〉 (2.10)
|6〉 = |12 , 1〉 (2.11)
where the amplitudes are determined by
sin Θ± = 1√1 + (Z± +R±)2 /2
(2.12)
Z± = µBB
A2S1/2
(gJ2S1/2+ gI)±
12 (2.13)
R± =√Z2± + 2. (2.14)
16
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
Bibliography
[1] Rothman, L. S., I. E. Gordon, Y. Babikov, A. Barbe, D. C. Benner,P. F. Bernath, M. Birk, L. Bizzocchi, V. Boudon, L. R. Brown,A. Campargue, K. Chance, E. A. Cohen, L. H. Coudert, V. M. Devi,B. J. Drouin, A. Fayt, J. M. Flaud, R. R. Gamache, J. J. Harrison,J. M. Hartmann, C. Hill, J. T. Hodges, D. Jacquemart, A. Jolly,J. Lamouroux, R. J. Le Roy, G. Li, D. A. Long, O. M. Lyulin, C. J.Mackie, S. T. Massie, S. Mikhailenko, H. S. P. Mueller, O. V.Naumenko, A. V. Nikitin, J. Orphal, V. Perevalov, A. Perrin,E. R. Polovtseva, C. Richard, M. A. H. Smith, E. Starikova,K. Sung, S. Tashkun, J. Tennyson, G. C. Toon, V. G. Tyuterev, andG. Wagner (2013) “The HITRAN2012 molecular spectroscopic database,”J. Quant. Spectrosc. Radiat. Transf., 130(SI), pp. 4–50.
[2] Goldenstein, C. S., V. A. Miller, R. M. Spearrin, and C. L. Strand(2017) “SpectraPlot.com: Integrated spectroscopic modeling of atomic andmolecular gases,” J. Quant. Spectrosc. Radiat. Transf., 200, pp. 249–257.
[3] Waseem, M., J. Yoshida, T. Saito, and T. Mukaiyama (2019) “Quanti-tative analysis of p-wave three-body losses via a cascade process,” Phys. Rev.A, 99, p. 052704.URL https://link.aps.org/doi/10.1103/PhysRevA.99.052704
[4] Nakasuji, T., J. Yoshida, and T. Mukaiyama (2013) “Experimentaldetermination of p-wave scattering parameters in ultracold 6Li atoms,” Phys.Rev. A, 88, p. 012710.URL https://link.aps.org/doi/10.1103/PhysRevA.88.012710
[5] Austen, L. (2012) Production of p-wave Feshbach molecules from an ultra-cold Fermi gas, Ph.D. thesis, UCL (University College London).URL https://discovery.ucl.ac.uk/id/eprint/1338129/
[6] Zhang, J., E. G. M. van Kempen, T. Bourdel, L. Khaykovich,J. Cubizolles, F. Chevy, M. Teichmann, L. Tarruell, S. J. J. M. F.Kokkelmans, and C. Salomon (2004) “P -wave Feshbach resonances of
134
ultracold 6Li,” Phys. Rev. A, 70, p. 030702.URL https://link.aps.org/doi/10.1103/PhysRevA.70.030702
[7] Schunck, C. H., M. W. Zwierlein, C. A. Stan, S. M. F. Raupach,W. Ketterle, A. Simoni, E. Tiesinga, C. J. Williams, and P. S.Julienne (2005) “Feshbach resonances in fermionic 6Li,” Phys. Rev. A, 71,p. 045601.URL https://link.aps.org/doi/10.1103/PhysRevA.71.045601
[8] Gurarie, V. and L. Radzihovsky (2007) “Resonantly-paired fermionicsuperfluids,” Annals of Physics, 322, pp. 2 – 119.
[9] Fuchs, J., C. Ticknor, P. Dyke, G. Veeravalli, E. Kuhnle, W. Row-lands, P. Hannaford, and C. J. Vale (2008) “Binding energies of 6Lip-wave Feshbach molecules,” Phys. Rev. A, 77, p. 053616.URL https://link.aps.org/doi/10.1103/PhysRevA.77.053616
[10] Gubbels, K. B. and H. T. C. Stoof (2007) “Theory for p-Wave FeshbachMolecules,” Phys. Rev. Lett., 99, p. 190406.URL https://link.aps.org/doi/10.1103/PhysRevLett.99.190406
[11] Anderson, M. H., J. R. Ensher, M. R. Matthews, C. E. Wieman,and E. A. Cornell (1995) “Observation of Bose-Einstein Condensation in aDilute Atomic Vapor,” Science, 269(5221), pp. 198–201, https://science.sciencemag.org/content/269/5221/198.full.pdf.URL https://science.sciencemag.org/content/269/5221/198
[12] Davis, K. B., M. O. Mewes, M. R. Andrews, N. J. van Druten,D. S. Durfee, D. M. Kurn, and W. Ketterle (1995) “Bose-EinsteinCondensation in a Gas of Sodium Atoms,” Phys. Rev. Lett., 75, pp. 3969–3973.URL https://link.aps.org/doi/10.1103/PhysRevLett.75.3969
[13] Bradley, C. C., C. A. Sackett, J. J. Tollett, and R. G. Hulet(1995) “Evidence of Bose-Einstein Condensation in an Atomic Gas withAttractive Interactions,” Phys. Rev. Lett., 75, pp. 1687–1690.URL https://link.aps.org/doi/10.1103/PhysRevLett.75.1687
[14] DeMarco, B. and D. S. Jin (1999) “Onset of Fermi Degeneracy in aTrapped Atomic Gas,” Science, 285(5434), pp. 1703–1706, https://science.sciencemag.org/content/285/5434/1703.full.pdf.URL https://science.sciencemag.org/content/285/5434/1703
[15] London, F. (1938) “The λ-phenomenon of liquid helium and the Bose-Einstein degeneracy,” Nature, 141(3571), pp. 643–644.
135
[16] Maiman, T. H. (1960) “Stimulated optical radiation in ruby,” nature,187(4736), pp. 493–494.
[17] Ashkin, A. (1970) “Acceleration and Trapping of Particles by RadiationPressure,” Phys. Rev. Lett., 24, pp. 156–159.URL https://link.aps.org/doi/10.1103/PhysRevLett.24.156
[18] Ashkin, A., J. M. Dziedzic, J. E. Bjorkholm, and S. Chu (1986) “Ob-servation of a single-beam gradient force optical trap for dielectric particles,”Opt. Lett., 11(5), pp. 288–290.URL http://ol.osa.org/abstract.cfm?URI=ol-11-5-288
[19] Moffitt, J. R., Y. R. Chemla, S. B. Smith, and C. Bustamante (2008)“Recent Advances in Optical Tweezers,” Annual Review of Biochemistry,77(1), pp. 205–228, pMID: 18307407, https://doi.org/10.1146/annurev.biochem.77.043007.090225.URL https://doi.org/10.1146/annurev.biochem.77.043007.090225
[20] Gao, D., W. Ding, M. Nieto-Vesperinas, X. Ding, M. Rahman,T. Zhang, C. Lim, and C.-W. Qiu (2017) “Optical manipulation from themicroscale to the nanoscale: fundamentals, advances and prospects,” Light:Science & Applications, 6(9), pp. e17039–e17039.
[21] Hänsch, T. W. and A. L. Schawlow (1975) “Cooling of gases by laserradiation,” Optics Communications, 13(1), pp. 68–69.
[22] Chu, S., L. Hollberg, J. E. Bjorkholm, A. Cable, and A. Ashkin(1985) “Three-dimensional viscous confinement and cooling of atoms byresonance radiation pressure,” Physical review letters, 55(1), p. 48.
[23] Ungar, P. J., D. S. Weiss, E. Riis, and S. Chu (1989) “Optical molassesand multilevel atoms: theory,” JOSA B, 6(11), pp. 2058–2071.
[24] Weiss, D. S., E. Riis, Y. Shevy, P. J. Ungar, and S. Chu (1989)“Optical molasses and multilevel atoms: experiment,” JOSA B, 6(11), pp.2072–2083.
[25] Migdall, A. L., J. V. Prodan, W. D. Phillips, T. H. Bergeman, andH. J. Metcalf (1985) “First Observation of Magnetically Trapped NeutralAtoms,” Phys. Rev. Lett., 54, pp. 2596–2599.URL https://link.aps.org/doi/10.1103/PhysRevLett.54.2596
[26] Raab, E. L., M. Prentiss, A. Cable, S. Chu, and D. E. Pritchard(1987) “Trapping of Neutral Sodium Atoms with Radiation Pressure,” Phys.Rev. Lett., 59, pp. 2631–2634.URL https://link.aps.org/doi/10.1103/PhysRevLett.59.2631
136
[27] Lett, P. D., R. N. Watts, C. I. Westbrook, W. D. Phillips, P. L.Gould, and H. J. Metcalf (1988) “Observation of Atoms Laser Cooledbelow the Doppler Limit,” Phys. Rev. Lett., 61, pp. 169–172.URL https://link.aps.org/doi/10.1103/PhysRevLett.61.169
[28] Dalibard, J. and C. Cohen-Tannoudji (1989) “Laser cooling below theDoppler limit by polarization gradients: simple theoretical models,” JOSA B,6(11), pp. 2023–2045.
[29] Davis, K. B., M.-O. Mewes, M. A. Joffe, M. R. Andrews, andW. Ketterle (1995) “Evaporative Cooling of Sodium Atoms,” Phys. Rev.Lett., 74, pp. 5202–5205.URL https://link.aps.org/doi/10.1103/PhysRevLett.74.5202
[30] Petrich, W., M. H. Anderson, J. R. Ensher, and E. A. Cornell(1995) “Stable, Tightly Confining Magnetic Trap for Evaporative Cooling ofNeutral Atoms,” Phys. Rev. Lett., 74, pp. 3352–3355.URL https://link.aps.org/doi/10.1103/PhysRevLett.74.3352
[31] Adams, C. S., H. J. Lee, N. Davidson, M. Kasevich, and S. Chu(1995) “Evaporative Cooling in a Crossed Dipole Trap,” Phys. Rev. Lett., 74,pp. 3577–3580.URL https://link.aps.org/doi/10.1103/PhysRevLett.74.3577
[32] Chin, C., R. Grimm, P. Julienne, and E. Tiesinga (2010) “Feshbachresonances in ultracold gases,” Rev. Mod. Phys., 82, pp. 1225–1286.URL https://link.aps.org/doi/10.1103/RevModPhys.82.1225
[33] Bardeen, J., L. N. Cooper, and J. R. Schrieffer (1957) “Theory ofsuperconductivity,” Physical review, 108(5), p. 1175.
[34] Stoof, H. T. C., M. Houbiers, C. A. Sackett, and R. G. Hulet(1996) “Superfluidity of Spin-Polarized 6Li,” Phys. Rev. Lett., 76, pp. 10–13.URL https://link.aps.org/doi/10.1103/PhysRevLett.76.10
[35] Holland, M., S. Kokkelmans, M. L. Chiofalo, and R. Walser (2001)“Resonance superfluidity in a quantum degenerate Fermi gas,” Physical reviewletters, 87(12), p. 120406.
[36] Ohashi, Y. and A. Griffin (2002) “BCS-BEC crossover in a gas of Fermiatoms with a Feshbach resonance,” Physical review letters, 89(13), p. 130402.
[37] O’hara, K., S. Hemmer, M. Gehm, S. Granade, and J. Thomas (2002)“Observation of a strongly interacting degenerate Fermi gas of atoms,” Science,298(5601), pp. 2179–2182.
137
[38] Zhou, L. and X. Cui (2017) “Stretching p-wave molecules by transverseconfinements,” Phys. Rev. A, 96, p. 030701.URL https://link.aps.org/doi/10.1103/PhysRevA.96.030701
[39] Marcum, A. (2019) Ultracold Fermions in Reduced Dimensions: Three-BodyRecombination, Tomonaga-Luttinger Liquids, and a Honeycomb Lattice, Ph.D.thesis, PSU (Penn State University).URL https://etda.libraries.psu.edu/catalog/16250asm264
[40] Ismail, A. M. (2018) Two-body Loss in a Fermi Gas Near a P-Wave FeshbachResonance, Ph.D. thesis, PSU (Penn State University).URL https://etda.libraries.psu.edu/catalog/16195abi104
[41] Gehm, M. E. (2003) “Properties of 6Li,” Jetlab,.
[42] Siegman, A. E. (1986) LASERS, university science books.
[43] Chen, Y., T. Huang, C. Kao, C. Wang, and S. Wang (1997) “Opti-mization in scaling fiber-coupled laser-diode end-pumped lasers to higherpower: Influence of thermal effect,” IEEE J. Quantum Electron., 33(8), pp.1424–1429.
[44] Chen, Y.-F., T. Liao, C. Kao, T. Huang, K. Lin, and S. Wang (1996)“Optimization of fiber-coupled laser-diode end-pumped lasers: influence ofpump-beam quality,” IEEE Journal of Quantum Electronics, 32(11), pp.2010–2016.
[45] Chen, Y.-F. (1999) “Design criteria for concentration optimization in scalingdiode end-pumped lasers to high powers: influence of thermal fracture,” IEEEJournal of Quantum Electronics, 35(2), pp. 234–239.
[46] Laporta, P. and M. Brussard (1991) “Design criteria for mode sizeoptimization in diode-pumped solid-state lasers,” IEEE Journal of QuantumElectronics, 27(10), pp. 2319–2326.
[47] Eismann, U., F. Gerbier, C. Canalias, A. Zukauskas, G. Trenec,J. Vigue, F. Chevy, and C. Salomon (2012) “An all-solid-state lasersource at 671 nm for cold-atom experiments with lithium,” Appl. Phys. B:Lasers Opt., 106(1), pp. 25–36.
[48] Taylor, J. R. (1972) Scattering Theory: The Quantum Theory on Nonrela-tivistic Collisions, John Wiley & Sons, Inc., New York.
[49] Shankar, R. (2012) Principles of quantum mechanics, Springer Science &Business Media.
138
[50] Lin, Z., K. Shimizu, M. S. Zhan, F. Shimizu, and H. Takuma (1991)“Laser cooling and trapping of Li,” Jpn. J. Appl. Phys., 30(7B), pp. L1324–L1326.
[51] Parsons, M. F., F. Huber, A. Mazurenko, C. S. Chiu, W. Setiawan,K. Wooley-Brown, S. Blatt, and M. Greiner (2015) “Site-ResolvedImaging of Fermionic 6Li in an Optical Lattice,” Phys. Rev. Lett., 114, p.213002.URL https://link.aps.org/doi/10.1103/PhysRevLett.114.213002
[52] Müller, H., S.-w. Chiow, Q. Long, S. Herrmann, and S. Chu (2008)“Atom Interferometry with up to 24-Photon-Momentum-Transfer Beam Split-ters,” Phys. Rev. Lett., 100, p. 180405.URL https://link.aps.org/doi/10.1103/PhysRevLett.100.180405
[53] Miffre, A., M. Jacquey, M. Büchner, G. Trénec, and J. Vigué(2006) “Atom interferometry measurement of the electric polarizability oflithium,” Eur. Phys. J. D, 38(2), pp. 353–365.
[54] Olivares, I., A. Duarte, E. Saravia, and F. Duarte (2002) “Lithiumisotope separation with tunable diode lasers,” Appl. Opt., 41(15), pp. 2973–2977.
[55] Bakr, W. S., J. I. Gillen, A. Peng, S. Foelling, and M. Greiner(2009) “A quantum gas microscope for detecting single atoms in a Hubbard-regime optical lattice,” Nature, 462(7269), pp. 74–77.
[56] Kinoshita, T., T. Wenger, and D. S. Weiss (2005) “All-optical Bose-Einstein condensation using a compressible crossed dipole trap,” Phys. Rev.A, 71, p. 011602.URL https://link.aps.org/doi/10.1103/PhysRevA.71.011602
[57] Grier, A. T., I. Ferrier-Barbut, B. S. Rem, M. Delehaye,L. Khaykovich, F. Chevy, and C. Salomon (2013) “Λ-enhanced sub-Doppler cooling of lithium atoms in D1 gray molasses,” Phys. Rev. A, 87, p.063411.URL https://link.aps.org/doi/10.1103/PhysRevA.87.063411
[58] Burchianti, A., G. Valtolina, J. A. Seman, E. Pace, M. De Pas,M. Inguscio, M. Zaccanti, and G. Roati (2014) “Efficient all-opticalproduction of large 6Li quantum gases using D1 gray-molasses cooling,” Phys.Rev. A, 90, p. 043408.URL https://link.aps.org/doi/10.1103/PhysRevA.90.043408
139
[59] Payne, S. A., L. K. Smith, R. J. Beach, B. H. T. Chai, J. H. Tassano,L. D. Deloach, W. L. Kway, R. W. Solarz, and W. F. Krupke (1994)“Properties of Cr:LiSrAlF6 crystals for laser operation,” Appl. Opt., 33(24),pp. 5526–5536.
[60] Hou, F. Y., L. Yu, X. J. Jia, Y. H. Zheng, C. D. Xie, and K. C. Peng(2011) “Experimental generation of optical non-classical states of light with1.34 µm wavelength,” Eur. Phys. J. D, 62(3), pp. 433–437.
[61] Du, C., S. Ruan, Y. Yu, and F. Zeng (2005) “6-W diode-end-pumpedNd:GdVO4/LBO quasi-continuous-wave red laser at 671 nm,” Opt. Express,13(6), pp. 2013–2018.
[62] Zhang, H., J. Liu, J. Wang, C. Wang, L. Zhu, Z. Shao, X. Meng,X. Hu, M. Jiang, and Y. Chow (2002) “Characterization of the lasercrystal Nd:GdVO4,” J. Opt. Soc. Am. B, 19(1), pp. 18–27.
[63] Jensen, T., V. G. Ostroumov, J. P. Meyn, G. Huber, A. I. Zagu-mennyi, and I. A. Shcherbakov (1994) “Spectroscopic characterizationand laser performance of diode-laser-pumped Nd-GdVO4,” Appl. Phys. B:Lasers Opt., 58(5), pp. 373–379.
[64] Camargo, F. A., T. Zanon-Willette, T. Badr, N. U. Wetter, andJ.-J. Zondy (2010) “Tunable single-frequency Nd:YVO4 BiB3O6 ring laserat 671 nm,” IEEE J. Quantum Electron., 46(5), pp. 804–809.
[65] McDonagh, L., R. Wallenstein, R. Knappe, and A. Nebel (2006)“High-efficiency 60 W TEM00 Nd:YVO4 oscillator pumped at 888 nm,” Opt.Lett., 31(22), pp. 3297–3299.
[66] Eismann, U., A. Bergschneider, F. Sievers, N. Kretzschmar, C. Sa-lomon, and F. Chevy (2013) “2.1-watts intracavity-frequency-doubled all-solid-state light source at 671 nm for laser cooling of lithium,” Opt. Express,21(7), pp. 9091–9102.
[67] Koch, P., F. Ruebel, J. Bartschke, and J. A. L’Huillier (2015)“5.7 W CW single-frequency laser at 671 nm by single-pass second harmonicgeneration of a 17.2 W injection-locked 1342 nm Nd:YVO4 ring laser usingperiodically poled MgO:LiNbO3,” Appl. Opt., 54(33), pp. 9954–9959.
[68] Cui, X.-Y., Q. Shen, M.-C. Yan, C. Zeng, T. Yuan, W.-Z. Zhang,X.-C. Yao, C.-Z. Peng, X. Jiang, Y.-A. Chen, and J.-W. Pan (2018)“High-power 671 nm laser by second-harmonic generation with 93% efficiencyin an external ring cavity,” Opt. Lett., 43(8), pp. 1666–1669.
140
[69] Agnesi, A., A. Guandalini, G. Reali, S. Dell’Acqua, and G. Pic-cinno (2004) “High-brightness 2.4-W continuous-wave Nd:GdVO4 laser at670 nm,” Opt. Lett., 29(1), pp. 56–58.
[70] Lü, Y. F., X. H. Zhang, J. Xia, X. D. Yin, A. F. Zhang, L. Bao,D. Wang, and H. Quan (2009) “Highly efficient intracavity frequency-doubled Nd:GdVO4-LBO red laser at 670 nm under direct 880 nm pumping,”Laser Phys., 19(12), pp. 2174–2178.
[71] Wang, Y., W. Li, L. Pan, J. Yu, and R. Zhang (2013) “Diode-end-pumped continuous wave single-longitudinal-mode Nd:GdVO4 laser at 1342nm,” Appl. Opt., 52(9), pp. 1987–1991.
[72] Wang, Y. T., R. H. Zhang, J. H. Li, W. J. Li, C. Tan, and B. L.Zhang (2014) “A diode-end-pumped continuous-wave single-longitudinal-mode Nd:GdVO4-LBO red laser at 670 nm,” Laser Phys., 24(3), p. 035001.
[73] Wang, Y. T., R. H. Zhang, J. H. Li, and W. J. Li (2015) “Power scalingof single-longitudinal-mode Nd:GdVO4 laser at 1342 nm,” Laser Phys., 25(6),p. 065003.
[74] DeShazer, L. (1994) “Vanadate crystals exploit diode-pump technology,”Laser Focus World, 30(2), p. 88.
[75] Bennetts, S., G. D. McDonald, K. S. Hardman, J. E. Debs, C. C. N.Kuhn, J. D. Close, and N. P. Robins (2014) “External cavity diode laserswith 5 kHz linewidth and 200 nm tuning range at 1.55 µm and methods forlinewidth measurement,” Opt. Express, 22(9), pp. 10642–10654.
[76] Gauthier, D. J., P. Narum, and R. W. Boyd (1986) “Simple, compact,high-performance permanent-magnet Faraday isolator,” Opt. Lett., 11(10),pp. 623–625.
[77] Siegman, A. E. (1986) Lasers, Revised ed., Univ. Science Books, CA.
[78] Chénais, S., F. Druon, S. Forget, F. Balembois, and P. Georges(2006) “On thermal effects in solid-state lasers: The case of ytterbium-dopedmaterials,” Prog. Quantum. Electron., 30(4), pp. 89–153.
[79] Délen, X., F. Balembois, O. Musset, and P. Georges (2011) “Charac-teristics of laser operation at 1064 nm in Nd:YVO4 under diode pumping at808 and 914 nm,” J. Opt. Soc. Am. B, 28(1), pp. 52–57.
[80] Brown, D. (1998) “Heat, fluorescence, and stimulated-emission power den-sities and fractions in Nd : YAG,” IEEE J. Quantum Electron., 34(3), pp.560–572.
141
[81] Okida, M., M. Itoh, T. Yatagai, H. Ogilvy, J. Piper, and T. Omatsu(2005) “Heat generation in Nd doped vanadate crystals with 1.34 µm laseraction,” Opt. Express, 13(13), pp. 4909–4915.
[82] Lenhardt, F., M. Nittmann, T. Bauer, J. Bartschke, and J. A.L’Huillier (2009) “High-power 888-nm-pumped Nd:YVO4 1342-nm oscilla-tor operating in the TEM00 mode,” Appl. Phys. B: Lasers Opt., 96(4), pp.803–807.
[83] Fornasiero, L., S. Kuck, T. Jensen, G. Huber, and B. Chai (1998)“Excited state absorption and stimulated emission of Nd3+ in crystals. Part2: YVO4, GdVO4, and Sr5(PO4)3F,” Appl. Phys. B: Lasers Opt., 67(5), pp.549–553.
[84] Drever, R. W. P., J. L. Hall, F. V. Kowalski, J. Hough, G. M.Ford, A. J. Munley, and H. Ward (1983) “Laser phase and frequencystabilization using an optical-resonator,” Appl. Phys. B, 31(2), pp. 97–105.
[85] Miake, Y., T. Mukaiyama, K. M. O’Hara, and S. Gensemer (2015)“A self-injected, diode-pumped, solid-state ring laser for laser cooling of Liatoms,” Rev. Sci. Instrum., 86(4), p. 043113.
[86] Cohen-Tannoudji, C., J. Dupont-Roc, and G. Grynberg (1998) Atom-photon interactions: basic processes and applications.
[87] Foot, C. J. et al. (2005) Atomic physics, vol. 7, Oxford University Press.
[88] Williams, J. R. (2010) UNIVERSAL FEW-BODY PHYSICS IN A THREE-COMPONENT FERMI GAS, Ph.D. thesis, PSU (Penn State University).URL https://etda.libraries.psu.edu/catalog/11299
[89] Zhang, Y. (2014) One-dimensional Fermi Gases with Rapid Control ofInteractions: Toward the Observation of Dicke States and Exotic SuperfluidPairing, Ph.D. thesis, PSU (Penn State University).URL https://etda.libraries.psu.edu/catalog/22661
[90] Stites, R. W. (2012) The rapid control of interactions in a two-componentFermi gas, Ph.D. thesis, PSU (Penn State University).URL https://etda.libraries.psu.edu/catalog/16395
[91] Gurarie, V., L. Radzihovsky, and A. V. Andreev (2005) “QuantumPhase Transitions across a p-Wave Feshbach Resonance,” Phys. Rev. Lett.,94, p. 230403.URL https://link.aps.org/doi/10.1103/PhysRevLett.94.230403
142
[92] Botelho, S. S. and C. A. R. S. d. Melo (2005) “Quantum PhaseTransition in the BCS-to-BEC Evolution of p-wave Fermi Gases,” Journal ofLow Temperature Physics, 140(5-6), p. 409–428.URL http://dx.doi.org/10.1007/s10909-005-7324-3
[93] Cheng, C.-H. and S.-K. Yip (2005) “Anisotropic Fermi Superfluid viap-Wave Feshbach Resonance,” Phys. Rev. Lett., 95, p. 070404.URL https://link.aps.org/doi/10.1103/PhysRevLett.95.070404
[94] Iskin, M. and C. A. R. Sá de Melo (2006) “Evolution from BCS to BECSuperfluidity in p-Wave Fermi Gases,” Phys. Rev. Lett., 96, p. 040402.URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040402
[95] Read, N. and D. Green (2000) “Paired states of fermions in two dimensionswith breaking of parity and time-reversal symmetries and the fractionalquantum Hall effect,” Phys. Rev. B, 61, pp. 10267–10297.URL https://link.aps.org/doi/10.1103/PhysRevB.61.10267
[96] Fedorov, A. K., V. I. Yudson, and G. V. Shlyapnikov (2017) “P -wavesuperfluidity of atomic lattice fermions,” Phys. Rev. A, 95, p. 043615.URL https://link.aps.org/doi/10.1103/PhysRevA.95.043615
[97] Tewari, S., S. Das Sarma, C. Nayak, C. Zhang, and P. Zoller (2007)“Quantum Computation using Vortices and Majorana Zero Modes of a px+ipySuperfluid of Fermionic Cold Atoms,” Phys. Rev. Lett., 98, p. 010506.URL https://link.aps.org/doi/10.1103/PhysRevLett.98.010506
[98] Girardeau, M. D. and E. M. Wright (2005) “Static and DynamicProperties of Trapped Fermionic Tonks-Girardeau Gases,” Phys. Rev. Lett.,95, p. 010406.URL https://link.aps.org/doi/10.1103/PhysRevLett.95.010406
[99] Bender, S. A., K. D. Erker, and B. E. Granger (2005) “ExponentiallyDecaying Correlations in a Gas of Strongly Interacting Spin-Polarized 1DFermions with Zero-Range Interactions,” Phys. Rev. Lett., 95, p. 230404.URL https://link.aps.org/doi/10.1103/PhysRevLett.95.230404
[100] Girardeau, M. D. and A. Minguzzi (2006) “Bosonization, Pairing, andSuperconductivity of the Fermionic Tonks-Girardeau Gas,” Phys. Rev. Lett.,96, p. 080404.URL https://link.aps.org/doi/10.1103/PhysRevLett.96.080404
[101] Cheon, T. and T. Shigehara (1999) “Fermion-Boson Duality of One-Dimensional Quantum Particles with Generalized Contact Interactions,” Phys.Rev. Lett., 82, pp. 2536–2539.URL https://link.aps.org/doi/10.1103/PhysRevLett.82.2536
143
[102] Imambekov, A., A. A. Lukyanov, L. I. Glazman, and V. Gritsev(2010) “Exact Solution for 1D Spin-Polarized Fermions with Resonant Inter-actions,” Phys. Rev. Lett., 104, p. 040402.URL https://link.aps.org/doi/10.1103/PhysRevLett.104.040402
[103] Pan, L., S. Chen, and X. Cui (2018) “Many-body stabilization of a resonantp-wave Fermi gas in one dimension,” Phys. Rev. A, 98, p. 011603.URL https://link.aps.org/doi/10.1103/PhysRevA.98.011603
[104] Kitaev, A. Y. (2001) “Unpaired Majorana fermions in quantum wires,”Physics-Uspekhi, 44(10S), pp. 131–136.URL https://doi.org/10.1070%2F1063-7869%2F44%2F10s%2Fs29
[105] Luciuk, C., S. Trotzky, S. Smale, Z. Yu, S. Zhang, and J. H. Thy-wissen (2016) “Evidence for universal relations describing a gas with p-waveinteractions,” Nature Physics, 12(6), pp. 599+.
[106] Ticknor, C., C. A. Regal, D. S. Jin, and J. L. Bohn (2004) “Multipletstructure of Feshbach resonances in nonzero partial waves,” Phys. Rev. A,69, p. 042712.URL https://link.aps.org/doi/10.1103/PhysRevA.69.042712
[107] Chevy, F., E. G. M. van Kempen, T. Bourdel, J. Zhang,L. Khaykovich, M. Teichmann, L. Tarruell, S. J. J. M. F. Kokkel-mans, and C. Salomon (2005) “Resonant scattering properties close to ap-wave Feshbach resonance,” Phys. Rev. A, 71, p. 062710.URL https://link.aps.org/doi/10.1103/PhysRevA.71.062710
[108] Regal, C. A., C. Ticknor, J. L. Bohn, and D. S. Jin (2003) “Tuningp-Wave Interactions in an Ultracold Fermi Gas of Atoms,” Phys. Rev. Lett.,90, p. 053201.URL https://link.aps.org/doi/10.1103/PhysRevLett.90.053201
[109] Waseem, M., J. Yoshida, T. Saito, and T. Mukaiyama (2018)“Unitarity-limited behavior of three-body collisions in a p-wave interactingFermi gas,” Phys. Rev. A, 98, p. 020702.URL https://link.aps.org/doi/10.1103/PhysRevA.98.020702
[110] Deh, B., C. Marzok, C. Zimmermann, and P. W. Courteille (2008)“Feshbach resonances in mixtures of ultracold 6Li and 87Rb gases,” Phys. Rev.A, 77, p. 010701.URL https://link.aps.org/doi/10.1103/PhysRevA.77.010701
[111] Suno, H., B. D. Esry, and C. H. Greene (2003) “Recombination of ThreeUltracold Fermionic Atoms,” Physical Review Letters, 90(5), p. 053202.
144
[112] Feshbach, H. (1962) “A unified theory of nuclear reactions. II,” Annals ofPhysics, 19(2), pp. 287 – 313.URL http://www.sciencedirect.com/science/article/pii/000349166290221X
[113] Köhler, T., K. Góral, and P. S. Julienne (2006) “Production of coldmolecules via magnetically tunable Feshbach resonances,” Rev. Mod. Phys.,78, pp. 1311–1361.URL https://link.aps.org/doi/10.1103/RevModPhys.78.1311
[114] Combescot, R. (2003) “Feshbach Resonance in Dense Ultracold FermiGases,” Phys. Rev. Lett., 91, p. 120401.URL https://link.aps.org/doi/10.1103/PhysRevLett.91.120401
[115] Bruun, G. M. (2004) “Universality of a two-component Fermi gas with aresonant interaction,” Phys. Rev. A, 70, p. 053602.URL https://link.aps.org/doi/10.1103/PhysRevA.70.053602
[116] Simonucci, S., P. Pieri, and G. Strinati (2005) “Broad vs. narrowFano-Feshbach resonances in the BCS-BEC crossover with trapped Fermiatoms,” EPL (Europhysics Letters), 69(5), p. 713.
[117] De Palo, S., M. L. Chiofalo, M. Holland, and S. Kokkelmans (2004)“Resonance effects on the crossover of bosonic to fermionic superfluidity,”Physics Letters A, 327(5-6), pp. 490–499.
[118] Partridge, G. B., K. E. Strecker, R. I. Kamar, M. W. Jack, andR. G. Hulet (2005) “Molecular Probe of Pairing in the BEC-BCS Crossover,”Phys. Rev. Lett., 95, p. 020404.URL https://link.aps.org/doi/10.1103/PhysRevLett.95.020404
[119] Werner, F., L. Tarruell, and Y. Castin (2009) “Number of closed-channel molecules in the BEC-BCS crossover,” The European Physical JournalB, 68(3), pp. 401–415.
[120] Granger, B. E. and D. Blume (2004) “Tuning the Interactions of Spin-Polarized Fermions Using Quasi-One-Dimensional Confinement,” Phys. Rev.Lett., 92, p. 133202.URL https://link.aps.org/doi/10.1103/PhysRevLett.92.133202
[121] Peng, S.-G., S. Tan, and K. Jiang (2014) “Manipulation of p-WaveScattering of Cold Atoms in Low Dimensions Using the Magnetic FieldVector,” Phys. Rev. Lett., 112, p. 250401.URL https://link.aps.org/doi/10.1103/PhysRevLett.112.250401
145
[122] Pricoupenko, L. (2008) “Resonant Scattering of Ultracold Atoms in LowDimensions,” Phys. Rev. Lett., 100, p. 170404.URL https://link.aps.org/doi/10.1103/PhysRevLett.100.170404
[123] Marcum, A. S., F. R. Fonta, A. M. Ismail, and K. M. O’Hara“Suppression of Three-Body Loss Near a p-Wave Resonance Due to Quasi-1DConfinement,” In preperation.
[124] Gaebler, J. P., J. T. Stewart, J. L. Bohn, and D. S. Jin (2007)“p-Wave Feshbach Molecules,” Phys. Rev. Lett., 98, p. 200403.URL https://link.aps.org/doi/10.1103/PhysRevLett.98.200403
[125] Sievers, F., N. Kretzschmar, D. R. Fernandes, D. Suchet, M. Ra-binovic, S. Wu, C. V. Parker, L. Khaykovich, C. Salomon, andF. Chevy (2015) “Simultaneous sub-Doppler laser cooling of fermionic Li 6and K 40 on the D 1 line: Theory and experiment,” Physical review A, 91(2),p. 023426.
[126] Burchianti, A., G. Valtolina, J. Seman, E. Pace, M. De Pas, M. In-guscio, M. Zaccanti, and G. Roati (2014) “Efficient all-optical productionof large Li 6 quantum gases using D 1 gray-molasses cooling,” Physical ReviewA, 90(4), p. 043408.
[127] Liao, Y.-a., A. S. C. Rittner, T. Paprotta, W. Li, G. B. Partridge,R. G. Hulet, S. K. Baur, and E. J. Mueller (2010) “Spin-imbalance ina one-dimensional Fermi gas,” Nature, 467(7315), pp. 567–569.
[128] Mehta, N. P., B. Esry, and C. H. Greene (2007) “Three-body recombi-nation in one dimension,” Physical Review A, 76(2), p. 022711.
[129] Suno, H., B. Esry, and C. H. Greene (2003) “Three-body recombinationof cold fermionic atoms,” New Journal of Physics, 5(1), p. 53.
[130] Hazlett, E. L., Y. Zhang, R. W. Stites, and K. M. O’Hara (2012)“Realization of a Resonant Fermi Gas with a Large Effective Range,” Phys.Rev. Lett., 108, p. 045304.URL https://link.aps.org/doi/10.1103/PhysRevLett.108.045304
[131] Napolitano, R., J. Weiner, C. J. Williams, and P. S. Julienne (1994)“Line Shapes of High Resolution Photoassociation Spectra of Optically CooledAtoms,” Phys. Rev. Lett., 73, pp. 1352–1355.URL https://link.aps.org/doi/10.1103/PhysRevLett.73.1352
[132] Mathey, L., E. Tiesinga, P. S. Julienne, and C. W. Clark (2009)“Collisional cooling of ultracold-atom ensembles using Feshbach resonances,”
146
Phys. Rev. A, 80, p. 030702.URL https://link.aps.org/doi/10.1103/PhysRevA.80.030702
[133] Mies, F. H., E. Tiesinga, and P. S. Julienne (2000) “Manipulationof Feshbach resonances in ultracold atomic collisions using time-dependentmagnetic fields,” Phys. Rev. A, 61, p. 022721.URL https://link.aps.org/doi/10.1103/PhysRevA.61.022721
[134] Yurovsky, V. A. and A. Ben-Reuven (2003) “Three-body loss of trappedultracold 87Rb atoms due to a Feshbach resonance,” Phys. Rev. A, 67, p.050701.URL https://link.aps.org/doi/10.1103/PhysRevA.67.050701
[135] Li, J., J. Liu, L. Luo, and B. Gao (2018) “Three-Body Recombinationnear a Narrow Feshbach Resonance in 6Li,” Phys. Rev. Lett., 120, p. 193402.URL https://link.aps.org/doi/10.1103/PhysRevLett.120.193402
[136] Han, Y.-J., Y.-H. Chan, W. Yi, A. Daley, S. Diehl, P. Zoller, andL.-M. Duan (2009) “Stabilization of the p-wave superfluid state in an opticallattice,” Physical review letters, 103(7), p. 070404.
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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