𝑝-wave Collisions in Ultracold FermionsbyFurkan Top
Submitted to the Department of Physics
in partial fulfillment of the requirements for the degree ofDoctor of Philosophy
at theMASSACHUSETTS INSTITUTE OF TECHNOLOGY
September 2020
c Massachusetts Institute of Technology 2020. All rights reserved.
○

Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Department of Physics
June 23, 2020

Certified by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Wolfgang Ketterle
John D. MacArthur Professor of Physics
Thesis Supervisor

Accepted by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Nergis Mavalvala
Associate Department Head of Physics

2

𝑝-wave Collisions in Ultracold FermionsbyFurkan Top
Submitted to the Department of Physics
on June 23, 2020, in partial fulfillment of therequirements for the degree ofDoctor of Philosophy

Abstract
In this thesis, we have created a single-component Fermi gas at unprecedented highdensities and characterized its collisional properties. In particular, we have measured
three-body loss coefficients and the p-wave background scattering length for the lowest
hyperfine state of 6 Li using a single beam optical dipole trap. We have achieved
densities on the order of 1015 1/cm3 . We have determined the 𝑝-wave background+1.3
scattering length to be |𝑎𝑏𝑔
𝑝 | = 39−1.6 𝑎0 using cross-dimensional thermalization. Aftermeasurement of non-resonant elastic and inelastic processes, we have used p-wave
collisions in the spin polarized sample for forced evaporative cooling, and have shown
its limitations. While insufficiency of low-velocity 𝑝-wave collisions prevents reachingan equilibrium Fermi-Dirac distribution, temperatures as low as 𝑇 /𝑇𝐹 = 0.42 wereobtained.
Thesis Supervisor: Wolfgang Ketterle
Title: John D. MacArthur Professor of Physics

3

4

Acknowledgments
Thank you Wolfgang, for this once-in-a-lifetime opportunity of working with you. I’ll
always remember the evening you came to lab, and take a look at our dark spot when
I was alone and having problems with the Na MOT. Since then, I wish I had a chance
to work with you in the lab like first generations of graduate students in the Ketterlegroup. I felt your support throughout the last six years at MIT, especially during theCovid-19 pandemic.
My labmate Yair comes next. Although we were a team of two, we ended up
setting up the most stable era of the BEC2 machine. Thank you for sailing with meduring the most challenging two years of this journey.
I had a chance to share an office space with Alan Jamison during the second halfof my PhD. Alan is one of the most brilliant minds I have ever seen. I learned a great
deal ranging from atoms to the darkest corners of America. Thank you Alan! I wish
we could spend more time together.
Li never hesitated to take some of his time and fix our problems with Cicero and
Atticus. It’s impossible to miss his positive attitude and enthusiasm. He is a lifetime
learner. Thank you Li! Julius has been with us for only two years. I remember him
many times following up with my questions hours after our first discussion. Thankyou Julius! With their know-how, Ivana and Nikki were always the first door to knock
when we get stuck in a technical problem. Thank you for your willingness to help!
Working in an experimental physics lab requires some machining. Mark Belanger
from the Edgerton student shop makes the machine shop a place of happiness withhis humor and good taste in music. He is a true teacher. He helped us with the designof our pneumatically controlled iris in the last turn of my thesis project. Using thisiris enhanced loading our lithium atoms into the optical trap significantly. Thank you
Mark! I’ll build my car one piece at a time.
Finally, I want to thank Cathy Modica from the physics department for always
looking after students’ well-being. You make MIT our home.

56This doctoral thesis has been examined by a Committee of the
Department of Physics as follows:

Professor Martin Zwierlein . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chairman, Thesis Committee
Thomas A. Frank (1977) Professor of Physics

Professor Wolfgang Ketterle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Thesis Supervisor
John D. MacArthur Professor of Physics

Professor Liang Fu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Member, Thesis Committee
Lawrence C. (1944) and Sarah W. Biedenharn Career Development
Associate Professor of Physics

8Contents
1 Introduction

21

2 The Experimental Apparatus

25

2.1

2.2

The New Microchannel Nozzle . . . . . . . . . . . . . . . . . . . . . .

27

2.1.1

Effusion from a Hole . . . . . . . . . . . . . . . . . . . . . . .

27

2.1.2

Design of the Nozzle . . . . . . . . . . . . . . . . . . . . . . .

28

2.1.3

Na Flux Measurement . . . . . . . . . . . . . . . . . . . . . .

31

2.1.4

Li Flux Measurement . . . . . . . . . . . . . . . . . . . . . . .

35

Experimental Techniques . . . . . . . . . . . . . . . . . . . . . . . . .

38

2.2.1

Calibration of our Magnets . . . . . . . . . . . . . . . . . . . .

38

2.2.2

Measuring the Saturation Intensity and Accurate Atom Counting 43

2.2.3|1⟩-|2⟩ Mixture Population Ratio Control . . . . . . . . . . . .

3 Optical Dipole Trap

47
51

3.1

Brief Theory of a Single Beam ODT. . . . . . . . . . . . . . . . . .

52

3.2

Near or Far Resonant Light for our Experiment? . . . . . . . . . . . .

53

3.3

Experimental Implementation . . . . . . . . . . . . . . . . . . . . . .

55

3.3.1

Lens Choices . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

3.3.2

Implementation . . . . . . . . . . . . . . . . . . . . . . . . . .

55

Characterization of the Trap . . . . . . . . . . . . . . . . . . . . . . .

59

3.4.1

Imaging IR Spot Size . . . . . . . . . . . . . . . . . . . . . . .

59

3.4.2

Measurement of Trapping Frequencies. . . . . . . . . . . . .

60

3.4.3

Measurement of the Trap Depth through Spilling . . . . . . .

68

3.4

9

3.5

3.6

3.4.4

Astigmatism . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

3.4.5

Anharmonic Correction to Density Averages . . . . . . . . . .

75

Tilt Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.5.1

Radial Tilt

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

78

3.5.2

Axial Tilt . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

80

Measurement of Heating Rate for |1⟩ . . . . . . . . . . . . . . . . . .

83

4 𝑝-wave Collisions
4.1

78

87

Collision Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

88

4.1.1

Impact Parameter and Scattering Cross-section . . . . . . . .

89

4.1.2

Quantum Terminology . . . . . . . . . . . . . . . . . . . . . .

89

4.1.3

Partial Wave Expansion and p-wave in Threshold . . . . . . .

90

4.2

Sample Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95

4.3

Measurement of 𝑎𝑏𝑔
𝑝 . . . . . . . . . . . . . . . . . . . . . . . . . . . .

96

4.3.1

Cross-dimensional Thermalization . . . . . . . . . . . . . . . .

96

4.3.2

Experimental Results . . . . . . . . . . . . . . . . . . . . . . .

97

4.4

4.5

𝑝-wave losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.4.1

Three-body loss in |1⟩ and 𝐿3 . . . . . . . . . . . . . . . . . . 101

4.4.2

𝑝-wave Dipolar Relaxation of |6⟩ . . . . . . . . . . . . . . . . . 105

𝑝-wave Evaporation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.5.1

Relative Lack of Low-velocity Collisions in 𝑝-wave . . . . . . . 113

4.5.2

Evaporation Strategy . . . . . . . . . . . . . . . . . . . . . . . 118

4.5.3

Evaporation Results . . . . . . . . . . . . . . . . . . . . . . . 118

5 Outlook

125

A Threshold Scalings

127

B Fitting functions for degenerate Fermions

129

C Laser Table AOM schemes

135

C.1 Li . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
10

C.2 Na . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
C.3 IR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

11

12

List of Figures
2-1 The cracked viewport . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

2-2 Description of the new micro-channel nozzle. Panel-(a) shows the housing to hold hypodermic needles. After stacking needles on two connected pieces as in Panel (b) and (c), the third piece is also clamped
to the other two. The clamping pressure prevents tubes from sliding
out. Panel-(e) shows the nozzle piece connected to the nozzle holder.
Off-centering of the new nozzle piece is intentional in order to match
position of the old nozzle hence alignment of the atomic beam to themachine.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
′

30

′

2-3 𝐹 = 2 to 𝐹 = 3 and 𝐹 = 2 to 𝐹 = 2 percentage absorption signals
when probe beam frequency is scanned. Green dashed line is two
gaussians fit to data. . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

2-4 Photodiode signals for two different probe powers and scanning ranges.
The first and second panels show 3.3% and 3.2% absorption on 𝐹 = 2
to 𝐹 ′ = 3 transition. The gray scope trace shows laser power without
atomic flux. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

2-5 Li atomic beam absorption measurement setup. Li cooling/trapping
laser is set to free scanning while the beam intensity is stabilized using
the AOM shown in this setup. Na flux measurement is also done via
shining a probe beam between differential pumping tubes. On theother hand, in the case of Na, the probe beam is generated using the
MOT absorption scanning setup(described in the text) while the Na
laser is locked. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13

36

2-6 Measurement of Li flux using absorption between differential pumping
tubes. The upper panel shows the phodiode voltage while laser isfreely scanning. The inverted copy of the absorption dip is also visible
in the downwards back scanning slope. The middle panel shows themodulation transfer spectroscopy error signal generated by the scan.
The dispersive feature with the biggest amplitude is the cross-oversignal used to lock the Li laser everyday. The bottom panel is thepercentage absorption signal. . . . . . . . . . . . . . . . . . . . . . . .

37

2-7 Shift of the magnetic trap center position created by inner coil gradientas bias of outer coils increase. The shift is proportional to the gradientvalue. Gradient per ampere value is extracted from the slope of data
and the set current value in coils measured by a flux gate. . . . . . .

41

2-8 Shift of the magnetic trap center for a fixed Side Axis bias as inner
quadrupole gradient is changed. The shift is inversely proportional
to the gradient value set by the inner coils control voltage. Bias per
ampere value is extracted from the slope of data and the set current
value in coils measured by a flux gate. . . . . . . . . . . . . . . . . .

42

2-9 Measurement of 𝐼𝑠𝑎𝑡 in units of the average count in the backgroundlaser image. Each black dot is an image and the dashed line is the fitof Equation 2.8 to the data. The fit gives 𝐼𝑠𝑎𝑡 = 36 average counts fora 5𝜇s pulse. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

2-10 An example measurement showing saturated missing count number in
the image due to optical pumping as the imaging pulse strength isincreased. The atom number is measured using the saturated value
and camera specific count to photon conversion factor. . . . . . . . .

46

2-11 Fitting the impurity percentage as the function of RF control voltage

49

3-1 Polarizibility advantage of the Ti:Sapphire light compared to its lifetime disadvantage in order to find feasible detuning range . . . . . . .
14

54

3-2 ODT beams with respect to vacuum chamber. ODT1 is the beam wecharacterize in this chapter and use in the main results of this thesis.
ODT2 is the crossing beam to increase the weak axial confinement of
ODT1. The slower beam is also shown as reference. CAD drawing istaken from Aviv Keshet’s PhD thesis[17].

. . . . . . . . . . . . . . .

56

3-3 the free space NuAmp output beam path

. . . . . . . . . . . . . . .

58

3-4 Gaussian 𝜎 width of the beam in two principle axis as the second
imaging lens is scanned. Two minima are displaced along the beam
propagation direction. We model the effect of astigmatism in Section
3.4.4 using this feature. . . . . . . . . . . . . . . . . . . . . . . . . . .

60

3-5 Shape Oscillation in Radial Kinetic Energy oscillates at twice the trap
frequency and damps with a time constant 𝜏 = 11𝜇s . . . . . . . . . .

63

3-6 Evolution of TOF kinetic energies during the thermalization time scale.
Initial total energy is 625𝜇K, final one 643𝜇K with axial bigger thanradial. There was 30% atom loss during this measurement . . . . . .

63

3-7 Radial trap frequency measurement for the deepest trap with 6.2W
power. Two panels show the shape oscillation in two radial axes. . . .

64

3-8 Measured radial trapping frequencies for different trap powers. The
dashed lines are expected 𝑃 scaling. . . . . . . . . . . . . . . . . . . .

64

3-9 Axial center-of-mass oscillation induced by gradient displacement . .

67

3-10 Factor 𝛼 = 𝑈𝑈0 as function of tilt parameter . . . . . . . . . . . . . . .

68

3-11 The second spilling experiment data and fit.

We extract ratio ofRayleigh range to Trap Depth from this measurement . . . . . . . . .

703-12 Trap in the Rayleigh range as the astigmatism Δ is increased from 0
to 4𝑧𝑅 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

72

3-13 Trap in the radial range at z=0 plane, as the astigmatism Δ is increased
from 0 to 4𝑧𝑅 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-14 Depth and Curvatures in the range Δ = (0, 2𝑧𝑅 ). . . . . . . . . . .

72
73

3-15 Increase in the axial and radial frequency ratio in the range Δ =(0, 2𝑧𝑅 ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15

73

3-16 Ratio of anharmonic and harmonic mean densities as a function offilling factor 𝜂 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

76

3-17 Ratio of anharmonic and harmonic mean squared densities as a function of filling factor 𝜂 . . . . . . . . . . . . . . . . . . . . . . . . . . .

77

3-18 Evolution of the trap with increasing tilt paramter . . . . . . . . . . .

79

3-19 Evolution of the effective trap depth and trapping frequencies withincreasing tilt parameter radial . . . . . . . . . . . . . . . . . . . . .

80

3-20 Evolution of the axial trapping potential for three different tilt parameters 82
3-21 Evolution of the effective trap depth and trapping frequencies withincreasing tilt parameter axial . . . . . . . . . . . . . . . . . . . . . .

82

3-22 Evolution of the effective trap depth and trapping frequencies withincreasing tilt parameter axial . . . . . . . . . . . . . . . . . . . . . .

83

3-23 Heating in 5.4W trap. Panel-(a) shows the atom number decay. Panel(b) displays the increase in the temperature during 50s hold. A linear
fit to the earlier part of the data gives a heating rate of 7.15𝜇K/s.

.

85

4-1 𝑉𝑒𝑓 𝑓 of equation-4.8 for 𝑙 = 1 and 𝑟 in (50, 500)𝑎0 using 6 Li parameters

92

4-2 Probability distribution of scattering into a direction for 𝑝 partial wave 94
4-3 Example thermalization data. Error bars are statistical standard deviation of the data. Thermalization rate is determined using the exponential fit given in the dashed line. The inlet shows the averagetemperature. There is no external heating. . . . . . . . . . . . . . . .

99

4-4 Measurement of the background 𝑝-wave scattering length. A linear
fit by weighing the data with inverse of variances (vertical error bars)
gives |𝑎𝑝𝑏𝑔 | = 39+1.3−1.6 𝑎0 . Dashed lines are |𝑎𝑝𝑏𝑔 | = 40.3𝑎0 and 37.4𝑎0
lines respectively representing 95% confidence bound of the fit. Thereis also a 20% systematic uncertainty on the horizontal axis due touncertainty in atom number counting. . . . . . . . . . . . . . . . . . . 100
16

4-5 Scaling of 𝐿3 with temperature. The black dashed line is a fit to 𝑇 2
power scaling. Gray dot-dashed lines show 𝑇 and 𝑇 3 scaling. Error bars
show one standard deviation uncertainty in the extracted fit parameter𝐿3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4-6 𝐿3 measurement with less anharmonic cloud. Panel-(a) shows timedependence of the atom number and the fit to Equation-4.33. Panel(b) shows the time dependence of the Temperatures. Temperature goesup by almost factor three mainly due to trap photon scattering. Thismeans a reduction of factor three in Γ3𝑏 , proving the importance ofconsidering time dependent rate in the analysis. . . . . . . . . . . . . 104
4-7 bosonic Cr fermionic Li dipolar loss parameter comparison as a functionof magnetic field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

4-8 Dipolar relaxation of state |6⟩ into state |1⟩. As state |6⟩ goes throughdipolar relaxation, we observe creation of |1⟩ impurity atoms.

. . . . 108

4-9 The ratio of good to bad collisions (Panel-(a)) and the elastic collision
rate near the 𝑝-wave Feshbach resonance . . . . . . . . . . . . . . . . 112
4-10 The ratio of good to bad collisions and elastic collision rates far awayfrom the 𝑝-wave Feshbach resonance. The solid line shows the ratio
for an harmonically trapped cloud with 𝑇 = 𝑇𝐹 generically assuming
𝜏𝑣𝑎𝑐𝑢𝑢𝑚 = 60 seconds. The dashed line is the elastic collision rate of
the same cloud. We explored the marked region in this work. . . . . . 113
4-11 𝑠 and 𝑝-wave collision comparison. Panel-(a) shows the rate of collisions involving an atom with speed 𝑣 with respect to the thermallyaveraged collision rate. Panel-(b) is the Panel-(a) weighted by theMaxwell-Boltzman speed distribution. Panel-(c) is the ratio of two
curves. Panel-(d) is the fraction of collisions up to 𝐸𝑘𝑖𝑛𝑒𝑡𝑖𝑐 = 𝜂𝑘𝐵 𝑇 . . 117
17

4-12 𝑝-wave evaporation. Panel (a) is average of 4 shots for a sample with
𝑇 /𝑇𝐹 = 0.42. Panel (b) displays the evolution of 𝑇 /𝑇𝐹 vs the atom
number. Panel (c) and (d) are gaussian fit to 1.7𝜎 radius wings ofthe atomic cloud in Panel(a). The gaussian fit clearly overestimates
the peak and expected Fermi-Dirac distribution, a sign of quantum
degeneracy. Fermi-Dirac profile (the red line) by the atom number,
temperature and trapping frequencies misses low velocity atoms, a signof lack of low velocity 𝑝-wave collisions.

. . . . . . . . . . . . . . . . 120

4-13 𝑝-wave evaporation. Radially averaged profile for the cloud in Figure
4-12 Panel-(a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
4-14 Growing fugacity discrepancy with depth of the cut . . . . . . . . . . 122
4-15 𝑝-wave evaporation. Radially average cloud profiles before and after introducing 𝑠-wave collisions using spin impurity. The red line shows the
Fermi-Dirac profile set by the atom number, temperature and trapping
frequencies, dots are the radially averaged OD values. . . . . . . . . . 123
5-1 Pauli Blocking of light scattering. Blue circles are a smaller(left) anda bigger(right) Fermi sea. Yellow arrows represent size of the recoil
momentum, same for both cases. As one can see from dashed circles,
overlap of the original and recoiled Fermi sea is much bigger for bigger
Fermi energies. Therefore, the suppression of light scattering increases
as Fermi Energy gets bigger. . . . . . . . . . . . . . . . . . . . . . . . 126
C-1 Layout of AOMs in the Li laser table . . . . . . . . . . . . . . . . . . 135
C-2 Li modulation transfer spectroscopy setup . . . . . . . . . . . . . . . 136
C-3 Layout of AOMs in the Na laser table . . . . . . . . . . . . . . . . . . 137
C-4 The final IR light setup . . . . . . . . . . . . . . . . . . . . . . . . . . 138

18

List of Tables
2.1

Locations of p-wave Feshbach resonances of the lowest hyperfine groundstate combinations from[27] . . . . . . . . . . . . . . . . . . . . . . .

40

2.2

Characteristics of magnets in BEC2 lab . . . . . . . . . . . . . . . . .

40

3.1

Kinetic (K.E) and Potential Energies (P.E) pre-release, in the momentof capture, after shape oscillations and at steady state19. . . . . . . .

61

20

Chapter 1
Introduction
There are two fundamental classes of particles: bosons and fermions. Bosons haveinteger spin, can occupy the same quantum state and can condense. On the other
hand, fermions have half-integer spin, obey Pauli exclusion principle and have to pair
up in order to condense.
Pauli exclusion principle prohibits fermions from being in the same quantum state,
and acts an effective spatial repulsion stemming from single particle statistics. The
pair correlation goes to zero at close distances, and 𝑠-wave interaction is fundamentally prohibited. Structure of all atoms, and properties of all materials in nature is amanifestation of this fundamental characteristic.
Ultracold degenerate quantum gases have been systems of choice for testing quantum mechanics, quantum statistics and quantum many-body physics for the last 25
years, because features of the tested system are well-controlled parameters, tailoredby the experimenter. For example, interactions between constituent atoms, confining potentials and temperature of the trapped degenerate gas are tunable in a broad
range.
Using our ultracold bosons and fermions, we can study physics happening indifferent scales in nature. The first observation of superfluidity was when liquid 4 He
was cooled below its critical temperature in 1938 by Kapitza, Allen and Misener.
4

He atoms are bosons. Under the critical temperature 2.2K, part of them undergo

Bose-Einstein condensation thus become superfluid. Associated densities are around21

1022 cm−3 . In contrast, quantization of circulation, a signature of superfluidity, in arotating 23 Na BEC was photographed in 2001[1]. Temperatures associated with thissystem is below 200𝑛K and densities are on the order of 1014 cm−3 . Despite ordersof magnitude difference, we study the same physics using ultracold atoms.
Another example we can give is the study of strongly interacting Fermi gases. We
can find it in neutron stars with a mass bigger than our sun’s squeezed into a ten mileradius giving densities of 1038 cm−3 . We can study the same physics using stronglyinteracting ultracold 6 Li atoms at orders of magnitude smaller densities around 1013
cm−3 [18].
In the planning phase of experiments described in this thesis, we set ourselves
to study optical properties of fermions. In particular, our main goal was to observe
Pauli blocking of light scattering in a trapped fermionic sample. After scattering
light, an atom wants end up in a final state within 2𝐸𝑅 of its original state. In caseof deeply degenerate fermions with 𝐸𝐹 ≫ 𝐸𝑅 , those final states are already occupied
except a thin layer of atoms around the fermi sea. As a result, the light scattering is
suppressed. The condition of having Fermi energies much bigger than recoil energy
sets the density scale in the problem to at least 1015 cm−3 , among highest achieved
with ultracold atoms trapped in an optical trap.
We decided to study this using a spin-polarized fermion cloud in an optical dipole
trap because it is the most simple and stable system we can generate in the lab. Mostof the time, it is considered non-interacting at low temperatures due to lack of 𝑠-wave
interactions and 𝑘𝐹8 scaling of 𝑝-wave elastic collision rates at low temperature. However, squeezing fermions to high densities brings 𝑝-wave interactions to an observableregime. Therefore we have to understand 𝑝-wave losses and 𝑝-wave elastic collisions.
In this thesis, we study 𝑝-wave collisions of fermions in a tight 1064nm optical
dipole trap. This project was the first time we cooled 6 Li to degeneracy in BEC2 labsince I joined the lab. We start by giving a general background of our experiment in
Chapter 2. In the following chapter, we describe the design and characterization of
our optical trap. In chapter 4, we discuss 𝑝-wave collisions and discuss main resultsof this thesis. Finally, we conclude this thesis with the outlook for the next BEC2
22

experiment. Other projects I contributed during my PhD ([21][19][29]) is describedin [20] and Boris Shteynas’ PhD thesis.

23

24

Chapter 2

The Experimental Apparatus

The current iteration of the BEC2 apparatus became operational in 2012. Detailsof the machine design can be found in Aviv Keshet’s PhD thesis[17]. It is a typical
Ketterle group atom cooling and trapping machine with an effusive dual-species oven,zeeman slower tube and steel vacuum chamber where science happens. The oven,capable of producing collimated 23 Na & 6 Li atomic beams, and double differential
pumping stage following the oven is a replica of Claudiu Stan’s design[32] except themain nozzle.

We improved the main nozzle design after a loss of ultra-high vacuum incident.
In early Summer 2018, the BEC2 machine had an oven change for replacing ourdepleted alkali sources. Then in mid-September, the pressure went up to 10−2 Torr in
all chambers of the machine. After some investigation, we realized that a small chunkof alkali fell onto a viewport between the oven and the first differential pumping tube
due to gravity while cleaning the excess alkali deposition after the main nozzle. Overtime, the alkali chunk reacted with glass and cracked the glass near its connection
to the metal as seen in Figure 2-1. We also found a major leak in the double-sided
flange holding the main nozzle.
25

Figure 2-1: The cracked viewport

During the rebaking of the vacuum chamber, we developed a microchannel arraynozzle to replace the old nozzle design. The new design improves the collimation,
therefore it prolongs the oven lifetime 25 times more. In other words, it reduces thefrequency of oven changes significantly. Furthermore, the amount of wasted alkali
and resulting deposition on the inner surfaces after the nozzle is also 25 times less.
This eliminates the possibility of a similar mistake and resulting vacuum loss.
We start this chapter by describing details of the new nozzle’s design. Then, we
26

report measurements of Na & Li atomic flux generated by the new nozzle. Measuring
the flux is important to document the performance. Additionally, it assured us thatthe oven was functioning properly while we could not trap atoms in the magneto
optical trap, and we were searching source of this problem after the bakeout.
In the following section, we go over calibrations crucial for the main results of thisthesis. First, we document characteristics of our magnets for the future reference.
We use these magnets to generate magnetic gradients and biases to control atoms’internal and external degrees of freedom. After the magnet characteristics, we present
calibrations related to imaging and atom number counting. These calibrations ruleout any systematic in our Fermi-Dirac fits(see section 4-3.3) due to imaging. Finally,in the last section, we describe how we control the |2⟩ impurity level in the |1⟩ − |2⟩
mixture. We wanted this control to compare the performance of 𝑝-wave collisions and𝑠-wave collisions through a small spin impurity minority(section 4-3.3).

2.1

The New Microchannel Nozzle

2.1.1

Effusion from a Hole

Effusion means escaping atoms can form an atomic beam and not flow hydrodynamically. Among atoms escaping through an effusive hole of negligible thickness and area
𝐴ℎ , for a particular solid angle Ω with angle 𝜃 defined with respect to the normal of
the hole area, all atoms with speed 𝑣 in volume 𝑑𝑉 = 𝑣𝑑𝑡𝐴ℎ cos 𝜃 will pass throughthe hole.
The number of atoms with speed 𝑣 in the mentioned volume is 𝑑𝑁 = 𝑛𝑑𝑉 𝑓𝑣 𝑑Ω
4𝜋
where 𝑛 is the atom density and 𝑓𝑣 is the the Maxwell-Boltzman speed distribution.
After averaging over the speed distribution, flux of atoms per solid angle is𝑑𝑁
1=
𝑛𝑣𝑚𝑒𝑎𝑛 𝐴ℎ cos 𝜃
𝑑𝑡𝑑Ω
4𝜋

(2.1)

The flux has a cos 𝜃 distribution which emphasizes the forward direction the mostcompared to all angles. Integrating over all solid angles in the right half space(forward
27

direction) gives a flux of1𝑑𝑁= 𝑛𝑣𝑚𝑒𝑎𝑛 𝐴ℎ
𝑑𝑡
4

(2.2)

In the case of a channel of finite length instead of a hole with negligible thickness,
the total flux becomes𝑑𝑁
1= 𝛼 𝑛𝑣𝑚𝑒𝑎𝑛 𝐴ℎ
𝑑𝑡
4

(2.3)

where 𝛼 is a constant determined by the channel geometry. In particular, for a long
cylindrical tube of length 𝑙 >> radius 𝑟, 𝛼 = 38 𝑟𝑙 [26].

2.1.2

Design of the Nozzle

In typical a Ketterle group machine, atoms roughly travel a meter in the zeeman
slower before being captured by a centimeter big magneto optical trap. This means,
to first order, among atoms exiting the oven only ones with smaller than 𝜃 = 10mrad
divergence angle will be captured by the magneto optical trap. Having an atomic
beam collimated as tight as this constraint would make the alignment of atomic beam
to the trapping volume cumbersome. As a result, the logical design consideration is
to choose a collimation angle where we do not waste the whole 𝜋 radian of atoms,
while having easy alignment of atomic beam to the machine.
Our old nozzle design has an aspect ratio of 0.5 with 10mm diameter tube on a
5mm thick stainless steel piece. This corresponds to a collimation at 𝜃 = 𝜋4 rad. With
this design, we had to change our Na source every 18 months.
In the new design, we use an array of stacked hypodermic needles as our collimatorlike in [28] with an easier design to manufacture compared to theirs. There are also
recirculating oven designs with very long lifespan. In these ovens, atoms that are
not captured by the trap are recycled. Overall, stacking hypodermic needles was theeasiest way to increase the atomic source lifetime without a major change in the oven
design. It allows us to have a similar area to the previous effusive hole, while taking
advantage of the better collimation of higher aspect ratio tubes. The forward flux at
𝜃 = 0 rad is the same for both designs to the first order. Therefore the lifetime should
be proportional to the aspect ratio and inversely proportional to the collimation angle.
28

MicroGroup cut 304𝐻25𝑋𝑋 model number 304 stainless steel hypodermic needles in an aspect ratio of 12.5 for us. These tubes have 500𝜇m outer diameter, 400𝜇m
inner diameter and 5mm length. The new aspect ratio gives 𝜃 = 80mrad collimation, enough to increase the lifetime by 25 while keeping the alignment of the beamreasonable.
The biggest problem in stacking an array of tubes is making sure that there is nodefect in stacking and tubes are parallel to the axis of the experiment. We overcame
this problem by machining an equilateral triangle housing as shown in Figure 2-2(a).
The 60𝑜 apex constraints tubes to pack hexagonally. The housing comprises threesymmetrical pieces screwed together to form the equilateral triangle in the middle ofa 35mm diameter 5mm thick stainless steel piece. After stacking 105 needles as in
Figure 2-2(b)(c)(d), the nozzle is bolted on the standard nozzle holder vacuum piece.

29

Figure 2-2: Description of the new micro-channel nozzle. Panel-(a) shows the housing
to hold hypodermic needles. After stacking needles on two connected pieces as inPanel (b) and (c), the third piece is also clamped to the other two. The clamping
pressure prevents tubes from sliding out. Panel-(e) shows the nozzle piece connectedto the nozzle holder. Off-centering of the new nozzle piece is intentional in order tomatch position of the old nozzle hence alignment of the atomic beam to the machine.

30

2.1.3

Na Flux Measurement

The absorption measurements between differential pumping tubes of the BEC2 machine shows a 3.3% peak probe light absorption on the 𝐹 = 2 to 𝐹 = 3 transitionof Na 𝐷2-line. The 1𝜎-width of this line is 10.5 MHz. Modelling our nozzle as aneffusive hole, this absorption corresponds to a 9.28 × 1014 Na atoms/sec atomic flux
out of the oven.
Figure 2-3 shows the absorption data. An intensity stabilized laser beam of power
1.6𝜇W and 0.8mm 1/𝑒2 radius probes the atomic beam perpendicularly. This laser
beam is generated by coupling another laser beam, which is 30MHz red detuned fromthe 𝐹 = 2 to 𝐹 ′ = 3 transition, to the MOT scanning setup with double passes. The
MOT scanning shifts the frequency of the input light by 2𝑓𝐻 − 4𝑓𝐿 where usually 𝑓𝐻
is a fixed frequency and 𝑓𝐿 is scanned. For this measurement 𝑓𝐻 is set to 133MHz
and 𝑓𝐿 is scanned in 50-80MHz range.
When the probe beam frequency was scanned, the peak absorption on 𝐹 = 2 to𝐹 = 3 was 3.33%. Furthermore, the peak absorption on 𝐹 = 2 to 𝐹 = 2 transition,
58.3 MHz away from the former transition, was measured as 1.32% (Figure 2-3). The
ratio of two peak absorption values 3.33/1.32 = 2.52 is close to relative transition
strengths of these two transitions 14/5 = 2.8. The spike around 𝑓𝐿 = 68MHz is an
artifact due to beam intensity stabilization as shown in Figure 2-4, which shows theabsorption without the atomic flux.
In the first panel of Figure 2-4, the probe frequency is scanned 120MHz and two
hyperfine transitions are visible. In the second panel, the scan range was only 88
MHz, and the probe power is stabilized at 16𝜇W with the same beam waist as theprevious one. For this panel, the center of the second hyperfine transition is outof scanning range. Gray curves are photodiode signals when the atomic shutter was
closed. All features except the one due to the beam intensity stabilization disappeared
as expected.
31

′

′

Figure 2-3: 𝐹 = 2 to 𝐹 = 3 and 𝐹 = 2 to 𝐹 = 2 percentage absorption signals whenprobe beam frequency is scanned. Green dashed line is two gaussians fit to data.

32

Figure 2-4: Photodiode signals for two different probe powers and scanning ranges.
The first and second panels show 3.3% and 3.2% absorption on 𝐹 = 2 to 𝐹 ′ = 3
transition. The gray scope trace shows laser power without atomic flux.

33ModelThe probe beam absorption is proportional to the local and density. We can relate
the atomic flux 𝐹 to the local beam density 𝑛𝑙𝑜𝑐𝑎𝑙 using
𝐹 = 𝑛𝑙𝑜𝑐𝑎𝑙 𝑣¯𝐴𝑏𝑒𝑎𝑚(2.4)

where 𝐴𝑏𝑒𝑎𝑚 = 16.62mm2 is the cross-section of an atomic beam on the plane, perpendicular to the beam direction. This cross-section is set by the size of our differential√ √︀pumping tubes. The average velocity in an atomic beam is 𝑣¯ = 34 𝜋 2𝑘𝑇 /𝑚[26]when using the temperature of mixing chambers 𝑇 = 415𝑜 C, 𝑣¯ = 938.6 m/s.

Reduction in Absorption Peak due to Line-Broadening
As the absorption feature broadens due to the transverse velocity distribution of theatomic beam, the peak is reduced in a way keeping the total area constant. So, we
have a peak absorption
Δ𝐼Γ
≈𝑛𝑙𝑜𝑐𝑎𝑙 𝜎0 𝑧𝑏𝑒𝑎𝑚𝐼Γ𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑(2.5)

where Γ is the natural FWHM of the transition and Γ𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑 = 2.355𝜎𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑
is the measured FWHM of the absorption. For the Na measurement, saturation
parameter 𝑠 is approximately zero, Γ = 10MHz, 𝜎𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑 ≈ 10.5MHz correspondingΓto a FWHM of 24.7MHz. Hence the reduction factor is approximately Γ𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑= 0.4.

𝜎0 is the absorption cross-section and 𝑧𝑏𝑒𝑎𝑚 = 4.6mm is limited by the differential
pumping tube.

The Absorption Cross-section′

The absorption cross-section of |𝐹 = 2, 𝑚𝐹 = +2⟩ to |𝐹 = 3, 𝑚𝐹 = +3⟩ transition is
3 2
𝜆589 . In our measurement, all 𝑚𝐹 states are populated equally. Due to lack𝜎𝑟𝑒𝑠 = 2𝜋

of a well-defined quantization axis, an atom can undergo 𝜎± and 𝜋 transitions with′equal probabilities. This reduces the cross-section for 𝐹 = 2 to 𝐹 = 3 transition to𝜎0 = 7/15𝜎𝑟𝑒𝑠 = 7.73 × 10−14 m2 .
34

Local Atom Density and Atomic Flux at the Nozzle

Using the measured value, the density of 𝐹 = 2 ground hyperfine state Na atoms is𝑛𝑙𝑜𝑐𝑎𝑙 = 2.3 × 1014 1/m3 . Thus, the 𝐹 = 2 sodium flux between differential pumping
tubes is 𝐹 = 3.6 × 1012 atoms/s. Furthermore, because of the thermal distribution,
only 5/8 of total atomic density should be 𝐹 = 2 atoms. This makes the flux out of
𝑙𝑛𝑑)2 = 1.3 × 1015 atoms/s where 𝑙𝑛𝑑 is the distance fromour nozzle 𝐹𝑛𝑜𝑧𝑧𝑙𝑒 = 58 𝐹 ( 𝑧𝑏𝑒𝑎𝑚/2
𝑙𝑛𝑑
nozzle to differential pumping tubes, The factor ( 𝑧𝑏𝑒𝑎𝑚)2 is the effect of relative solid/2angle cut by differential pumping tubes.

2.1.4

Li Flux Measurement

The peak absorption between the differential pumping tubes of the BEC2 machine on
the 𝐹 = 3/2 hyperfine manifold of 6 𝐿𝑖 using a probe light with 𝐼/𝐼𝑠𝑎𝑡 ≪ 1 is 0.66%.
One 𝜎 Gaussian width of the absorption feature is 20.7MHz. The peak absorption
for the 𝐹 = 1/2 hyperfine manifold is 0.29% and 𝜎 = 27.6MHz. The ratio of two
measured peak absorption values 2.3 are close to the ratio of the two hyperfine state
populations in a thermal distribution 2.
Figure 2-5 describes the measurement process. The laser is set to scan 480MHz
while the probe beam intensity is stabilized using PID control through an AOM.
Figure 2-6 top panel shows the raw absorption signal. The upward slope during thescan window is an imperfect intensity stabilization due to a wide laser frequency
scan window. Turning the atomic beam shutter off makes absorption dips disappear.
The absorption features are also visible in the downward sloping backscan of thelaser. Figure 2-6 middle panel shows the modulation transfer spectroscopy error
signal demonstrating the laser scan. The left feature with two peaks is the signal
corresponding to 𝐹 = 1/2 hyperfine manifold.
35

Absorption Signal PhotodiodeAtomic beam
d=4.6mm
Differential pumping tubes

PhotodiodeBeam sampler

PBS

AOM

Fiber coupler
Figure 2-5: Li atomic beam absorption measurement setup. Li cooling/trapping laser
is set to free scanning while the beam intensity is stabilized using the AOM shownin this setup. Na flux measurement is also done via shining a probe beam between
differential pumping tubes. On the other hand, in the case of Na, the probe beam isgenerated using the MOT absorption scanning setup(described in the text) while the
Na laser is locked.

36

1.885
0.2

0.4

0.2

0.4

Percent absorption

Error Signal

Phodiode Voltage[V]

1.89

0.1

0
0.2

0.22

0.24

0.26

0.28

0.3

0.32

0.34

0.36

0.38

Figure 2-6: Measurement of Li flux using absorption between differential pumping
tubes. The upper panel shows the phodiode voltage while laser is freely scanning.
The inverted copy of the absorption dip is also visible in the downwards back scanningslope. The middle panel shows the modulation transfer spectroscopy error signal
generated by the scan. The dispersive feature with the biggest amplitude is the crossover signal used to lock the Li laser everyday. The bottom panel is the percentage
absorption signal.

37

The 𝐹 = 3/2 hyperfine manifold absorption peak is the sum of three unresolvable
absorption processes belonging to 𝐹 = 3/2 → 𝐹 ′ = 5/2, 𝐹 = 3/2 → 𝐹 ′ = 3/2 and𝐹 = 3/2 → 𝐹 ′ = 1/2. The natural linewidth of 6 Li is 5.87MHz. In addition, theexcited state hyperfine splitting on 𝐷2 line is 4.4MHz. To the first order, we canconsider the 𝐹 = 3/2 hyperfine peak we observe as a single peak with sum of all
3 2
𝜆671 = 1.1 × 10−13 m2 smeared 1.75 times due to excitedfeatures 𝜎0 = 𝜎𝑟𝑒𝑠 /2 = 4𝜋

state hyperfine broadening.
Using equation 2.5, the local density of 𝐹 = 3/2 hypefine state is 𝑛𝑙𝑜𝑐𝑎𝑙 = 0.65 ×
1014 m−3 . Thus the flux of 𝐹 = 3/2 atoms between differential pumping tubes is
𝐹 = 1.9 × 1012 atoms/s. 2/3 of the total atoms should be in the 𝐹 = 3/2 hyperfine
𝑙𝑛𝑑state. Therefore 𝐹𝑛𝑜𝑧𝑧𝑙𝑒 = 23 𝐹 ( 𝑧𝑏𝑒𝑎𝑚)2 = 0.66 × 1015 atoms/s.
/2

2.2

Experimental Techniques

2.2.1

Calibration of our Magnets

We use home made coils wrapped around various parts of the apparatus to generatemagnetic gradients and biases. The major workhorse magnetic field generators in
the experiment are inner and outer pair coils placed on top and bottom buckets ofthe experiment. For details of their geometries, reader can consult to Aviv Keshet’s
PhD thesis[17]. They are driven by an h-bridge circuit allowing us to put them in
Helmholtz or anti-Helmoltz configurations to generate a magnetic bias or a quadrupole
field. A 500A power supply for each coil pair results in close to 1000 G/cm maximum
magnetic gradient on the tight axis along the gravity. Additionally, we use a pairof coils called "side axis coils" for generating bias field to create a quantization axis
for optical pumping and tilting our optical dipole trap on the same axis. Finally,last year, we wound an imaging bias coil around the top and bottom bucket flanges
generating a bias field in the axis of gravity. This section describes calibration of all
the existing magnets and documents in Table 2.2.
The term Feshbach resonance describes the resonance enhancement of interac38

tion of ultracold atoms when the collision energy matches the energy of a molecularbound state. The latter can be tuned by changing the magnetic field. As a result,interaction strength is controlled by changing the magnetic field. The lossy 𝑝-wave
Feshbach resonances are well-known[35][27]. Experimentally measured locations ofthese resonances for combinations of the lowest hyperfine ground state are given in
Table 2.1.
We find the location of |1⟩-|1⟩ Feschbach resonance as we scan the magnetic field
and monitor atom losses. Then we use the identified location to calibrate the bias
field generated by our outer coils. The measured magnetic bias generation capabilityof outer coils is 3.53 G/A.
After having a precise value for the bias we apply, we measure the gradient per
ampere capability of our inner coil with this calibrated bias field. Specifically, we
monitor the shift in the position of the magnetic trap cloud for a fixed gradient
generated by inner coils as a function of outer bias given in Figure 2-7. The measured
value is 1.25 G/cm/A in the gravity direction, and it should be half of this value inthe plane perpendicular to the gravity axis.
Using the calibrated inner gradient value, we can calibrate our imaging bias andside axis bias coils by monitoring the shift of the magnetically trapped cloud for a
fixed bias as a function of the gradient strength as in Figure 2-8. Side Axis coils
generate 0.4 G/A bias field. The same parameter for the imaging bias is 0.11 G/A.
Finally, we calibrate the outer gradient by using a fixed imaging bias and measuring the shift of the magnetically trapped cloud similar to calibration of inner coils
gradient. The outer coils generate 0.852 G/cm/A in the gravity direction.

39

Resonance|2⟩-|2⟩|1⟩-|2⟩|1⟩-|1⟩𝐵
215G
185.09G
159.14G

Table 2.1: Locations of p-wave Feshbach resonances of the lowest hyperfine groundstate combinations from[27]MagnetGradient per A
Inner coils
1.25 G/cm/A (in ⃗𝑔 axis)
Outer coils
0.85 G/cm/A (in ⃗𝑔 axis)
Side bias coils
Imaging bias coils

Bias per A
3.53 G/A
0.4 G/A
0.11 G/A

Table 2.2: Characteristics of magnets in BEC2 lab

40

Magtrap Position [mm]

3.5
3
2.5
2
1.5
1
0.5
0

5

10

15

20

25

Outer Bias [G]Figure 2-7: Shift of the magnetic trap center position created by inner coil gradient asbias of outer coils increase. The shift is proportional to the gradient value. Gradientper ampere value is extracted from the slope of data and the set current value in coils
measured by a flux gate.

41

4.2

Y Position [mm]

4

3.8

3.6

3.4

3.2
0.5

1

1.5

2

2.5

3

3.5

Inner coils control voltage[V]Figure 2-8: Shift of the magnetic trap center for a fixed Side Axis bias as inner
quadrupole gradient is changed. The shift is inversely proportional to the gradientvalue set by the inner coils control voltage. Bias per ampere value is extracted fromthe slope of data and the set current value in coils measured by a flux gate.

422.2.2

Measuring the Saturation Intensity and Accurate AtomCounting

When the Rabi frequency of the radiation is strong compared to the natural decay rateΓ of the excited state, a collection of two-level atoms scatters photons at a maximum
rate of Γ/2. This effect is known as saturation. The number of scattered photons
should be the same as the number of photons absorbed from the driving radiation.
This allows us to see that the absorption cross-section diminishes and goes to zero as
driving radiation intensity increases to give a saturated rate of Γ/2.
Imaging artifacts due to saturation can disguise as quantum degeneracy in FermiDirac fits to the time-of-flight pictures by altering the profile of distribution of atoms.
Here, we determine 𝐼𝑠𝑎𝑡 experimentally in terms of the number of counts in the experimental background laser beam images by measuring the peak optical density(OD)
after the sample preparation as a function of blasted photon number. We modify the
Beer’s law by taking the saturation into account
𝑑𝐼
𝑛𝜎0
=−
𝐼
𝑑𝑧
1 + 𝐼/𝐼𝑠𝑎𝑡(2.6)

where 𝐼𝑠𝑎𝑡 is the saturation intensity and 𝜎0 is the absorption cross-section. In caseof no saturation, we would measure an optical density of 𝑂𝐷𝑠=0 = 𝑛𝜎0 Δ𝑧 where Δ𝑧
is thickness of the sample along the probe beam propagation direction. Then
∫︁ 𝐼𝑓
𝐼𝑖

𝑑𝐼
𝐼
(1 +
) = −𝑂𝐷𝑠=0
𝐼
𝐼𝑠𝑎𝑡

(2.7)

𝐼𝑖
𝑂𝐷𝑠=0 − 𝑂𝐷=
𝐼𝑠𝑎𝑡
1 − 𝑒−𝑂𝐷(2.8)

since 𝐼𝑓 /𝐼𝑖 = 𝑒−𝑂𝐷 , we have

Measuring the 𝑂𝐷 as a function of 𝐼𝑖 in the units of the average count in the
background laser image, and fitting the measurement to Equation 2.8 determines the𝐼𝑠𝑎𝑡 in the units of average count in the image. An example measurement using a 5𝜇s
43imaging pulse is given in Figure 2-9. For our ethernet Pixelfly camera, the saturation
𝐼< 0.05intensity is 36 average counts in the image for this 5𝜇s pulse. We stayed at 𝐼𝑠𝑎𝑡and ended up using a 1.8𝜇W, 15𝜇 imaging pulse with a gaussian 𝜎 = 1.6mm for our
Fermi-Dirac fits.

Average Count in Image
20

60

40

80

100

120

0.3

Peak measured OD

0.4

0.5

0.6

0.7

0.8

0.9

Figure 2-9: Measurement of 𝐼𝑠𝑎𝑡 in units of the average count in the background laser
image. Each black dot is an image and the dashed line is the fit of Equation 2.8 tothe data. The fit gives 𝐼𝑠𝑎𝑡 = 36 average counts for a 5𝜇s pulse.

44

Having an accurate atom number count is essential for estimating the Fermi energyof a trapped fermion cloud. The usual way of getting an atom number count is tocalculate the OD sum by integrating the OD distribution in the time-of-flight image
and convert this sum to an atom number via multiplying by real pixel area anddividing by the absorption cross-section. This requires accurate knowledge of the real
pixel area thus magnification of the imaging system. We confirmed magnification of
our imaging systems by taking a picture of an airforce target in the imaging plane.

In addition to the usual way, we also measured the atom number by measuring the
maximum number of photons missing from the laser beam as we increase intensityof the incident imaging beam. A state |1⟩ atom illuminated by 𝜎+ polarized lightfalls back to the 𝐹 = 1/2 state 5 out of 9 photon scattering event. This means each
atom on average will scatter 2.25 photons before entire population is transferred to
3/2 hyperfine manifold. The described optical pumping effect saturates the numberof missing photons in the absorption image and allows us to accurately determine
number of atoms. It is named ’titration of atoms by photons’ by Wolfgang Ketterle.

An example measurement is shown in Figure 2-10. By using the plateaued missing
number of counts, photon to count ratio and average photon scattering per atom
before optical pumping, we get an accurate estimate of total number of atoms. We
measured the photon to count conversion factor of Pixelfly camera as 11.97 photons
per count by comparing the estimated number of photons in an imaging pulse and thetotal number of counts in the image for the same pulse. The measured value matches
the 0.3 quantum efficiency around 670nm and given electron/count conversion factorof 3.8. Finally, the atom number counted using this method agrees with the number
count of the OD sum within 10%.
45

10 5

18

Missing count number

15

12

9

6

3

0
0

30

60

90

Average count in without atom image (Probe laser flux)

Figure 2-10: An example measurement showing saturated missing count number in
the image due to optical pumping as the imaging pulse strength is increased. The
atom number is measured using the saturated value and camera specific count tophoton conversion factor.

46

2.2.3|1⟩-|2⟩ Mixture Population Ratio Control

The convention for having 𝑠-wave collisions with fermions is to use two different atoms,
two different isotopes of the same atom or two different hyperfine states of the sameatom. Among mixtures of different hyperfine states of 6 Li, |1⟩-|2⟩ mixture(mixture of
two lowest hyperfine states) is the most preferred one owing to the wide-range tunability of interactions using Feshbach resonances and lack of strong decay mechanisms.
We use |1⟩-|2⟩ mixture in order to introduce 𝑠-wave collisions to our sample where
𝑝-wave collisions were limited because of the relative lack of low-velocity collisions(seeSection 4.3.1). We chose a diabatic Landau-Zener sweep in state |1⟩ cloud in orderto have partial transfer to |2⟩.
The Landau-Zener transition probability is2

˙

𝑃 = 1 − 𝑒−2𝜋Ω /𝛿(2.9)

where the Rabi frequency Ω is proportional to the magnetic field of the RF 𝐵𝑟𝑓
which depends on the current in the loop thus linearly proportional to the voltage on
the loop. Assuming that the gain of the amplifier is independent of the seed and it isa constant gain voltage amplifier,2

𝑃 = 1 − 𝑒−𝐴 𝑉𝑟𝑓(2.10)

Assume we have 𝑁 atoms, 𝑁1 = (1 − 𝑃 )𝑁 of them is in |1⟩ with absorption crosssection 𝜎1 and 𝑃 𝑁 is in |2⟩ with absorption cross section 𝜎2 . Then the atom number
calculated using the integrated OD of the image and 𝜎1 will have the functional form

𝑁 (𝑃 ) = 𝑁0 [(1 − 𝑃 ) + 𝑃 𝛼](2.11)

where 𝛼 = 𝜎2 /𝜎1 . Fitting the Equation-2.4 to measured atom number as a function of the RF seed gives Figure 2-11. We control the impurity percentage 1 − 𝑁1 /𝑁
by changing the RF seed. We chose to have 10% impurity in order to have as much 𝑠47

wave collisions as we can without changing the cloud profile in time-of-flight picturesdue to impurity population.

48

105

Atoms Number

4.5

2.5
0

40

80

RF seed [mV]Figure 2-11: Fitting the impurity percentage as the function of RF control voltage

49

50

Chapter 3
Optical Dipole Trap
In this chapter, we describe and characterize the optical dipole trap (ODT) setup
designed to trap 6 Li atoms. Our major target for this trap was achieving high densities
thus high Fermi energies. At these high densities, the study of 𝑝-wave collisions
become important. With the addition of deep degeneracy( 𝑇𝑇𝐹 ≪ 1), these Fermi
energies (much bigger than the photon recoil energy) can also be used to observe thePauli blocking of light scattering in a trapped quantum degenerate fermionic cloud[30]
We start with introducing a single beam ODT in Section-3.1. High densities
require a tight confinement. Optical dipole traps with tight confinement can be
implemented using a high-power far-detuned light beam or relatively less power nearresonant beam. Both approaches have their limitations. We address these limitationsin Section-3.2.
After deciding which detuning to use, the second step to strong confinement is atightly focused laser beam. Moreover, we tried a cross-dipole trap for compensating
a single beam’s weak axial confinement and creating tight confinement in all 3 axes.
Loading a spin polarized sample of fermions to a dimple created by two tightly focused beams crossing at 45𝑜 turned out to be challenging. The small volume overlap,
sensitivity of alignment of the crossing and lack of collisional loading due to slow
𝑝-wave collision rates prevented us from getting a boost in density using the second
beam. General overview of implementation of our optical trap is in Section-3.3.
Once the trap is built, the following step is to characterize it. Section-3.4 goes51

over the trap characterization process. In the following Section-3.5, we introduce a
way to control the trap depth without loosing significant confinement as a tool to
use in evaporative cooling. Finally, we report the measurement of the heating rate in
Section-3.6.

3.1

Brief Theory of a Single Beam ODT

We considered using a far-detuned (1064nm) or a near-resonant (around 681nm
Ti:Sapphire) light. For 1064𝑛𝑚 and 681𝑛𝑚 lights, the detunings are ≈ 160THz
and ≈ 6THz respectively. Since these detunings are much bigger than the fine structure splitting of 6 Li Δ𝐹 𝑆 = 10GHz, we can treat the atom as a two-level system anduse following equations[11] to calculate the dipole trap depth and scattering rate fora laser beam with intensity distribution 𝐼(𝑟),𝑈𝑑𝑖𝑝 (𝑟) = −

3𝜋𝑐2
1
1
Γ(
+
)𝐼(𝑟)
3
2𝜔0 𝜔0 − 𝜔 𝜔0 + 𝜔= 𝛼𝐼(𝑟)Γ𝑠𝑐 (𝑟) =

(3.1)
(3.2)

1
3𝜋𝑐2 𝜔 3 2
1
( )Γ(
+
)2 𝐼(𝑟)32ℎ̄𝜔0 𝜔0
𝜔0 − 𝜔 𝜔0 + 𝜔= 𝛽𝐼(𝑟)

(3.3)
(3.4)

where the intensity of a gaussian beam propagating in 𝑧 is given by
𝐼(𝑟) =

2𝑃 −2𝑟2 /𝑤2
𝑒
𝜋𝑤2

(3.5)

𝑃 is the laser power, 𝑟 is the radial distance from center of the beam, 𝑤 =√︀
𝑤0 1 + (𝜆𝑧/𝜋𝑤02 )2 and 𝑧 is the axial distance from the focal point of the beam. We
can use harmonic approximation around the trap center
52

𝛼2𝑃
2𝑟2
(1
−
)
𝜋𝑤2
𝑤2
𝜆𝑧 2 2𝑟2
𝛼2𝑃(1−() − 2)
≈
𝜋𝑤02
𝜋𝑤02
𝑤0

𝑈𝑑𝑖𝑝 (𝑟) ≈

(3.6)
(3.7)

If we define 𝑈0 = 𝛼2𝑃, this is a harmonic oscillator with trapping frequencies
𝜋𝑤02
√︂
√︁
4𝑈0
2𝑈0
. We can further define an effective frequency
𝜔𝑟 = 𝑚𝑤
2 and 𝜔𝑧 =
𝜋𝑤2
0

𝑚(

𝜆

0 )2

𝜔𝑒𝑓 𝑓 = (𝜔𝑟2 𝜔𝑧 )1/3

(3.8)
√︃

24/3 𝜆1/3= ( 5/6 )
𝜋

𝛼𝑃
14/3

(3.9)

𝑚𝑤0

Some important effective trap frequency 𝜔𝑒𝑓 𝑓 scalings are𝜔𝑒𝑓 𝑓 ∝

3.2√
𝑃𝜔𝑒𝑓 𝑓 ∝

1
7/3

(3.10)

𝑤0

Near or Far Resonant Light for our Experiment?

For the near-resonant trap, the lifetime of the trapped sample will be limited by thenear-resonant trap photon scattering rate Γ𝑇 𝑖:𝑆𝑎𝑝𝑝ℎ𝑖𝑟𝑒 . We can define the trapping
frequency over the laser photon scattering frequency as a figure of merit for theexperiment.

4/3 1/3

𝜔𝑒𝑓 𝑓
Γ𝑇 𝑖:𝑆𝑎𝑝𝑝ℎ𝑖𝑟𝑒=

𝜆)( 2 𝜋5/6

√︁

2𝑃
𝛽 𝜋𝑤
2

𝛼𝑃
7/3
𝑚𝑤0

(3.11)

0This ratio should at least be around 10 in order for the experiment to be feasible.
We can also monitor the power advantage of the near-resonant light by comparing
polarizibilities at two wavelengths 𝛼𝑇 𝑖:𝑆𝑎𝑝𝑝ℎ𝑖𝑟𝑒 to 𝛼1064 . At the end of the day, weonly invest in near-resonant light if it provides a significant (𝛼𝑇 𝑖:𝑆𝑎𝑝𝑝ℎ𝑖𝑟𝑒 /𝛼1064 >100)53

advantage over using IR light because we have an IR arsenal in the lab, whereas
we had to bring the near-resonant light from BEC3. If we plot Equation-3.11 and𝛼𝑇 𝑖:𝑆𝑎𝑝𝑝ℎ𝑖𝑟𝑒 /𝛼1064 together for a 5𝜇m and 0.3𝑊 near-resonant trap as a function ofdetuning, we get Figure-3-1.

400
300
ωeff
ΓTi:Sapphire

200

αTi:Sapphire
α1064

100

2

4

6

δ in THz

Figure 3-1: Polarizibility advantage of the Ti:Sapphire light compared to its lifetime
disadvantage in order to find feasible detuning range

Using the criteria we described, the figure shows that the near-resonant light isonly advantegous in the detuning range 𝛿 = (0.3, 2.7)THz. Our near-resonant lightbudget was around 0.3W, an equivalent of 120W IR at 0.3THz detuning.
The degeneracy required by the Pauli Blocking experiment demands further cooling in ODT after symphatetic cooling in the magnetic trap. A near-resonant trap isless suitable for this purpose because of the limited lifetime. As a result, building an
IR trap was necessart for transferring atoms from the magnetic trap and subsequentevaporation. Considering our two 40W Nufern NuAmp amplifiers and 20W IPG 1064
laser, we decide to use our own IR light for mentioned target experiments instead ofinvesting in bringing the near-resonant light from next door. The near-resonant light
can still be used to squeeze the cloud further as the last step of the experiment if an
extra compression is important.
54

3.3

Experimental Implementation

3.3.1

Lens Choices

Achieving the highest Fermi energy possible requires focusing the trapping beam tothe smallest spot size we can generate within our experimental capabilities. The
shortest focal length lens we could place before viewports was 𝑓 = 250mm. Filling
this lens with a 𝐷 = 40mm diameter beam should give a diffraction limited spot
size of 𝑤 = 2𝜆𝑓= 4.2𝜇m. We chose the Thorlabs AC508-250-C-ML achromat as𝜋𝐷
our focusing lens considering the low 𝐹 number. We used a Galilean telescope madeby and Thorlabs AC508-500-C-ML(𝑓 = 500mm) achromat and Thorlabs LC4210C(𝑓 = −25mm) plano-concave lenses in order to expand the initial small laser beam
to 40mm diameter.

3.3.2

Implementation

Figure 3-2 shows our vacuum chamber. The port-4b has the biggest viewport area
and was only used for optical pumping. As a result, we chose the axis of 4b-4a as our
main ODT axis. We characterize this trap in following sections. ODT2 in the 3d-3aaxis is the second crossing beam in order to increase the weak axial confinement of
the first beam. In order to place the focusing lens for this second beam, we put the
ODT focusing lens in the MOT beam’s path. We correct the MOT beam beforehandby changing its size and divergence. Loading the dimple created by crossing of two
beams turned out to be time consuming and we only used ODT1 for achieving main
results of this thesis.
55

ODT 1ODT 2Slower Beams

Figure 3-2: ODT beams with respect to vacuum chamber. ODT1 is the beam wecharacterize in this chapter and use in the main results of this thesis. ODT2 is the
crossing beam to increase the weak axial confinement of ODT1. The slower beam isalso shown as reference. CAD drawing is taken from Aviv Keshet’s PhD thesis[17].
In the current version of the optical dipole trap setup, we establish independent
control over each beam by fiberizing and PID controlling each beam independently.
The AOM scheme is given in Appendix-C3. ODT1 light is delivered through a photonic bandgap fiber(LMA-PM-15) with a high power Alphanov connectors. In contrast, ODT2 light goes through a Thorlabs-PM980-XP fiber with a regular FC/APC
connector on the laser table side. Although there is room in our power budget, theODT2 light delivered to the experiment table is limited to 1W due to fiber’s damage threshold. Finally, instead of using a regular collimator and following Galilean
telescope as for ODT1, we use a commercial Schäfter + Kirchhoff collimator(60FCSMA-T-4-A100-03) for ODT2 giving a 24mm beam at the output. This increases our56

spot size but provides convenience.
Our mantra from the beginning was to keep the setup simple and this implied
controlling both beams together. For the initial iteration of the trap, we PID controlled the NuAmp output using a high-power water-cooled AOM (AAOptoelectronics
MQ40-A3-L1064-W). The first order of the AOM was delivered in free-space to theexperiment table(Figure 3-3). During the near 1.5m travel, the beam is transferredfrom one level of the optical table to a higher one using a periscope setup. On themain optical table, the beam was expanded using a single telescope and split into twoarms using a half waveplate and a cube. We relied on perpendicular polarizations toprevent interference of two ODT beams. Soon after trapping atoms with this initial
setup, we see that the focus of this beam is jittering spatially in the beam propagation
direction. Although, we attribute this to the pointing noise of the Nufern output, we
do not investigate the source of this jitter carefully due to time constraints.

57

Chamber
Isolator

Figure 3-3: the free space NuAmp output beam path

58

3.4

Characterization of the Trap

Fermi energy of a harmonically trapped Fermi gas is 𝐸𝐹 = ℎ̄¯𝜔 (6𝑁 )1/3 where 𝑁 is thetotal atom number and 𝜔¯ = (𝜔𝑥 𝜔𝑦 𝜔𝑧 )1/3 is the average trapping frequency. We alreadyemphasized the importance of an accurate atom number counting in estimation of our
Fermi energy (see Section-2.2.2). In addition to this, the accurate estimation of the
Fermi energy also requires understanding the trap and measuring trapping frequencies
properly.
The first characterization step after setting up the new trap is to measure the
IR spot size at the focus using an imaging system. We see two spatially separated
minima in two principle axes of the beam, a the first clue for astigmatism in our trap
as described in Section 3.4.1.
Following the measurement of the spot size, we characterize the trapping confinement felt by atoms in Section 3.4.2. For measuring trapping frequencies, we use shapeand center-of-mass oscillations of the cloud after pushing the cloud out of equilibrium.
These measurements ends up giving a ratio of radial to axial trapping frequency higher
than expected. This is another clue for astigmatism.
We measure the trap depth directly in Section 3.4.3 for a better understanding of
possible astigmatism. In section 3.4.4, we analyze the effect of astigmatism on trap
depth and trapping frequencies. Finally we close the section with an estimation of
the effect of deviations from harmonic potential on the mean density.

3.4.1

Imaging IR Spot SizeWe built an imaging system consisting a 50mm Thorlabs achromat in the side bucket(port 4-b of Figure 3-2) and a 250mm Thorlabs achromat as the second lens in order toimage the IR focus spot size. Each lens in the imaging system is equipped with control
on 3 position and 2 tilt degrees of freedom to make sure the IR beam is centered on
the lens, and is perpendicular to the beam propagation direction. This makes sure we
minimize the aberrations. We image the IR beam by moving the second lens of theimaging setup. Figure 3-4 shows the separation of minima of two principle axes of59the beam along the beam propagation direction. If it is not an artifact of the imaging
system, the result points towards astigmatism in the ODT beam.

Figure 3-4: Gaussian 𝜎 width of the beam in two principle axis as the second imaging
lens is scanned. Two minima are displaced along the beam propagation direction. We
model the effect of astigmatism in Section 3.4.4 using this feature.

3.4.2

Measurement of Trapping Frequencies

Shape Oscillations and Thermalization After Sudden Turn-off and On

In order to measure ODT frequencies, we suddenly turn the trap off for a short periodof time compared to radial trapping period 2𝜋/𝜔𝑟 . Atoms freely expand during theoff duration. When the trap is turned back on, the potential energy gain is bigger
on the tight confinement axis. This creates an imbalance between potential andkinetic energies in the moment of capture. Furthermore, there is an energy imbalance
between two tight and loose confinement axes. Evolution of energies is summarizedin Table 3.1.
60

Pre Rel.

CaptureAfter Sh.Os
𝜔 2 𝑡2𝑜𝑓 𝑓

Radial P.E.

𝐸𝑖 /6

(1 + 𝜔 2 𝑡2𝑜𝑓 𝑓 )𝐸𝑖 /6

Radial K.E

𝐸𝑖 /6

𝐸𝑖 /6

2
𝜔 2 𝑡2𝑜𝑓 𝑓(1 + 2 )𝐸𝑖 /6

Axial P.E

𝐸𝑖 /6

≈ 𝐸𝑖 /6

𝐸𝑖 /6

Axial K.E

𝐸𝑖 /6

𝐸𝑖 /6

𝐸𝑖 /6

(1 +

)𝐸𝑖 /6

Steady-State𝜔 2 𝑡2𝑜𝑓 𝑓)𝐸𝑖 /63
22
𝜔 𝑡(1 + 3𝑜𝑓 𝑓 )𝐸𝑖 /6(1 +𝜔 2 𝑡2𝑜𝑓 𝑓)𝐸𝑖 /63
22
𝜔 𝑡(1 + 3𝑜𝑓 𝑓 )𝐸𝑖 /6(1 +Table 3.1: Kinetic (K.E) and Potential Energies (P.E) pre-release, in the moment of
capture, after shape oscillations and at steady state

If we do not have any external heating, we can make a simple model of this process.
Define total radial and axial energies 𝐸𝑟 ,𝐸𝑎 and total initial energy 𝐸𝑖 . Then,
𝐸𝑟 = 𝐸𝑟.𝑘𝑖𝑛 + 𝐸𝑟.𝑝𝑜𝑡(3.12)

𝐸𝑎 = 𝐸𝑎.𝑘𝑖𝑛 + 𝐸𝑎.𝑝𝑜𝑡(3.13)

Δ𝐸𝑐𝑑 = (𝐸𝑟 − 𝐸𝑎 )(3.14)

where Δ𝐸𝑐𝑑 is the cross-dimensional energy difference. Total energy should beconserved. In the moment of capture, the energy is𝜔 2 𝑡2𝑜𝑓 𝑓)𝐸𝑖(2𝐸𝑟 + 𝐸𝑎 ) = (1 +3

(3.15)

where 𝜔 is the radial trapping frequency and 𝑡𝑜𝑓 𝑓 is the trap off duration. Elastic
collisions will make the energy flow cross-dimensionally until both dimensions havethe same energy. In other words,
˙ 𝑐𝑑 = −Γ𝑡ℎ𝑒𝑟𝑚 Δ𝐸𝑐𝑑
Δ𝐸

(3.16)

𝜔 2 𝑡2𝑜𝑓 𝑓 −Γ𝑡ℎ𝑒𝑟𝑚 𝑡
Δ𝐸𝑐𝑑 (𝑡) = 𝐸𝑖𝑒6

(3.17)

𝜔 2 𝑡2𝑜𝑓 𝑓 −Γ𝑡ℎ𝑒𝑟𝑚 𝑡
1 𝜔 2 𝑡2𝑜𝑓 𝑓
𝐸𝑟 = 𝐸𝑖 ( +) + 𝐸𝑖𝑒
3918(3.18)

Solving this gives,

Therefore,61Now, let’s investigate what is happening within a dimension. We can again define
an intra-dimensional energy difference(𝐸𝑟.𝑝𝑜𝑡 − 𝐸𝑟.𝑘𝑖𝑛 ) = Δ𝐸𝑖𝑑

(3.19)

This difference will show shape oscillations with 𝜔𝑠ℎ = 2𝜔𝑡𝑟𝑎𝑝 damped at a rate
Γ𝑑𝑎𝑚𝑝 due to anharmonicities.
˙ 𝑖𝑑 = −(Γ𝑑𝑎𝑚𝑝 + 𝑖𝜔𝑠ℎ𝑎𝑝𝑒 )Δ𝐸𝑖𝑑
Δ𝐸(3.20)

𝜔 2 𝑡2𝑜𝑓 𝑓
𝐸𝑖 𝑒−(Γ𝑑𝑎𝑚𝑝 𝑡+𝑖𝜔𝑠ℎ 𝑡)
Δ𝐸𝑖𝑑 (𝑡) =6(3.21)and the solution is

In TOF pictures, we look at the kinetic energy distribution of the cloud.
𝐸𝑟 = 𝐸𝑟.𝑝𝑜𝑡 + 𝐸𝑟.𝑘𝑖𝑛(3.22)

Δ𝐸𝑖𝑑 (𝑡) = 𝐸𝑟 − 2𝐸𝑟.𝑘𝑖𝑛 (𝑡)(3.23)

So, the TOF signal has the form
𝜔 2 𝑡2𝑜𝑓 𝑓
𝜔 2 𝑡2𝑜𝑓 𝑓 −(Γ𝑑𝑎𝑚𝑝 𝑡+𝑖𝜔𝑠ℎ 𝑡)𝐸𝑖−Γ𝑡ℎ𝑒𝑟𝑚 𝑡
𝐸𝑟.𝑘𝑖𝑛 (𝑡) = [(1 +)(1 + 𝑒
)−
𝑒
]
4
3
3

(3.24)

There are two separate time scales. Fast and damped shape oscillations at 2 times
trapping frequencies and slow thermalization through elastic collisions. An example
of our fast time scale signal of the deepest trap and 2𝜇s turn-off is in Figure 3-5.
Following Figure 3-6 shows the long time scale energy transfer.
62

250

Kinetic Energy [ K]

200

150

100

50

0
0

10

20

30
Hold time [us]

40

50

60

Figure 3-5: Shape Oscillation in Radial Kinetic Energy oscillates at twice the trap

Axial Kinetic Energy [ K]frequency and damps with a time constant 𝜏 = 11𝜇s
250

200

150

0

100

200

300

400

500

600

700

500

600

700

Radial Kinetic Energy [ K]

Hold time [ms]
250

200

150

0

100

200

300

400

Hold time [ms]Figure 3-6: Evolution of TOF kinetic energies during the thermalization time scale.
Initial total energy is 625𝜇K, final one 643𝜇K with axial bigger than radial. Therewas 30% atom loss during this measurement63

We measured radial trapping frequencies for various laser powers (𝑃 ) using thistechnique. The signal for our deepest trap with 6.2W power is given in Figure 3-7.
Measured radial trapping frequencies are 𝜔𝑟𝑥 = 102kHz and 𝜔𝑟𝑦 = 94kHz. Measured√
values across the trap power spectrum and the expected 𝑃 scaling is shown in Figure
3-8.
(a)

(b)

0.3

Size Y [mm]Size X [mm]

0.28

0.2
0

5

10

15

20

0.18

25

0

5

Hold in [us]

10

15

20

25

Hold in [us]Figure 3-7: Radial trap frequency measurement for the deepest trap with 6.2W power.
Two panels show the shape oscillation in two radial axes.

104

Trap Frequency [kHz]

11

5
2

4

6

Trap Power [W]Figure 3-8: Measured radial trapping frequencies for different trap powers. The
dashed lines are expected 𝑃 scaling.
64

If we apply this technique to our densest clouds, as in Figure 3-5 and 3-6, we
also see a total energy increase during the thermalization process. So we have to add
heating to our simple model.
Assuming heating rates on radial and axial directions are 𝐻𝑟 and 𝐻𝑎 , the mainequations become

𝐸𝑟 − 𝐸𝑎 = Δ𝐸𝑐𝑑

(3.25)

2𝐸𝑟 + 𝐸𝑎 = 𝐸(3.26)

We can describe the time evolution the cross-dimensional energy difference

˙ 𝑐𝑑 = −Γ𝑡ℎ Δ𝐸𝑐𝑑 + 𝐻𝑟 − 𝐻𝑎Δ𝐸

(3.27)𝑎)This equation has a steady state solution of Δ𝐸𝑐𝑑𝑠𝑠 = (𝐻𝑟Γ−𝐻. So any imbalance𝑡ℎin the heating 𝐻𝑟 − 𝐻𝑎 ̸= 0 should give us a finite difference in energies of two axes.
Moreover, the Γ𝑡ℎ is also time dependent for 𝑝-wave collisions since it scales withatom number and temperature. The total energy 𝐸 does not increase infinitely overtime because after an increase in temperatures, ratio of trap depth to temperature 𝜂
becomes low enough to increase the evaporation and stabilize the temperature.
Finally, for data with significant atom loss and heating, time dependent thermalization rate should be considered in the fitting routine while extracting the thermalization rate.

˙ 𝑐𝑑 = −Γ𝑡ℎ (𝑡)Δ𝐸𝑐𝑑 + 𝐻
Δ𝐸
where 𝐻 represents the net heating
∫︀ 𝑡

′

′

Multiplying both sides by 𝑒 0 Γ𝑡ℎ (𝑡 )𝑑𝑡 give65

(3.28)

∫︀ 𝑡

′

∫︀ 𝑡

′

′

′

˙ 𝑐𝑑 = 𝑒 0 Γ𝑡ℎ (𝑡 )𝑑𝑡 (−Γ𝑡ℎ (𝑡)Δ𝐸𝑐𝑑 + 𝐻)
𝑒 0 Γ𝑡ℎ (𝑡 )𝑑𝑡 Δ𝐸
∫︀ 𝑡

′

′

∫︀ 𝑡

′

′

𝑑(Δ𝐸𝑐𝑑 (𝑡)𝑒 0 Γ𝑡ℎ (𝑡 )𝑑𝑡 ) = 𝐻𝑒 0 Γ𝑡ℎ (𝑡 )𝑑𝑡 𝑑𝑡
∫︁ ∫︀
∫︀ 𝑡
𝑡
′ )𝑑𝑡′′′Γ(𝑡𝑡ℎ
Δ𝐸𝑐𝑑 (𝑡)𝑒 0= 𝐻 𝑒 0 Γ𝑡ℎ (𝑡 )𝑑𝑡 𝑑𝑡 + 𝐶∫︁ 𝑡 ∫︀ *∫︀
𝑡′′− 0𝑡 Γ𝑡ℎ (𝑡′ )𝑑𝑡′
𝑒 0 Γ𝑡ℎ (𝑡 )𝑑𝑡 𝑑𝑡* + 𝐶]
[𝐻
Δ𝐸𝑐𝑑 (𝑡) = 𝑒∫︁ 𝑡 0 ∫︀ 𝑡* Γ (𝑡′ )
𝑡ℎ
Γ (0)
𝑑𝑡′
˜
Δ𝐸𝑐𝑑 (𝑡) = 𝑒−Γ𝑡ℎ (0)𝑡 [𝐻𝑒 𝑡ℎ 0 Γ𝑡ℎ (0) 𝑑𝑡* + 𝐶]
∫︁0 𝑡
˜
˜′
Δ𝐸𝑐𝑑 (𝑡) = 𝑒−Γ𝑡ℎ (0)𝑡 [𝐻𝑒Γ𝑡ℎ (0)𝑡 𝑑𝑡˜′ + 𝐶]

(3.29)(3.30)(3.31)(3.32)(3.33)(3.34)

0where 𝑡˜ =∫︀ 𝑡 Γ𝑡ℎ (𝑡′ )0 Γ𝑡ℎ (0)𝑑𝑡′ . Fitting the cross-dimentional energy difference to Equation-

3.34 gives the initial rate Γ𝑡ℎ (0).
If we did not have the heating, we would just have∫︀ 𝑡′

Δ𝐸𝑐𝑑 (𝑡) = Δ𝐸𝑐𝑑 (0)𝑒− 0 Γ𝑡ℎ (𝑡 )𝑑𝑡= Δ𝐸𝑐𝑑 (0)𝑒

∫︀ 𝑡 Γ𝑡ℎ (𝑡′ )−Γ𝑡ℎ (0) 0 Γ
˜= Δ𝐸𝑐𝑑 (0)𝑒−Γ𝑡ℎ (0)𝑡where 𝑡˜ =∫︀ 𝑡 Γ𝑡ℎ (𝑡′ )0 Γ𝑡ℎ (0)

𝑑𝑡′ .

66

′

𝑡ℎ (0)(3.35)
𝑑𝑡′(3.36)(3.37)

Magnetic Force
We can induce a center-of-mass displacement by adiabatically ramping up a magnetic
gradient. Turning this magnetic gradient off non-adiabatically induces a center-ofmass oscillation of the cloud. We measure axial trapping frequency of our ODTusing this method. The signal is in Figure 3-9. Measured trapping frequency is𝜔𝑧 = 2𝜋 × 1.99 kHz for a 6.1W trap. This value is smaller than the value predictedusing the Gaussian beam spot size set by the measured radial frequency and the beam
power.

Y Position [mm]Figure 3-9: Axial center-of-mass oscillation induced by gradient displacement
2.995

2.975
0

0.6

1.2

Hold after gradient off [msec]

67

1.8

3.4.3

Measurement of the Trap Depth through Spilling

The trap depth is also an important parameter to know when dealing with evaporative
cooling. Since we have a discrepancy between the radial and axial trap frequencies,
we want an independent measurement of the trap depth.
The number of atoms up to an energy in a 3D Harmonic trap is∫︁
𝛽 3 𝐸 2 −𝛽ℰ
𝒩 (𝐸) = 𝑁ℰ 𝑒 𝑑ℰ2 0
𝑁= (2 + 𝑒−𝛽𝐸 (−2 − 𝛽𝐸(2 + 𝛽𝐸)))2(3.38)(3.39)

Section 3.5 presents a method to reduce the depth of an optical trap using amagnetic force. Figure 3-10 shows the depth reduction 𝛼 as a function of the defined
tilt parameter 𝜉 (defined in Section 3.5) proportional to the strength of the magnetic
force and the ratio of Rayleigh range 𝑧𝑅 = 𝜋𝑤02 /𝜆 to the trap depth 𝑈0 .

Figure 3-10: Factor 𝛼 = 𝑈𝑈0 as function of tilt parameter

U
U0
1
0.8
0.6
0.4
0.2
0.2

0.4

0.6
68

0.8

1

ξ

We can fit this curve to an 8th order polynomial of the form

𝛼𝑓 =

8
∑︁

𝑎𝑖 𝜉 𝑖(3.40)

𝑖=0

where 𝑎𝑖 are fit coefficients.

The analytical form of the density of states for a single beam ODT is calculated
in [31]. Harmonic approximation describes the behavior fairly accurately up to athird of the trap depth but the actual density of states deviates significantly for highenergies. For measurements in this section, we stayed where 𝛽𝐸 ≪ 1 and harmonic
approximation presents the reality well. In the bottom of trap, number of atoms up
to energy 𝐸 is approximately
𝒩 (𝐸) = 𝑁 (𝛽𝐸)2(3.41)

We perform two types of experiments to measure the trap depth. In our firstexperiment, we reduce the optical trap’s power fast, spill atoms and ramp it back up
fast for 3 different magnetic force strengths varied by changing the applied bias field.
We take time-of-flight pictures out of this full trap for counting remaining atoms. In
the second experiment, we monitor the number of atoms up to an energy by spilling
atoms via fast reduction of optical potential to a smaller control voltage in presenceof a strong magnetic gradient without ramping the optical trap at the end. We takean image after turning off the magnetic fields.

In all experiments, we fit the remaining atom number to equation 3.41, where the𝐶energy level we probe is 𝐸 = 𝑉𝑉𝐶0
𝑈0 𝛼(𝛾2 𝑉𝐶𝑧𝑅 ) and 𝑉𝐶 is the scanned parameter ODT
𝑉𝐶0

𝑈0

control voltage in Cicero, 𝑉𝐶0 = 3.5𝑉 is the control voltage for the deepest trap and𝛾2 is a known constant dependent on the magnetic gradient and bias we applied.
69

Figure 3-11: The second spilling experiment data and fit. We extract ratio of Rayleighrange to Trap Depth from this measurement
104

Remaining Atom Number

2.5

0
0.3

0.4
ODT Control Voltage [V]0.5

Fit to the second experiment data gives 𝑧𝑈𝑅0 = 2.912 ± 0.014 × 10−6 cm/𝜇K for the
6.2W ODT. The first experiment also has similar results of 𝑧𝑈𝑅0 1 = 1.9727±0.029×10−6
cm/𝜇K, 𝑧𝑈𝑅0 2 = 2.9085 ± 0.02 × 10−6 cm/𝜇K, 𝑧𝑈𝑅0 3 = 3.072 ± 0.026 × 10−6 cm/𝜇K for
increasing magnetic force strengths. In this analysis, we ignored the effect of magnetic
gradient to the density of states in the potential.

703.4.4

Astigmatism

The ratio of the measured axial and radial trapping frequencies is bigger than ratio
predicted by a gaussian beam. Furthermore, we saw spatially separated focuses for
two radial directions of the beam when we imaged our IR spot for the trap. These
point to astigmatism. Here, we give a quantitative model for the effect of astigmatismon the trapping potential depth and trapping frequencies.

The optical trapping potential created by a gaussian beam with waist 𝑤𝑥 , 𝑤𝑦 andtwo radial focuses separated by Δ meters is

1

1

√︁
𝑈 (𝑥, 𝑦, 𝑧) = 𝑈0 √︁𝑒2
1 + ( 𝑧𝑧𝑅 )2 1 + ( 𝑧+Δ
)
𝑧

−2𝑥2
2 (1+( 𝑧 )2 )
𝑤𝑥
𝑧𝑅

𝑒

−2𝑦 2
2 (1+( 𝑧+Δ )2 )𝑤𝑦𝑧𝑅

(3.42)

𝑅

Parametrizing 𝑥,𝑦 in units of 𝑤𝑥 ,𝑤𝑦 and Δ,𝑧 in 𝑧𝑅 gives

2

2

−2𝑦
−2𝑥
1
1
𝑈 (𝑥, 𝑦, 𝑧)2
1+(𝑧− Δ
)2 1+(𝑧+ Δ
2
2)
√︁
√︁
=
𝑒
𝑒
𝑈0
1 + (𝑧 − Δ )2 1 + (𝑧 + Δ )22

(3.43)

2

Figure-3-12 shows the potential as a function of Δ in the beam propagation direction. For Δ < 2, there is a single minimum. At Δ = 2, curvature of the minima
disappear. When Δ is bigger than 2, two minima appear. Radial profile of thepotential is given in Figure-3-13.
71

U
U0

-4

2

-2

zR

4

Δ=0
Δ=2
Δ=4

Figure 3-12: Trap in the Rayleigh range as the astigmatism Δ is increased from 0 to
4𝑧𝑅
U
U0

-2

1

-1

2

r in w

Δ=0
Δ=2
Δ=4

Figure 3-13: Trap in the radial range at z=0 plane, as the astigmatism Δ is increased
from 0 to 4𝑧𝑅72

We can also calculate the change in radial and axial trapping frequencies due toastigmatism. Change of depth and curvatures is given in Figure-3-14. Increase of theratio of trapping frequencies is in Figure-3-15.

1

U
U0

0.5

ωzΔ
ωz0
ωrΔ
ωr0

1

2

Δ in zR

Figure 3-14: Depth and Curvatures in the range Δ = (0, 2𝑧𝑅 )4ωrΔ ωz0ωr0 ωzΔ2

1

2

Δ in zR

Figure 3-15: Increase in the axial and radial frequency ratio in the range Δ = (0, 2𝑧𝑅 )73

For a single beam ODT without any anomalies, if power of the beam is 𝑃 , wehave
2𝑃𝑈0 = 𝛼 2
𝜋𝑤
√︃ 0
4𝑈0
𝜔𝑅 =
𝑚𝑤02
√︃
2𝑈0
𝜔𝑧 =
𝑚𝑧𝑅2

(3.44)(3.45)(3.46)

A way to look at these equations is: 𝑃 = 5.95W and measured 𝜔𝑅 = 89kHz fixes
a spot size of 𝑤0 = 7.97𝜇m and depth of 𝑈0 = 3.62mK. Using these numbers, thecalculated 𝜔𝑧 = 2.69kHz does not match the measured one 2.019kHz. The expected
trap frequency ratio using these numbers is 𝜔𝑅 /𝜔𝑧 = 33.3 but we measured a ratio of𝜔𝑅 /𝜔𝑧 = 44.3.
Assuming we only have astigmatism, measured ratio of 𝜔𝑅 /𝜔𝑧 indicates Δ =1.05𝑧𝑅 . This means, if we did not have any astigmatism, our actual axial frequency
would be 𝜔𝑧 = 3.43kHz and 𝜔𝑅 = 114.28kHz, corresponding to 7.05𝜇m spot size and4.62mK trap depth.

743.4.5Anharmonic Correction to Density Averages

A trapped Bose-Einstein condensate occupies bottom of the trapping optical potential. In contrast, fermions occupy a larger volume due to Pauli exclusion principle.
After occupation of certain fraction of the trap depth, the harmonic approximation is
not a good representation of the reality. Here, we calculate the ratio of the harmonic
and anharmonic densities as a function of the filling factor 𝜂 = 𝑈0 /𝑘𝐵 𝑇 .
The Boltzmann spatial distribution is𝑛(⃗𝑟) = 𝐶𝑒−𝛽𝑈 (⃗𝑟)

(3.47)Atom number constraint gives,
𝐶 = ∫︀

𝑁
𝑒−𝛽𝑈 (⃗𝑟) 𝑑𝑉(3.48)

Then mean density is
𝐶 2 𝑒−2𝛽𝑈 (⃗𝑟) 𝑑𝑉𝑁∫︀ −2𝛽𝑈(⃗𝑟)𝑒𝑑𝑉= 𝑁 ∫︀ −𝛽𝑈 (⃗𝑟)
[ 𝑒
𝑑𝑉 ]2
∫︀

𝑛𝑚𝑒𝑎𝑛 =

(3.49)(3.50)The tilt due to gravity reduces the trap depth in the axial direction to 0.98𝑈0(See section 3-5 for calculation of the depth reduction due to tilt). For a single beam
ODT, trap depth drops to 2% at ≈ 1.5𝑤0 and ≈ 7.5𝑧𝑅 .
We can parametrize the single beam optical trap in units of 𝑤0 and 𝑧𝑟 .
2𝑟 2

𝜂 𝑒 (1+𝑧2 )𝑈 (𝑟, 𝑧) = −𝛽 1 + 𝑧2(3.51)

Harmonically approximating this potential gives,
𝑈ℎ𝑎𝑟𝑚 (𝑟, 𝑧) =75

𝜂 2(𝑧 + 2𝑟2 )𝛽(3.52)

We numerically evaluate the harmonic and anharmonic densities according to∫︀ 𝑧=7.5 ∫︀ 𝑟=1.5 −2𝛽𝑈 (𝑟,𝑧)𝑒2𝜋𝑟𝑑𝑟𝑑𝑧𝑟=0𝑛𝑚,𝑎𝑛ℎ𝑎𝑟𝑚𝑜𝑛𝑖𝑐 = ∫︀𝑧=−7.5
∫︀
𝑟=1.5
𝑧=7.5
[ 𝑧=−7.5 𝑟=0 𝑒−𝛽𝑈 (𝑟,𝑧) 2𝜋𝑟𝑑𝑟𝑑𝑧]2∫︀ 𝑧=7.5 ∫︀ 𝑟=1.5 −2𝛽𝑈ℎ𝑎𝑟𝑚 (𝑟,𝑧) 2𝜋𝑟𝑑𝑟𝑑𝑧𝑒𝑧=−7.5 𝑟=0𝑛𝑚,ℎ𝑎𝑟𝑚𝑜𝑛𝑖𝑐 = ∫︀ 𝑧=7.5 ∫︀ 𝑟=1.5[ 𝑧=−7.5 𝑟=0 𝑒−𝛽𝑈ℎ𝑎𝑟𝑚 (𝑟,𝑧) 2𝜋𝑟𝑑𝑟𝑑𝑧]2(3.53)(3.54)We then take the ratio 𝑛𝑚,𝑎𝑛ℎ𝑎𝑟𝑚𝑜𝑛𝑖𝑐 /𝑛𝑚,ℎ𝑎𝑟𝑚𝑜𝑛𝑖𝑐 . The result is given in Figure
3-16.

nanharmonic
nharmonic
0.7

0
1

5

10

15

η

Figure 3-16: Ratio of anharmonic and harmonic mean densities as a function of filling
factor 𝜂

76

Similarly,

𝐶 3 𝑒−3𝛽𝑈 (⃗𝑟) 𝑑𝑉
𝑁−3𝛽𝑈 (⃗𝑟)𝑒𝑑𝑉= 𝑁 2 ∫︀ −𝛽𝑈 (⃗𝑟)
[ 𝑒
𝑑𝑉 ]3
∫︀

(𝑛2 )𝑚𝑒𝑎𝑛 =(3.55)(3.56)

Result of integrating over the same boundary is given in Figure 3-17.

n2 anharmonicn2 harmonic
0.55

0
1

5

10

15

η

Figure 3-17: Ratio of anharmonic and harmonic mean squared densities as a functionof filling factor 𝜂

77

3.5

Tilt Analysis

In this section, we analyze the feasibility of evaporating atoms in an optical trap byapplying a magnetic force without significant loss in trap confinement. This technique
was first introduced in [16]. Here, we are following a similar idea with the exceptionof neglecting gravity. The gravitational force is 3G/cm for the lowest hyperfine stateof 6 Li at low magnetic fields. It shifts the bottom of our potential negligibly and it is
two orders of magnitude smaller than gradients we use.
First, we show that we do not have the experimental capability to reduce thetrap depth significantly via applying the force in the radial confinement direction. In
the following section, we investigate the scenario with tilting the trap in the beam
propagation direction of a single beam optical dipole trap. The analysis shows that
this axial tilt has a similar performance to the radial tilt while giving a bigger control
capability over the potential.

3.5.1

Radial Tilt

A crossed optical dipole trap potential with two beams of waist 𝑤 propagating on 𝑥
and 𝑦 directions and a magnetic gradient on the 𝑧 direction is2(𝑥2 +𝑧 2 )2(𝑦 2 +𝑧 2 )1
𝑈 = −𝑚𝐹 𝑔𝐹 𝜇𝐵 𝐵 ′ 𝑧 − 𝑈0 (𝑒− 𝑤2 + 𝑒− 𝑤2 )2

(3.57)

Let’s parametrize the trap and express all spatial measures in units of 𝑤. We
can define a tilt parameter 𝜉 in range (0,1) such that when it is one, confinementof the trap is zero. On the axis of magnetic gradient (𝑥, 𝑦 = 0), the trap can beapproximated as:

𝑈 = −𝑚𝐹 𝑔𝐹 𝜇𝐵 𝐵 ′ 𝑤 𝑧 − 𝑈0 𝑒−2𝑧
The slope and curvature of the potential is78

2(3.58)

𝑑𝑈
2= −𝑚𝐹 𝑔𝐹 𝜇𝐵 𝐵 ′ 𝑤 + 4𝑈0 𝑧𝑒−2𝑧
𝑑𝑧
𝑑2 𝑈
2
2
= 𝑈0 (4𝑒−2𝑧 − (4𝑧)2 𝑒−2𝑧 )2𝑑𝑧(3.59)(3.60)

A stable trap requires positive curvature around a minimum. 𝑈 has a positive
curvature between in the range 𝑧 = (−0.5, 0.5). Therefore, it becomes unstable whenminima shifts beyond 𝑧 = 0.5. In other words, when
𝑒1/2 𝑚𝐹 𝑔𝐹 𝜇𝐵 𝐵 ′ 𝑤=12𝑈0
1/2 𝑚 𝑔 𝜇 𝐵 ′ 𝑤
𝐹 𝐹 𝐵Now, we can define 𝜉 = 𝑒(3.61)

and write the trap as

2𝑈0

𝑈
2
= −2𝑒−1/2 𝜉𝑧 − 𝑒−2𝑧𝑈0(3.62)

U
U0

ξ=0

-2

1

-1

2

z in w

ξ=0.6
ξ=1

-1

Figure 3-18: Evolution of the trap with increasing tilt paramter
The evolution of the trap bottom and trap depth with respect to tilt parameter 𝜉
is given in Figure-3-18. For tilt parameters between 0 and 1, we can define an effective
depth for the trap as the difference in potential energy of two extrema. Furthermore,79

we can evaluate the trap frequencies of the tilted trap using the second derivative(curvature) at the shifted minimum point. The numerical evaluation of effective trap
depth and trapping frequencies as a function 𝜉 is given in Figure-3-19

1
0.8
Ueff
U0

0.6

ωz
ωz0

0.4

ωx,y
ωx0

0.2
0.2

0.4

0.6

0.8

1

ξ

Figure 3-19: Evolution of the effective trap depth and trapping frequencies withincreasing tilt parameter radial
For the deepest trap we can achieve in the experiment 𝑈0 = 3.6mK, the strongest
gradient we can apply is 450G/cm. We can push the Breit-Rabi slope up to 𝑚𝐹 𝑔𝐹 = 1.
Considering out effective waist is 8𝜇m, biggest tilt parameter we can achieve radiallyis 𝜉 = 0.0055 corresponding to negligible reduction in the trap depth. We haveto modify the technique in order to make it useful for our experiment. In the next
section, we analyze reducing the trap depth in the beam propagation direction insteadof using a radial tilt.

3.5.2

Axial Tilt

Axial potential of a single beam ODT is a Lorentzian whereas the radial potential
is a Gaussian. Therefore it was not clear what to expect as a result of applying amagnetic tilt without doing the following analysis. A single-beam optical dipole trap
potential with a gradient applied along the beam propagation direction is
80

2

2

−2(𝑥 +𝑦 )1
𝑤2 (1+( 𝑧𝑧 )2 )𝑅− 𝑔𝐹 𝑚𝐹 𝜇𝐵 𝐵 ′ 𝑧
𝑈 (𝑥, 𝑦, 𝑧) = −𝑈0𝑒
1 + ( 𝑧𝑧𝑅 )2(3.63)

Let’s parametrize the trap and express 𝑧 coordinate in the unit of 𝑧𝑅 and 𝑥, 𝑦
coordinates in units of 𝑤 and use a similar analysis as the previous section.

−2(𝑥2 +𝑦 2 )𝑈 (𝑥, 𝑦, 𝑧)
𝑔𝐹 𝑚𝐹 𝜇𝐵 𝐵 ′ 𝑧𝑅
1(1+𝑧 2 )
−
=−
𝑧
𝑒
𝑈0
1 + 𝑧2𝑈0(3.64)

Again, on the axis of magnetic gradient the trap is

1
𝑔𝐹 𝑚𝐹 𝜇𝐵 𝐵 ′ 𝑧𝑅
𝑈 (0, 0, 𝑧)
=−
𝑧
−
𝑈0
1 + 𝑧2𝑈0(3.65)The slope and curvature of the potential on this axis𝑑𝑈
𝑔𝐹 𝑚𝐹 𝜇𝐵 𝐵 ′ 𝑧𝑅
2𝑧
=−
+
𝑑𝑧
𝑈0
(1 + 𝑧 2 )2
2
(2𝑧)2
𝑑2 𝑈
=
(1
−
)
𝑑𝑧 2
(1 + 𝑧 2 )2
(1 + 𝑧 2 )(3.66)(3.67)

𝑈 has a positive curvature between in the range 𝑧 = (− √13 , √13 ). Therefore, it
becomes unstable when the minimum shifts beyond 𝑧 = √13 . Then
8 𝑔 𝑚 𝜇 𝐵 ′ 𝑧𝑅
√ 𝐹 𝐹 𝐵=1
𝑈0
3 3

(3.68)

With the proper definition of the tilt parameter the trapping potential is√2(𝑥2 +𝑦 2 )𝑈 (𝑥, 𝑦, 𝑧)
1
3
3
−
2
2
(1+𝑧 )=𝑒−𝜉𝑧
𝑈0
1 + 𝑧28
81(3.69)

U
U0

ξ=0

-2

1

-1

2

z in zR

ξ=0.6
ξ=1

-1

Figure 3-20: Evolution of the axial trapping potential for three different tilt parameters

1
0.8
Ueff
U0

0.6

ωz
ωz0

0.4

ωx,y
ωx0

0.2
0.2

0.4

0.6

0.8

1

ξ

Figure 3-21: Evolution of the effective trap depth and trapping frequencies withincreasing tilt parameter axial

Figure 3-20 shows the effect of tilting on the trapping potential. Doing same
numerical analysis for the effective trap depth and trapping frequencies as in theprevious section gives Figure-3-21.
82

We can also compare radial and axial tilt by plotting effective trapping frequency𝜔¯ = (𝜔𝑥 𝜔𝑦 𝜔𝑧 )1/3 vs trap depth in Figure-3-22.
ω
ω0

1
0.8
0.6

Radial Tilt

0.4

Axial Tilt
Simple Decompression

0.2

0.001

0.01

0.1

0.3

1

Ueff
U0

Figure 3-22: Evolution of the effective trap depth and trapping frequencies withincreasing tilt parameter axial

Radial and Axial tilting have similar performance. They both manage to keep
60% of the confinement while reducing the trap depth to zero. Contrarily, the commonly used method of simple decompression diminishes the confinement hence theevaporation performance.

3.6

Measurement of Heating Rate for |1⟩

We measured the heating rate in the 5.4𝑊 trap by preparing a low atom number
sample and scanning the hold time up to 50s. Figure 3-23 displays the atom number
loss and increase in the temperature. Heating rate for this power is 7.15𝜇K/s. Following the analysis and Equation-30a in [11], this should correspond to a trap depthof 1𝑚K whereas our estimated trap depth is 3.1mK. In other words, our measured
heating rate is 3 times less than the calculated trap photon scattering heating rate.
Increase in the temperature starts to bend down (Figure 3-23 Panel-(b)) because
heating decreases when temperature approaches the trap depth[11]. Furthermore,83

evaporation also becomes more prominent as trap depth to temperature ratio drops
due to heating.

84

(a)

(b)

10 5

3

700

Atoms Number

Temperature in uK

RadialAxial
Fit to the pre-evaporation part

0
0

25

Hold in s

50

400

100
0

25

50

Hold in s

Figure 3-23: Heating in 5.4W trap. Panel-(a) shows the atom number decay. Panel(b) displays the increase in the temperature during 50s hold. A linear fit to the earlier
part of the data gives a heating rate of 7.15𝜇K/s.

85

86

Chapter 4
𝑝-wave Collisions
Ultracold trapped gases are dilute since average interparticle separation set by thedensity of the gas is much bigger than the length scale set by interaction. The diluteness makes two-body interactions enough to describe the physics while ultracold
temperatures simplify the representation of these two-body interactions further to asingle parameter like scattering length. This simplification give us an easier description of many-body properties of ultracold trapped gases.
Moreover, evaporative cooling is a fundamental part of any cold atom experimentand collisions play an essential role in it. This method proceeds by discarding particles
above a certain energy from the thermal particle distribution. Then, the remaining
particles equilibrate at a lower temperature through elastic collisions. This thermalization step should be much faster than loss mechanisms in the system in order tohave an efficient evaporation. Besides setting limits to the performance of the forced
evaporation, loss mechanisms also limit the achievable densities in ultracold trappedsamples.
In this chapter, we study 𝑝-wave collisions in ultracold trapped fermions. We startby reviewing the scattering problem and the threshold behaviour of 𝑝-wave collisions
as introduced in many quantum mechanics textbooks. For a more rigorous treatment of the problem, the reader can turn to Scattering Theory by John Taylor [33].
Following the introduction, Section 4.2 presents the sample preparation process to
be followed by experiments described in subsequent sections. Section 4.3 reports the87measurement of background 𝑝-wave scattering length 𝑎𝑏𝑔
𝑝 . In Section 4.4, we describe
𝑝-wave inelastic losses. Finally in Section 4.5, we discuss forced 𝑝-wave evaporation
using 𝑝-wave elastic collisions, and show its limitations.

4.1

Collision Problem

The Hamiltonian for two distinguishable, interacting particles is

𝐻=

𝑝21
𝑝2
+ 1 + 𝑉 (|⃗𝑟1 − 𝑟⃗2 |)
2𝑚 2𝑚

(4.1)

where 𝑝1 and 𝑝2 are momenta of two particles, and 𝑉 (𝑟) is an interaction potential
only depending on the relative displacement of two particles. Instead of individual
coordinates, we can change the representation of the problem into center-of-mass andrelative coordinates. In this representation, the Hamiltonian becomes

𝐻 = 𝐻𝑐𝑚 + 𝐻𝑟𝑒𝑙
𝑝2𝑐𝑚
4𝑚
𝑝2𝑟𝑒𝑙
𝐻𝑟𝑒𝑙 =+ 𝑉 (|𝑟⃗𝑟𝑒𝑙 |)
𝑚

𝐻𝑐𝑚 =

(4.2)
(4.3)
(4.4)

where 𝑝⃗𝑐𝑚 = 𝑝⃗1 + 𝑝⃗2 and 𝑝⃗𝑟𝑒𝑙 = 𝑝⃗1 − 𝑝⃗2 and 𝑟⃗𝑟𝑒𝑙 = 𝑟⃗1 − 𝑟⃗2 .
This new Hamiltonian is separable in center-of-mass and relative coordinates.
The center-of-mass acts like a free particle with mass 2𝑚, and the relative coordinate
Hamiltonian is an 𝑚/2 particle scattering through a central potential 𝑉 (𝑟). Furthermore, the relative Hamiltonian commutes with 𝐿2 and 𝐿𝑧 . As in elementary quantum
mechanics, this will allow us to expand the relevant functions in linear combinationof all angular momentums, and work in the spherical basis.
88

4.1.1

Impact Parameter and Scattering Cross-section

In this section, we introduce two important classical parameters. These classicalconcepts will give us insight into the threshold behavior of 𝑝-wave scattering. First
one is the impact parameter 𝑏. It describes the distance of closest approach to thetarget when the potential (interaction) is not present.
Another important concept is the scattering cross-section. Classically, one canthink of it as an effective area seen by the incoming flux of particles. In general, we
are also interested in the angular distribution of outgoing particles thus we define𝑑𝜎a differential cross-section 𝑑Ω
representing the contribution of each solid angle. We

can measure this quantity experimentally by using the number of atoms detected ina solid angle per unit time and dividing the measured value by the initial flux ofparticles.

4.1.2

Quantum Terminology

Far away from the short-ranged interaction potential as 𝑟 → ∞, the dominant partof the potential is the centrifugal barrier. The wavefunction in this range becomes a
combination of incoming plane wave and outgoing spherical waves.
𝜓𝑟→∞ = 𝑒𝑖𝑘𝑧 + 𝑓 (𝑘, 𝜃, 𝜑)

𝑒𝑖𝑘𝑟
𝑟

(4.5)

where 𝑧 is direction of the incoming plane wave. The amplitude of the spherical wave
𝑓 (𝑘, 𝜃, 𝜑) is called the scattering amplitude, and has dimension of length. Since the
Hamiltonian is rotationally invariant, the scattering amplitude is independent of 𝜑.
In other words, the rotational invariance makes scattering amplitude only dependon the scattering energy and the scattering angle. Conservation of probability flux
establishes the scattering cross-section 𝜎 in terms of the scattering amplitude 𝑓 (𝑘, 𝜃),𝑑𝜎= |𝑓 (𝑘, 𝜃)|2
𝑑Ω

(4.6)

The wavefunction of identical particles has to be properly symmetrized. In this89

setting, scattering with 𝜃 and 𝜋 − 𝜃 is not distinguishable.
Proper symmetrization of the cross-section for identical fermions is𝑑𝜎= |𝑓 (𝑘, 𝜃) − 𝑓 (𝑘, 𝜋 − 𝜃)|2
𝑑Ω

(4.7)

Here the range of theta is limited to (0, 𝜋/2) in order to prevent double counting.
Identical fermions cannot scatter into 90 degrees regardless of the collision energy orinteraction potential.

4.1.3

Partial Wave Expansion and p-wave in Threshold

The rotational symmetry of the Hamiltonian allows us to expand the scattering amplitude in spherical basis. This partial wave expansion of is𝑓 (𝑘, 𝜃) =∞∑︁

(2𝑙 + 1)𝑓𝑙 (𝑘)𝑃𝑙 (cos 𝜃)

(4.8)

𝑙=0

But if the particles are identical and spin-polarized fermions, proper symmetrization requires𝑓 (𝑘, 𝜃) =∞∑︁

(2𝑙 + 1)𝑓𝑙 (𝑘)[𝑃𝑙 (cos 𝜃) − 𝑃𝑙 (− cos 𝜃)]

(4.9)

𝑙=0

for even 𝑙, 𝑃𝑙 (cos 𝜃) − 𝑃𝑙 (− cos 𝜃) = 0 thus even partial waves do not contributeto the scattering cross-section.

Experimentally Relevant Partial Wave6

Li atoms in different ground hyperfine states interact through a linear combinationof singlet and triplet potentials[24]. Each of them is a combination of a dominant
repulsive part due to Pauli repulsion at short distances(< 7𝑎0 ) and a Van der Waals(𝐶6 /𝑟6 ) tail at long distances. Van der Waals interaction between two atoms stems
from the fact that the dipole moment of the electron distribution has zero point quantum fluctuations. These fluctuations induce correlated dipole moments in the second
atom, and interaction of these two dipoles has the 1/𝑟6 characteristic dependence.
90After partial wave expansion, the effective potential for the radial part of thepartial wave is

𝑉𝑒𝑓 𝑓 = 𝑉 (𝑟) +ℎ̄2 𝑙(𝑙 + 1)2𝜇 𝑟2

(4.10)

where 𝜇 = 𝑚/2 is the reduced mass. For long distances (> 10𝑎0), the dominantinteraction potential is the Van der Waals potential
𝐶6ℎ̄2 𝑙(𝑙 + 1)≈− 6 +
𝑟2𝜇 𝑟2(4.11)

2
6= ( ℎ̄26𝜇𝐶)1/2 with a centrifugal barrierThis potential has a maximum at 𝑟𝑚𝑎𝑥,𝑙𝑙(𝑙+1)−1/2

height 𝑈𝑐𝑓 = 3√1 6 𝐶6𝜇)−3/2 with respect to zero. Figure 4-1 presents the Li( ℎ̄2 𝑙(𝑙+1)𝑝-wave barrier with a height of 8.1mK. This barrier prevents particles with non-zero
angular momentum getting close to each other just as in the planetary binary motion.
Intuitively, we can expect 𝑝-wave scattering to disappear gradually for energies below
this barrier height.

Van der Waals attraction, 𝐶6 , coefficient for Li is 1389 Hartree 𝑎60 [8][22]. We can
6 1/4) . The 𝐶6
define a length scale 𝛽6 for the Van der Waals potential as 𝛽6 = ( 𝑚𝐶ℎ̄2

coefficient corresponds to a Van der Walls length of 𝛽6 = 62.4𝑎0 . In contrast, thetypical inter-particle spacing in our experiment is 1/𝑘𝐹 ≈ 380𝑎0 (for maximum Fermi
energies we achieved, goes down to 190𝑎0 ).
91

Veff (mK)
8

4

0
50

250

450

r(a0 )

Figure 4-1: 𝑉𝑒𝑓 𝑓 of equation-4.8 for 𝑙 = 1 and 𝑟 in (50, 500)𝑎0 using 6 Li parameters

92

In general, if the collision energy 𝐸 is less than ℎ̄2 /𝑚𝛽62 , the most significant contribution comes from the 𝑠-wave contribution and scattering is isotropic for bosons
and distinguishable particles. On the other hand, for identical fermions interacting
through Van der Waals potential elastic cross-section becomes zero at zero temperature.
The value of the 𝐿2 for p-wave collisions is ℎ̄2 𝑙(𝑙 + 1). In the classical sense,√︀
impact parameter of these collisions for a momentum 𝑝 would be ℎ̄ 𝑙(𝑙 + 1)/𝑝. In
other words, 𝑝-wave collisions are never head-on and have an impact parameter on
the order of thermal deBroglie wavelength. For very-low temperatures deBroglie
wavelength is bigger than range of the interaction potential thus atoms do not feeleach other. This cartoon picture is another way of seeing particles having less 𝑝-wave
collisions as the collision energy gets smaller than the 𝑝-wave barrier.
On the other hand, no matter how small the cross-section is, the relevant partial
wave for spin polarized fermions is 𝑝-wave since 𝑠-wave collisions are prohibited.
We are interested in the threshold behavior of elastic and inelastic 𝑝-wave collisions
i.e functional form of the disappearance of 𝑝-wave scattering cross-section and loss
coefficients as collision energy approaches zero.
Angular Distribution of 𝑝-wave and 𝜎𝑝 at Threshold
Since 𝑃1 (cos 𝜃) = −𝑃1 (− cos 𝜃), for p-wave partial wave of equation-4.10 becomes𝑓 (𝑘, 𝜃) = 6𝑓𝑝 (𝑘)𝑃1 (cos 𝜃)= 6𝑓𝑝 (𝑘) cos 𝜃(4.12)(4.13)(4.14)

Then the total p-wave cross-section is𝑑𝜎𝑝= 36|𝑓𝑝 (𝑘)|2 cos2 𝜃𝑑Ω
93(4.15)

The angular distribution of the scattering has the well-known dumbell shape(cos2
distribution) given in Figure 4-2. Integrating over all solid angles gives

2

∫︁ 2𝜋

𝜎𝑝 = 36|𝑓𝑝 (𝑘)|∫︁ 𝜋/2𝑑𝜑0cos2 𝜃 sin 𝜃𝑑𝜃(4.16)

0= 24𝜋|𝑓𝑝 (𝑘)|2(4.17)

Figure 4-2: Probability distribution of scattering into a direction for 𝑝 partial wave
For potentials falling of faster than 1/𝑟5 , the elastic 𝑝-wave scattering length 𝑎𝑏𝑔
𝑝is well defined. As 𝑘 → 0, square of the elastic p-wave scattering amplitude becomes|𝑓𝑝 (𝑘)|2 = 𝑎6𝑝 𝑘 4 [33]. As a result, threshold behavior of elastic 𝑝-wave cross-section for
identical fermions is

𝜎𝑝 (𝑘) = 24𝜋|𝑓𝑝 (𝑘)|2 = 24𝜋𝑎6𝑝 𝑘 4

94(4.18)

This threshold behavior strongly depends on how the interaction potential decays.
For example, dipole-dipole interaction (1/𝑟3 ) has a slower decay compared to Van der
Waals interaction. In dipolar scattering of identical fermions, all partial waves contribute to the scattering amplitude even at low energies and scattering cross-section
becomes universal and does not vanish at 𝑇 = 0. This allowed direct evaporative cooling of fermions to high quantum degeneracy through forced dipolar evaporation[2].
Inelastic three-body collisions are usually characterized with the rate constant 𝐿3
instead of cross-section. Threshold scaling of this rate constant for spin polarized
identical fermions is 𝐿3 ∼ 𝑎8𝑝 𝑘 4 [10]. This scaling is observed experimentally in [34].
We also qualitatively saw this behavior in our trapped sample(see Section 3.5.2).
A summary of threshold scalings of all collisions relevant to our experiment is in
Appendix-A.

4.2

Sample Preparation

The experiment starts with laser cooling of 23 Na and 6 Li in a double species MOT. Following the MOT stage, Li(Na) is optically pumped to the stretched state |3/2, +3/2⟩(|2, +2⟩) in its upper hyperfine state, and captured in a plugged quadrupole magnetic
trap[5]. In the magnetic trap, Li is sympathetically cooled through forced microwave
evaporation of Na atoms[12].
During the last evaporation ramp, a single beam optical 1064nm dipole trap(ODT)
with a variable spot size is ramped up. Na-Li collisions keeps cooling the Li cloud
while loading Li atoms into ODT collisionally. The focal spot size of the ODT beam
is controlled by changing the open aperture of a pneumatically controlled iris. At
the end of the evaporation, the quadrupole field is turned off, and Li is transferred
to |1/2, +1/2⟩ state using a Landau-Zener sweep. Following the spin flip, the iris is
opened to its full aperture in half a second for reducing the spot size, while the beam
power in the optical dipole trap ramped down to 30% of its maximum power in orderto reduce three-body losses due to high densities.
Finally, ODT power is ramped up to the maximum power of 6.2W. The cloud is95

held in this tight and deep trap for 30ms. At this point, there are 5.32 million Li atoms
at 303𝜇K temperature and 𝑇 /𝑇𝐹 = 0.75. The density of the cloud is 1.3 × 1015 cm−3 ,and trapping frequencies are (𝜔𝑟𝑥 , 𝜔𝑟𝑦 , 𝜔𝑧 ) = (91.5, 102, 2) kHz. The amount of possible impurity Na atoms or Li atoms in states |3/2, +3/2⟩ and |1/2, −1/2⟩ is measured
to be less than 0.01%, limited by noise in absorption images.

4.3

Measurement of 𝑎𝑏𝑔
𝑝

4.3.1

Cross-dimensional Thermalization

A common way of measuring the thermally averaged collision rate is the crossdimensional thermalization. In this method, the energy of one of the axes is changed
suddenly. Subsequent equilibration of energy in each axis through elastic collisions
gives a thermalization time scale. This thermalization rate is directly proportional to
the thermally averaged collision rate.
The energy of one axis can be changed by changing trapping frequency of an axis
adiabatically. This method was beyond the reach of our experimental capabilities.
A second method is turning the trap off for a very short time. Free expansion of
the cloud during the trap-off time introduces a bigger potential energy in the radial
directions of the optical dipole trap. This method and ways of analyzing its signal
is described in Section-3.4.2. Initially, we used this method right after the sample
preparation. At that stage, the density is 1015 cm−3 and 𝑈0 /𝐸𝐹 = 9. High threebody loss, evaporation and heating during the thermalization introduce systematics
in the data and make the data analysis cumbersome.
Finally, to avoid these systematics, the atom number is lowered to 10% by reducing
the trap power to three different values around a tenth of the full value, and suddenly
tilting the trapping potential via applying a magnetic force in the beam propagation
direction. After fast spilling of atoms, magnetic fields are turned off and the opticaltrap power is suddenly ramped back up to the full value. These result in an atom
cloud naturally out of thermal equilibrium with temperature difference between axes
96

up to 35% since atoms escape in the beam propagation direction but the elastic
collision rate is not high enough to keep the cloud in thermal equilibrium.

4.3.2

Experimental Results

An example thermalization measurement is given in Figure 4-3. The temperature
difference does not decay to zero because the trap turn-off is not completely suddenwith respect to the radial trap frequencies, which creates cooling due to adiabatic
decompression resulting in a systematic 5% temperature difference. Fitting an exponential to the decay of the temperature difference of the equilibrating axes gives thethermalization rates shown in Figure 4-4). Axis changing collisions close to 90𝑜 are
the most important thermalization. As a result, 𝑝-wave , on average, requires more
collisions to thermalize compared to 𝑠-wave due to anisotropic disctribution of thedifferential cross-section. Monte-Carlo simulations show that, on average, an atom
has 𝛼 = 4.1 𝑝-wave collisions while thermalizing[7]. Therefore, the thermalization
rate isΓ𝑡ℎ𝑒𝑟𝑚 = Γ𝑐𝑜𝑙 /𝛼(4.19)

where the thermally averaged p-wave collision rate Γ𝑐𝑜𝑙 isΓ𝑐𝑜𝑙 = 𝑛𝑚𝑒𝑎𝑛 ⟨𝐾2 ⟩(4.20)= 𝑛𝑚𝑒𝑎𝑛 ⟨𝜎𝑝 (𝑘)𝑣𝑟𝑒𝑙𝑎𝑡𝑖𝑣𝑒 ⟩
√︂
∫︁
8
−3/2
= 𝑛𝑚𝑒𝑎𝑛(𝑘𝐵 𝑇 )
24𝜋 |𝑓𝑝 (𝑘)|2 𝐸𝑒−𝛽𝐸 𝑑𝐸
𝜇𝜋(4.21)(4.22)

Let’s focus on the integral,
∫︁

2

−𝛽𝐸

|𝑓𝑝 (𝑘)| 𝐸𝑒

4𝜇2 3 −𝛽𝐸
𝐸 𝑒
𝑑𝐸
ℎ̄4
4𝜇2 24= 𝑎6𝑝 4 5
ℎ̄ 𝛽
∫︁

𝑑𝐸 =

97

𝑎6𝑝

(4.23)(4.24)

Then,√ 𝑎6𝑝 𝜇3/2
5/2
Γ𝑐𝑜𝑙 = 1152 2𝜋
4 𝑛𝑚𝑒𝑎𝑛 (𝑘𝐵 𝑇 )ℎ̄(4.25)

where 𝜇 = 𝑚/2 is the reduced mass. 𝑛𝑚𝑒𝑎𝑛 is calculated using harmonic approximation i.e measured number of atoms and trapping frequencies.
Using the measured thermalization rates, the background 𝑝-wave scattering length+1.3
is determined to be |𝑎𝑝𝑏𝑔 | = 39−1.6
𝑎0 , where 𝑎0 is Bohr radius, by fitting Equation-

4.26 to our thermalization data (Figure-4-4). The measured value is well withintheoretical calculations ranging from −41𝑎0 to −33.1𝑎0 , and agrees with the value
deduced by measuring 𝑎3𝑝 Δ𝐵 around the |1⟩ − |1⟩ Feshbach resonance and using thetheoretically predicted width Δ𝐵 ([23] and references therein).

98Figure 4-3: Example thermalization data. Error bars are statistical standard deviation of the data. Thermalization rate is determined using the exponential fit givenin the dashed line. The inlet shows the average temperature. There is no external
heating.
[ K]
Average temperature

T/T average

0.4

120

105
0

0.2

0.3

0.6

Hold in [s]00.3

Hold time [s]

99

0.6

Figure 4-4: Measurement of the background 𝑝-wave scattering length. A linear fit
by weighing the data with inverse of variances (vertical error bars) gives |𝑎𝑝𝑏𝑔 | =39+1.3−1.6 𝑎0 . Dashed lines are |𝑎𝑝𝑏𝑔 | = 40.3𝑎0 and 37.4𝑎0 lines respectively representing
95% confidence bound of the fit. There is also a 20% systematic uncertainty on thehorizontal axis due to uncertainty in atom number counting.

Thermalization rate [1/s]

15

10

5

0
0

2

4

6

nmean T 5/2 [K 5/2 m -3 ]

100

8

10
10 10

4.4

𝑝-wave losses

4.4.1

Three-body loss in |1⟩ and 𝐿3

The three-body event rate 𝐿3 has a strong temperature scaling (𝑇 2 )[10]. Our data
always show heating during the three-body rate measurements. Therefore, our 𝐿3 isalso time dependent and we have to consider this in our data analysis. The three-bodyloss equation is𝑁˙= −𝐿3 (𝑡)⟨𝑛2 ⟩𝑁= −𝐿3 (𝑡)(4.26)

𝑁2√
24 3𝜋 3 𝜎𝑥2 𝜎𝑦2 𝜎𝑧2(4.27)

where we used mean squared density for a harmonically trapped cloud and 𝜎𝑖 =√︀𝑘𝐵 𝑇 /𝑚𝜔𝑖2 . We can define a volume 𝑉 2 = 𝜎𝑥2 𝜎𝑦2 𝜎𝑧2 . Then the equation becomes𝑁˙
1
𝐿3 (𝑡)√=−
𝑁3
24 3𝜋 3 𝑉 2 (𝑡)∫︁ 𝑡
1𝐿3 (𝑡′ ) ′
1
1
√
𝑑𝑡 +
=
2
2
′
3
2𝑁
2𝑁02
24 3𝜋 0 𝑉 (𝑡 )(4.28)(4.29)

If we define Γ3𝑏 (𝑡) = 𝑉𝐿32(𝑡)
,
(𝑡)

∫︁ 𝑡
1
Γ3𝑏 (𝑡′ ) ′
1
1
√
=
Γ
(0)
𝑑𝑡
+
3𝑏
2𝑁 2
2𝑁02
24 3𝜋 30 Γ3𝑏 (0)111= √
Γ3𝑏 (0)𝑡˜ +
2
2𝑁
2𝑁02
24 3𝜋 3√︃
1
𝑁=1√
Γ (0)𝑡˜ + 𝑁12
12 3𝜋 3 3𝑏(4.30)(4.31)(4.32)

0where we scaled the time points as 𝑡˜ =∫︀ 𝑡 Γ3𝑏 (𝑡′ )0 Γ3𝑏 (0)𝑑𝑡′ considering the change of therate over time. We fit our atom number decay curves to equation 4.33.
101

The first set of measurements of 𝐿3 are taken in the following way. After sample
preparation, the optical trap is decompressed to various final powers, and atom loss
curves are measured in order to determine the 𝑝-wave three-body loss coefficient 𝐿3
at different temperatures (Figure-4-5). Effects of anharmonicities is not included in
the data analysis. 𝐿3 is measured as 1.2 × 10−30 cm6 /𝑠 at 313𝜇K. We did not consider
reduction of the density due to anharmonicity in our analysis but we know that rightafter sample preparation, especially for shallower traps, anharmonicity effects areimportant.
This led us to a second measurement where we spill majority of atoms as in
the thermalization measurement and measure the three-body decay curve still using
Equation-4.33. Data is given in Figure 4-6. 𝐿3 extracted from the fit is 3.87 × 10−31
at 126𝜇K temperature. This is 2 times faster than the rate predicted using themeasurement of Figure 4-5 and temperature scaling.

102Three-body loss coefficient L3 [cm6 /s]

13

10-31

8.5

4
220

270

320

Temperature [ K]Figure 4-5: Scaling of 𝐿3 with temperature. The black dashed line is a fit to 𝑇 2 power
scaling. Gray dot-dashed lines show 𝑇 and 𝑇 3 scaling. Error bars show one standard
deviation uncertainty in the extracted fit parameter 𝐿3 .

103

(a)
10 5

7

Atom Number

6
5
4
3
2
1
0

6

12

(b)
450

Temperature [uK]Radial
Axial

275

100
0

10

20

Time in [s]Figure 4-6: 𝐿3 measurement with less anharmonic cloud. Panel-(a) shows time dependence of the atom number and the fit to Equation-4.33. Panel-(b) shows the timedependence of the Temperatures. Temperature goes up by almost factor three mainly
due to trap photon scattering. This means a reduction of factor three in Γ3𝑏 , proving
the importance of considering time dependent
104 rate in the analysis.

4.4.2

𝑝-wave Dipolar Relaxation of |6⟩

In this section, we calculate the rate and magnetic field dependence of elastic andinelastic dipolar scattering of 6 Li using first order born approximation following approaches in [25][14]. Due to 1/𝑟3 nature of the dipolar interaction, contribution of
all partial waves to the elastic cross-section is important even at zero energy but thedominant partial wave for spin polarized |6⟩ cloud is 𝑝-wave. Elastic cross-section has
a universal value
𝜎𝑑𝑖𝑝,𝑒𝑙 = 6.702

𝜋𝜇0 𝜇2 𝑚[3]
2 ℎ2(4.33)

Stretched state |6⟩ of Li has a 𝜇𝐵𝑜ℎ𝑟 magnetic moment. The value of the elastic
dipolar scattering cross-section for this state is 9.95 × 10−19 cm2 negligibly small forour density regime.
The dipolar interaction of dipoles 𝐹⃗1 and 𝐹⃗2 has the form
𝑉𝑑𝑑 =𝜇0 (𝑔𝑓 𝜇𝐵𝑜ℎ𝑟 )2 𝐹⃗1 · 𝐹⃗2 − 3(𝐹⃗1 · 𝑟^)(𝐹⃗2 · 𝑟^)4𝜋
𝑟3(4.34)

This interaction can also be written in terms of spin flipping operators 𝐹+ and𝐹− . In that representation, it is easier to see that one or both of atoms can flip theirspins. As a result, inelastic processes for two |6⟩ atoms described by this interaction
potential are
1|6, 6⟩ → √ (|6, 5⟩ + |5, 6⟩)2(4.35)|6, 6⟩ → |5, 5⟩(4.36)

and

These spin-flip processes release excess Zeeman energies Δ𝐸 = 𝑔𝐹 𝜇𝐵𝑜ℎ𝑟 𝐵 and2𝑔𝐹 𝜇𝐵𝑜ℎ𝑟 𝐵 respectively. The released energy is shared by atoms as increase in theirkinetic energy.
Cross-sections for these two inelastic processes are
105

𝑘𝑓 1
8 3 𝜇0 (𝑔𝐹 𝜇𝐵 )2 𝑚 2
𝐹 () [1 − ℎ(𝑘𝑓 1 /𝑘𝑖 )]
2
15
𝑘𝑖
4𝜋ℎ̄
2
8
𝜇0 (𝑔𝐹 𝜇𝐵 ) 𝑚 2
𝑘𝑓 2
𝜎2 = 𝐹 2 () [1 − ℎ(𝑘𝑓 2 /𝑘𝑖 )]
2
15
𝑘𝑖
4𝜋ℎ̄
𝜎1 =

(4.37)(4.38)

whereℎ̄2 𝑘𝑓2ℎ̄2 𝑘𝑖2 Δ𝐸
=
+
2𝑚
2𝑚
2

(4.39)(1 − 𝑥)2
1 3 (1 − 𝑥2 )2
log()ℎ(𝑥) = − −2 8 𝑥(1 + 𝑥2 )(1 + 𝑥)2(4.40)

andWe can define an inelastic loss parameter 𝛽 = 2𝜎𝑗 ℎ̄ 𝑘𝑚𝑖 as in [25], and compare themagnetic field dependence of dipolar inelastic loss for bosonic 52 Cr and fermionic 6 Li
as 𝑘𝑖,𝐶𝑟 → 0(BEC limit) and 𝐸𝐹,𝐿𝑖 ≈ 100𝜇𝐾(order of magnitude fermi energy in our
experiment) using the theory we described.

LossParameter [cm3 /s]
10-11
10-12

Cr 52

10-13

Li 6

10-14
10-15
0.01

0.10

1

10

100

B [Gauss]

Figure 4-7: bosonic Cr fermionic Li dipolar loss parameter comparison as a functionof magnetic field
As shown in Figure-4-7, dipolar relaxation can be supressed for fermions by using
higher magnetic fields whereas the behavior is opposite for bosons. At zero field,
106

Chromium is has more than three orders of magnitude loss compared to 6 Li owing to
its high mass and magnetic moment.

Dipolar relaxation of |1⟩-|2⟩ mixture

The dipolar relaxation of |1⟩-|2⟩ mixture is a reversed version of the 𝜎1 process in |6⟩.
In other words, it has a 𝑝-wave exit channel. We can calculate the cross-section justby noting the change in 𝑔𝐹 𝑚𝐹 from 1 in case of |6⟩ to 1/3 in case of |1⟩.

|6⟩ sample at high densities

When we hold |6⟩ atoms at high density in our deepest trap, we observe generation of|1⟩ impurities over time as shown in Figure 4-8. We confirmed that the peak amountof impurity state |1⟩ atoms and the timing of the peak is density dependent. Althoughmany spin states are generated through dipolar and spin-exhange relaxations, |6⟩ −|1⟩ mixture is stable. The |1⟩ impurity is mainly generated through 𝑝-wave dipolar
relaxation of |6⟩ followed by a fast[15] spin-exchange relaxation to state |1⟩ i.e
1|6, 6⟩ → √ (|6, 5⟩ + |5, 6⟩)2(4.41)→ |5, 5⟩(4.42)|6, 5⟩ → |6, 1⟩(4.43)

This result is intriguing because hyperfine changing spin flips like |65⟩ → |61⟩ in
6

Li releases > 10mK energy and flipped atoms usually escape from the trap. Since

we have a 3.6mK deep trap and high density, we maintain spin-flipped atoms throughcollisions in our trap.
107

Number of atoms in state |6

State 6
State 1

3

8

6
2
4
1

0

2

10

-3

10

-2

10

-1

Hold time [s]

10

0

10

Number of atoms in state |1

104

106

0

1

Figure 4-8: Dipolar relaxation of state |6⟩ into state |1⟩. As state |6⟩ goes throughdipolar relaxation, we observe creation of |1⟩ impurity atoms.

108Summary of relevant scattering and loss parameters for 6 Li
ParameterValue

𝜎𝑒𝑙,𝑑𝑖𝑝,|6⟩10−18 cm2

𝜎𝑒𝑙,𝑑𝑖𝑝,|1⟩

1.2 × 10−20 cm2

𝜎𝑖𝑛𝑒𝑙,𝑑𝑖𝑝,|6⟩

1.1 × 10−18 cm2 @ 𝐵 = 0, 𝑘𝑖 = 100𝜇K

𝜎𝑖𝑛𝑒𝑙,𝑑𝑖𝑝,|1⟩−|2⟩

2.45 × 10−20 cm2 @ 𝐵 = 0, 𝑘𝑖 = 100𝜇K

𝐿3,|1⟩

3.87 × 10−31 cm6 /s @ 135𝜇K

109

4.5

𝑝-wave Evaporation

We measured elastic and inelastic 𝑝-wave collision parameters for state |1⟩. We have totake a look at the ratio of good to bad collisions before starting 𝑝-wave evaporation
since this value plays a crucial role in evaporation efficiency. For a harmonicallytrapped gas, the mean and mean squared densities are𝑁

𝑛𝑚𝑒𝑎𝑛 =

8𝜋 3/2 ( 𝑘𝐵𝑚𝑇 )3/2
3/2
𝑘𝐵 𝑚3/2 𝑇𝐹3=
3 3/2 3/2𝑇𝜔¯348ℎ̄ 𝜋𝑁2(𝑛2 )𝑚𝑒𝑎𝑛 = √𝜔¯6𝑘𝐵 𝑇 3324 3𝜋 ( 𝑚 )3𝑚3 𝑇𝐹6𝑘𝐵√=
864 3𝜋 3 ℎ̄6 𝑇 3(4.44)(4.45)(4.46)(4.47)

Mukaiyama[34] uses the notationℎ̄ 4 8/3
𝑘 𝑉
𝑚 𝑇 𝐵
2
9 𝑚𝑘𝐵
8/3 2
=𝐶
3 𝑉𝐵 𝑇4 ℎ̄

𝐿3 (𝑇 ) = 𝐶(4.48)(4.49)

for the threshold scaling of the three-body loss coefficient where 𝐶 = 𝐶0 (1+𝛽𝑘𝑒 𝑘𝑇2 𝑉𝐵 )𝛾 ,√︁
𝛽 = 9, 𝛾 = 14 and 𝑘𝑇 = 3𝑚𝑘𝐵 𝑇 /2ℎ̄2 is the mean wave number of atoms. His
measured value is 𝐶 = 2 × 106 near the |1⟩ − |1⟩ Feschbach resonance. We measured 𝐶 = 1 × 104 at near-zero magnetic field where we use our measured valueof 𝑉𝐵 = 𝑎3𝑝 = (39𝑎0 )3 for the background 𝑝-wave scattering length and 𝐿3 value at
135𝜇K. Then away from the 𝑝-wave Feshbach resonance, the ratio of good to bad
110collisions is√
𝑛𝑚𝑒𝑎𝑛 288 𝜋𝑎6𝑝 𝑚3/2 (𝑘𝐵 𝑇 )5/2 /ℎ̄4
Γ𝑒𝑙
=
Γ𝑖𝑛𝑒𝑙
(𝑛2 )𝑚𝑒𝑎𝑛 𝐿3 (𝑇 )
2
𝑇
∼ 3
𝐸𝐹
1
∼ 2
𝑎𝑝

(4.50)(4.51)(4.52)For a cloud with 𝑇 /𝑇𝐹 = 1, the ratio scales like 1/𝑇𝐹 . We further definedΓ𝑒𝑙𝑎𝑠𝑡𝑖𝑐
Γ𝑒𝑙𝑎𝑠𝑡𝑖𝑐=
Γ𝑙𝑜𝑠𝑠
Γ𝑖𝑛𝑒𝑙𝑎𝑠𝑡𝑖𝑐 + 1/𝜏𝑣𝑎𝑐𝑢𝑢𝑚(4.53)

assuming generically 𝜏𝑣𝑎𝑐𝑢𝑢𝑚 = 60 seconds. We can compare this new parameter close
to and far away from resonance. Resonant results are given in Figure 4-9. Our rates
far away from resonance is in Figure 4-10.

111

Figure 4-9: The ratio of good to bad collisions (Panel-(a)) and the elastic collision
rate near the 𝑝-wave Feshbach resonance

112

15

Density [10 / cm3 ]0.1 0.3

1

3
4

200

10

150

10

2

10
100

10
50

Γelastic [1/s]Γelastic / Γ loss

3

1
0

200

400

600

800

1000

TF [ K]
Figure 4-10: The ratio of good to bad collisions and elastic collision rates far away fromthe 𝑝-wave Feshbach resonance. The solid line shows the ratio for an harmonicallytrapped cloud with 𝑇 = 𝑇𝐹 generically assuming 𝜏𝑣𝑎𝑐𝑢𝑢𝑚 = 60 seconds. The dashedline is the elastic collision rate of the same cloud. We explored the marked region inthis work.

4.5.1

Relative Lack of Low-velocity Collisions in 𝑝-wave

Due to disappearing cross-section for low-energy 𝑝-wave collisions, relative proportionof 𝑝-wave collisions involving one low-velocity atom is less compared to 𝑠-wave collisons. This effect increases the number of 𝑝-wave elastic collisions required to recover
the Maxwell-Boltzman distribution after a cut of the tail during evaporative coolingcompared to 𝑠-wave.
Rate of 𝑙th partial wave collisions involving an atom with speed 𝑣 is∫︁ ∫︁Γ𝑙,|𝑣⃗1 |=𝑣 = 𝑛𝑚𝑒𝑎𝑛

𝜎𝑙 |𝑣⃗1 − 𝑣⃗2 |𝑓𝑣1 𝑓𝑣2 𝑑3 𝑣⃗1 𝑑3 𝑣⃗2113(4.54)

where 𝜎𝑙 is the collision cross-section. Since we are forcing 𝑣1 to take a particularvalue, properly normalized velocity distributions are

𝑚 2

𝑚

2

2

2

𝑓𝑣1 = 𝑒𝛽 2 𝑣 𝛿(𝑣1𝑥 − 𝑣𝑥 )𝛿(𝑣1𝑦 − 𝑣𝑦 )𝛿(𝑣1𝑧 − 𝑣𝑧 )𝑒−𝛽 2 (𝑣1𝑥 +𝑣1𝑦 +𝑣1𝑧 )𝑓𝑣2 = (

𝛽𝑚 3/2 −𝛽 𝑚 (𝑣2𝑥
2 +𝑣 2 +𝑣 2 )
2𝑧
2𝑦
) 𝑒 2
2𝜋

(4.55)(4.56)

Integrating over 𝑣1 gives,∫︁Γ𝑙,|𝑣⃗1 |=𝑣 = 𝑛𝑚𝑒𝑎𝑛

𝜎𝑙 |⃗𝑣 − 𝑣⃗2 |𝑓𝑣2 𝑑3 𝑣2(4.57)For low-energy s-wave collisions, scattering cross-section is independent of energy.
So,𝛽𝑚 3/2
Γ𝑠,|𝑣⃗1 |=𝑣 = 𝑛𝑚𝑒𝑎𝑛 𝜎𝑠 () 2𝜋
2𝜋

∫︁ ∞ ∫︁ 𝜋 √︁𝑚 2
𝑣 2 + 𝑣22 − 2𝑣𝑣2 cos 𝜃 sin 𝜃𝑑𝜃𝑒−𝛽 2 𝑣2 𝑣22 𝑑𝑣2
00(4.58)
Let’s change the variables for an easier numerical integration. 𝛽 𝑚2 𝑣22 = 𝑢2 , 𝛽 𝑚2 𝑣2 𝑑𝑣2 =𝑢𝑑𝑢 and 𝛽 𝑚2 𝑣 2 = 𝑢˜2 . The integral becomes√︂ √︂∫︁ ∞ ∫︁ 𝜋 √812𝑢˜2 + 𝑢2 − 2˜𝑢𝑢 cos 𝜃 sin 𝜃𝑑𝜃𝑒−𝑢 𝑢2 𝑑𝑢 (4.59)= 𝑛𝑚𝑒𝑎𝑛 𝜎𝑠𝜋 𝛽𝑚 00
For 𝑣 = 0, we can give an analytical expression for the rate easily.
√︂ √︂∫︁ ∞ ∫︁ 𝜋
81
2= 𝑛𝑚𝑒𝑎𝑛 𝜎𝑠sin 𝜃𝑑𝜃𝑒−𝑢 𝑢3 𝑑𝑢𝜋 𝛽𝑚 00√︂ √︂
81= 𝑛𝑚𝑒𝑎𝑛 𝜎𝑠𝜋 𝛽𝑚(4.60)(4.61)

4𝑚
4
𝑝-wave cross section is energy dependent. Using 𝜎𝑝 = 12𝜋𝑎6𝑝 16ℎ̄
4 𝑣𝑟𝑒𝑙 ,

114

4 ∫︁
6 3𝑚
Γ𝑝,|𝑣⃗1 |=𝑣 = 𝑛𝑚𝑒𝑎𝑛 𝑎𝑝 𝜋 4 |𝑣 − 𝑣⃗2 |5 𝑓𝑣2 𝑑3 𝑣2(4.62)4ℎ̄∫︁ ∞ ∫︁ 𝜋
4𝑚 2
6 3𝑚 𝛽𝑚 3/2= 𝑛𝑚𝑒𝑎𝑛 𝑎𝑝 𝜋 4 () 2𝜋(𝑣 2 + 𝑣22 − 2𝑣𝑣2 sin 𝜃)5/2 cos 𝜃𝑑𝜃𝑒−𝛽 2 𝑣2 𝑣22 𝑑𝑣2
4ℎ̄ 2𝜋00(4.63)

Using same change of variables as before 𝛽 𝑚2 𝑣22 = 𝑢2 , 𝛽 𝑚2 𝑣2 𝑑𝑣2 = 𝑢𝑑𝑢 and 𝛽 𝑚2 𝑣 2 = 𝑢˜2∫︁ ∞ ∫︁ 𝜋
√
𝑚3/2
2
(˜
𝑢2 + 𝑢2 − 2˜𝑢𝑢 cos 𝜃)5/2 sin 𝜃𝑑𝜃𝑒−𝑢 𝑢2 𝑑𝑢
2𝜋 5/2 4𝛽 ℎ̄ 00= 𝑛𝑚𝑒𝑎𝑛 𝑎6𝑝 6(4.64)
again for 𝑣 = 0,
∫︁ ∞ ∫︁ 𝜋√
𝑚3/2
22𝜋 5/2 4𝑠𝑖𝑛𝜃𝑑𝜃𝑒−𝑢 𝑢7 𝑑𝑢𝛽 ℎ̄ 003/2√𝑚= 𝑛𝑚𝑒𝑎𝑛 𝑎6𝑝 36 2𝜋 5/2 4𝛽 ℎ̄= 𝑛𝑚𝑒𝑎𝑛 𝑎6𝑝 6(4.65)(4.66)As a quick sanity check to our calculation, we have to be able to recover thethermally averaged collision rate when we integrate Γ𝑙,|𝑣⃗1 |=𝑣 over all possible speeds.
Maxwell-Boltzman speed distribution is
√︂
𝑓𝑣 =

𝛽
2
2
(𝛽𝑚)3/2 𝑣 2 𝑒− 2 𝑚𝑣
𝜋(4.67)

Therefore,

∫︁ ∞
Γ𝑙 =

Γ𝑙,|𝑣⃗1 |=𝑣 𝑓𝑣 𝑑𝑣(4.68)

0

In particular,√︂Γ𝑠 = 𝑛𝑚𝑒𝑎𝑛 𝜎𝑠
115

16
𝜋𝛽𝑚

(4.69)

we recovered famous 𝑛𝜎𝑣 expression for 𝑠-wave collision rate
√ 𝑚3/2
Γ𝑝 = 𝑛𝑚𝑒𝑎𝑛 𝑎6𝑝 288 𝜋 5/2 4
𝛽 ℎ̄(4.70)

same as we calculated in Equation-7.26. The factor 2 difference is because in thiscalculation we assumed particles are distinguishable.

Going back to equations 4.34 and 4.39 and using equations 4.42 and 4.43, we
can see that 𝑠-wave has 4 times more collisions with a zero speed atom compared to𝑝-wave.

We can define relevant dimensionless quantities√ ∫︁ ∞ ∫︁ 𝜋
Γ𝑝,|𝑣⃗1 |=𝑣
22=(˜
𝑢2 + 𝑢2 − 2˜𝑢𝑢 cos 𝜃)5/2 sin 𝜃𝑑𝜃𝑒−𝑢 𝑢2 𝑑𝑢
Γ𝑝
48 0
0

(4.71)

andΓ𝑠,|𝑣⃗1 |=𝑣
1
=√
Γ𝑠
2

∫︁ ∞ ∫︁ 𝜋 √02𝑢˜2 + 𝑢2 − 2˜𝑢𝑢 cos 𝜃 sin 𝜃𝑑𝜃𝑒−𝑢 𝑢2 𝑑𝑢(4.72)

0where 𝛽 𝑚2 𝑣 2 = 𝑢˜2 . Figure-4-11 panel-(a) plots these across the velocity spectrum.
√︁𝐵𝑇
Ratio of the two curves is given in panel-(b). Considering the mean speed is 8𝑘𝜋𝑚
, onaverage, 𝑠-wave has 3 times more collisions for speeds below average speed. Detailedbalance tells us that the rate of collisions generating low-speed particles should besame as the rate of collisions involving a low-speed particle at the input. As a resultof our discussion in this section, we can see that after cutting the tail of velocity
distribution in evaporative cooling, 𝑝-wave collisions take more time to populate thelow velocities and recover the thermal distribution.
116

(c)

(a)
10

4
p- waveΓ v ,s Γ v ,p/ΓsΓp

Γv \ Γ

s- wave5002Speed [ (40

8 k B T 1/2
) ]
πm

0

2

4

8 k B T 1/2) ]Speed [ (
πm

(d)

(b)
Weigted Collision Rate

2

1

0.8
s- wavep- wave

0.4

0.5

0
0

2

p-waves-wave040

8 k B T 1/2
Speed [ (
) ]
πm

2

4

6

8

10

η

Figure 4-11: 𝑠 and 𝑝-wave collision comparison. Panel-(a) shows the rate of collisions
involving an atom with speed 𝑣 with respect to the thermally averaged collision rate.
Panel-(b) is the Panel-(a) weighted by the Maxwell-Boltzman speed distribution.
Panel-(c) is the ratio of two curves. Panel-(d) is the fraction of collisions up to
𝐸𝑘𝑖𝑛𝑒𝑡𝑖𝑐 = 𝜂𝑘𝐵 𝑇 .

117

4.5.2

Evaporation Strategy

Measured ratio of elastic to inelastic collisions for our tightest confinements go as low
as 14. For this reason, we evaporate atoms in a trap one fifth of maximum available
laser power where the trap depth to temperature ratio is 𝜂 = 7. Furthermore, thesmaller trap depth gives us full control over the potential depth using a magnetic
force.
If the energy distribution is truncated at 𝜂 = 3. After thermalization, close to halfof the initial atom number is lost, temperature goes down by a factor 2 and phasespace density goes up by factor 3[6]. Keeping our low good to bad ratio in mind, we
attempt a simple evaporation ramp where we reduce the trap depth and spill atoms
using a magnetic force.

4.5.3

Evaporation Results

We used a square root magnetic force ramp considering reduction in the elastic collision rate as atom number and temperature diminishes. Scanning the final trap depth
yields results in Figure 4-12. We can extract the temperature of cloud by fitting a
gaussian to wings of the cloud. Moreover, we can estimate the Fermi energy using
measured number of atoms and trapping frequencies. Using temperature and calculated Fermi energy, we get 𝑇 /𝑇𝐹 . Evolution of 𝑇 /𝑇𝐹 for this ramp is given in Figure
4-12 Panel-(b). We see reduction in 𝑇 /𝑇𝐹 hence increase in peak phase space density.
This is the first time observation of phase space density increase using direct p-wave
evaporation with short range interactions.
Figure 4-12 Panel-(a) displays the average of 4 absorption pictures for the ramp
with lowest 𝑇 /𝑇𝐹 . Cuts through 𝑥 and 𝑦 centers of the cloud is given in Panel-(a).
Gaussian fits to the 1.7𝜎 of the cloud clearly over estimates the peak. This is a signof quantum degeneracy. The red curve shows the expected Fermi-Dirac distribution
for measured temperature and number of atoms. Finally, radially averaged profile forthe same cloud is in Figure 4-15.
The cloud is missing low-velocity atoms expected by the Fermi-Dirac distribution
118

therefore it is out of thermal equilibrium. Fermi-Dirac fits to the measured absorption
profile gives fugacity values higher than expected fugacity set by measured 𝑇 /𝑇𝐹 .
The gap between fitted fugacity value and expected increases with the depth of theramp(Figure 4-14).
Other evaporation ramps as slow as 20 initial elastic collision times does not end up
in a thermal Fermi-Dirac distribution. We attributed this to non-thermal Fermi-Dirac
distribution caused by lack of low-velocity involving 𝑝-wave collisions as described in
the previous section. In order to confirm, we introduced a 10% spin impurity rightafter the evaporation ramp. 𝑠-wave collisions with the impurity corrects the velocity
distribution and measured fugacity value matches with the fugacity predicted using
the TOF cloud size within error bars as shown in Figure 4-15.

119

(a)

2

(c)

x profile

OD

0.25 mm

x
y

0

0.67

0
0

OD

(b)

y profile

2

OD

T/T F

0.7

0.55

0.4
0.4

1.61 mm(d)

1.4
Atom Number

2.4

0
0

1.61 mm6

10

Figure 4-12: 𝑝-wave evaporation. Panel (a) is average of 4 shots for a sample with
𝑇 /𝑇𝐹 = 0.42. Panel (b) displays the evolution of 𝑇 /𝑇𝐹 vs the atom number. Panel(c) and (d) are gaussian fit to 1.7𝜎 radius wings of the atomic cloud in Panel(a). The
gaussian fit clearly overestimates the peak and expected Fermi-Dirac distribution, asign of quantum degeneracy. Fermi-Dirac profile (the red line) by the atom number,
temperature and trapping frequencies misses low velocity atoms, a sign of lack of low
velocity 𝑝-wave collisions.

120

Radially averaged profile

Optical density

1.8

00.48mm00.48mmFigure 4-13: 𝑝-wave evaporation. Radially averaged profile for the cloud in Figure
4-12 Panel-(a)121Ratio of T/T F from fucatiy to from Fermi fit sizes

2

1.8

1.6

1.4

1.2

1

0.8
100

150

200

250

300

Average Temperature from Gaussian Fit sizes [uK]Figure 4-14: Growing fugacity discrepancy with depth of the cut

122

350

Before mixing

After mixing

Optical density

1.25

Optical density

1.25

00.48mm00.48mm00.48mm00.48mmFigure 4-15: 𝑝-wave evaporation. Radially average cloud profiles before and after
introducing 𝑠-wave collisions using spin impurity. The red line shows the Fermi-Dirac
profile set by the atom number, temperature and trapping frequencies, dots are theradially averaged OD values.

123

124

Chapter 5
OutlookIn this thesis, we studied 𝑝-wave collisions using the advantage of high densities
created by a tight optical trap. A natural extension of the current capability in thelab would be observing Pauli blocking of light scattering[30]. The blocking effect is
illustrated in Figure 5-1. Required Fermi energies are already reached, the missing
step is cooling the cloud into deep degeneracy.

125

ky

ky

krecoilkx

kx

Figure 5-1: Pauli Blocking of light scattering. Blue circles are a smaller(left) and abigger(right) Fermi sea. Yellow arrows represent size of the recoil momentum, samefor both cases. As one can see from dashed circles, overlap of the original and recoiled
Fermi sea is much bigger for bigger Fermi energies. Therefore, the suppression of lightscattering increases as Fermi Energy gets bigger.

126

Appendix A
Threshold Scalings

127

128

𝑛

𝑛 scaling𝑛dipolar inelastic

Collision Type𝑠-wave elastic

three-body recombination-𝑛dipolar elastic𝑛2

𝑎8𝑝 [10]𝑛2

three-body recombination𝑘, 𝐵 = 0-, 𝐵 = 0

𝑎6𝑠 , 𝑎 > 0
[9]

𝑘

𝑘
2

𝑘 scaling𝑁˙= −𝐺2 ⟨𝑛⟩ − 𝐿3 ⟨𝑛2 ⟩ − 𝜏0𝑁∫︀where ⟨𝑛𝑎 ⟩ = 𝑑3 𝑟 𝑛𝑎+1 /𝑁|𝑎2.455|, 𝑎 < 0𝑠𝑎𝑠𝑎 scaling|1⟩-|2⟩ mixture, Threshold Scalings

𝑘2, 𝐵 ≫ 0

√1 , 𝐵 ≫ 0𝐵𝑘𝑘 4 [10]

𝑘5

𝑎6𝑝

𝑛

𝑝-wave elastic

𝑘 scaling𝑎 scaling𝑛 scaling

Collision Type

Spin Polarized Sample, Threshold Scalings[13]∼ 𝑛2 |𝑎2.455|𝑘 2 , 𝑎 < 0𝑠∼ 𝑛2 𝑎6𝑠 𝑘 2 , 𝑎 > 0∼ 𝑛𝑎𝑠 𝑘rate

∼ 𝑛𝑘, 𝐵 = 0√ , 𝐵 ≫ 0
∼ 𝑛𝑘
𝐵

2

∼ 𝑛𝜎𝑒𝑙.𝑑𝑖𝑝 𝑘[4]∼ 𝑛2 𝑎8𝑝 𝑘 4∼ 𝑛𝑎6𝑝 𝑘 5

rateAppendix B
Fitting functions for degenerate
Fermions
Introduction
Occupation of the phase space is1
12𝑝+ 12 𝑚𝜔 2 𝑟2 −𝜇)ℎ3 𝑒𝛽( 2𝑚
+1

𝑓=

(B.1)

Then the density distribution is1𝑛(𝑟) = 3
ℎ

∫︁

1
𝑒

𝑝2
+ 12 𝑚𝜔 2 𝑟2 −𝜇)
𝛽( 2𝑚

𝑑3 𝑝

(B.2)

𝑑3 𝑟𝑑𝑝𝑧(B.3)+1

2D momentum distribution is1
𝑝(𝑥, 𝑦) = 3
ℎ

∫︁ ∫︁

1
𝑝2
𝛽( 2𝑚
+ 12 𝑚𝜔 2 𝑟2 −𝜇)𝑒+1Let’s introduce the Fermi integral,

∫︁ ∞

𝑥

𝐿𝑖𝑗+1 (−𝑒 ) = Γ(𝑗 + 1)0𝑡𝑗
𝑑𝑡
𝑒𝑡−𝑥 + 1

(B.4)
(B.5)

129

and
𝐿𝑖𝑛 (𝑧) =

∞
∑︁
𝑧𝑘

(B.6)

𝑘𝑛
𝑘=1

Let’s also calculate two integrals we’ll use them over and over.
First one is:∫︁ +∞𝛽(𝜇−𝛼𝑥2 )𝐿𝑖𝑛 [−𝑒

∫︁ +∞
∞
∑︁
2
𝛽𝜇 𝑛
𝑑𝑥] =
𝑒−𝑛𝛽𝛼𝑥 𝑑𝑥
(−𝑒 )

−∞

=

𝑘=1

−∞

∞
∑︁
(−𝑒𝛽𝜇 )𝑛𝜋 1/2(𝑘𝛽𝛼)1/2

𝑘𝑛

𝑘=1

√︂
=

(B.7)
(B.8)

𝜋
𝐿𝑖 1 [−𝑒𝛽𝜇 ]𝛽𝛼 𝑛+ 2

(B.9)

The second one is:∫︁ ∞2

𝐿𝑖𝑘 [−𝑒𝛽(𝜇−𝛼𝑥 ) ]𝑥2 𝑑𝑥(B.10)

0and we can convert it to the first one,∫︁ ∞𝛽(𝜇−𝛼𝑥2 )𝐿𝑖𝑘 [−𝑒∫︁ +∞ ∫︁ +∞ ∫︁ +∞2]𝑟 𝑑𝑟 =
−∞

0

−∞

2

2

2 2

𝐿𝑖𝑘 [−𝑒𝛽(𝜇−𝛼(𝑥 +𝑦 +𝑧 ) ]𝑑𝑥𝑑𝑦𝑑𝑧 (B.11)−∞

𝜋= ( )3/2 𝐿𝑖𝑛+ 3 [−𝑒𝛽𝜇 ]2𝛽𝛼(B.12)

Momentum and Density Distributions

Starting from equation 3.34,
∫︁ +∞
2
2
(𝑝2
𝑥 +𝑝𝑦 +𝑝𝑧 )2𝜋
3/2)
𝐿𝑖3/2 [−𝑒𝛽(𝜇− 2𝑚 ]𝑑𝑝𝑧
𝑝(𝑥, 𝑦) = − ( 2 2
ℎ𝜔
¯ 𝛽𝑚
−∞
2
(𝑝2
𝑥 +𝑝𝑦 )2𝜋
𝜋2𝑚 1/2= −( 2 2)3/2 () 𝐿𝑖2 [−𝑒𝛽(𝜇− 2𝑚 ) ]ℎ 𝜔 𝛽𝑚𝛽(𝑝2 +𝑝2 )12𝜋 2𝛽(𝜇− 𝑥2𝑚 𝑦 )
=−
(
)
𝐿𝑖
[−𝑒
]
2
𝑚ℎ3 𝜔 3 𝛽Using equation 3.33 and 35, the density profile is:130(B.13)(B.14)(B.15)

𝑛(𝑟) = −(1
2𝜋𝑚 3/2
2 2) 𝐿𝑖3/2 [−𝑒𝛽(𝜇− 2 𝑚𝜔 𝑟 ) ]2ℎ𝛽(B.16)

Then similar to momentum case above, 2D density profile is:𝑛(𝑥, 𝑦) = −(1
2𝜋𝑚 3/2 2𝜋 1/2
2 2
2 2
) (
) 𝐿𝑖2 [−𝑒𝛽(𝜇− 2 𝑚(𝜔𝑥 𝑥 +𝜔𝑦 𝑦 )) ]
2
2
ℎ𝛽
𝑚𝜔𝑧

(B.17)

Number constraint gives
2𝜋𝑚
𝑁 = −( 2 )3/2
ℎ𝛽

∫︁ ∞

1

2 2

𝐿𝑖3/2 [−𝑒𝛽(𝜇− 2 𝑚𝜔 𝑟 ) ]4𝜋𝑟2 𝑑𝑟0∫︀ ∞ 𝑘𝛽(𝜇− 1 𝑚𝜔2 𝑟2 ) 2
∞
∑︁
2
𝑒
𝑟 𝑑𝑟
4𝜋
2𝜋𝑚
0
= (−1)𝑘+1 ( 2 )3/2
3/2
ℎ𝛽
𝑘
𝑘=1
√︂
∞
∑︁
1
𝜋
𝑘+1 2𝜋𝑚 3/2= (−1) ( 2 )4𝜋𝑒𝑘𝛽𝜇ℎ𝛽2 (𝛽𝑚𝜔 2 )3/2 𝑘 3
𝑘=1
= −(

1 3
) 𝐿𝑖3 [−𝑒𝛽𝜇 ]ℎ̄𝜔𝛽

(B.18)(B.19)(B.20)(B.21)

2D TOF profile

The time of flight expansion is a convolution. Particles starting at coordinate 𝑟⃗0 will
end up in 𝑟⃗0 + 𝑚𝑝⃗ 𝑡 after flight duration 𝑡. We can also see this as all the particles at
position ⃗𝑟 will be the ones with original momentum 𝑚(⃗𝑟 − 𝑟⃗0 )/𝑡 and position 𝑟0 .
Then the density after flying 𝑡 is

1 𝑚2𝑛(⃗𝑟) = 3 2
ℎ 𝑡

∫︁

1 𝑚2
= 3 2
ℎ 𝑡

∫︁

1
(𝑚(⃗
𝑟 −𝑟⃗0 )/𝑡)2𝛽(+ 12 𝑚¯𝜔 2 𝑟02 −𝜇)
2𝑚

𝑒

𝑑3 𝑟0

1
𝛽 𝑚2

𝑒 2𝑡

∑︀

𝑟𝑖 −𝑟⃗0𝑖 )2 +(𝜔𝑖 𝑡)2 𝑟𝑖2 ) −𝛽𝜇
𝑖=𝑥,𝑦,𝑧 ((⃗
𝑒
+1

131

(B.22)+1
𝑑3 𝑟0(B.23)Name 𝛼𝑖2 = 1 + (𝜔𝑖 𝑡)2
1 𝑚2
= 3 2
ℎ 𝑡

∫︁

1
𝛽

𝑒

𝑚 ∑︀

2𝑡2

⃗
𝑟𝑖 2
0𝑖 − 𝛼 )𝑖 (𝛼𝑖 𝑟⃗
𝑖

∑︀ 2
1
𝛽 𝑚2
𝑖 𝑟𝑖 (1− 𝛼2 )
2𝑡
𝑖

𝑒

2 ∫︁

=

1 𝑚
ℎ3 𝑡2

= −(

𝑑3 𝑟0

(B.24)

𝑑3 𝑟0(B.25)

𝑒−𝛽𝜇 + 1

1
∑︀
⃗
𝑟𝑖 2
𝛽 𝑚2
0𝑖 − 𝛼 )𝑖 (𝛼𝑖 𝑟⃗
2𝑡
𝑖

𝑒

𝛽𝑚
𝜔 2 𝑟2 /(1+(𝜔𝑖 𝑡)2 )
2 𝑖

𝑒
1

𝑒−𝛽𝜇 + 1𝑚 ∑︀
2𝜋𝑚 3/2
2 2
2) √︁
𝐿𝑖3/2 [−𝑒−𝛽 2 𝑖 𝜔𝑖 𝑟𝑖 /(1+(𝜔𝑖 𝑡) ) 𝑒𝛽𝜇 ]2ℎ𝛽(1 + 𝜔 2 𝑡2 )(1 + 𝜔 2 𝑡2 )(1 + 𝜔 2 𝑡2 )

𝑥

𝑦

𝑧

(B.26)
Integrating this expression in z-axis gives
𝑚
𝑚
1 2𝜋
2 2
2
2 2
2
𝑛(𝑥, 𝑦) = − 3 ( )2 √︁
𝐿𝑖2 [−𝑒−𝛽 2 (𝜔𝑥 𝑥 /(1+(𝜔𝑥 𝑡) )+𝜔𝑦 𝑦 /(1+(𝜔𝑦 𝑡) ) 𝑒𝛽𝜇 ]ℎ 𝛽 𝜔 (1 + 𝜔 2 𝑡2 )(1 + 𝜔 2 𝑡2 )

𝑧

𝑥

𝑦

(B.27)
if 𝜔𝑖 𝑡 ≫ 1 the expression simplifies to

𝑚
2
2
𝑚
1 2𝜋
𝐿𝑖2 [−𝑒−𝛽 2𝑡2 (𝑥 +𝑦 ) 𝑒𝛽𝜇 ]𝑛(𝑥, 𝑦) = − 3 ( )22ℎ 𝛽 𝜔𝑧 𝜔𝑦 𝜔𝑥 𝑡

(B.28)The peak density is:
1 2𝜋𝑚
𝑛𝑝𝑒𝑎𝑘 = − 3 ( )2 √︁
𝐿𝑖2 [−𝑒𝛽𝜇 ]ℎ 𝛽 𝜔 (1 + 𝜔 2 𝑡2 )(1 + 𝜔 2 𝑡2 )𝑧𝑥

(B.29)𝑦

As 𝑇 → 0 (𝛽 → ∞), let’s use

lim 𝐿𝑖𝑛 [−𝑒𝑥 ] =𝑥→∞𝑥𝑛Γ(𝑛 + 1)

(B.30)

Then the peak is
1 2𝜋
𝑚
(𝛽𝜇)2
lim 𝑛𝑝𝑒𝑎𝑘 = − 3 ( )2 √︁𝛽→∞ℎ 𝛽 𝜔 (1 + 𝜔 2 𝑡2 )(1 + 𝜔 2 𝑡2 ) Γ(3)𝑧𝑥

(B.31)

𝑦

2

=−

(2𝜋)
𝑚
(𝐸𝐹 )2√︁ℎ3 𝜔 (1 + 𝜔 2 𝑡2 )(1 + 𝜔 2 𝑡2 ) Γ(3)
𝑧

𝑥

132

𝑦

(B.32)

Independent of the temperature. On the other hand, we are fitting to a functionof form
𝑚

2 2

2

2 2

2

𝑛(𝑥, 𝑦) = −𝐶 𝐿𝑖2 [−𝑒−𝛽 2 (𝜔𝑥 𝑥 /(1+(𝜔𝑥 𝑡) )+𝜔𝑦 𝑦 /(1+(𝜔𝑦 𝑡) ) 𝑒𝛽𝜇 ](B.33)

where𝐶=𝑚
1 2𝜋 2√︁( )3ℎ 𝛽 𝜔 (1 + 𝜔 2 𝑡2 )(1 + 𝜔 2 𝑡2 )𝑧𝑥

(B.34)𝑦So the fit coefficient have information about the temperature.
If TOF is long enough in both directions 𝜔𝑖 𝑡 ≫ 1, we get a circularly symmetricprofile1 2𝜋𝑚−𝛽 𝑚2 𝑟2 𝛽𝜇2𝑡𝑛(𝑟) = − 3 ( )2
𝐿𝑖
[−𝑒
𝑒 ]
2
ℎ 𝛽 𝜔𝑧 𝜔𝑥 𝜔𝑦 𝑡2(B.35)

Our fitting function is
2

2

2𝐶2

2𝐶3

− 𝑥 2− 𝑦 2

𝑂𝐷(𝑥, 𝑦) = −𝐶1 𝐿𝑖2 [−𝑒𝐶4 ](B.36)

where
𝐶4 = 𝑒𝛽𝜇(B.37)

𝑘𝐵 𝑇(1 + 𝜔𝑦2 𝑡2 )
𝑚𝜔𝑦2
𝑘𝐵 𝑇
𝐶2 =
(1 + 𝜔𝑥2 𝑡2 )𝑚𝜔𝑥2
1𝑚
𝐶1 = 𝜎𝑎𝑏𝑠 3 (2𝜋𝑘𝐵 𝑇 )2 √︁ℎ𝜔 (1 + 𝜔 2 𝑡2 )(1 + 𝜔 2 𝑡2 )
𝐶3 =

𝑧

𝑥

(B.38)(B.39)(B.40)𝑦Temperature we get from each of these should be the same
Integrated Line Profile
2D TOF profile is
2

2

𝑦
𝑥
𝛽𝜇 − 2𝜎𝑦2 − 2𝜎𝑥2𝑛(𝑥, 𝑦) = 𝐴𝐿𝑖2 [−𝑒 𝑒]Integrating this over one of the axes to get the line profile gives
133(B.41)

√︁
2− 𝑥= 𝐴 2𝜋𝜎𝑦2 𝐿𝑖5/2 [−𝑒𝛽𝜇 𝑒 2𝜎𝑥2 ]134(B.42)

Appendix C
Laser Table AOM schemes
C.1

Li

Li Laser Table AOM Scheme
Optical Pumping

Spectroscopy

Imaging
228MHz500MHzSlower
AOM1

Toptica TA 100

0

Repump
228

-500
-500
+SB

500MHzSlower
AOM2

0

-1

-1000
+SB

70%
Fiber Coupler

-1

+228

80MHz

Double-pass
3/2

MOT
TTL

+1

-1
EOM

80MHz
0

44%
Fiber Coupler

Repumper
TTL

Double-pass
1/2

-1

55%
Fiber Coupler
16mW

Fiber Coupler

35%
TA

Fiber Coupler

Figure C-1: Layout of AOMs in the Li laser table135

Li Laser Lock Setup

Fiber Coupler
PhDiode

Pump
Probe

λ/4

λ/2

λ/2

PhDiodeLi Cell+1

λ/2

117MHz
AOM

EOM
λ/2

λ/4

Figure C-2: Li modulation transfer spectroscopy setup

136

Na

Na Laser Table AOM Scheme

AOM3

Imaging2->3 SH+1

0

MOT SH
589 Laser

Fiber Coupler

1.719GHz

855MHz -1

AOM1

AOM2

0

+1

RepumpTTL

0EOM

1.7GHz Slower SH
Imaging
103.1MHz

209MHz98.28MHz+1
Repump SH+1

F=1 SH+1 192MHz -1
Imaging
99MHz

F=2 SH
Fiber Coupler

Figure C-3: Layout of AOMs in the Na laser table

137

Optical Pumping

2->2 DoublePass

Side-Axis
TTL

MOT TTL

Side-Axis SH
0

87MHz

74MHz

29.2MHz

C.2

C.3

IR

IR AOM Scheme

-1

AOM 2

+1

0

IPG 1064 20W

AOM 1

Figure C-4: The final IR light setup

138

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