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# Manipulation of the nuclear spin states of 87 Sr in degenerate SU(N)-symmetric Fermi gases

Andrea Litvinov

► To cite this version:

Andrea Litvinov. Manipulation of the nuclear spin states of 87 Sr in degenerate SU(N)-symmetric Fermi gases. Quantum Physics [quant-ph]. Sorbonne Paris Nord, 2023. English. NNT: tel-04201342

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https://sorbonne-paris-nord.hal.science/tel-04201342

Submitted on 9 Sep 2023

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UNIVERSITE PARIS XIII - SORBONNE PARIS NORD
Ecole Doctorale Sciences, Technologies, Santé Galilée

# Manipulation des états de spin nucléaires du $^{87}$Sr dans des gaz de Fermi dégénérés en symétrie SU(N)

## Manipulation of the nuclear spin states of $^{87}$Sr in degenerate SU(N)-symmetric Fermi gases

### THESE DE DOCTORAT

présentée par

Andréa LITVINOV

Laboratoire de Physique des Lasers

Pour l'obtention du grade de
DOCTEUR EN PHYSIQUE

Soutenue le 10 Janvier 2023 devant le jury d'examen composé de:

<table><tr><td>FERLAINO Francesca, IQOQI</td><td>Présidente du Jury</td></tr><tr><td>BOUCHOULE Isabelle, LCF</td><td>Rapportrice</td></tr><tr><td>BEUGNON Jérôme, LKB</td><td>Rapporteur</td></tr><tr><td>CHEVY Frédéric, ENS</td><td>Examenateur</td></tr><tr><td>DUBESSY Romain, LPL</td><td>Examenateur</td></tr><tr><td>ROBERT-DE-SAINT-VINCENT Martin, LPL</td><td>Examenateur</td></tr><tr><td>LABURTHE-TOLRA Bruno, LPL</td><td>Directeur de thèse</td></tr><tr><td>GORCEIX Olivier, LPL</td><td>Co-directeur de thèse</td></tr></table>
# Contents

<table><tbody><tr><td><b>1 Fermi gases of <sup>87</sup>Sr in 3D optical lattices - Experimental Setup</b></td><td><b>13</b></td></tr><tr><td>1.1 Cooling <sup>87</sup>Sr to Fermi degeneracy</td><td>14</td></tr><tr><td>1.1.1 Energy levels</td><td>14</td></tr><tr><td>1.1.2 Oven - Transverse cooling and Zeeman slower</td><td>16</td></tr><tr><td>1.1.3 <sup>1</sup>S<sub>0</sub> ↔ <sup>1</sup>P<sub>1</sub> broadband MOT - <sup>3</sup>P<sub>2</sub> shelving</td><td>17</td></tr><tr><td>1.1.4 Repumping metastable <sup>3</sup>P<sub>2</sub></td><td>18</td></tr><tr><td>1.1.5 <sup>1</sup>S<sub>0</sub> ↔ <sup>3</sup>P<sub>1</sub> narrow MOT</td><td>19</td></tr><tr><td>1.1.6 Optical dipole trap loading</td><td>22</td></tr><tr><td>1.1.7 Forced evaporation</td><td>27</td></tr><tr><td>1.2 Optical lattices</td><td>27</td></tr><tr><td>1.2.1 1064 nm 1D lattice</td><td>30</td></tr><tr><td>1.2.2 532 nm 2D lattice</td><td>35</td></tr><tr><td><b>2 Absorption imaging of objects smaller than the resolution limit</b></td><td><b>37</b></td></tr><tr><td>2.1 Motivations</td><td>38</td></tr><tr><td>2.1.1 Measuring the interaction energy of an SU(10) degenerate Fermi gas</td><td>38</td></tr><tr><td>2.1.2 Distorsion of imaged in-situ density</td><td>39</td></tr><tr><td>2.1.3 Incorrect estimation of the pixelated local density</td><td>40</td></tr><tr><td>2.2 Non-linear correction on the measurement of the sub-resolved local density</td><td>42</td></tr><tr><td>2.2.1 Parametrization of the density dependent correction</td><td>43</td></tr><tr><td>2.2.2 Calibration of the non-linear correction</td><td>45</td></tr><tr><td>2.3 Experimental demonstration</td><td>47</td></tr></tbody></table>
<table><tr><td>2.3.1 Production of degenerate Fermi gases expanding along one direction only</td><td>47</td></tr><tr><td>2.3.2 Recovery of the unresolved transverse sizes σ<sub>x</sub> of elongated gases</td><td>47</td></tr><tr><td>2.3.3 Recovery of the distorted longitudinal density profiles</td><td>52</td></tr><tr><td>2.4 Measurement attempt of the mean field interaction energy in a SU(10) degenerate Fermi gas</td><td>55</td></tr></table>

<table><tr><td>3 Preparation and measurement of the spin populations in a Fermi gas of <sup>87</sup>Sr</td><td>59</td></tr><tr><td>3.1 Measurement of the spin populations</td><td>60</td></tr><tr><td>3.1.1 Experimental procedure</td><td>61</td></tr><tr><td>3.1.2 Adiabatic following - Dressed-states picture</td><td>63</td></tr><tr><td>3.1.3 Spin sensitivity</td><td>68</td></tr><tr><td>3.1.4 Spin dependent momentum transfer in SU(10) manifold</td><td>70</td></tr><tr><td>3.1.5 Simultaneous measurement of four spin populations</td><td>75</td></tr><tr><td>3.1.6 Intensity window for improved adiabaticity</td><td>77</td></tr><tr><td>3.2 Selective preparation of polarized Fermi sea of <sup>87</sup>Sr</td><td>79</td></tr><tr><td>3.2.1 Optical pumping</td><td>79</td></tr><tr><td>3.2.2 Controlling the number of spin components from 1 to 10</td><td>88</td></tr><tr><td>3.2.3 Spin purification to prepare a polarized Fermi sea</td><td>94</td></tr><tr><td>3.2.4 Polarized Fermi sea</td><td>96</td></tr></table>

<table><tr><td>4 Coherent manipulation of the nuclear spin states of <sup>87</sup>Sr</td><td>99</td></tr><tr><td>4.1 Degeneracy lift associated with spin dependent light shift</td><td>101</td></tr><tr><td>4.1.1 Tensor dynamic polarizability</td><td>101</td></tr><tr><td>4.1.2 Scattering rate</td><td>105</td></tr><tr><td>4.1.3 Tensor light shift associated with the intercombination line</td><td>108</td></tr><tr><td>4.2 Engineering the degeneracy lift of <sup>87</sup>Sr ground state manifold</td><td>112</td></tr><tr><td>4.2.1 Experimental design</td><td>112</td></tr><tr><td>4.2.2 Measurement of the spin dependent light shifts</td><td>112</td></tr><tr><td>4.2.3 Measurement of the scattering rate</td><td>115</td></tr><tr><td>4.2.4 Spin depolarization</td><td>118</td></tr></table>
<table>
  <tbody>
    <tr>
      <td>4.3</td>
      <td>Selective global manipulation of the spin states</td>
      <td>118</td>
    </tr>
    <tr>
      <td>4.3.1</td>
      <td>Scheme overview</td>
      <td>119</td>
    </tr>
    <tr>
      <td>4.3.2</td>
      <td>Spectrum of the Raman transitions in the <sup>1</sup>S<sub>0</sub> manifold</td>
      <td>119</td>
    </tr>
    <tr>
      <td>4.3.3</td>
      <td>Engineering selective spin flips</td>
      <td>125</td>
    </tr>
    <tr>
      <td>4.4</td>
      <td>Site selective adiabatic passage to prepare spin textures - Outlook</td>
      <td>129</td>
    </tr>
    <tr>
      <td>4.4.1</td>
      <td>General Idea</td>
      <td>129</td>
    </tr>
    <tr>
      <td>4.4.2</td>
      <td>Geometry of the lattices</td>
      <td>130</td>
    </tr>
    <tr>
      <td>4.4.3</td>
      <td>Site dependent Raman resonances</td>
      <td>133</td>
    </tr>
    <tr>
      <td>4.4.4</td>
      <td>Predicted texture writing</td>
      <td>134</td>
    </tr>
    <tr>
      <td>5</td>
      <td>2D Fermi gases in the lowest band of a low recoil 1D lattice</td>
      <td>137</td>
    </tr>
    <tr>
      <td>5.1</td>
      <td>Limit to the atom number for fermions in the lowest band</td>
      <td>139</td>
    </tr>
    <tr>
      <td>5.2</td>
      <td>Adiabatic loading of the lowest band: collision requirements</td>
      <td>140</td>
    </tr>
    <tr>
      <td>5.3</td>
      <td>Loading experiments and measurement of the band populations with a Boltzmann approximation</td>
      <td>141</td>
    </tr>
    <tr>
      <td>5.4</td>
      <td>Band populations in the semi classical approximation for a 2D gas of Fermions</td>
      <td>146</td>
    </tr>
    <tr>
      <td>5.4.1</td>
      <td>Derivation of the number of atoms in each band</td>
      <td>146</td>
    </tr>
    <tr>
      <td>5.4.2</td>
      <td>Accounting for multiple layers in the lattice</td>
      <td>149</td>
    </tr>
    <tr>
      <td>5.4.3</td>
      <td>Fit function with time of flight imaging</td>
      <td>151</td>
    </tr>
    <tr>
      <td>5.5</td>
      <td>Semi-classical measurement of the bands populations of several independent 2D layers</td>
      <td>153</td>
    </tr>
    <tr>
      <td>5.5.1</td>
      <td>Analysis of a single image</td>
      <td>153</td>
    </tr>
    <tr>
      <td>5.5.2</td>
      <td>Systematic study as a function of the evaporation depth, for SU(3) and SU(10) gases</td>
      <td>157</td>
    </tr>
    <tr>
      <td>5.5.3</td>
      <td>Preparation of degenerate Fermi 2D gases with improved adiabaticity</td>
      <td>159</td>
    </tr>
    <tr>
      <td>5.6</td>
      <td>Conclusion and outlook</td>
      <td>161</td>
    </tr>
    <tr>
      <td>A</td>
      <td>Basic theory for d-dimensional Fermi gases in harmonic traps</td>
      <td>167</td>
    </tr>
    <tr>
      <td>A.1</td>
      <td>Density of states and Fermi energy</td>
      <td>167</td>
    </tr>
    <tr>
      <td>A.1.1</td>
      <td>3D harmonic trap tightly confined along one direction only</td>
      <td>168</td>
    </tr>
  </tbody>
</table>
<table><tbody><tr><td>A.2 Density and chemical potential, exact derivation in the semi-classical regime</td><td>170</td></tr><tr><td>A.3 Thermodynamics and entropy, derivation in the semi-classical regime</td><td>171</td></tr><tr><td>A.4 Thermodynamics, entropy and chemical potential, degenerate approximations</td><td>174</td></tr><tr><td>A.5 Thermodynamics, entropy and chemical potential, classical gas approximations</td><td>177</td></tr><tr><td><b>B Basic theory for optical lattices</b></td><td><b>179</b></td></tr><tr><td>B.1 Eigenstates and energy bands</td><td>179</td></tr><tr><td>B.2 Adiabatic lattice loading with fermions</td><td>182</td></tr><tr><td><b>C Adiabatic passages in Λ scheme</b></td><td><b>185</b></td></tr><tr><td>C.1 Two ground states</td><td>185</td></tr><tr><td>C.2 Three states manifold</td><td>188</td></tr><tr><td>C.3 Experimental optimization of adiabatic passage</td><td>191</td></tr><tr><td><b>D Guided expansion - parallaxe - anticonfinement</b></td><td><b>195</b></td></tr><tr><td>D.1 Derivation</td><td>196</td></tr><tr><td>D.1.1 Effective gravity</td><td>196</td></tr><tr><td>D.1.2 Center of mass</td><td>197</td></tr><tr><td>D.1.3 Cloud width</td><td>198</td></tr><tr><td>D.2 Measurements</td><td>199</td></tr></tbody></table>
Introduction

Emergent properties arise from collective systems whose whole is not reductible to
its individual components [Lew75]. Pioneering discoveries on superconductivity and
superfluidity illustrate the complexity of such systems for which experimental research
prevailed in the realm of quantum physics. The sudden vanishing resistance of mercury
at 4.2 K, first observed by Heike Kamerlingh Onnes in 1911 [DK10], is an outstanding
example of collective phenomena as superconductivity was just discovered. Later, the
observation of abnormally low viscosity in ultra cold helium 4 [Kap38; AM38] marked
the discovery of superfluidity.

*Degenerate quantum gases* have become an ideal platform for the study of collective
phenomena. The high degree of control and tunability of lasers and magnetic fields
gives the possibility to shape arbitray potentials [GWO00; BDZ08; LZB19] and to tune
inter-atomic interactions [Chi+10; FJL00]. On the one hand, the first realization of a
Bose-Einstein condensate [And+95] paved the way for extensive research on superfluids,
from the study of the hydrodynamic oscillations in a Bose gas [Sta+98], to the real-
ization of phase coherent arrays of superfluid dropplets [Nor+21]. On the other hand,
it was also demonstrated that gases of fermions can be cooled to quantum degeneracy
[DJ99]. Following the original idea of Feynman [Fey82], proposals have suggested the
realization of quantum simulators for the Fermi-Hubbard model [Hof+02]. Repulsive
fermions in an optical lattice allow to realize an ideal and tunable version of the Hubbard
model [Hub63; Hub64], a paradigm for the multitude of strong correlation problems in
condensed matter physics. The experimental trapping of fermions in optical lattices
[Köh+05], and later entering into the strongly correlated regime [Sch+08; Jör+08], are
the first steps for the study of quantum magnetism with fermions [BDZ08].
*Active research on the Fermi-Hubbard model* led to the realization of numerous quantum simulators [BDZ08; Ess10; GB17]. Typically, two spin states of alkali fermions such as potassium 40 [Che+15; Dre+17], or litium 6 [Gre+16; Bol+16; Par+16], are trapped in optical lattices in the strongly interacting regime. Most of these experiments are suited to the analogy with spin half solid states physics, in the so-called SU(2) symmetry in the case of spin-independent interaction properties. Spin ordering and antiferromagnetic correlations, emerging from the collective spin system, are probed with single site resolution using most advanced atomic microscopes [Bak+09].

*Beyond spin half physics*, proposals suggested the study of quantum magnetism in enlarged SU(N) symmetry [CHU09; HGR09; Gor+10; CR14], with no solid states analog. Involving an enhanced number of configurations due to the large spin degree of freedom, exotic collective phenomena are expected. For instance, SU(N)-symmetric exchange interaction shall result in frustrated spin ordering [HGR09]. Experimental realizations of the SU(N) Hubbard model yielded the measurements of the antiferromagnetic correlations [Oza+18; Tai+20] with alkaline-earth like ytterbium, and showed the enhanced correlations compared to that ones of SU(2) spin systems. Furthermore, these experiments with SU(N) gases show evidence of enhanced large-spin Pomeranchuk cooling [Tai+12], which is a practical advantage to reach stronger spin-correlations at constant entropy, and to approach singlet states [Rey+07]. Regarding ultra cold quantum gases of alkaline-earth like fermions, SU(N) symmetry arises from the decoupling of the nuclear spin with the electronic degrees of freedom [Gor+10]. Most importantly, this symmetry results in the conservation of the nuclear spin of degree freedom. Moreover, these species have a large nuclear spin, and strontium 87, with largest nuclear spin $I = 9/2$ in the atomic ground state $^1S_0$, is an ideal candidate for the study of magnetism in enlarged SU(N) symmetry.

*Strontium 87* degenerate quantum gases were first realized in 2010 [DeS+10]. The mHz clock transition $^1S_0 \leftrightarrow ^3P_0$, associated with weak magnetic sensitivity, raised particular interest for the realization of optical lattice clocks [Boy07; Mic+09; Mic10], and proved high performances with up to $10^{-18}$ precision [Blo+14]. We are most interested in the 7.4 kHz narrow intercombination line $^1S_0 \leftrightarrow ^3P_1$, with associated hyperfine
splitting orders of magnitude larger than the linewidth. This transition offers ideal conditions for the realization of spin orbit coupling schemes [GD10; LJS11] with minimal spontaneous emission [CR14]. For instance, it has been demonstrated that spin components can be separated using conservative spin dependent forces [SGS11]. In lattices, strong artificial magnetic fields can be tailored at the optical wavelength scale with light, realizing spin dependent superlattices [GB17; Sch+20]. Neutral atoms moving in such lattices can acquire a geometric phase [Aid+11], simulating the Aharonov-Bhom effect [AB59].

Our experiment is designed to realize the 2D Fermi-Hubbard model with strontium 87 with enlarged SU(N) symmetry. It is specifically planned to study the dynamics of spin entanglement. Using a spin dependent optical superlattice [Li+16; Hei+20] associated with the intercombination line, a texture of alternating spins without interactions [Sun+18] can be prepared, realizing a Néel spin order [Née48]. By slowly reducing the depth of the superlattice, spin dynamics are enabled by the time dependent spin orbit coupling. It will be possible to slowly approach the regime where many-body physics are driven by super-exchange interactions, within a spin-independent lattice. The spin system shall, at least locally, adiabatically approach the singlet state [Sun+21]. The (quasi-)long range spin correlations, that can result effectively in squeezing of the alternate magnetization [Com+22], will be detected using the same spin dependent optical superlattice. The goal of my PhD was to set up the tools to carry out this experiment, first focusing on the SU(2) case, which is the most simple to demonstrate the practicability of the proposed schemes.

## Thesis overview

The first chapter reviews our protocol to prepare degenerate Fermi gas of strontium 87. I show that the loading of our optical dipole trap can be significantly improved by increasing the capture volume. This is realized by superimposing another set of far off detuned laser beams five times larger than the dipole trap used for evaporation. Twice the number of atoms is loaded into the dipole trap, which facilitates the preparation of Fermi seas with higher phase space densities. Typically, we can produce degenerate
Fermi gases at $T/T_F \simeq 0.2$ with 40 000 atoms and 10 spin components.

These degenerate gases are next loaded in lattices, made of a combination of a 1D lattice with 2 µm site spacing, with a 2D square lattice with 270 nm site spacing. I applied the quantum magnifier scheme demonstrated in [Ast+21] to spatially separate the different sites of the 1D lattice, and infer the number of atoms in each layer. The obtained spatial resolution is of the order of approximately 220 nm along the lattice eigen axis. Finally, I show that we are able to load the 2D lattice with approximately 95% strontium 87 atoms into the first Brillouin zone.

In the second chapter, I present our method to measure local densities with absorption imaging of objects smaller than the resolution limit of our imaging setup [Lit+21]. The method is based on the local Beer Lambert absorption law, and it takes into account the effect of fast local variations that are averaged by the detector (inducing strong bias). This method is demonstrated on elongated Fermi gases, from which we infer the unresolved transverse size, as small as one fourth of our imaging resolution limit, and can surpass the diffraction limit, or other limits due to aberrations. Then, I discuss an application of this method to attempt to measure the mean field interaction energy of a SU(10) degenerate Fermi gas of $^{87}$Sr.

The next chapters focus on the implementation of the key protocols that will be used: for the preparation of spin textures from single spin component band insulators, for the adiabatic approach to the singlet state, and for the measurement of the alternate magnetization.

In the third chapter, I first discuss our demonstrated method [Bat+20] to measure the nuclear spin populations with a spin orbit coupling scheme associated with the intercombination line [SGS11]. The robustness of our method is provided by adiabatic following of a quasi-dark state, ensuring minimal spontaneous emission. Using the strong difference of magnetic sensitivty of the coupled states, we selectively transfer, with a resonant passage, well defined momentum recoils to well defined spin states, with a simple retro-reflected laser beam. The overall efficiency is yet 85%, up to now limited by available light power. I show then that we can selectively prepare a polarized Fermi sea of $^{87}$Sr, without requiring sympathetic cooling by another atomic
species [Tey+10]. A comparison with the preparation of a degenerate ten spins mixture shows that our scheme doesn't affect the phase space density of the evaporated gases. Finally, I estimate that the Fermi sea is polarized with 90 ± 10% fiability accounting for the overall efficiency of our detection scheme.

In the fourth chapter, I demonstrate an adiabatic scheme to selectively and coher-
ently manipulate the nuclear spin states of $^{87}$Sr with light. Taking advantage of the
tensor light shift associated with the narrow intercombination line, we lift the degener-
acy of the two photon Raman transitions within the $^{1}$S$_{0}$ ground spin states manifold.
The spins are coherently flipped with an adiabatic passage through the resonance of a
selected Raman transition. This scheme is demonstrated in bulk gases. I then present
expectations for the fidelity of preparation of alternate spin textures, based on the same
principle. With a time dependent resolution of the Lindblad equation, I estimate that
the spin fidelity of the spin texture writting is 85% with respect to an alternate spin
pattern, in the actual conditions, and should raise to 97% with ongoing experimental
improvements.

In the fifth chapter, I focus on the loading of the lowest band of our 1D lattice
with large site spacing 2 µm. We are able to prepare several independent 2D Fermi
gases with approximately 99% atoms in the lowest band and strictly positive chemical
potential. By comparing two methods, I show that the adiabatic following of the 3D
to 2D dimensionnality cross over is better ensured by maintening a sufficient collision
rate rather than by reaching very deep evaporation and low temperature prior to the
lattice loading. I also developped a thermodynamic model to analyze our time-of-flight
imaging in order to provide a proper estimate of the degeneracy parameter in each of
the 2D layers. We find an optimal µ/ε<sub>F</sub> ≥ 0.4, signaling that the 2D gases are strongly
degenerate.
# Chapter 1

## Fermi gases of $^{87}$Sr in 3D optical lattices - Experimental Setup

Ten years after the first proposal [WE89], it was demonstrated in the late 90’s, on ytterbium first [Kuw+99] then on strontium [Kat+99; Muk+03], that it is possible to produce colder and denser atomic clouds with narrow line laser cooling associated with the intercombination line, than with broadband cooling. The Doppler limit is proportional to the inverse lifetime of the coupled excited state allowing to reach lower temperatures. The radiation trapping limit [WSW90; SWW91; CCL98] arising from multiple light scattering is lower for the longer lived excited states, allowing to reach higher densities [Ben+17]. While the use of large magnetic traps with alkali atoms was quickly spreading to produce degenerate gases with collision assisted evaporation, closed shell strontium with zero dipole moment in its ground state cannot be loaded in magnetic traps. The high phase space densities reached thanks to narrow line cooling are compatible with the use of far off resonant optical dipole traps [IIK00; GWO00], in that the 1/e size of the magneto-optical trap is on the order of magnitude of the waist of the lasers. Since then, the production of degenerate Fermi gases of $^{87}$Sr [DeS+10; SGS13; SSK14; Ste13] has spread [DeS+10; Tey+10; SGS11; Son+20], but remains difficult with few experiments worldwide.

This chapter presents an overview of our setup and experimental procedures to prepare degenerate Fermi gases of strontium that we trap in 3D optical lattices.
Atoms produced in an oven at 490°C and Zeeman slowed are captured in a broad-band magneto-optical trap (MOT) associated with the 30 MHz wide transition $^1S_0 \leftrightarrow ^1P_1$ at 461 nm, cooling atoms down to the Doppler limit $T_D \simeq 1$ mK. The transition is not closed so that atoms are accumulated and magnetically trapped in metastable state $^3P_2$. We repump them in the ground state with the 403 nm transition $^3P_2 \leftrightarrow ^3D_2$, to begin a second MOT stage. The second MOT is associated with the 7.4 kHz narrow intercombination line $^1S_0 \leftrightarrow ^3P_1$. This narrow MOT allows us to cool fermionic strontium down to $T \simeq 3$ µK, which is higher than the typical temperature $T < 1$ µK reached for bosonic species of strontium [SSK14]. The gas is then loaded in an optical dipole trap to start forced evaporation, which is our last step of phase space compression to reach Fermi degeneracy, down to approximately $T \simeq 0.15 T_F$ in our experiment. Finally, atoms are adiabatically loaded in 3D optical lattices, made of the combination of a vertical 1D lattice with large site spacing and a square 2D lattice.

I will first present the most relevant energy levels for laser cooling. Then I will focus on the main features of each cooling stage, to ensure a sufficient understanding of the experimental discussions in the next chapters. A complete description of the setup can be found in Pierre Bataille's thesis [Bat22] which reviews all technical specifications.

# 1.1 Cooling $^{87}$Sr to Fermi degeneracy

## 1.1.1 Energy levels

Strontium atom is an alkaline-earth species with four naturally stable isotopes. Three of them, $^{84}$Sr, $^{86}$Sr, $^{88}$Sr are bosonic isotopes all with I=0 nuclear spin, and the only fermionic isotope with 7% natural abundance $^{87}$Sr has I=9/2 large nuclear spin, which is the largest of stable alkaline-earth species. Our platform makes use of fermionic $^{87}$Sr only, and hence I will focus on this isotope.

The ground state $^1S_0$ of strontium has closed-shell electronic structure [Kr]5s$^2$, and zero electric spin. The transition at 461 nm connects $^1S_0$ to the singlet state $^1P_1$ which has a very short life time 5 ns [Wer+92; Yas+06], corresponding to a broad linewidth $\Gamma/2\pi$ of 30 MHz, as shown on figure 1.1. This broad band transition is used
<figure><img src="image_2.png" /><figcaption>Figure 1.1: Energy levels and optical transitions (solid double arrowed lines) associated with electric dipoles (E1), for fermionic strontium 87. The dotted lines indicate spontaneous decay paths: $^1P_1$ state has 1:20 000 probability to decay toward $^1D_2$, and the latter has 3:4 chance to decay to $^3P_1$, and 1:4 chance to decay to metastable $^3P_2$. The three energy levels $F = 7/2, 9/2, 11/2$ are the three hyperfine states of $^3P_1$. The energy splitting and optical transitions are not to scale.</figcaption></figure>

for capturing atoms out of our oven and for absorption imaging. From $^1P_1$, 1 out of 20
000 atoms can decay to the singlet $^1D_2$ state, and then atoms have 1:4 probability to decay to $^3P_2$ or 3:4 probability to decay to $^3P_1$.

While $^3P_1$ triplet state can decay directly to the ground state $^1S_0$ with electric dipole transition (E1) at rate 7.4 kHz, the $^3P_2$ triplet state is a long lived metastable state. The decay paths to lower energy states require high order processes such as magnetic quadrupole (M2) transitions, and the associated life time has been measured up to 500 s [YK04]. Hence, on experimental time scale, $^1S_0 \leftrightarrow ^3P_1$ is an open cycle and with optical excitation at 461 nm, atoms are continuously shelved from the ground state into $^3P_2$ at rate $\Gamma_{shelving} \approx \Gamma_{^1P_1}/2 \times 1.25 \, 10^{-5}$.

The total electronic spin is not conserved on singlet-triplet transitions such as the intercombination line $^1S_0 \leftrightarrow ^3P_1$, hence a one photon decay path associated with an electron spin flip should be forbidden, and these excited states should be stable. However, higher-order processes reduce the life-time of these excited states and may allow for single photon electric dipole (E1) transitions. For instance, $^3P_1$ is mixed with broad-band $^1P_1$ state through spin-orbit interaction (SOC), such that the life-time of $^3P_1$ is reduced to 25 µs, corresponding to linewidth $\Gamma/2\pi = 7.4$ kHz. This is much smaller than typical MHz broad transitions resulting in reduced spontaneous emission, but still large compared to clock transitions. This typically means that the intercombination line can be used at advantage both for dissipative processes and for coherent processes with associated reduced spontaneous emission.

### 1.1.2 Oven - Transverse cooling and Zeeman slower

A deposit of pure solid strontium lies inside a stainless steel reservoir. It is heated by radiation with a network of isolated heating resistances. The strontium reservoir is then attached to a nest of microtubes to ensure a velocity selection on the hot atomic vapor, so that fewer atoms with large transverse velocities can leave the oven.

The temperature of the oven is set with continuous voltage control, and it is monitored with one thermocouple probe, at contact with the external surface of the reservoir. In our experiment, we typically set the temperature of the oven at 760 K
with input power 110 W which corresponds to an outgoing collimated flux of atoms
$\dot{N} \approx 6.10^{12}$ atoms/s, with Boltzman velocity distribution and most probable axial ve-
locity $v \approx 500$ m/s. With this operating mode, the estimated lifetime of the oven is
nearly 5 years in our experiment. The strontium deposit has been reloaded only once
since the experiment launch in 2014.

The hot atomic vapor is optically collimated with 2D optical molasses at the exit of the oven [SVH05]. Two orthogonal laser beams are retroflected along the transverse velocities of the atomic flux. Transverse cooling is ensured with anisotropic beams, the largest dimension being along the atomic flux to ensure maximal interaction time with the atoms, and the frequency of the lasers is set red detuned $\Delta = -\Gamma/2$ from the $^1S_0 \leftrightarrow ^1P_1$ transition. This stage ensures an increase of the number of atoms later captured by a factor of 3 in our experiment. The axial flux of atoms in then slowed with a 50 cm long Zeeman slower [PM82], from $v \approx 500$ m/s down to the capture velocity $v_c \approx 20$ m/s of our broadband MOT.

The frequency of the blue lasers is set from a master laser with extended cavity, whose frequency is locked on a hot atomic flux with modulation transfer spectroscopy [MKC08; Lee+21]. The optical light used for transverse cooling and the zeeman slower are generated by slave laser diodes whose frequency is set and stabilized by injected light from the master laser into the slave diode cavities [Bat22]. The same procedure is done for a slave dedicated to the MOT.

1.1.3 $^1S_0 \leftrightarrow ^1P_1$ broadband MOT - $^3P_2$ shelving

The atoms are then captured in a 3D MOT on the broad-band transition at 461 nm, and cooled down to the Doppler limit $T_D \approx 1$ mK. The magnetic coils in anti-Helmoltz configuration are supplied by 170 Amper during the MOT stage, producing a magnetic quadrupole with strong field gradient $\nabla B = 51$ G/cm. The $^1S_0 \leftrightarrow ^1P_1$ transition is not cyclic, and atoms are continuously shelved at rate $\Gamma_{\text{shelving}} \approx \Gamma_{^1P_1}/2 \times 1.25 \ 10^{-5}$ from the ground state into the state $^3P_2$. This metastable state has a large magnetic dipole moment, so that metastable atoms stay trapped in the magnetic quadrupole.
The metastable state has long lifetime 9 min [YK04], but into the magnetic trap it
is lowered by the background collisions, down to nearly 10 s. In our experiment, we
typically set the time for loading atoms in $^3P_2$ to 5 s, and it can be tuned from 2 to
10 seconds depending on the best balance between signal to noise ratio and time of
sequence for the on-going experiment. After 10 seconds of loading, the populations
reach steady state and no more atoms can be accumulated in $^3P_2$. After then, the blue
MOT is turned off.

1.1.4 Repumping metastable $^3P_2$

The temperature of the atoms accumulated in $^3P_2$ is approximately the Doppler tem-
perature of the broadband MOT, which is too hot to be directly loaded in our optical
dipole trap. To pursue optical cooling until the gas is sufficiently cold, we use a second
narrow MOT stage, which requires that atoms are in the ground state. Hence, we use
the optical transition $^3P_2 \leftrightarrow 5s6d^3D_2$ at 403 nm [SS14] to repump atoms back in the
ground state with a two photon radiative decay through $^3P_1$, as sketched in figure 1.1.
When shelved in metastable $^3P_2$, $^{87}$Sr atoms are spread in different hyperfine states,
and the $^3P_2 \leftrightarrow ^3D_2$ hyperfine resonances are resolved in the GHz range. Hence, to max-
imize the efficiency of the rempumping, we modulate the frequency of the repumper
laser to scan through the three most populated hyperfine resonances, and this is done
by modulating the voltage of the piezo-electric crystal of the cavity of the laser. The re-
sulting amplitude of the laser frequency modulation is 3 GHz in our experiment, which
allows to repump up to 40 million atoms after loading the metastable during 5 seconds
with oven heated at 760 K.

Although rempumping through other $5snd^3D_2$ states can be used, for instance
$n=5$ [Pol+05; SGS13] at 497 nm or $n=4$ [Mic+09; Aka+21] where the decay paths
are favorable to minimize atom loss, we stick to using the 403 nm line for its easier
implementation and low cost laser. It is also possible to continuously repump atoms
during the broadband MOT stage in order to reach lower temperatures, at the cost
of atom loss [SSK14]. Since the number of atoms is critical for our experiment to
produce deeply degenerate Fermi gas, we rather repump atoms to the ground state
after turning off the blue MOT. Note that it was recently demonstrated [Aka+21]
that it is possible to take advantage of the cyclic transition $^3P_2 \leftrightarrow 5s4d^3D_3$ at 3 µm
inside the magnetic quadrupole, and produce ultra-cold gas at $T \approx 200$ nK with one
dimensional polarization gradient cooling.

1.1.5 $^1S_0 \leftrightarrow ^3P_1$ narrow MOT

The MOT

The second and most critical optical cooling stage in our experiment is associated
with the 7.4 kHz intercombination line $^1S_0 \leftrightarrow ^3P_1$ at 689 nm. While the achievable
temperature is limited in a broadband MOT due to high Doppler temperature, and
the density is limited due to the effective repulsive force associated with multiple light
scattering [WSW90; SWW91; CCL98], MOT associated with the intercombination line
allows to produce much colder and denser atomic clouds [Kat+99], since the Doppler
temperature is as cold as $T_D = 177$ nK for strontium, which is very convenient for
further loading optical dipole traps.

Within $^3P_1$, the hyperfine states are highly resolved with $\Delta_{HFS}/\Gamma_{^3P_1} > 100\,000$, and the MOT laser beams are choosen red detuned from the $^3P_1$, $F = 11/2$ hyperfine state such that $m_F = \pm 9/2$ spin states can be trapped with $\sigma_\pm$ laser beams. For this transition the Doppler limit is $T_D = 177$ nK while the associated recoil temperature $T_R = 462$ nK, defined as $T_R = \hbar^2k^2/2m$ where $k = 2\pi/\lambda$, so that MOT cooling is limited by the recoil energy of one photon and not by the Doppler effect. This allows to directly cool atoms in the $\mu$K range with standard MOT without using sub-Doppler cooling techniques, which is very convenient. However, the capture velocity is limited by the narrow linewidth, such that artificial broadening is needed to capture repumped atoms at 1 mK, either with frequency modulation or with power broadening. Hence, the 689 nm laser light is initially at full power, nearly 12 mW distributed over the 3 MOT axes, and its frequency is triangularly modulated over 4 MHz at rate 20 kHz. Simultaneously, the input current of the MOT coils is lowered to 3.5 Ampers for a weak magnetic field gradient $\nabla B \approx 1$ G/cm. At this stage, we capture up to 40 millions atoms at $T \approx 25$ $\mu$K. Then the modulation is reduced in two steps down to zero and
the intensity down to a few $I_{sat} \approx 3~\mu\text{W/cm}^2$.

For this narrow line cooling, and any optical scheme associated with the intercom-
bination line, high laser stability and finesse are required. Hence, the frequency of our
master laser is locked onto an ultra-stable cavity using a Pound-Drever-Hall feedback
signal [Dre+83]. The laser spectrum is as narrow as 1 kHz which is sufficient for the
targeted 7.4 kHz linewidth. The optical beams used for the narrow MOT come from
slave laser diodes, whose frequency is set and stabilized by injecting light from the
master laser into the slave diode cavity [Bat+20].

Stirring laser

The narrow MOT for $^{87}$Sr requires a second laser to prevent strong atom loss [SSK14]. With high nuclear spin $I = 9/2$ and long lifetime $\tau \approx 20~\mu$s of the excited state $^3P_1$, atoms can remain in excited spin states transparent to light for a sufficiently long duration such that they leave the trap. This is illustrated on figure 1.2, left. If considering a single spin state for simplicity, here $m_F = 5/2$, the dissipative force is localised on the left hand side of the zero field position. Actually, this scheme is symmetric in $m_F$, accordingly with the Zeeman shift, and considering negative spin states, the dissipative force is localised on the right hand side of the zero field position. Hence, atoms would need to flip from $m_F$ to $-m_F$ when crossing the zero field to feel a global restoring force, as shown by the average over all spin states as the blue dashed line on the top figure. While magnetic atoms such as alkalis can flip spin when crossing the trap center and always remain in a stable trap configuration, alkaline earth atoms with zero magnetic moment need light assistance to change spin states.

It was then proposed [Muk+03] to use a second laser beam for the narrow MOT stage
to randomize the spin states such that every atom feels a restoring force on average,
minimizing the probability of atom loss. This second laser, so called *stirring* laser
for the $^{87}$Sr narrow MOT, has its setup in MOT configuration, and is actually super-
imposed to the MOT laser, and its frequency is set red detuned from the $^3P_1$, $F = 9/2$
hyperfine state. Figure 1.2, right, shows the spatial dependence of the STIR saturation
parameter, here defined as:
<figure><img src="image_3.png" /><figcaption>Figure 1.2: Instability of the narrow red MOT for large spin $^{87}$Sr isotope. Left: spatial dependence of the restoring force (a.u.) for the narrow MOT associated with the intercombination line $|^1S_0\rangle \leftrightarrow |^3P_1, F = 11/2\rangle$. The ground spin state $m_F = 5/2$ is coupled to Zeeman shifted excited 3/2 and 7/2 spins of $F = 11/2$ hyperfine state with Lande factor $g_F \approx 0.36$. For a given detuning (here $\Delta = -4\Gamma$), the corresponding restoring force is plotted on the upper figure as the solid line. The restoring force averaged over all the spin states is plotted as the dashed line. Right: spatial dependence of the saturation parameter (a.u.) associated with $\sigma_+$ and $\sigma_-$ excitation on the STIR line $|^1S_0\rangle \leftrightarrow |^3P_1, F = 9/2\rangle$ with detuning $\Delta = -\Gamma$, as solid line for the ground spin 5/2, and dashed line for the average over all ground spin states.</figcaption></figure>

$$s(r) = \frac{\Omega^2}{4\Delta^2 + \Gamma^2} \qquad (1.1)$$

where $\Delta = \delta - m_F g_F \mu_B B$ for atoms at rest, and $\Omega^2 = (\Gamma^2/2) \times (I/I_{sat})$ for both $\sigma_+$ and $\sigma_-$ excitation. Since the $g_F$ factor is 4.5 smaller than that one of $^3P_1$, $F = 11/2$ state, atom-light interaction is much less localized than the MOT line so that atoms are no more transparent to light and hence have the possibility to change spin states. With an efficient randomization of the spin states associated with the STIR line, all atoms feel an averaging restoring force (dashed line, upper left figure 1.2), and remain confined into the MOT. The intensity and frequency of the stirring laser are set with the
same conditions than the MOT laser, starting with a frequency modulation of 6 MHz
and strong intensity, and with final single frequency at a few $I_{sat}$ with no modulation.

In the final narrow MOT stage with field gradient $\nabla B \simeq 2.5$ G/cm and detuning
$\Delta \simeq \Gamma_{3P_1}$ for both stirring and MOT lasers, the bulk gas of $^{87}$Sr extends over nearly
600 µm with 7 million atoms at $T \simeq 3$ µK. The next critical step to reach Fermi
degeneracy is to transfer trapped atoms from the narrow MOT to the optical dipole
trap to perform forced evaporation.

1.1.6 Optical dipole trap loading

The transfer of atoms from the narrow MOT in our optical dipole trap (ODT) [GWO00]
is associated with a few critical difficulties. First, there is the strong difference of size
of the two traps. With simple geometry considerations, a straight volume mapping of
the narrow MOT bulk gas onto the ODT shape would mean that less than 1% of the
atoms would be transfered from one trap to the other. To increase to loading efficiency,
the MOT is further compressed [I1K00] simultaneously to loading the ODT. However,
inside the ODT, the differential light shift associated with the intercombination line
has to be managed to prevent strong atom loss. Because the ground state is more
red-shifted than the excited state $^3P_1$, the $^1S_0 \leftrightarrow ^3P_1$ transition acquires a blue shift
much stronger than $\Gamma_{3P_1}$. Morover, the light shift varies spatially as a function of
the intensity profile of the ODT, which is much smaller than the size of the narrow
MOT. Hence, if we choose the MOT light to be slightly red detuned at the center of
the dipole trap in presence of the light shift, it is blue detuned at the edge of the ODT
and atoms are expelled. I will here first introduce the properties of the ODT and the
space dependent resonance for the intercombination line, then I will quickly describe
our method to circumvent those two major difficulties and maximize the phase space
density when loading the ODT.

*Geometry:* our optical dipole trap is made of two crossed far-off red detuned laser beams at 1070 nm. The first beam is nearly horizontal with small tilt $\simeq 1^\circ$, and its shape is elliptic with large horizontal waist $\simeq 150~\mu$m and small vertical waist $\simeq 80~\mu$m
at the atoms position, for a strong vertical confinement. The second beam is nearly vertical with angle ≈ 30° with respect to gravity, and its shape is isotropic with waist ≈ 80 µm at the atoms position, for strong confinement in the horizontal plane. The crossing of the two beams defines a rather spherical volume of radius ≈ 80 µm, which is nearly 500 times smaller than the size of the 600 µm wide narrow MOT before loading. The depth of initial trap at loading is 50 µK at the crossing, accounting for gravity sag, with power distribution P=9.6 W in the horizontal beam and P=4.7 W in the vertical beam.

*Differential light shift:* atoms are transfered from the narrow MOT associated with the $^1S_0 \leftrightarrow {}^3P_1$ transition at 689 nm, to the ODT at 1070 nm. For far-off detuned atom-light interaction and non-resolved hyperfine structure, the light shift is determined only by the rotationnally invariant contribution to the dynamic polarizability and reads as the dipole potential which can be computed from resolution of the optical Bloch equations [CDG98]:

$$U_{dip} = -\frac{\hbar|\Omega|^2}{4\bar{\Delta}} \qquad (1.2)$$

Here, $1/\bar{\Delta} = 1/(\omega_L - \omega_0) - 1/(\omega_L + \omega_0)$, such that both the resonant and anti-resonant contributions to the atom-light dipole interaction are considered. $\Omega$ is the Rabi coupling frequency, which is easily computed from the inverse lifetime $\Gamma$ and light intensity $I$:

$$\Omega^2 = \frac{\Gamma^2 I}{2 I_{sat}} \qquad (1.3)$$

where $I_{sat}$ is the saturation intensity of the considered optical transition.

Regarding the light shift of $^1S_0$, the contribution from $5snp^1P$ states, $n < 11$, accounts for more than 99% according to Thomas-Reiche-Kuhn sum rule of oscillator strengths [Lud08]. The light shift can then be computed from the sum of the relative dipole potentials $U_{ak} \propto A_{ak}/\bar{\Delta}_{ak}$ where $A_{ak}$ is the transition probability, according to the known values of the transition probabilities and wavelengths [Wer+92; SN10]. The light shift of triplet state $5s5p^3P_1$ can be properly defined by considering
<figure><img src="image_4.png" /><figcaption>Figure 1.3: Differential light shift of the $^1S_0 \leftrightarrow ^3P_1$ intercombination line during the loading of the optical dipole trap at 1070 nm. Black solid lines indicate the energy levels light shifted and the red solid line indicates the MOT light frequency with the ground state space dependent offset. For this frequency, the MOT light is red detuned at the bottom of the trap and blue detuned outside.</figcaption></figure>

the triplet-triplet transitions involving excited $^3S$, $^3P$, and $^3D$ states, with according
oscillator strengths and line wavelengths [Wer+92; SN10]. The shift $\delta$ of the transi-
tion $^1S_0 \leftrightarrow ^3P_1$ is then simply the difference between the two introduced light shifts
$\hbar\delta = U_{dip}(^3P_1) - U_{dip}(^1S_0)$. In our experiment, $U_{dip}(^1S_0) \approx -1.5$ MHz for the ground
state and $U_{dip}(^3P_1) \approx -1.2$ MHz, such that the intercombination line is shifted by
nearly 300 kHz to the blue at the bottom of the trap, and is spatially proportional to
the intensity shape of the dipole trap laser beams. This is shown on figure 1.3.
ODT loading: the 1070 nm laser beams are quickly turned on after the narrow MOT stage, and atoms in continuous motion are attracted by the dipole trap of depth 50 µK. In the bottom of the ODT, the MOT resonance is so much shifted that it can be neglected and atoms simply run their phase space trajectory for conservative dipole interaction with no capture nor compression. Hence, we want to shift the light frequency to the blue to get closer to resonance at the bottom of the trap. This way, atoms are slowed down and captured, and we can obtain further phase space compression with MOT light induced dissipation inside ODT. Note that because of the differential light shift, atoms outside the ODT see blue detuned light. We then slowly ramp the frequency of the MOT light by 170 kHz during 120 ms, further compressing the position selectivity of cooling inside the differential light shift, and increasing phase space density. The intensity of the MOT light is simultaneously and slowly turned down to zero, so that is it as low as possible when getting close to resonance in the high atomic density region at the bottom of the dipole trap. This way, a balance is found between dissipative cooling and heating at the outskirts. With this method, we are able to load up to 1.4 million atoms at 3.8 µK inside the ODT, which corresponds to nearly 25% atoms transfered from the MOT to ODT with depth to temperature ratio $U/k_BT \simeq 13$, with density $n_0 \simeq 1.15 \times 10^{20} \text{atoms.m}^{-3}$.

*Enhanced capture volume:* with this setup, we are limited by the capture volume of the ODT compared to the volume of the narrow MOT. To increase the number of atoms captured, we take advantage of a later used far-off detuned optical lattice to increase the capture volume of the dipole trap, as shown on figure 1.4. We use a 1D optical lattice at 1064nm (introduced in section 1.2.1) with larger vertical waist 100 µm and horizontal waists 170 µm, such that the capture volume is more than 5 times larger. This lattice is one dimensional with vertical eigen axis, defining layers of large $170 \times 170 \mu m^2$ dipole traps with approximately 500 kHz depth, spaced by 2 µm.

This 1D lattice is turned on quickly and simultaneously to the ODT. The combination of the ODT and lattice makes a deeper trap reaching almost 90 $rm\mu K$, and we are able to load up to 2.6 million atoms with temperature 7.2 µK, which corresponds to nearly 43% atoms transfered from the MOT to ODT with depth to temperature ratio $U_{ODT}/k_BT \simeq 12.5$. This way, twice the number of atoms are loaded,
<figure><img src="image_5.png" /><figcaption>Figure 1.4: Enhanced capture volume of the optical dipole trap. Left: optical dipole trap loading without 1064 nm lattice, in situ (top) and after time of flight (bottom), absorbtion images. Middle: enhanced loading using the 1064 nm lattice. Top right: vertical cut of the saturated in situ absorbtion images, the dashed line is the in-situ density on the saturated region infered from a Boltzmann distribution and the low density wings. Bottom right: integrated optical depth from the time of flight images (bottom) used to measure the number of loaded atoms and temperatures with gaussian fits for Boltzmann gases in harmonic traps.</figcaption></figure>

while the density $n_0 \simeq 3.3 \times 10^{19}$ is approximately unchanged, accounting for the weak increase of confinement $\omega'^2 = \omega_{ODT}^2 + \omega_{1064}^2$, and $U/k_B T$ 1.3 times smaller. At this step, the collision rate is approximate one collision per atom per millisecond.

Note that the efficiency of loading could be further improved with additional lasers. An ODT at magic wavelength 840 nm for the intercombination line [Muk+03] circumvents the differential AC Stark shift and hence allows to have homogeneous MOT cooling inside the dipole trap. Also, it is possible to use a transparancy beam [Son+20], detuned by a few GHz from the $^3P_1 \leftrightarrow ^3S_1$ transition, resulting in a localized light shift
for the intercombination line. Atoms inside the transparency beam are protected from MOT light, and the energy is dissipated through collisions to outside atoms which are continuously optically cooled, allowing to further pass the radiation pressure limit and reach even higher densities and lower temperatures before evaporation.

1.1.7 Forced evaporation

The gas is finally cooled down to Fermi degeneracy, see figure 1.5, with forced evapo-
ration. The 1D optical lattice used to help loading the optical dipole trap is smoothly
turned off during the first 2.5 seconds of evaporation. This way, atoms are transfered
from vertically stacked 2D harmonic traps to a three dimensional (3D) harmonic trap,
which permits a more efficient evaporation [LRW96] within higher dimensionnality. In
the meantime, the power of the horizontal ODT beam is lowered from 11.6 W to 2.4
W and from 4.7 W to 2 W for the vertical beam. This corresponds to a reduction of
the geometric mean of the trap frequency $\bar{\omega} = (\omega_x\omega_y\omega_z)^{1/3}$ from 275 Hz to 140 Hz. The
powers of the dipole trap laser beams are then reduced much more slowly at the end
of the evaporation, the horizontal beam from 2.4 W to 1.2 W and the vertical beam
from 2 W to 0.84 W, during 5 seconds. All in all, in our experiment, forced evaporation
runs over 8 to 10 seconds, and the gas reaches degenerate regime with 10 Fermi seas at
$T \le 0.5 T_F$ with approximately 7500 atoms per spin state, and the phase space density
$\rho \propto N\bar{\omega}^3/T^3$ is increased by three orders of magnitude, with respect to the dipole trap
loading, as shown on figure 1.5. Note that our evaporation efficiency is limited by the
background collisions, with typical time $\tau \simeq 8$ s which is approximately the duration of
evaporation.

1.2 Optical lattices

The objective of our experiment is to realize a quantum simulator for the Fermi Hubbard
model. In our experiment, this is realized by trapping spinor atoms at the sites of several
independent 2D optical lattices which are vertically stacked. The resulting 3D optical
lattice is presented in figure 1.6. It is made of a vertical 1D lattice at 1064 nm, defining
horizontal 2D layers at its maxima, and a 2D lattice at 532 nm shaping 2D square
<figure><img src="image_6.png" /><figcaption>Figure 1.5: Forced evaporation dynamics. The temperatures and number of atoms are measured from time of flight images, and the trapping frequencies of the dipole trap are inferred from an independent calibration of the trap geometry. At each evaporation step, the gas thermalizes during 330 ms before time of flight. In inset, the phase space density $\rho$ increase with respect to the initial gas $\rho_0$ at ODT loading, where the phase space density is computed as $N\bar{\omega}^3/T^3$.</figcaption></figure>

lattices within each horizontal layer. The combination of the two lattices is a set of horizontal 2D square lattices with sites spacing 256 nm separated by $\simeq 2\mu$m. The tunneling within the 2D bravais lattices is controlled by the intensity of the 532 nm laser beams, while the tunneling from layer to layer is separately controlled by the depth of the 1064 nm lattice. The tunneling from layer to layer is negligible on the timescale of to the 2D spin dynamics.

In this section, I first introduce the geometry of the vertical 1D lattice at 1064 nm, and our method to measure the depth of the low recoil lattice. Moreover, several sites of the 1D lattice are loaded in our experiment, typically 8 layers, and I discuss our
<figure><img src="image_7.png" /><figcaption>Figure 1.6: Optical lattices setup. Two lasers beams at 532 nm with $\theta = 16°$ elevation are retroflected to shape a 2D lattice, and the 1D lattice is made of two interfering laser beams at 1064 nm separated by $\theta = 32°$ elevation. The resulting lattices are layers of 2D lattices separated by 2 µm, with sites spacing of 256 nm.</figcaption></figure>

application of the quantum magnifier scheme [Ast+21] to measure the distribution of
atoms in the several layers. Then, I introduce the geometry of the 2D lattice.
1.2.1 1064 nm 1D lattice

Geometry

The 1D lattice is made from the interference of two laser beams at 1064 nm with wavevectors $\vec{k}_1, \vec{k}_2$. In a spherical basis $(r, \theta, \phi)$, where elevation angle $\theta$ and azimuthal angle $\phi$ are defined on upper left inset of figure 1.6, the wavevector $\vec{k}$ of each laser beam is here defined as:

$$ \vec{k} = \frac{2\pi}{\lambda} \begin{pmatrix} \cos\phi \sin\theta \\ \sin\phi \sin\theta \\ \cos\theta \end{pmatrix} \qquad (1.4) $$

The two laser beams at 1064 nm have same azimuthal angle $\phi$ and elevation angle $\theta = \pm 16^\circ$ so that they are separated by $\delta\theta = 32^\circ$ elevation and $\vec{k}_1 \cdot \vec{k}_2 = k_1 k_2 \cos(32^\circ)$. The stationary wave resulting from the interference of the two laser beams is a vertical 1D lattice with eigen-axis $\vec{a}$ along $\vec{k}_1 - \vec{k}_2$, and the sites spacing is $a = \lambda/2 \sin(16^\circ) \simeq 2 \mu m$. The two beams are controlled by the same AOM and separated right before crossing at the position of the Fermi gas, ensuring that the phase coherence between the interfering beams is maximally maintained.

Depth and band gap calibration of low recoil 1D lattice

Lattice depth calibration is usually done with diffraction calibration [Den+02]. In the case of low recoil lattices, the diffraction orders cannot be resolved with time of flight imaging since they are separated by only a few nK, 7 nK in the case of 2 $\mu$m sites spacing for $^{87}$Sr, which is much lower than the lowest momentum spread that can be reasonable attained for degenerate Fermi gases of strontium, ($\simeq 20$ nK with approximately 20 000 atoms in our experiment). To circumvent this issue, we rather observe the time evolution of the gas momentum distribution inside the lattice potential [Huc+09] after suddenly switching it on. The low recoil lattice is pulsed on bulk gas, long enough such that the atoms are accelerated into the lattice potential. Atoms initially laying next to the negative interferences of the lattice will acquire maximal acceleration when reaching the bottom of the sites. When the lattice is turned off, with time of flight, then the atoms will expand with the kinetic energy acquired from the
<figure><img src="image_8.png" /><figcaption>Figure 1.7: Experimental images for the depth calibration of a low recoil 1D lattice. For each duration of the lattice pulse, in bottom axis, a time of flight image is recorded.</figcaption></figure>

lattice depth. If the pulse time is long enough, then atoms must have had enough time to explore the whole depth of the lattice sites, and then the maximal velocity measured in time of flight maps the depth of the lattice.

Figure 1.7 shows experimental images of the scheme for different pulse times. If the pulse time is long enough, then the atoms have explored the whole depth, which happens from 60 µs in this measurement. Figure 1.8 shows a cut of the optical depth from figure 1.7 along the longitudinal axis for each time of flight, from which it is possible to detect the atoms with maximal velocity. At 60 µs of pulse, the maximal separation in the imaged density is nearly $dx \simeq 100$ px, such that the maximal velocity acquired is $v_{max} = \frac{dx}{2t_{tof}}$. This gives the maximum acquired kinetic energy, which gives the measurement of the depth $V_0$ of the lattice:

$$V_0 = \frac{1}{2}mv_{max}^2 \qquad (1.5)$$

From this measurement, the trap depth is 113 kHz, which corresponds to $V_0 = 791 E_R$. The band gap can then be directly inferred from $\hbar\omega_{BG} = 2\sqrt{V_0E_R}$, which is 8 kHz in this case.
<figure><img src="image_9.png" /><figcaption>Figure 1.8: Longitudinal cut profile through the experimental images of the depth calibration shown in figure 1.7, for each pulse time, shown in left axis. There is no axis showing the density.</figcaption></figure>

Distribution of atoms in the several layers

To measure the number of atoms in each layer, I applied the quantum magnifier scheme recently demonstrated by [Ast+21]. This scheme relies on exchanging the position and momentum distributions of the atoms with a $\pi/2$ phase space rotation, and this is realized in a conservative harmonic potential of period T every $\delta t = T/4$. In the experiment, this is done by suddenly turning on and off a harmonic potential during one fourth of the period T = $2\pi/\omega$ where $\omega$ is the trapping frequency along the magnification direction, vertical in this case, and this way, initial positions {$x_i$} of trapped atoms converted to initial momentums {$p_i$}. Then, the gas is released for a time of flight measurement. Since the measured positions map the initial momentum distribution, this is actually a measurement of the spatial distribution of the atoms in the lattice
<figure><img src="image_10.png" /></figure>

Figure 1.9: Magnification of the sites of the 1D lattice using the quantum magnifier scheme [Ast+21] along the vertical axis. Left: experimental absorption image of the magnification of the sites, with approximately 85 × 10³ atoms at $T/T_F \simeq 0.3$ before loading. The color bar represents the optical depth per pixel surface. Center: fit of the integrated optical depth to measure the populations into the different 2D layers. The relative population measured in each layer is noted as a percentage above each peak picturing a layer, in red, big font. The bins represent a guess of the populations with respect to the size $\sigma$ of the initial gas and the sites spacing $a$ of the lattice. Here, $\sigma \simeq 3.1~\mu m$ and $a \simeq 2~\mu m$. The relative population guessed in each layer is noted as a percentage above each peak picturing a layer, in dark blue, small font. Right: loading of two layers only.

$\{x_i\}$ thanks to the harmonic pulse. If the eigenaxis of the harmonic pulse is perfectly colinear with the eigenaxis of the bravais lattice, then this measurement results in a magnification $M = \omega\tau$ of the bravais lattice, where $\omega$ is the harmonic frequency and $\tau$ the time of flight.

In our experiment, I used the horizontal laser beam used for the dipole trap with strong trapping frequency essentially along the vertical axis, $\omega_z \simeq 2\pi \times 450$ Hz, which is parallel to the lattice eigenaxis. With a $T/4$ pulse and 10 ms of time of flight, the vertical magnification is $M \simeq 30$ for sites spacing $a \simeq 2~\mu m$ of the 1D lattice, which correponds to approximatively 9 pixels of size 6.45 $\mu m$ in our camera. The resulting experimental absorption imaging is shown in figure 1.9, left, where we can see that the 2D layers are well resolved. Figure 1.9, center shows the integrated optical depth along
the axis of the layers, such that it is possible to measure the number of atoms in each
layer. From this data, I deduce the relative populations in each site, as noted above
the integrated density of each layer in red, big font. The measure let us know that
approximately 50% of atoms are into the 2 equally populated middle layers, and about
30% in the two next equally populated layers. This measure is compared with a guess
of the populations, infered from the size $\sigma_z = \sqrt{k_B T/m\omega_z^2}$ of the initial Boltzmann gas
and the sites spacing $a$. This indicates that the atoms barely move while being loaded
into the 1D lattice, as a consequence of the long period of the potential that rapidly
kills tunneling at bar lattice depth. The initial density is sliced into bins mapped on the
lattice sites, as shown on the figure, and then the relative number of atoms are infered
from the binned density. The result, for a gas of initial size $\sigma_z \approx 3.1$ µm, corresponding
to an initial 3D trap with frequency $\omega_z/2\pi \approx 130$ Hz at temperature $T \approx 65$ nK, is
texted above the bins in dark blue, small font. The guess fits very well with an average
deviation of less than 1%. In the guess, the lattice sites are shifted by $a/5$ to match
the asymmetry in the populations of the sites. Finally, we show that the number of
loaded layers can be tuned, as it depends on the initial extension of the 3D gas. Hence,
adiabatic compression of the gas along the lattice eigen-axis reduces the number of
lattice sites that are loaded, and figure 1.9, right, shows that we are able to load two
layers only, which moreoever have approximately the same number of atoms.

Layers independence

The layer to layer tunneling is suppressed thanks to the very low recoil, i.e. very
large sites spacing *a*, of the lattice. Strontium atoms in the ground state have zero
eletric dipole moment, and the magnetic dipolar interaction of the nuclei are negligieable
[Gor+10]. The 2D layers defined by the 1D lattice are then completly independent. This
is actually a very interesting feature since it allows to simultaneously run independent
quantum simulations in each layer, and hence increases the experiment capability to
acquire much data and increase the quality of statistical analysis.
### 1.2.2 532 nm 2D lattice

#### Geometry

The 2D lattice is made of two retro-reflected laser beams with waists 150 µm at the atoms position and powers $P_1$ and $P_2$. The intensities are controlled with two independent AOMs detuned by 200 MHz to prevent interferences. The azimuthal angle $\delta\phi$ between the two laser beams is exactly 90° and both have $\theta = 16°$ elevation. The spherical coordinates of the wavevectors of the two beams can then be written as $\vec{k}_1 = (k, \theta_1 = +16°, \phi_1 = \phi)$ and $\vec{k}_2 = (k, \theta_2 = \theta_1, \phi_1 = \phi + 90°)$, where $k = 2\pi/\lambda$, $\lambda = 532$ nm. The intensity of resulting stationary wave is

$$I(\vec{r}) = 4P_1 \sin(\vec{k}_1 \cdot \vec{r})^2 + 4P_2 \sin(\vec{k}_2 \cdot \vec{r})^2 \quad (1.6)$$

Within each 2D layer defined by the maxima of the 1D lattice at 1064 nm, the projected 2D bravais lattice can be then written as:

$$\vec{R}_{m,n} = m \frac{\lambda}{2 \cos \theta} \vec{a}_1 + n \frac{\lambda}{2 \cos \theta} \vec{a}_2 \quad (1.7)$$

where $\theta = 16°$. Here, $m$ and $n$ are the sites indices of the bravais lattice and $a_1$, $a_2$ the eigen-axis, respective normalized projections of $k_1$ and $k_2$ along the (Oxy) plan defined on figure 1.6.

Note that $k_1$ and $k_2$ are not orthogonal, $\vec{k}_1 \cdot \vec{k}_2 \propto \cos(\theta)^2 \neq 0$, but the projection of the 2D lattice on the horizontal planes defined by the 1D lattice maxima is a proper 2D square lattice. Indeed, $\vec{a}_1 = (1, \theta = 0, \phi_1 = \phi)$ and $\vec{a}_2 = (1, \theta_2 = 0, \phi_1 = \phi + 90°)$ so that $\vec{a}_1 \cdot \vec{a}_2 = 0$. Moreover, the $\delta = 200$ MHz detuning between the two lattice arms is negligible with respect to the light frequency $\nu_l = c/\lambda$, i.e. $\delta/\nu_l \simeq 3 \cdot 10^{-7}$, so that the lattice can be considered square, with sites spacing $a = 276$ nm.

#### Ground band population

To measure the number of atoms in the ground band of the 2D lattice, I first identified the first Brillouin zone (FBZ) with a diffraction measurement. The 2D lattice is quickly turned on and off during a few microseconds on the bulk gas, and the diffraction orders
<figure><img src="image_11.png" /><figcaption>Figure 1.10: Absorption images of atomic clouds after diffraction (a) and band mapping (b) of the 2D lattice at 532 nm. (a) Simultaneous diffraction of the atomic cloud by the two arms of the 2D lattice. The first Brillouin zone (FBZ) at $\pm 1\hbar\vec{k}_R$ is graphically inferred from the first diffraction orders at $\pm 2\hbar\vec{k}$ momentum recoils. The non diffracted population is the brightest central cloud. (b) Measurement of the population in the ground of band of the 2D lattice with 2D band mapping after adiabatic loading of the lattice, where the FBZ is mapped from the diffraction measurement.</figcaption></figure>

are identified after a time of flight. The first diffraction orders of each arm of the 2D lattice give a measurement $\pm 2\hbar\vec{k}_R$, where $\vec{k}_R$ is the momentum recoil of each lattices, which further allows to identify the FBZ. This is shown on figure 1.10, (a), where the FBZ is identified as the white rectangle at $\pm 1\hbar\vec{k}_R$ of each lattice. Then, the number of atoms in the FBZ after adiabatic loading of the 2D lattice can be measured from a band-mapping [Köh+05]. In our experiment, this is done by slowly turning off the 2D lattice within 2 ms before time of flight. By mapping the FBZ identified with lattice diffraction onto the band mapping, it is possible to count the number of atoms loaded in the FBZ. With this measurement, I infer that 91% to 97% atoms are loaded in the ground band of the lattice, allowing for $\pm 1$ pixel uncertainty on the identification of the FBZ.
# Chapter 2

## Absorption imaging of objects smaller than the resolution limit

In our experiment and more generally in the whole community, we study physics in cold and ultra cold gases which are typically tiny objects not much larger than a micrometer. Such objects are sometimes not resolved by standard imaging setups with low numerical aperture, and we typically consider that the data provided by *in-situ* absorption imaging cannot be used in that case. In this chapter, I present our method [Lit+21] to recover the actual size of our *in-situ* object even when this is below our imaging resolution limited by either low numerical aperture, pixelation, or aberrations such as out of focus imaging. It relies on an estimate of the fraction of absorbed photons inferred from two absorption pictures, with and without atoms, which depends non-linearly on the local density of atoms when the optical depth is large enough. By using a convenient ansatz on the density profile, and an independent measurement of the number of atoms, we show that the actual size of the sample can be known. We have tested our method on absorption images of elongated gases, unresolved along the transverse axis only, in which case a straight application of the integrated Beer-Lambert law leads to strong distortions of the measured local density, so that even the longitudinal axis, assumed resolved, is distorted. Our results show that we are able to measure RMS sizes as small as one fourth of our resolution limit, and we confirm the validity of our method by comparing our measurements to expected sizes from equi-partion of energy. Finally, I will show that this method also permits to recover the density profile along the elongated
axis. We verified this result with a comparison to expected density profiles from both Boltzmann statistics for hot gases and Fermi statistics for degenerate gases.

## 2.1 Motivations

### 2.1.1 Measuring the interaction energy of an SU(10) degenerate Fermi gas

This work was first motivated by an experiment to measure the enhanced mean-field interaction in the weakly interacting regime of a SU(10) degenerate Fermi gas [Son+20] of fermionic $^{87}$Sr, which has 10 degenerate ground spin states and spin-independent scattering length $a_S = 97 a_0$, in Bohr radius units. In a two components Fermi gas with equal populations, the mean-field energy $E_{int}^{SU(2)}/E_{ho} \simeq k_F a_s/3$ [VS99; MPS02] is small, where $k_F$ is the Fermi momentum, $a_s$ the scattering length and $E_{ho}$ the oscillator energy. However, a Fermi gas with enlarged SU(N) symmetry [CHU09; Gor+10; Tai+12] has increased interaction energy $E_{int}^{SU(N)} = (N-1)E_{int}^{SU(2)}$ due to reducted role of the Pauli exclusion principle. When a trapped ultra cold quantum gas is released by switching off the trap, the local density decreases during expansion and hence so does the mean-field energy $E_{int}$, which is progressively converted to kinetic energy due to energy conservation. A measurement of the velocity distribution after time of flight provides the release energy $E_R$ [Bou04], which is the sum of the initial kinetic energy prior to the release $E_K$, with the mean-field energy [GPS08]. In a given direction, the release energy is then given by

$$E_{R,\text{tof}}^{1D} = \frac{1}{3}E_K + \frac{1}{3}E_{\text{int}} \qquad (2.1)$$

In our experiment with fermionic $^{87}$Sr, we wanted to further enhance the mean-field signal with a 1D expansion, i.e. by allowing the gas to expand along one axis only, so that the full mean-field energy is converted to kinetic energy into this axis only. In this case, the release energy along a given direction is then given by:

$$E_R^{1D} = \frac{1}{3}E_K + E_{\text{int}} \qquad (2.2)$$
The mean field energy can finally be infered by comparing these two quantities, i.e. the release energy measured from a 1D expansion and measured independently from a time of flight. This method has the advantage that it is twice as sensitive as a single time of flight measurement of the release energy to infer the mean field energy.

## 2.1.2 Distorsion of imaged in-situ density

We first did the measurement of the release energy with a 1D expansion. The gas, initially trapped in a 3D harmonic potential, is suddenly released along one direction only while it remains confined in the two other directions. After expanding along this single direction, we take in-situ absorption images of the elongated gas inside the 2D confinement, which is presented in figure 2.1 (a). From this measurement, we identify several problems. First, the imaged gas is confined into the 2D trap, and figure 2.1 (b) shows that the density distribution along the short axis, which is along the 2D confinement, is pixelated as it is spread over a few pixels only, and the negative signals indicate imaging aberrations such as light diffraction on the atomic cloud, and out of focus imaging. Moreover, from the imaged density of atoms $n(i, j)$, measured from the standard application of the Beer Lambert law on each pixel (i,j):

$$n(i, j) = -\frac{1}{\sigma_0} \log \frac{I(i, j)}{I_0(i, j)} \qquad (2.3)$$

we obtain an integrated density profile along the elongated axis which shows significant deviations from the expected gaussian distributions at temperatures $T > T_F$ as shown in figure 2.1 (c). Here, $\sigma_0 = 3\lambda^2/2\pi$ is the resonant atom-light absorption cross-section, and $I(I_0)$ the recorded light intensity with (without) atoms. Finally, the number of measured atoms on the 1D expansion measurement, $N_{at}^{1D} \simeq 9 \times 10^3$ did not match ones of the time of flight experiments $N_{at}^{tof} \simeq 14 \times 10^3$, which are shown on figure 2.1 (d). This is explained because the density profile is not resolved along the short axis, resulting in pixelation, and the local density of atoms cannot be directly inferred from the Beer-Lambert law used for absorption imaging because of its non linearity. Hence, using the Beer-Lambert law induces non-linear error on the estimated local atomic density, which results on a distorted image, and any straight estimate of the release energy is mistaken, even along the a priori resolved elongated axis.
<figure><img src="image_12.png" /><figcaption>Figure 2.1: Distorsion of optical depth with in-situ absorption imaging of an elongate gas inside a 2D confinement. (a) Optical depth measured with absorption imaging and the Beer Lambert law. The grey solide lines indicate horizontal, and vertical, cuts shown on figures (b), and (c.1), respectively. (b) Each vertical bin corresponds to a single pixel of the horizontal cut. (c.1) Vertical cut of the optical depth in solid black line, and gaussian fit in dashed red line. The residuals of the fit, normalized by the signal maximum amplitude, are shown on figure (c.2). (d.1) Time of flight measurement, and (d.2) residuals of the gaussian fit from which we measure the number of atoms.</figcaption></figure>

### 2.1.3 Incorrect estimation of the pixelated local density

The local intensity loss of a weakly saturating laser beam propagating through an atomic cloud is proportionnal to the atom-light absorption cross section $\sigma$ and the local density of atoms $n$. This is the well known Beer-Lambert law, and for resonant light propagating along Oz, it is written as:

$$ \frac{dI}{I} = -n\sigma_0 dz \qquad (2.4) $$

Assuming that the length of the gas along the propagation axis is smaller than the
Rayleigh length of the imaging laser beam, diffraction effects along propagation can be neglected, and the x,y,z intensity variations are independent. Then, the optical depth $OD(x, y) = \sigma_0 \int n(x, y, z)dz$ is defined after integration of the Beer-Lambert law along the propagation axis:

$$OD(x, y) = -\log \frac{I(x, y)}{I_0(x, y)} \qquad (2.5)$$

where $I$ is the light intensity after passing through the atomic cloud, and $I_0$ is the light intensity without gas. The column density $\tilde{n}(x, y)$ is then directly inferred from

$$\tilde{n}(x, y) = \frac{OD(x, y)}{\sigma_0} \qquad (2.6)$$

On experiments, the light intensity is recorded on cameras with pixel size $a$. The recorded power $P(i, j)$ on pixel $(i, j)$ is then the light intensity integrated over the surface $a^2$ of a pixel at location $(i,j)$:

$$P(i, j) = \iint_{pixel(i,j)} I(x, y)dxdy \qquad (2.7)$$

where $pixel(i, j) = [\{ia, (i + 1)a\} \times \{ja, (j + 1)a\}]$ is the domain of integration. If the intensity profiles vary slowly enough over the pixel length $a$, i.e. $dI/dx \ll I/a$ and $dI/dy \ll I/a$, then the double sum in equation 2.7 can be approximated to $P(i, j) \approx a^2 I(ai, aj)$. In this case, the local density of atoms, average over pixel $(i, j)$, can be measured with the optical depth that is inferred from the averaged local intensity using equation 2.5

$$\langle \tilde{n} \rangle_{(i,j)} \sigma_0 \equiv \langle OD \rangle_{(i,j)} \approx -\log \frac{P(i, j)}{P_0(i, j)} \qquad (2.8)$$

However, if the gas is too small so that the intensity profile varies quickly with respect to the pixel size, the approximation for the Beer-Lambert in equation 2.8 doesn't hold. In this case, the logarithm of the averaged absorbed light is not equal to the averaged optical depth:

$$\log \frac{P(i, j)}{P_0(i, j)} \neq \frac{1}{a^2} \iint_{pixel(i,j)} \log \frac{I(x, y)}{I_0(x, y)} dxdy \qquad (2.9)$$
which is due to the non-linearity of the logarithm. Hence, it is generally not possible
to infer the local density of atoms by using the integrated Beer-Lambert law with the
recorded light power on each pixel, and this is what we observed on the 1D expansion
measurement shown on figure 2.1. It is interesting to note here that even if only one of
the two dimensions are not resolved, it is enough to forbid the approximation 2.8 and
the data remains in the non valid case similarly to equation 2.9.

This shows that a straight use of the integrated Beer-Lambert law to measure the
optical depth is mistaken and leads to distorted density profiles in case of pixelation,
but it is also true if the resolution limit, larger than the imaged object, is due to the
diffraction limit for instance. This effect is even more signicant for large optical depths
with quickly varying profiles, but our problem is different from the total absorption
regime [Rei+07]; we consider here that there is light collected at every pixel, accordingly
to our measurement presented on figure 2.1. In this regime, the error on the measured
number of atoms can be compensated for by rescaling the absorption cross section
[Est+06], giving a first order correction on the measurement of the local density of
atoms. However, here we show that the error on the measurement of the local density
of atoms is non linear, so that it is important to consider an effective absorption cross
section which depends on the density σ₀ → σ̄(n̄). I will now show how to apply this
correction.

## 2.2 Non-linear correction on the measurement of the sub-resolved local density

In this section, I describe how to account for the resolution problem by making use
of the fact that when light passes through an atomic cloud, the amount of absorbed
photons depends on the density of atoms and the depth of the crossed cloud. The first
met atoms absorb some quantity of light, so that the next atoms are in their shadow
and hence are not exposed to the same intensity. Hence, for a fixed number of atoms
in a given light beam, the total number of absorbed photons depends only on the size
of the atomic cloud, since the shadowing effect increases for larger densities. I will
now show that relying on this principle, it is possible to dodge the resolution limit and
infer the local density from the number of missing photons, thanks to an independent measurement, e.g. the total number of atoms. This way, we infer both the size of the gas along the non resolved axis 0x, and the longitudinal density profile along the elongated axis Oy.

## 2.2.1 Parametrization of the density dependent correction

We focus here on elongated gases with a slowly varying longitudinal profile, as shown on figure 2.1, (c.1), and non-resolved density profile along the transverse axis, as shown on figure 2.1, (b), accounting for possible light diffraction on the tightly confined axis and defocusing. Let's first derive an equation for the total number of absorbed photons as a function of the local column density $\tilde{n}(x, y)$. From two images of the imaging beam, one with atoms and recorded power $P(i, j)$ at pixel $(i, j)$, and one without atoms and recorded power $P_0(i, j)$, we define the ratio of absorbed photons ($P_0(i, j) - P(i, j)$) / $P_0(i, j)$. Note that here, we defined the domain of integration as the pixel size $a^2$, similarly to equation 2.7. Let's now define

$$R_{ph}(j) = \sum_{i} \frac{P_{0}(i, j) - P(i, j)}{P_{0}(i, j)} \qquad (2.10)$$

the ratio of absorbed photons recorded on the camera along the elongated axis Oy (index j) which is not problematic with respect to the imaging limitation. The integration along a pixel line in the transverse axis Ox (index i) permits this reduction of dimensionality, which moreover allows to not account for the light diffraction along the short axis Ox [Arm+10]. Then, by inserting definition 2.7, one gets:

$$ 
\begin{aligned}
R_{ph}(j) &= \sum_{i} \frac{1}{P_0(i, j)} \iint_{pixel(i,j)} \left( I_0(x, y) - I(x, y) \right) dx dy \\
&= \sum_{i} \iint_{pixel(i,j)} \frac{I_0(x, y)}{P_0(i, j)} \left( 1 - \frac{I(x, y)}{I_0(x, y)} \right) dx dy
\end{aligned} 
\qquad (2.11) 
$$

Even if the object is small, the imaging light beam has to be large enough such that the light intensity is homogeneous over the surface of a pixel. In this case, $P_0(i, j) =$
$a^2 I_0(x, y)$. Then, by inserting the local column density $\tilde{n}(x, y)$ defined in equation 2.6, one gets:

$$R_{ph}(j) = \sum_{i} \frac{1}{a^2} \int_{ai}^{(a+1)i} dx \int_{aj}^{(a+1)j} dy \left( 1 - \exp\left(-\sigma_0 \int n(x, y, z) dz\right) \right) \quad (2.12)$$

Far from the cloud of atoms, no photons are absorbed, such that the sum over the short axis $\sum_i \int_{ai}^{(a+1)i} [...] dx$ can be extended to infinity $\int_{\mathbb{R}} [...] dx$ where the ratio of absorbed photons is zero. Also, in our case, the gas is very elongated along axis Oy, and hence it is rather homogeneous over a pixel length along Oy, such that $\int_{aj}^{(a+1)j} f[n(x, y, z)] dy \approx a f[n(x, aj, z)]$, where f can be any slow varying function of the atomic density $n(x, y, z)$. With these simplifications, equation 2.12 can be rewritten as:

$$R_{ph}(j) = \frac{1}{a} \int_{\mathbb{R}} \left( 1 - \exp(-\sigma_0 \tilde{n}(x, aj)) \right) dx \quad (2.13)$$

Equation 2.13 is our first equation relating the ratio of absorbed photons which is the actual data, with respect to the column density of atoms, which is what we want to infer. Until now, the following approximation have been made. First, the atomic gas is elongated enough along axis Oy so that the variations of the density profile along this axis are negligible along one pixel length. Also, we supposed that the light intensity of the imaging beam is homogeneous over the surface of a pixel, and, the diffraction of the probe light can be neglected over the depth of the sample along its propagation direction.

Now, let's parametrize the sub-resolution feature, which is the size of the cloud along the unresolved transverse axis. To do this, we introduce an ansatz on the local density of the gas, and it is relevant to use a gaussian ansatz for thermal gases:

$$\tilde{n}(x, aj) = e^{-\frac{x^2}{2\sigma_x^2}} \tilde{n}(0, aj) \quad (2.14)$$

where $\sigma_x$ is the transverse size of the cloud, and $\tilde{n}(0, aj)$ the peak column density along axis Oy. By inserting the gaussian ansatz in equation 2.13, and with change of variable $u = x/\sigma_x$, one gets:
$$
\begin{align}
R_{ph}(j) &= \frac{\sigma_x}{a} \int_{\mathbb{R}} \left( 1 - \exp\left(-\sigma_0 \tilde{n}(0, aj)e^{-u^2/2}\right) \right) dx \tag{2.15} \\
&\equiv \frac{\sigma_x}{a} F(\sigma_0 \tilde{n}(0, aj)) \nonumber
\end{align}
$$

which defines the transfer function $F$. This last equation relates the quantity $R_{ph}(j)$ to the local density of atoms that we want to retrieve, parametrized by the size $\sigma_x$. $F$ is monotonous, hence it can be inverted to have a direct reading of the local density:

$$
\begin{align}
\tilde{n}(0, aj) &= \frac{1}{\sigma_0} F^{-1}\left(\frac{a}{\sigma_x} R_{ph}(j)\right) \tag{2.16} \\
&\equiv \frac{1}{\sigma_0} OD_{eff}(j) \nonumber
\end{align}
$$

This highlights that the expected local density is obtained from the establishment of an effective optical depth $OD_{eff}(j)$, dependent on the measurement $R_{ph}(j)$, and parametrized by the size $\sigma_x$. Actually, this is equivalent to the establishment of an effective absorption cross section $\sigma_{eff}$ by the following transformation:

$$
\tilde{n}(0, aj) = \frac{1}{\sigma_0} OD_{eff}(j) \Rightarrow \tilde{n}(0, aj) = \frac{1}{\sigma_{eff}} OD(j) \quad (2.17)
$$

where $OD = -\log P/P_0$ is the straight application of the Beer-Lamber law (see figure 2.1), contrary to $OD_{eff}$ which is inferred from our method. The expected local density of atoms is then equivalently inferred by introducing either an effective optical depth, or an effective absorption cross-section, both dependent on the measurement $R_{ph}(j)$, i.e. on the local density of atoms itself. In both cases, this correction is parametrized by the unknown size $\sigma_x$, hence a supplementary constraint is required.

2.2.2 Calibration of the non-linear correction

Several constraints can be used, typically the parameters defining the statistics of a thermal cloud, and here we choose an independent measurement of the number of atoms. A gas is prepared in the same conditions than the 1D expansion experiment and it is imaged after a time of flight, i.e. released along the three directions so that
it expands enough to be resolved and equation 2.6 can be used. In this case, the total number of atoms is obtained from a direct measurement of the optical depth as defined in equation 2.5:

$$N_{at}^{tof} = \frac{a^2}{\sigma_0} \sum_{i,j} OD^{3D}(i, j) \qquad (2.18)$$

where the *tof* annotation refers to the independent measurement with a time of flight. Regarding the 1D expansion experiment, the total number of atoms can be computed from the column density $\tilde{n}(0, aj)$ and transverse size $\sigma_x$, according to the gaussian ansatz 2.14, and using equation 2.16, the number of atoms finally reads:

$$N_{at} = \sqrt{2\pi}\sigma_x \frac{a}{\sigma_0} \sum_j F^{-1}\left(\frac{a}{\sigma_x}R_{ph}(j)\right) \qquad (2.19)$$

Here, the pixel size $a$ appears because the number of atoms is obtained from the discrete sum over pixels, which has to be rescaled to the physical pixel size. Equation 2.19 relates the unresolved transverse size $\sigma_x$ to the measured number of missing photons, through the constrained number of atoms. Finally, the size $\sigma_x$ that we measure is obtained by matching the number of atoms independently measured with a time of flight and with a 1D expansion, which is done by solving:

$$N_{at}^{tof} - \sqrt{2\pi}\sigma_x \frac{a}{\sigma_0} \sum_j F^{-1}\left(\frac{a}{\sigma_x}R_{ph}(j)\right) = 0 \qquad (2.20)$$

The establishment of this equation concludes the method to restore the density profiles from the distorted images of the atomic clouds. Note that the dependence on the atom-light absorption cross-section $\sigma_0$ is the same for both derivations of the number of atoms, such that it is canceled from the cost function. Hence, the estimation of the size $\sigma_x$ is independent of $\sigma_0$, which is an advantage because this method is then insensitive to fluctuations which could arise, for instance from optical pumping effects during the imaging.
## 2.3 Experimental demonstration

### 2.3.1 Production of degenerate Fermi gases expanding along one direction only

The gas is cooled by forced evaporation in our optical dipole trap as presented in chapter 1. The optical dipole trap (ODT) is made of two far red-detuned laser beams, one horizontal, and the other one makes a 30° angle with respect to gravity axis. After the gas has reached thermal equilibrium, we let the gas expand only in one direction by switching off the horizontal beam. The gas then falls along the mostly vertical beam. After a certain time of expansion along one dimension, the data is taken with absorption imaging, while the gas is still confined in the two perpendicular directions. The imaging setup has magnification 1 with an objective of focal length $f = 150 \text{ mm}$ and diameter 2 inches, for a numerical aperture NA=0.08. The resulting diffraction limit is $2\mu\text{m}$, much lower than the size $6.45 \times 6.45 \mu\text{m}$ of the square pixels of our camera, which is our ultimate resolution limit. Figure 2.2 shows absorption images of the channeled expansion for different times of channeling. This procedure was done for different final temperatures of the thermal gas before release. To measure the expected number of atoms, we use a second set of pictures which is actually a standard time of flight measurement. We let the gas expand in the three directions by turning off both ODT beams, then we take an absorption image. The 3D gas is prepared in the same conditions than the elongated gases, such that the number of atoms are the same in both images. Our imaging setup has magnification 1 and the pixels of the camera have dimension $6.5 \mu\text{m} \times 6.5 \mu\text{m}$.

### 2.3.2 Recovery of the unresolved transverse sizes $\sigma_x$ of elongated gases

For each image of the elongated gases, we retrieve the longitudinal profile of the ratio of absorbed photons defined in equation 2.10. Then assuming a size $\sigma_x$, we compute the corresponding number of atoms as defined in equation 2.19. It is possible to build a full curve by repeating this process for different values of $\sigma_x$, as shown on figure 2.3. In this figure, each one of the three curves corresponds to a single image, cooled to
<figure><img src="image_13.png" /><figcaption>Figure 2.2: Absorption images of channeled expansion. The time of channeling before imaging is written at the top of each image. The horizontal and vertical axis are given in camera pixels scale, as captured during the experiment. The colorbar is the optical depth. The dimension of one pixel is 6.5 × 6.5 µm² and the imaging setup has magnification 1. All these images belong to a same set of data with temperature $T \simeq 0.35 T_F$.</figcaption></figure>
<figure><img src="image_14.png" /><figcaption>Figure 2.3: Size σₓ, in pixel units, of the elongated gas along the non resolved dimension Ox, with respect to the number of atoms, computed from the inverse of equation 2.19. Each curve indicates the analysis of a single experimental image of an elongated gas, for three different temperatures indicated in Fermi temperature units. The vertical dotted lines indicate the number of atoms measured from the images of 3D gases, with its uncertainty. The horizontal dashed lines indicate the corresponding size for the unresolved elongated gases.</figcaption></figure>

the indicated degeneracy. The temperature and degeneracy are inferred from the 3D gases. Using the reference measurement of $N_{at}^{tof}$ from the 3D time of flight images, and its associated uncertainty with shot to shot fluctuations, we get the corresponding transverse size $σₓ$ and its confidence interval. This is pictured by the dashed lines on figure 2.3. We see here the principle of the analysis: for a given number of absorbed photons, more atoms means a smaller gas, and this is consistent with the shadowing effect described above.

The measured transverse sizes $σₓ$ are reported in figure 2.4 as a function of the time of channeled expansion, for every temperatures after evaporation.

First we can see that $σₓ$ doesn't vary much during the channeled expansion. For
<figure><img src="image_15.png" /><figcaption>Figure 2.4: Transverse size $\sigma_x$ in pixel units ($1px = 6.45 \mu m$ with respect to the time of channeled expansion time, for different degeneracies $T/T_F$, indicated in the legend. Each point corresponds to a statistical average over nearly ten images. The errorbars contain both the confidence interval of the inferred $\sigma_x$ for each image as well as the standard deviation over the different experimental realizations. The solid lines link the points of the same set of data, i.e. with same degeneracy, for visual comfort. The degeneracy is measured on the reference images with 3D time of flight.</figcaption></figure>

instance, the size of gases with temperature $T \approx 0.5 T_F$ vary within a $0.6 \pm 0.05$ px window, which corresponds to nearly 10% variations, and colder gases vary as much if not less. However, the hottest gases at $T \ge 0.7 T_F$ have their transverse sizes $\sigma_x$ somehow raised during the channeled expansion by nearly 10% to 15%. During the channeled expansion, the gas falls along nearly 100 $\mu m$ of the guiding beam, whose frequency for the transverse confinement evolves from 210 Hz at the initial position of the gas, to 145 Hz after 20 ms of channeling, according to our estimations. Atoms adiabatically follow the shape of the beam, due to the slowly varying Hamiltonian, and the transverse size should raise by 20%. We note the robustness of our method, considering that the measurement of the transverse sizes $\sigma_x$ is stable with respect to the time of channeling which correspond to different elongations along $Oy$. Even at 2 ms of
<figure><img src="image_16.png" /><figcaption>Figure 2.5: Transverse size $\sigma_x$ with respect to the kinetic energy of the gas. Each box corresponds to a different position in the guiding beam, accordingly to the time of channeled expansion, which is written in the top left corner of each box. Each black dot corresponds to an average over ten images of the inferred transverse sizes. The associated errorbars contain both the confidence interval of the inferred $\sigma_x$ for each image as well as the standard deviation over the different experimental realizations. The expected transverse size from equi-partition of energy is plotted as solid black line for each time of channeling.</figcaption></figure>

channeling where the gas is smallest along *Oy* as shown on figure 2.2, the inferred sizes are consistent with the other measurements, which shows that the longitudinal profile varies smoothly enough with respect to the pixel size and the approximation made in equation 2.13 is correct.
The smallest measured size (gaussian rms) is as low as one fourth of a pixel, which corresponds to 1.6 µm at the lowest measured temperatures in this experiment, $T \approx 30$ nK at $T \approx 0.15 T_F$. This is smaller than the typical distortions introduced by diffraction of the imaging beam and out-of-focus measurements, and much smaller than our resolution limit imposed by the 6.45 µm wide camera pixels.

To verify that our measurements of $\sigma_x$ are consistent, we compare the result of our method to theoretical predictions. Because of equi-partition of energy, we expect that $\frac{1}{2}m\omega^2\sigma_x^2 = E_K$, where $E_K \equiv \frac{1}{2}m \langle v_x \rangle^2$, neglecting inter-atomic interactions. The kinetic energy can be measured from the reference images of gases expanding in 3D, and the trap frequency can be independently measured. It is then possible to have an independent prediction of the in-situ transverse size of our elongated gases, with no free parameter. The comparison between our measurement and the prediction is plotted on figure 2.5 independently for each time of channeling. At high temperatures, and long time of expansion, there is a tendency of overestimating $\sigma_x$ with our method compared to the prediction, although both estimations remain in agreement within errorbars.

### 2.3.3 Recovery of the distorted longitudinal density profiles

We now are interested in the longitudinal density profile. I have shown in section 2.1.2 and 2.1.3 that absorption imaging distorts the local density of non-resolved atomic clouds. In figure 2.1 it is shown that even if only one axis (Ox) is non-resolved, the density profile inferred from the Beer-Lambert law is wrong along the *assumed-resolved* axis (Oy). However, according to equation (2.15), it is now possible to properly infer $\tilde{n}(0, aj)$ with our method, once the transverse size has been measured. Comparing the recovered density profile with Boltzmann and Fermi predictions for thermal gases, at respectively $T/T_F \simeq 1$ and $T/T_F \ll 1$, further confirms the validity of our method.

Figure 2.6, left, shows the longitudinal density profile recovered with our method, compared to the density profile directly inferred from the logarithm of the fraction of absorbed light on every pixel, as defined in equation (2.6). With $\sigma_x$ known, our method gives $\tilde{n}(0, aj)$ which is the peak column density along the elongated axis in atoms.m$^{-2}$
<figure><img src="image_17.png" /></figure>

Figure 2.6: Left: Density profile along (Oy) for a hot thermal gas at $T \approx T_F$. Left: in black dashed line, the integrated density profile obtained with equation 2.6 and in red dashed line the corresponding Boltzman fit. The hashed region corresponds to recovered density with our method with the associated uncertainty, and the red solide line is the associated Boltzmann fit. Bottom: fit residuals, the top one corresponds to the dashed lines, while the bottom one corresponds to our method. Right: Density profile along (Oy) for a degenerate Fermi gas at $T \approx 0.15 T_F$. The blue solid line is the predicted density profile for a Fermi degenerate gas, with no free parameters. The degeneracy, temperature, and trap frequencies are measured independently on the images of 3D expanding gases. The residuals on bottom figure show the agreement between the prediction and our method. For both left and right, the channeling time is 18 ms.

units. It is renormalized in atoms.m⁻¹ to be comparable with the actual data which is the integrated density profile, i.e. for the recovered profile $n_{1D}(y) = \sqrt{2\pi}\sigma_x\tilde{n}(0, y)$. In this figure, the density profiles correspond to gases that have channeled during 18 ms, and we expect that they reflect the momentum distribution of the gas before starting their expansion, with initial temperature $T \approx T_F$. At this temperature, the Boltzmann statistics still very well describe the momentum distribution of a thermal gas, and our fits are done with gaussian functions. While the density inferred from equation 2.6 deviates from the Boltzmann statistics, our method recovers a density profile in agreement with the expected statistics, as shown by the bottom fit residuals on figure 2.6, left.

Further verification is done on a degenerate Fermi gas. The recovered density is
compared to what is expected by Fermi statistics, with no free parameters. The result
is shown on figure 2.6 right. Here, the prediction is the solid blue line, and the density
measurement made with integrated Beer-Lambert law is the black dashed line, which is
obviously wrong. The density profile recovered by our method is plotted as the hashed
region, and it matches with the prediction, as shown by the residuals, plotted at the
bottom.

Note that the channeled expansion is performed along a laser beam that makes an
angle with the imaging plane (see appendix D), our measurement of the elongated gas
is affected by parallax. Also, the divergence of the laser beam results in a position de-
pendent force felt by the atoms as an anti-confinement. For the analysis shown in figure
2.6, the parallax and anticonfinement are compensated for with a single multiplicative
correction parameter C to the cloud size for a given expansion time. More details for
the analysis of the parallax and anticonfinement can be found in appendix D.

For both hot gas and degenerate gas, the residuals are small but there remains
strong pixel to pixel density fluctuations after the correction, particularly at the lowest
momenta. This is due to the non-linearity of the density dependent correction: at
higher local densities, small variations of the light absorption result in strong variations
of the density, because of the previously introduced shadowing effect.

To conclude, this demonstrates that with our method it is possible to measure
the size of atomic clouds as low as one fourth of our resolution limit. This method
also gives the possibility to recover density profiles initially distorted by pixelation,
diffraction and out of focus imaging, associated with absorption imaging. Here, we
used a gaussian ansatz on the cloud shape, and an independant measurement of the
total number of atoms to calibrate the effective optical depth that let us infer the
local density. Other ansatz are of course possible, however no unexpected features can
be retrieved from this method, since different shapes can lead to the same number
of absorbed photons. I also applied this method with a Fermi ansatz for the density
distribution along the transverse axis. The procedure is exactly the same except that
it necessitates independent calibration of the chemical potential and degeneracy, which
is easily done from the reference images of 3D expanding gases, and the results are also
in good agreement with different predictions. Note that although our method allows
to retrieve features below the imaging resolution, its fundamental limit is the imaging light wavelength $\lambda$ as it sets the scattering cross section of light by the atoms. Finally, we expect that with this method, features smaller than the imaging resolution within clouds otherwise larger than the imaging resolution can be measured. For example, using the appropriate ansatz, it could be generalized to measure the size of vortex cores within a superfluid, that are typically below the imaging resolution, or other hydrodynamic structures, such as solitons.

## 2.4 Measurement attempt of the mean field interaction energy in a SU(10) degenerate Fermi gas

First motivated by the measurement of the mean-field interaction energy in SU(10) degenerate Fermi gas of fermionic $^{87}$Sr in the weakly interacting regime, we want to enhance the mean-field signal of an expanding degenerate gas by allowing the gas to expand along one direction only. As shown earlier in section 2.1.1 the release energy along one axis of a 3D expansion is given by:

$$E_R^{3D} = \frac{1}{3}E_K + \frac{1}{3}E_{int} \qquad (2.21)$$

and the release energy of the gases expanding along one direction only is given by:

$$E_R^{1D} = \frac{1}{3}E_K + E_{int} \qquad (2.22)$$

which leads to a stronger impact from the interactions to the release energy than in 3D. The release energy is inferred from the measurement of the RMS radius $\sigma_y$ of the expanding clouds in ballistic expansion $\sigma^2(t) = \sigma(0)^2 + a^2t^2$, where $a^2 = 2E_R/m$. Regarding the 1D expansion and the pixelated absorption images, the measurement of the RMS radius of the cloud can now be realized, after the recovery of the density profile of the elongated gases which is discussed in previous section. Figure 2.7, left, reports the measurement of the longitudinal size $\sigma$ along Oy (see figure 2.1) with respect to the time of expansion, both for the channeling gases and the gases released in 3D, and for different degeneracies. Here, all the corrections have been applied for the channeling gases, i.e. the recovery of the density profiles, as well as accounting for the parallax and
<figure><img src="image_18.png" /><figcaption>Figure 2.7: Measurement of the release energy of expanding gases. Left: RMS size $\sigma_y$ in pixel unit (1 px = 6.45 µm) of the gases along the longitudinal axis $Oy$, with respect to the time of expansion. Dashed lines are ballistic fits $\sigma(t)^2 = \sigma(0)^2 + a^2t^2$ for the expansion of the reference gases imaged after time of flight, and solid line for the 1D channeled expansions. Every $\sigma_y$ of the channeled expansions are rescaled by a single multiplicative correction of the parallax and anticonfinement (see appendix D). Right: fitted release energies $E_R$ from the ballistic expansion, with respect to $T/T_F$. In inset: ratio of the release energies $E_R^{1D}/E_R^{3D}$.</figcaption></figure>

anti-confinement (see appendix D) with a single multiplicative correction parameter. The release energy is then inferred from the slope of the expansion, given that we measure the sizes after a long enough time so that $\sigma(0)^2 \ll a^2t^2$, and the result is reported on figure 2.7. The inset compares the release energies along the longitudinal axis $Oy$ (see figure 2.1) for the 1D and 3D expansions, and it shows that $E_R^{1D}$ is smaller than that of the 3D measurement for low $T/T_F$. This is not expected since the mean field interaction should significantly increase the release energy in 1D compared to 3D, particularly at the smallest temperatures.

We finally infer the kinetic energy and mean-field interaction energy which are
completely determined from the measurement of the release energies in 1D and 3D expansions,

$$ 
\begin{align} 
E_{int} &= \frac{3}{2}(E_R^{1D} - E_R^{3D}) \tag{2.23} \\ 
E_K &= \frac{3}{2}(3E_R^{3D} - E_R^{1D}) \nonumber 
\end{align} 
$$

and figure 2.8, (a) presents the result of our measurement of $E_{int}/E_K$ with respect to $T/T_F$. Our result shows a decrease of the interaction energy at low $T/T_F$, which is not in agreement with the expected behavior. In the weakly interacting regime, the strength of the mean-field interactions for a SU(N) Fermi gas can be numerically computed from [MPS02]:

$$ E_{int}^{SU(N)} = (N - 1) \frac{g}{4} \int d^3 r \, n^2(r) \qquad (2.24) $$

and

$$ E_K = \int d^3 p \frac{p^2}{2m} n(p) \qquad (2.25) $$

and the prediction, shown on figure 2.8, (a), solid line, confirms that we expect a raise of $E_{int}/E_K$ at low $T/T_F$.

The statistical noise of our result is as large as the signal we are looking for, and the tendency is in contradiction with the prediction. This result is not conclusive, and we suspect imperfect thermalization when the gases reach $T < 0.5 T_F$. This assumption is verified with an analysis of the aspect ratio of the gases expanding in 3D, as shown on figure 2.8 (b.1), with respect to the time of flight duration, for different degeneracies. A non-interacting Fermi gas with $T \approx 0.75 T_F$ has isotropic velocity distribution and the aspect ratio $\sigma_x/\sigma_y$ of the expanding gas is 1 [BR97], as shown on figure 2.8, (b.1), similarly to a classical gas. Then, according to our measurement, the aspect ratio for a degenerate gas at $T \approx 0.17 T_F$ is inverted at $tof \approx 4$ ms and finally converges to approximately 1.12. This was observed in [Son+20] with approximately 50 000 atoms per spin state as a signature of the mean field interactions [OHa+02; PGS03; JPS04].
<figure><img src="image_19.png" /><figcaption>Figure 2.8: (a) Measurement attempt of the interaction energy with 1D channeled expansions, inferred from equations 2.23 and our measurement of the release energies presented on figure 2.7, in function of $T/T_F$. Solid lines show predictions of the mean field energy from numerical resolution of equations 2.24, 2.25. (b.1) Inversion of the aspect ratio $\sigma_x/\sigma_y$ of the gases expanding in 3D with respect to the duration of the expansion and (b.2) increase of the aspect ratio with decreasing $T/T_F$. On figure (b.2), the aspect ratio is measured at fixed tof = 20 ms.</figcaption></figure>

However, from simulations based on [MPS02], we predict that with approximately 2000 atoms per spin state in a trap of geometric mean frequency 160 Hz, which is our case, the aspect ratio shouldn't exceed 1%. Finally, figure 2.8, (b.2) shows our measurement of the aspect ratio as a function of $T/T_F$ at fixed tof = 20 ms, and it exceeds 1% for all measurements at $T < 0.5 T_F$. This indicates that in this data, thermalization is imperfect at the end of evaporation for the most degenerate gases, and no reliable measurement of the mean-field interaction energy could be realized.
Chapter 3

Preparation and measurement of
the spin populations in a Fermi gas
of $^{87}$Sr

The narrow intercombination line of strontium 87 offers ideal conditions for the realiza-
tion of spin orbit coupling schemes [GD10; LJS11], with minimal spontaneous emission
[CR14]. Associated with spin resolved transitions, it is possible to measure and manip-
ulate the spin populations. For instance, a well established technique named Optical
Stern-Gerlach (OSG) [Sle+92] makes use of the spin-dependent dipole force associated
with the hyperfine structure of the intercombination line to spatially separate spin
components of alkaline-earth like atoms. It has been demonstrated on both strontium
[SGS11] and ytterbium [Tai+10], to measure the populations in each spin state. The
ground state ¹S₀ of fermionic ⁸⁷Sr has an electronic closed-shell i.e. *J* = 0, hence the
electronic sensitivity to magnetic fields is zero, and there only remains the magnetic
sensitivity of the nucleus which is as low as the electron to proton mass ratio relatively
to the electronic g-factor. It is then not practical to use magnetic fields to manipulate
the 10 degenerate ground spin states.

Excited triplet states such as $^3P_1$ with non zero momentum have a much stronger
magnetic susceptibility than the ground state $^1S_0$. Regarding $^3P_1$ state with $\Gamma/2\pi =$
7.4 kHz, the degeneracy lift of the excited spin states is easily larger than the linewidth:
with g-factor $g_J \simeq 0.27$, one can reach MHz Zeeman shifts with modest magnetic fields
of a few Gauss. In this case, the resonances of the $^1S_0 \leftrightarrow ^3P_1$ intercombination line acquire a spin dependence, which enables spin selective optical transitions. The number of spin components can then be tuned with standard optical pumping schemes [Tey+10; Pag+14; Son+20], further enabling the study of SU(N) many-body physics with tunable N. Although narrow MOT cooling may induce spontaneous polarization of a SU(N) Fermi gas, as demonstrated with erbium [Fri+12], specific schemes allow to selectively prepare polarized Fermi gases either with dissipative blast [He+20] only or associated with spin dependent light shifts [Son+20].

In this chapter, I first discuss our demonstrated method [Bat+20] to measure the nuclear spin populations with a spin orbit coupling scheme associated with the intercombination line [SGS11]. The robustness of our method is provided by adiabatic following of a quasi-dark state, ensuring minimal spontaneous emission. Using the strong difference of magnetic sensitivity of the coupled states, we selectively transfer, with a resonant passage, well defined momentum recoils to well defined spin states, with a simple retro-reflected laser beam. The overall efficiency is yet 85%, up to now limited by available light power. I will then show how we are able to tune the number of spin components in our Fermi gas of $^{87}$Sr. Using the high spin sensitivity of the excited state $^3P_1$ associated with optical pumping [Tey+10; Oza+18; He+20], we prepare SU(N) fermi gas with tunable N, which leads to our preparation of a polarized Fermi sea with a spin selective blast. Finally, I estimate that the Fermi sea is polarized with $90 \pm 10\%$ fiability accounting for the overall efficiency of our detection scheme.

## 3.1 Measurement of the spin populations

This spin dependent momentum transfer scheme inherently relies on spin-orbit coupling (SOC) [GD10; LJS11] enabled by a polarization lattice. With a zeeman degeneracy lift of the $^1S_0 \leftrightarrow ^3P_1$, $F = 11/2$ spin spin transitions, two highly saturating counter-propagating laser beams with same frequency and well defined $\sigma_+$ and $\sigma_-$ resonantly couple two selected ground spin states with an excited state, ensuring a strong spin sensitivity. The atoms in the selected spins are connected to a form a quasi-dark state $|\Psi_C\rangle$, ensuring reduced spontaneous emission. The dark state is adiabatically
rotated by ramping the frequency of the lasers, ensuring robustness. In the end, the
final states have acquired exactly two momentum recoils by absorption of one photon
from one beam and emission of one photon into the other beam. Ideally, the perfect
correlation between the momentum and the polarization of the laser beams ensures that
the two selected spins are diffracted in opposite directions, and after a time of flight,
the diffracted spin states are separated from the rest of the cloud, and their populations
can be measured with absorption imaging.

In this section, I first present our experimental demonstration of this scheme, and I
will then discuss the adiabatic rotation of the quasi-dark state $|\Psi_c\rangle$ in the dressed-states
picture. With a focus on the possible spin sensitivity loss due to defects on the lasers
polarization, we demonstrate the reliability of our measurement, which we verified on
all the spins of the SU(10) ground state manyfold. I will then show that our scheme
can be generalized to simultaneously measure the populations of more than two spin
states with a proper sequence of adiabatic rotations. Finally, we further increased the
probability of success of the momentum transfer by improving the connection to the
quasi-dark state $|\Psi_c\rangle$, which we practically do with a Blackman-like intensity window
simultaneously to the frequency.

**3.1.1 Experimental procedure**

After preparing a degenerate Fermi gas (see chapter 1), we first lift the degeneracy of the
$^3P_1$, $F = 11/2$ Zeeman sub-levels with a homogeneous magnetic field of amplitude $|\mathbf{B}| = 16$ G, as shown on figure 3.1, (a). The excited spin states, with magnetic susceptibility 380 kHz/G are then separated by $g_F\mu_B|\mathbf{B}| \simeq 6$ MHz, which is three orders of magnitude larger than the linewidth $\Gamma_{^3P_1}/2\pi = 7.4$ kHz. The ground spin states are barely affected with magnetic susceptibility as low as 0.2 kHz/G [Boy07, Bat+20]. Here, $g_F$ is the g-factor of the hyperfine state $F = 11/2$, and $\mu_B$ the Bohr magneton. The $^1S_0 \leftrightarrow ^3P_1$, $F = 11/2$ optical transitions are then spin resolved. Two highly saturating counter-propagating laser beams with same frequency and well defined polarizations $\sigma_+$ and $\sigma_-$ are turned on, shaping a polarization lattice, with $I \simeq 1000 I_{sat}$, $I_{sat} \simeq 3\mu W/cm^2$. For this, we use the vertical red MOT beams. Our measurement of the circularity of
<figure><img src="image_20.png" /><figcaption>Figure 3.1: Spin dependent momentum transfer to measure the spin populations in a SU(N) degenerate gas. (a) The degeneracy of the hyperfine excited state $^3P_1$, $F = 11/2$ is lifted by 6 MHz with a 16 G homogeneous magnetic field, with is much larger than the 7.4 kHz linewidth. Two counter-propagating laser beams with same frequency and well defined polarization $\sigma_-$ and $\sigma_+$ selectively couple two ground spin states $|m_F - 1\rangle$ and $|m_F + 1\rangle$ to the excited state $|m_F\rangle$. The frequency of the light excitation is ramped through resonance at rate $\dot{\delta}$ to realize the momentum transfer. (b) Integrated optical depth, from absorption image, of the selected spin states $|m_F \pm 1\rangle$ which are separated from the rest of the could after time of flight.</figcaption></figure>
the laser beams is 99 % in intensity. With appropriate near-resonant frequency, two selected ground spin states $|^1S_0, m_F - 1\rangle$ and $|^1S_0, m_F + 1\rangle$ are coupled to the excited state $|^3P_1, F = 11/2, m_F\rangle$, defining an isolated three level system in Λ configuration, as shown in figure 3.1 (a). The magnetic field used to lift the degeneracy of the $^1S_0 \leftrightarrow ^3P_1, F = 11/2$ spin spin transitions combined with optical couplings with well defined $\sigma_\pm$ polarizations ensures the strong spin selectivity. Then, we perform a frequency sweep through the resonance from $\delta \simeq +700$ kHz to $\delta \simeq -700$ kHz during 200 µs. Here, $\delta = \omega - \omega_0$ is the detuning to the excited state. Atoms in $|^1S_0, m_F - 1\rangle$ absorb one photon from the $\sigma_+$ laser beam and re-emit one photon into the counter propagating $\sigma_-$ laser beam, acquiring exactly two photon momentum recoils $+2\hbar\vec{k}_R$. The opposite happens for atoms initially in $|^1S_0, m_F + 1\rangle$, acquiring exactly minus two photon momentum recoils $-2\hbar\vec{k}_R$. During the momentum transfer, atoms are held against gravity with our horizontal beam for ODT. The selected spin states $|^1S_0, m_F \mp 1\rangle$ with acquired $\pm 2\hbar\vec{k}_R$ momentum recoils are then separated from the rest of the cloud after time of flight, and the populations in each spin state are measured from absorption images, as shown on figure 3.1 (b).

The momentum transfer is ensured by adiabatic following of a quasi dark state $|\Psi_C\rangle$. Its rotation from the initial state $|m_F \mp 1, 0\hbar\vec{k}\rangle$ to the final state $|m_F \pm 1, \pm 2\hbar\vec{k}\rangle$ is realized by the frequency ramp, which crosses the resonance with the excited state. I will now discuss the conditions for success of the adiabatic following of the quasi dark state and the associated reduced spontaneous emission, despite the resonant coupling.

### 3.1.2 Adiabatic following - Dressed-states picture

It is convenient to look at this three level system in Λ configuration in the dressed states picture [DC85]. The initial ground state $|1\rangle = |^1S_0, m_F - 1, 0\hbar\vec{k}\rangle$ with approximately zero momentum is coupled to excited state $|2\rangle = |^3P_1, F = 11/2, m_F, 1\hbar\vec{k}\rangle$ with one momentum recoil by $\sigma_+$ excitation with associated rabi frequency $\Omega_+$. The excited state is coupled to the final ground state $|3\rangle = |^1S_0, m_F + 1, 2\hbar\vec{k}\rangle$ with two momentum recoils by $\sigma_-$ excitation with associated rabi frequency $\Omega_-$. In this dressed states basis $\{|1\rangle, |2\rangle, |3\rangle\}$, shown on figure 3.2 (a), the hamiltonian can be written as:
<figure><img src="image_21.png" /><figcaption>Figure 3.2: Dressed atom picture for the spin dependent momentum transfer scheme. (a): dressed states spectrum, where $\Delta = \hbar k^2/2m$ is the photon recoil frequency, $\delta = \omega - \omega_0$ is the detuning, and $\Omega_{\pm}$ the rabi frequencies associated with $\sigma_{\pm}$ excitations. (b): the dashed lines are the energies of the dressed states $|1\rangle, |2\rangle, |3\rangle$, and solid lines the eigen energies of hamiltonian defined in equation 3.1, with respect to the detuning $\delta$. Eigenstate $|\Psi_C\rangle$ is defined in equation 3.2. (c): $|\Psi_C\rangle$ populations along the three dressed states $|1\rangle, |2\rangle, \text{ket3}$. For figures (b) and (c), parameters used are $\Delta = 4.8$ kHz and $\Omega_{\pm} = 55$ kHz.</figcaption></figure>

$$ \hat{H}(\delta) = \hbar \begin{pmatrix} 0 & \Omega_+/2 & 0 \\ \Omega_+/2 & \delta - \Delta & \Omega_-/2 \\ 0 & \Omega_-/2 & 4\Delta \end{pmatrix} \qquad (3.1) $$

where $\delta = \omega - \omega_0$ is the light detuning with respect to the resonance with the excited state in this three spins manyfold, and $\Delta = \hbar k^2/2m$ is the recoil frequency. The light coupling between those three dressed states gives rise to three eigenstates, whose spectrum is shown on figure 3.2, (b). We are mainly interested in the eigen-state, here named $|\Psi_C\rangle$, that connects to initial state $|1\rangle$ and final state $|3\rangle$. With first-order perturbative expansion $\Delta \ll \Omega$ from hamiltonian 3.1 at $\delta = 0$, it can be shown that:

$$ |\Psi_C(\delta = 0)\rangle = \frac{1}{\sqrt{\Omega_+^2 + \Omega_-^2}} (\Omega_- |1\rangle - \Omega_+ |3\rangle) - \frac{\Omega_+ \Omega_-}{\Omega^2} \frac{8\Delta}{\Omega} |2\rangle \quad (3.2) $$

The rate of spontaneous emission, inferred from the population in the excited state $|\langle 2|\Psi_C\rangle|^2$, reads:
$$ \Gamma_{s.e.} \approx \Gamma_{3P_1} (2\sqrt{2}\frac{\Delta}{\tilde{\Omega}})^2 \qquad (3.3) $$

where $\tilde{\Omega}^2 = \Omega_+\Omega_-$. With experimental parameters, $\Omega \pm /2\pi \approx CG_{\pm}220$ kHz and $\Delta/2\pi \approx 4.8$ kHz, where the $CG_{\pm}$ are the Clebsch-Gordan coefficients associated with $\sigma_{\pm}$ transitions. For a 200 µs ramp duration, equation 3.3 gives an upper-bound of spontaneous emission between 15% and 30%, depending on the selected spins for the momentum transfer. Most importantly, equation 3.3 shows that spontaneous emission is reduced by the light excitation, and hence, stronger resonant coupling results in less light scattering.

Far from resonance when $|\delta| \gg \Omega_{\pm}$, the excited state can be eliminated, and the hamiltonian 3.1 is then reduced to:

$$ \hat{H}(\delta) = \hbar \begin{pmatrix} 0 & \Omega_+^* \Omega_- / 2\delta \\ \Omega_+ \Omega_-^* / 2\delta & 4\Delta \end{pmatrix} \qquad (3.4) $$

in $|1\rangle, |3\rangle$ basis. In this perturbative regime, diagonalization of this hamiltonian shows that two eigen-states $|+\rangle, |-\rangle$ read:

$$ \begin{aligned} |+\rangle &\approx |3\rangle + \frac{\Omega_+ \Omega_-}{4\delta \Delta} |1\rangle \\ |-\rangle &\approx -|1\rangle + \frac{\Omega_+ \Omega_-}{4\delta \Delta} |3\rangle \end{aligned} \qquad (3.5) $$

We want the atoms to connect to $|\Psi_C\rangle$. The phase relation between $|1\rangle$ and $|3\rangle$ imposes that for large positive detunings, $|-\rangle \approx |\Psi_C\rangle \approx |1\rangle$ and for large negative detunings, $|+\rangle \approx |\Psi_C\rangle \approx |3\rangle$. Hence, the adiatic connection to the dark-state is possible if the initial and final detunings $|\delta_{i,f}|$ are large enough such that they respect the condition:

$$ \frac{\Omega_+ \Omega_-}{4 |\delta_{i,f}| \Delta} \ll 1 \qquad (3.6) $$

so that the initial and final states are almost identical to $|\Psi_C\rangle$. Figure 3.2, (c), shows that $|\langle 3|\Psi_C\rangle|^2 \approx 1$ for $\delta \ll 0$ and $|\langle 1|\Psi_C\rangle|^2 \approx 1$ for $\delta \gg 0$. Moreover, this
shows that the detuning has to be ramped from positive to negative such that the selected atoms initially at rest acquire two momentum recoils. With a detuning ramp in the other direction, it can be shown with the same method that atoms initially at rest follow a bright state which leads to the excited state and the momentum transfer fails.

*Adiabatic following* is ensured if the speed of rotation of $|\Psi_C\rangle$ determined by the speed of the detuning ramp, is much lower than the coupling strength, which reads:

$$ \dot{\delta} \ll \Omega^2 \qquad (3.7) $$

This criterion can be understood in the frame of nuclear magnetic resonance. A spin in precession will stay colinear to a rotating magnetic field if the rotation speed is slow with respect to the larmor frequency, such that the axis of precession stays in phase with the magnetic field. This condition is analog to $\dot{\delta} \ll \Omega^2$ regarding Raman adiabatic passage, so that the basis rotation is slow enough and the atoms stay in phase with the followed eigen-state rotating from one spin state to the other. The success of the adiabatic passage is ensured by fulfilling conditions 3.6 and 3.7.

*Transfer dynamics and spontaneous emission:* the dynamics of the transfer are compared to numerical simulations of the master equation including spontaneous emission. Figure 3.3 shows the experimental efficiency for the $|-3/2, 0\hbar k\rangle \rightarrow |1/2, 2\hbar k\rangle$ momentum transfer (black dots), as a function of the duration of the detuning ramp. While the simulation is in agreement with our experiment and verifies that a 200 µs ramp duration fulfills the Landau-Zener condition for adiabatic following defined in equation 3.7, it also shows that spontaneous emission (red solid line) limits the efficiency of the transfer, compared to no spontaneous emission (black solid line). The probability of success of the adiabatic following as a function of the detuning sweep rate $\dot{\delta}$ is finally compared to the Landau-Zener scaling (dashed-line on figure 3.3) $P_{LZ}(\Omega_+) \times P_{LZ}(\Omega_-)$ [CH86] for the two-photons $\Lambda$ scheme, resulting from the relative values of $\Omega_+, \Omega_-$, and $\Delta$, where $P_{LZ}$ is defined as:
<figure><img src="image_22.png" /><figcaption>Figure 3.3: Efficiency of the adiabatic passage in function of the duration of the detuning ramp, for spin dependent momentum transfer from $|-3/2, 0\hbar\vec{k}\rangle$ to $|1/2, 2\hbar\vec{k}\rangle$, with a detuning ramp over 1.4 MHz, assuming an initial 10% population in $-3/2$. The experimental result (black dots) is compared to numerical simulations (solid lines) with no free parameters of the master equation including spontaneous emission (red), neglecting spontaneous emission (black), and to the Landau-Zener scaling (dashed line) $P_{LZ}(\Omega_+) \times P_{LZ}(\Omega_-)$.</figcaption></figure>

$$P_{LZ}(\Omega) \approx 1 - \exp\left(-2\pi \frac{\Omega^2}{4\delta}\right) \qquad (3.8)$$

The lower Clebsch-Gordan thus gives the higher limit on the detuning sweep rate. If this limit is exceeded, different consequences arise depending on which arm of the momentum transfer has the lowest CG: associated with the first absorbed photon, atoms fail the transfer and stay at rest, while large spontaneous emission results from low CG associated with the second emitted photon.

At 200 µs, approximately 10 to 15% atoms fail the coherent momentum transfer, according to both equation 3.3 and numerical simulation. Our experimental limit to
achieve better transfer efficiency is set by the light intensity, limited by the large waist
of the MOT beams, as large as approximately 1 cm. An increase of the Rabi coupling
by a factor 2 would reduce spontaneous emission to 5% and increase the efficiency of
the coherent transfer up to 95%. This would require to increase the light intensity by
a factor 4 and the width of the detuning ramp as much, according to the conditions
for adiabatic transfer defined in equation 3.6. Ensuring spin selectivity requires then to
increase the degeneracy lift of the excited state manifold $^3P_1$, $F = 11/2$ by increasing the
magnetic field to $|\mathbf{B}| \simeq 60$ G. Although weakly sensitive, the zeeman lift of the ground
state with susceptibility 185 Hz/G would then exceed the recoil energy associated with
the momentum transfer. If not compensated for, for instance with a doppler shift
associated with a short free fall in our vertical scheme, this would result in an asymmetry
of the central frequency of the detuning ramp for the momentum transfer of selected
$|m_F - 1\rangle$ and $|m_F + 1\rangle$ spin states.

3.1.3 Spin sensitivity

The spin sensitivity is ensured by the association of the degeneracy lift of the $^1S_0 \leftrightarrow ^3P_1$, $F=11/2$ spin transitions, to the well defined wavectors $\pm \vec{k}$ and $\sigma \pm$ polarizations of the couplings. Hence, there are two possible sources of sensitivity loss, either from defects on the circularity of the laser beams polarization, or from misalignment of the counter-propagating laser beams and the magnetic field, which harms the correlation between the initial spin state and the direction of the momentum transfer. In the first case, spin states may be diffracted in the opposite direction to that assumed, while in the second case, the emergence of $\pi$ couplings allows the diffraction to spin states that should not be diffracted. To estimate the consequences of the polarization defects, we consider that the Rabi frequency resulting from a circularity defect $\epsilon \simeq 0.1$ (measured 99 % circularity in intensity) scales as $\epsilon\Omega$ and from a magnetic field misalignment $\theta \simeq 6^\circ$ as $\theta\Omega$. A helpful approach is to considered separatly the possible absorption processes for the unintended momentum transfers, and infer their probabilities of success from a Landau-Zener scaling $P_{LZ}(\epsilon\Omega)$ and $P_{LZ}(\theta\Omega)$, as defined in equation 3.8, with respective Rabi Frequencies. With a sweep rate satisfying 95 % probability of success for the intended momentum transfer (with $\epsilon = 0$ and $\theta = 0$), the unintended transfers
<figure><img src="image_23.png" /><figcaption>Figure 3.4: Measurement of the spin sensitivity with optical pumping prior to spin-dependent momentum transfer on the $|1/2\rangle \leftrightarrow |-3/2\rangle$ diffraction line. (a): pumping of $1/2$ and $-3/2$ spin states to verify $\pi$ defects in polarization. (b): pumping of $-3/2$ only, to verify defects in circularity.</figcaption></figure>

are reduced to approximately 3% accounting for the non zero polarization defects.

To verify the spin sensitivity, we use an optical pumping scheme to empty the populations of selected spin states (see section 3.2) prior to the momentum transfer. Then, we realize a momentum transfer targeting at the emptied spin states. If atoms are separated from the cloud after time of flight, they were initially populating non-targeted spin states, and they have acquired momentum because of the polarization defects.

To evaluate the $\pi$ defect associated with the alignment of the magnetic field, we empty the populations of both spin states $1/2$ and $-3/2$, and we realize a momentum transfer on the associated $|1/2\rangle \leftrightarrow |-3/2\rangle$ diffraction line. Then, we measure the atomic density at the expected locations of the separated clouds after time of flight, as shown on figure 3.4 (a.1), (a.2). The measured populations are zero, which is consistent with the prior pumping. Furthermore, no atoms from $-1/2$ spin state are diffracted, which verifies that defects on $\pi$ polarization can be neglected.

We also verify the quality of the circularity by selectively pumping one of the two
<figure><img src="image_24.png" /></figure>

Figure 3.5: Spectrum for spin dependent momentum transfer over $^1S_0$ manyfold. Each point corresponds to diffracted fraction of atoms with respect to the central frequency of the detuning ramp which is 1.4 MHz wide, with duration 330 µs. Top figure shows the lower diffracted cloud with $-2\hbar k$ momentum transfer (see figure 3.1), while bottom figure shows upper diffracted cloud with associated $+2\hbar k$ momentum transfer. The blue filled regions correspond to measurable asymmetry higher than 1 MHz in the optimal central frequency for the two diffracted spin populations.

targeted spin states prior to the diffraction measurement. Figure 3.4, (b.1), shows the measurement of the $|1/2\rangle \leftrightarrow |-3/2\rangle$ diffraction line after pumping -3/2. The measurement of the populations, see figure (b.2), shows that one cloud only is separated from the rest of the atomic gas. This shows that the momentum acquired by targeted spin state 1/2 is well defined, and verifies that defects in circularity can be neglected. These two measurements hence verify our estimations on the $\sigma_+$ and $\pi$ polarization defects, and show that our spin sensitivity is robust.

### 3.1.4 Spin dependent momentum transfer in SU(10) manifold

With a complete spectrum of the spin dependent momentum transfer over the SU(10) manifold of the $^1S_0$ ground state, shown on figure 3.5, we find the optimal central
frequency of the detuning ramps and the corresponding measured spin populations $N(m_F)$ for every spin state. A total of 8 diffraction lines are identified in this spectrum, each of them diffracting two spin states $|m_F - 1\rangle$, $|m_F + 1\rangle$, in opposite directions. From the sum over all spin states of the diffracted atoms at optimal central frequencies, we infer that a total of approximately 85% atoms are diffracted. This result is consistent with an average 10 to 15% momentum transfer fail due to spontaneous emission, as discussed in 3.1.2, and that all spin states are equally populated with approximately 10% atoms per spin state.

This estimation neglects the $|-9/2\rangle \leftrightarrow |-5/2\rangle$ diffraction line, which peaks at 4% diffracted atoms. We consider this measurement irregular because of two arguments. First, the non-repetability of the measurement of the $|-5/2\rangle$ population compared to the $|-5/2\rangle \leftrightarrow |-1/2\rangle$ diffraction line shows an anomaly. Moreover, while the strong asymmetry in the Clebsch-Gordan associated with $\sigma_+$ and $\sigma_-$ couplings, see figure 3.6, for this diffraction line could explain a low efficiency of the momentum transfer, it is not observed in the $|+9/2\rangle \leftrightarrow |+5/2\rangle$ diffraction line, which encounters the same asymmetry. Also, the measurement of the $|+5/2\rangle$ spin population is repeatable compared to the $|+5/2\rangle \leftrightarrow |+1/2\rangle$ line, showing that the asymmetry in Clebsch-Gordan coefficients is not responsible for the anomaly in the $|-9/2\rangle \leftrightarrow |-5/2\rangle$ diffraction line. Hence, this measurement is not yet understood, but we suspect a lately spotted hardware failure (heating of direct digital synthesizers (DDS)) which can be responsible for the lowered efficiency and that has yet to be confirmed after repair. Until then, the sensitivity of the measure remains satisfying for a reliable estimation of the spin populations for all states but $|-9/2\rangle$.

**Asymmetric Clebsch-Gordan coefficients:** The diffraction line $|+9/2\rangle \leftrightarrow |+5/2\rangle$ shows that a strong asymmetry in the Clebsch-Gordan coefficients (CG) associated with the two arms of the momentum transfer, here as high as $CG_+/CG_- \simeq 6$, is not detrimental to the sensitivity on the populations measurement, as long as the timescale for adiabatic passage is respected for the lowest CG. However, consequences arise regarding the optimal detuning ramp. The blue regions on figure 3.5 highlight the measured diffraction lines with strong asymmetry in the CGs. Here, we observe that the maximum
<figure><img src="image_25.png" /></figure>

<figure><img src="image_26.png" /></figure>

<figure><img src="image_27.png" /><figcaption>Figure 3.6: Clebsch Gordan coefficients squared associated with the $^1S_0 \leftrightarrow ^3P_1$, $F = 11/2$ transitions. Top, transition associated with +1 orbital momentum, on middle 0 orbital momentum, and bottom -1 orbital momentum. The width of the lines are directly proportional to the Clebsch Gordan coefficients, which are noted squared aside each line.</figcaption></figure>
efficiency of the momentum transfer for each of the two targeted spin states correspond to significantly different detunings. For instance, the optimal central frequencies for the diffraction of states $|7/2\rangle$ and $|3/2\rangle$ are approximately 14.5 MHz and 16 MHz, which is comparable to the width of the detuning ramp. Figure 3.7, (a.1) shows the strong asymmetry in the CGs in the $\Lambda$ scheme associated with the $|+9/2\rangle \leftrightarrow |+5/2\rangle$ diffraction line. The energy of the followed eigen-state, (red solid line on figures 3.7 (a.2) and (a.3)), is consequently displaced asymmetrically for each direction of the momentum transfer. This suggests that given the optimal center for the detuning ramp cannot be the same in both cases. Moreover, the $|+9/2\rangle$ spin state with associated weak CG is in an electromagnetically induced transparency (EIT) [Fri+12] configuration, such that the quasi-dark state $|\Psi_C\rangle$ is mostly along $|+9/2\rangle$ at $\delta = 0$:

$$|\Psi_C\rangle = \frac{1}{\sqrt{\Omega_+^2 + \Omega_-^2}} (\Omega_- |+5/2\rangle - \Omega_+ |+9/2\rangle) \quad (3.9)$$

where $\Omega_+/\Omega_- \simeq 6$. This suggests that the rotation of $|\Psi_C\rangle$ is either finished at $\delta = 0$, either not yet begun, and hence it cannot be center around the resonance. More precisely, equation 3.9 imposes that $|\Psi_C\rangle$ rotates from $|+9/2, 0\hbar\vec{k}\rangle$ to $|+5/2, 2\hbar\vec{k}\rangle$ when $\delta < 0$, accounting for the positive to negative detunings ramp $\dot{\delta} < 0$, whereas the opposite rotation for the opposite momentum transfer from $|+5/2, 0\hbar\vec{k}\rangle$ to $|+9/2, 2\hbar\vec{k}\rangle$ occurs when $\delta > 0$. Figure 3.7, (a.4), shows that the central frequency of the detuning ramp for the adiabatic passage, that I here define as $\delta_{center}$ so that $|\Psi_C(\delta_{center})\rangle \simeq 1/\sqrt{2}(|m_F \mp 1\rangle - |m_F \pm 1\rangle)$, is strongly shifted from the resonance $\delta = 0$. Regarding the $|+9/2\rangle \leftrightarrow |+5/2\rangle$ diffraction line, The numerical computation with experimental parameters shown on figure 3.7, (a.4), shows the separation of the central frequency for each direction of the momentum transfer is 900 kHz, in agreement to the measured spectrum (see figure 3.5), and most importantly, it is almost as large as the detuning ramp which is 1.4 MHz. Hence, the adiabatic connexion of the initial and final spin states to the followed eigen-state $|\Psi_C\rangle$ cannot be insured for both directions of the momentum transfer, which explains the asymmetry of the optimal central frequency for the detuning ramp.
<figure><img src="image_28.png" /></figure>

Figure 3.7: Focus on the asymmetry of the Clebsch-Gordan (CG) coefficients associated with the spin dependent momentum transfer. (a.1) Λ scheme for the $|+9/2\rangle \leftrightarrow |+5/2\rangle$ transfer with strong asymmetric CGs, respectively 0.14 and 0.84. (a.2.3): Momentum transfer in the eigen-energies picture, from $|+5/2, 0\hbar\vec{k}\rangle \rightarrow |+9/2, 2\hbar\vec{k}\rangle$ (a.2) and $|+9/2, 0\hbar\vec{k}\rangle \rightarrow |+5/2, 2\hbar\vec{k}\rangle$ (a.3), with respect to the same detuning ramp. The energy displacement of the followed eigen-state $|\Psi_C\rangle$ (red solid line) with respect to the dressed states (dashed lines) is highly asymmetric around the central frequency of the detuning ramp, as a result of the CGs asymmetry. (b) Momentum transfer for the $|+1/2\rangle \leftrightarrow |-3/2\rangle$ line with approximately equal CGs, respectively 0.54 and 0.44. (a.4) and (b.4): projections of $|\Psi_C\rangle$ along the dressed-states for each momentum transfer, as a function of $\delta$, here centered around the resonance with the zeeman shifted excited state.

This feature can be compared to the case $\Omega_- \simeq \Omega_+$ where the followed eigen-state is in a balanced superposition of the two spin states $|\Psi_C\rangle \simeq 1/\sqrt{2}(|1\rangle - |3\rangle)$, at resonance
$\delta = 0$. In this case, the central frequency of the rotation is obviously close enough to the resonance such that both directions of the momentum transfer are efficient for the same detuning ramp of 1.4 MHz. Figure 3.7, (b), shows that the for the $|+1/2\rangle \leftrightarrow |-3/2\rangle$ diffraction line with balanced CGs 0.54 and 0.44, the energy displacement of $|\Psi_C\rangle$ is negligible over the detuning ramp, and the separation of central frequencies for each direction of the transfer is only 10% of the detuning ramp. Hence, both directions of the momentum transfer are optimally realized with the same detuning ramp.

### 3.1.5 Simultaneous measurement of four spin populations

We demonstrated a generalization of our scheme to simultaneously measure more than two spin populations. Figure 3.8 shows an experimental measurement of four spin populations, $-7/2$, $-3/2$, $+1/2$, and $+5/2$, with a single realization. This is done by transferring four different momenta to each one of the selected spin states, respectively $+2\hbar\vec{k}$, $+4\hbar\vec{k}$, $-4\hbar\vec{k}$, and $-2\hbar\vec{k}$, such that they are all separated after time of flight. For this measurement, we realize a sequence of three adiabatic passages, shown on figure 3.8, (a). The first passage on diffraction line $|-3/2\rangle \leftrightarrow |+1/2\rangle$ is the same as introduced before:

$$ \begin{aligned} |-3/2, 0\hbar\vec{k}\rangle &\rightarrow |+1/2, +2\hbar\vec{k}\rangle \\ |+1/2, 0\hbar\vec{k}\rangle &\rightarrow |-3/2, -2\hbar\vec{k}\rangle \end{aligned} \qquad (3.10) $$

while the second passage $|+1/2\rangle \leftrightarrow |+5/2\rangle$ transfers two more momentum recoils to the final states of the first passage:

$$ \begin{aligned} |+1/2, +2\hbar\vec{k}\rangle &\rightarrow |+5/2, +4\hbar\vec{k}\rangle \\ |+5/2, 0\hbar\vec{k}\rangle &\rightarrow |+1/2, -2\hbar\vec{k}\rangle \end{aligned} \qquad (3.11) $$

as well as the third passage on diffraction line $|-7/2\rangle \leftrightarrow |-3/2\rangle$:

$$ \begin{aligned} |-3/2, -2\hbar\vec{k}\rangle &\rightarrow |-7/2, -4\hbar\vec{k}\rangle \\ |-7/2, 0\hbar\vec{k}\rangle &\rightarrow |-3/2, +2\hbar\vec{k}\rangle \end{aligned} \qquad (3.12) $$
<figure><img src="image_29.png" /></figure>

Figure 3.8: Simultaneous measurement of four spin populations with the spin dependent momentum transfer scheme. (a) Sequence of adiabatic passages. The second and third passages transfer two momentum recoils to the final states of the first passage, such that they final states have acquired a total of four momentum recoils, with associated recoil frequency $16\Delta$. (b) Absorption image after time of flight of the four diffracted spin states. (c) Integrated optical depth from the absorption image (b) to measure the spin populations. The red and blue solid lines are fits of the diffracted atomic clouds with a Boltzmann distribution.

From absorption images of the diffracted spin states, we measure the number of
atoms in each selected spin state, as shown on figures 3.8, (b) and (c). The first obser-
vation is that the population of atoms which have experienced two adiabatic passages,
i.e. $|+1/2\rangle$ and $|-3/2\rangle$ is weaker than the populations associated with $\pm2\hbar k$ recoils.
Obviously, the probability of success of two consecutive adiabatic passages, which scales
as the product of each individual probability of success, is lower than for one single pas-
sage. Moreover, the eigen-state $|\Psi_C\rangle$ associated with the transfer to the final states with
acquired four recoil momenta and 16 recoil energy must have a larger rate of sponta-
neous emission according to equation 3.9, further reducing the probability of success of
the second adiabatic passage. Hence, diffracted clouds at $\pm2\hbar k$ are contaminated by
the populations that failed passage to $\pm4\hbar k$.

Although having limited efficiency, this measurement that can yet be optimized
demonstrates the possibility to simultaneously measure the populations and associated
resolved momentum distributions of four different spin states. Further improvements may enable the simultaneous measurement of the populations of the ten spin states, matching the performances of OSG [SGS11; Tai+10].

### 3.1.6 Intensity window for improved adiabaticity

While the realization of an efficient adiabatic passage requires a slow enough rotation of the eigen-basis with respect to the coupling intensities, the initial and final parameters of the passage have to be chosen so that atoms properly connect to the eigen-state of interest $|\Psi_C\rangle$. In the case of the spin dependent momentum transfer, those conditions, defined in equations 3.6 and 3.7, are written as:

$$ \dot{\delta} \ll \Omega^2 \qquad (3.13) $$

$$ |\delta_{i,f}| \gg \frac{\Omega^2}{4\Delta} \qquad (3.14) $$

In the ideal case, respecting both conditions can be done with a very slow sweep rate and a large detuning ramp. While the speed is limited by processes realized at longer time scales such as unintended momentum transfers from polarization defects, and spontaneous emission, the span of the detuning ramp is limited by the neighboring resonances. With $\Omega/2\pi = 220$ kHz and $\Delta/2\pi = 4.8$ kHz, the detuning ramp should start much farther than approximately 1.5 MHz from the resonance, accounting for the CGs, while the excited spin states are separated by 6 MHz. Hence, preserving spin selectivity with a narrower detuning ramp is preferable, and it is actually not incompatible with a proper connection to the eigen-state of interest, which can be done by slowly increasing the Rabi coupling.

In the spin-dependent momentum transfer picture, the basis rotation is induced by the variation of the detuning. However, a variation of the Rabi coupling rotates the basis as well, starting from the atomic states at zero coupling. Consequently, a slow increase of the Rabi coupling at the beginning of the ramp, and slow decrease at the end, maximizes the connection of the atoms to $|\Psi_C\rangle$. We increased the efficiency of the adiabatic passage by approximately 5%, as shown on figure 3.9, by superimposing an intensity window over the detuning ramp. We compare the measurement of the
<figure><img src="image_30.png" /></figure>

Figure 3.9: Intensity window to enhance the efficiency of the adiabatic following. (a) $|1/2\rangle \leftrightarrow |-3/2\rangle$ diffraction line with squared intensity window, and (b) with an inverted parabolic intensity window. The intensity ramps are shown on the upper insets, the black solid is the measured intensity during the frequency ramp, the red dashed line is an inverted parabola $\propto -(t-t_0)^2/\Delta t^2$ with $\Delta t = 36\mu s$. Each image (a.1) and (b.1) are averages over 7 measurements.

populations of the diffracted clouds with a squared intensity ramp, shown on figure (a),
and with a smoothed intensity ramp, shown on figure (b), which practically corresponds
to an inverted parabola. Averaged over 7 measurements for each case, we measure
that the efficiency of the momentum transfer is then increased by approximatly 5%,
confirming the improvement on the connection to the followed eigenstate.

In this section I presented our method to selectively measure the population in every
spin state of the $^1S_0$ manifold with a spin dependent momentum transfer scheme. This
tool permits us to monitor manipulation schemes of the spin populations that I present
in the next section, and which we ultimately use to prepare polarized Fermi seas of
$^{87}$Sr.

## 3.2 Selective preparation of polarized Fermi sea of $^{87}$Sr

We apply a magnetic field to produce a Zeeman degeneracy lift of the $^3P_1$, $F = 11/2$ Zeeman states. We thus can resonantly couple a selected ground spin state to an excited state with a pulse of light with well defined $\sigma_-$ polarization to realize spin selective optical pumpings [Tey+10; Pag+14; Oza+18; Son+20]. First, I discuss our method to empty the population of one spin state and the associated heating, and then the preparation of two components Fermi mixtures. Finally, I will show how we can prepare polarized Fermi seas with this method.

### 3.2.1 Optical pumping

#### Experimental procedure

After loading the optical dipole trap, and before forced evaporation (see chapter 1), we lift the degeneracy of the $^3P_1$, $F = 11/2$ Zeeman sub-levels with a homogeneous magnetic field of amplitude $|\mathbf{B}| = 6.8$ G, as shown on figure 3.10, top. The excited spin states, with magnetic susceptibility 380 kHz/G are then separated by $g_F\mu_B|\mathbf{B}| \simeq 2.6$ MHz. The $^1S_0 \leftrightarrow ^3P_1$, $F = 11/2$ optical transitions are spin resolved, and we pulse a strongly saturating laser beam, $I \simeq 70I_{sat}$ with $I_{sat} \simeq 3\mu\text{W/cm}^2$, with well defined polarization $\sigma_-$, resonant with a $|^1S_0, m_F\rangle \leftrightarrow |^3P_1, m_F - 1\rangle$ transition, to pump the population out of the targeted $|m_F\rangle$ spin state. Figure 3.10, middle, shows the depletion of the each pumped spin population, measured with the spin dependent momentum transfer scheme, with respect to the frequency of the pump, with pulse duration 30 ms. This spectrum lets us infer the resonance of every $|^1S_0, m_F\rangle \leftrightarrow |^3P_1, m_F - 1\rangle$ transition with a Lorentz fit:

$$L(f) = N_0 \left( 1 - \frac{1}{1 + \left( \frac{f-f_0}{\Gamma/2} \right)^2} \right) \qquad (3.15)$$

of the populations depletion. Here, $y_0$ is the baseline, $f$ the frequency of the pump
and $f_0$ the resonance. The measured resonances $f_0$ are reported on figure 3.10, bottom,
for every spin state, as well as the width of the depletion. The resonances are sepa-
rated by 2.6 MHz which is in agreement with the expected Zeeman degeneracy lift with
magnetic field $|\mathbf{B}| = 6.8$ G.

This measurement shows that we are able to empty the population of any selected
spin state of the ground state manyfold with a pulse of well polarized $\sigma_-$ light at
appropriate frequency. Due to spontaneous emission, the population of the targeted
$|m_F\rangle$ spin state is transfered to $|m_F - 1\rangle$ and $|m_F - 2\rangle$ spin states, and the emission
of one spontaneous emission with $\pi$ or $\sigma_+$ polarization is enough to flip the spin of the
atom. Once an atom populates a new spin state, it will not see the excitation anymore
because of the degeneracy lift and the light polarization, and thus will remain stable in
the new spin state. Note that the $\sigma_-$ excitation on $-9/2$ is cyclic, such that atoms in
$|^3P_1, F = 11/2, m_F = -11/2\rangle$ cannot de-excite anywhere but back on $-9/2$. Note that
transfering this spin population would require to reverse the magnetic field.

**Frequency and polarization sensitivity**

While experiencing Rabi oscillations between the ground and the excited nuclear spin
states, an atom has a probability to spontaneously emit a photon with random polar-
ization and relax to any of the connected ground states, and the excitation has to be
applied for a long enough duration such that the probability of spontaneous emission
increases and the spin flip is realized. Figure 3.11 reports our measurements of the
+9/2, +7/2, -1/2, and -7/2, spin population depletion (black dots, indicated at bottom
right corner of each figure) with respect to the duration of the $\sigma_-$ light pulse. The
population depletions are fit with exponential decays $\exp(-t/\tau)$ (dashed lines) and the
infered decay times $\tau$ are all in the order of one millisecond (written at bottom left cor-
ner of each figure). The data is compared to numerical time integration of the Lindblad
equation shown by the solid black lines, accounting for possible detuning and circularity
defect that will be discussed in the following.

*Frequency sensitivity:* in the saturating regime *I* ≃ 70 *I*<sub>sat</sub>, one would expect that the
rate of spontaneous emission governs the timescale of optical pumping, which would
<figure><img src="image_31.png" /><figcaption>Figure 3.10: Experimental population transfer within the nuclear ground spin states manifold. Top: optical pumping scheme with $\sigma_-$ resonant coupling, where $\omega_{3P_1}$ is the $^3P_1$, $F = 11/2 \leftrightarrow 1S_0$ resonance for all spin states without magnetic field. Middle: spectroscopy for $\sigma_-$ excitation in the saturating regime for all spin states. The depletion of the population of each spin is measured with respect to the frequency of the pump. The black solid line is a lorentzian fit as defined in 3.15 to measure the optical pumping resonance $\omega_{op}$, assuming 10% initial population. Bottom: resonances (black circles) for every nuclear spin states.</figcaption></figure>

be $(\Gamma_{3P_1}/2)^{-1}$, i.e. tens of microseconds for $\Gamma_{3P_1} = 7.4$ kHz, which is much faster
than our measurement. This scaling is correct in the case $\Omega \gg \Gamma$, $\delta$ such that the
<figure><img src="image_32.png" /></figure>

Figure 3.11: Spin populations depletion with respect to the duration of the $\sigma_-$ light pulse. We measure the number of atoms remaining in the target spin state (black dots), respectively +9/2, +7/2, -1/2, and -7/2, from top to bottom figure, and the associated spins with the spin-dependent momentum transfer scheme, respectively +5/2, +3/2, +3/2, -3/2 (white dots). Each spin population depletion is fit with an exponential decay $\exp(-t/\tau)$ (dashed line), and the fitted decay time $\tau$ is noted on the bottom left corner of each figure. Black solid lines: numerical integration of the Lindblad equation with experimental parameteral $I = 70 I_{sat}$ accounting for a global perturbative defect in light circularity ($2.5 \times 10^{-3} \times I$) for all spin states, and tunable detuning.

probability for each atom to the populate the excited state is close to 1/2. However, if the light excitation is detuned from resonance, the amplitude of the Rabi oscillation is reduced, hence is the population of the excited state, which reads in the rotating wave
approximation (RWA) [MS99]:

$$P_e = \frac{1}{2} \frac{s}{1+s} \qquad (3.16)$$

where s is the saturation parameter defined as:

$$s = \frac{2\Omega^2}{\Gamma^2 + 4\delta^2} \qquad (3.17)$$

The radiative decay time in the non-resonant case can be directly inferred, and it reads, in units of the resonant radiative decay time:

$$\frac{\tau(\delta)}{\tau_0} = 1 + \left(\frac{2\delta}{\Gamma'}\right)^2 \qquad (3.18)$$

where $\Gamma' = \Gamma_{3P_1} \sqrt{1 + CG^2I/I_{sat}}$ is the power broadened linewidth of the transition, with associated Clebsch-Gordan (CG), and $\tau_0 \simeq 70$ µs. If considering the depletion of $m_F = +9/2$, with $I \simeq 70 I_{sat}$ and $CG \simeq 0.135$, a detuning $\delta = 4\Gamma_{3P_1}$ increases the decay rate to approximately 2.2 ms, which is in good agreement with the measured decay time, and numerical time integration of the Lindblad equation, accounting for the detuning, confirms this result, as shown by black solid lines on figure 3.11. The increase of decay time of the pumping of the other spin states +7/2, -1/2, and -7/2, is similarly explaining with respective detunings $7\Gamma_{3P_1}$, $8\Gamma_{3P_1}$, and $8\Gamma_{3P_1}$. Regarding our measurement, those detunings are relevant since we calibrated the optimal pumping frequency with a 100 kHz $\simeq 13\Gamma_{3P_1}/2\pi$ step. This also confirms that pumping is robust against frequency drifts which only extends the duration of the light pulse to a few milliseconds, and hence can be easily compensated for. Another possibility is to sweep the frequency of the optical pumping, and this gives the same results than extended pulse durations.

**Polarization sensitivity:** the number of atoms are measured with the spin-dependent momentum transfer scheme, hence a second spin population is measured simultaneously, as shown by the white dots (indicated top right corner of each figure). The pumping of spin states +9/2 and +7/2 are measured with the diffraction lines +9/2 $\leftrightarrow$ +5/2 and +7/2 $\leftrightarrow$ +3/2, which means that one of the two final states is measured simultaneously
with the depletion of initial state. However, the pumping of spin states -1/2 and -7/2 are measured with the diffraction lines -1/2 ↔ +3/2 and -7/2 ↔ -3/2, and the second measured state cannot be a final state of σ_ optical pumping, and the measurement of these populations, which is stable all over the duration of the pulse, confirms the robustness in spin sensitivity.

The first two measurements show that the populations of +5/2 and +3/2 are increased simultaneously with the depletion of +9/2 and +7/2, respectively, which is expected for optical pumping. However, for longer pulse duration, higher than 10 ms, the populations of final states +5/2 and +3/2 start decreasing. In our case, we explain this decay by the imperfection of the circularity of the light. Let's consider the pumping of +9/2 with σ_ excitation. A negligible fraction of σ_ light is enough to excite +5/2, and in a two level picture, the associated rate of spontaneous emission reads:

$$ \Gamma_{se} = \Gamma_{3P_1} \frac{s}{2} \qquad (3.19) $$

where s ≪ 1 is defined in [3.17], accounting for the polarization defect ε along σ_ so that:

$$ \Omega^2 = \frac{\Gamma^2}{2} \varepsilon^2 \frac{I}{I_{sat}} \qquad (3.20) $$

and a defect as small as ε² = 2.5 × 10⁻³, i.e. Iσ_+/Iσ_− = ε², which is hardly measurable with a powermeter, is enough to be responsible for a depletion of the +5/2 population with associated decay time τ ≃ 12.5 ms. On figure 3.11, the time integration of the Lindblad equation, shown by the black solid lines, accounts for this circularity defect ε² = 2.5 × 10⁻³, and is enough to explain the population depletion of +5/2 and +3/2 that we measure, as small as it is. This shows that pumping is very sensitive to the polarization, indeed a 1% defect in intensity, 5 times larger than we infer, would be enough to lower the decay time of |+5/2⟩ to 3 ms, and would break the spin sensitivity at our time scales. It is then critical that the polarization is optimized on the non-depletion of m_F - 2 population for σ_ optical pumping.

Another possible explanation would be that these states are also excited by the σ_
pulse, but the 2.6 MHz zeeman degeneracy lift of the excited states ensures the spin selectivity, and the verified equation 3.18 confirms that with such a detuning as large as the degeneracy lift, the associated decay time is as large as hundreds of milliseconds, which would not be observable on the measured time scales. Note that a defect in $\pi$ polarization could not explain the depletion of spin states $|+5/2\rangle$ and $|+3/2\rangle$ because of the Zeeman degeneracy lift, and considering the non-zero $\sigma_+$ defect, the stationary state of the three coupled ground states would obviously have non zero population in the target state. Even if we don't reach a stationary state, as shown by the populations of $+5/3$ and $+3/2$, our measurement shows that the target state is completely depleted, which shows that a defect in $\pi$ polarization is completely negligible, and this confirms a good alignment of the magnetic field with the propagation of the light.

## Associated heating

Since this scheme relies on spontaneous emission, the atoms obviously heat during the process. When experiencing the Rabi oscillations, atoms can relax to any of the connected ground states, including the initial state. This means that the atoms may need to emit more than one spontaneous photon before changing spin state. The probabilities for relaxation are exactly the square of the Clebsch-Gordan coefficients (CG), which are defined as $C_{j_1q_1,j_2q_2}^{kq} = \langle j_1q_1, j_2q_2|kq\rangle$, where $|j_1q_1\rangle = |^3P_1, F = 11/2, m_{Fe}\rangle$ and $|kq\rangle = |^1S_0, F = 9/2, m_{Fi}\rangle$. For each spontaneously emitted photon, let's define

$$p_{fail} = C_{m_{Fe}, 1-1}^{m_{Fe}} \qquad (3.21)$$

the probability for the atom to relax in the initial state, and

$$p_{success} = 1 - p_{fail} \qquad (3.22)$$

the probability to relax in one of the two possible final states. This is a binomial process, with associated probabilities for success $p_{success}$ and fail $p_{fail}$, and then the average number of tries for first success, here changing spin population, is simply $1/p_{success}$, which is exactly the average number of spontaneously emitted photon needed to change spin population $\langle N_{sp.em.} \rangle = 1/p_{success}$.
Let's now discuss the heating associated with optical pumping and resulting from the multiple momentum exchanges, say $n$, between an atom and the photons from the light excitation. An isolated atom at rest realizes a Brownian motion in the momentum space, consequently to the $n$ consecutive momentum recoils acquired with the absorption - spontaneous emission process. It results in an acceleration along the light propagation with acquired $n^2 E_R$ 1D kinetic energy, where $E_R = \hbar^2 k^2 / 2m$ is the recoil energy, and a three dimensional diffusion in momentum resulting from the spontaneously emitted photons with acquired $nE_R$ 3D kinetic energy. Let's quickly give a proof of this well known result. The momentum variation $\delta \vec{k}$ associated with multiple processes of one absorption of a photon from the excitation with momentum $\hbar \vec{k}_L$ and one spontaneous emission with momentum $\hbar \vec{k}_{se,i}$ and random direction $i$ reads:

$$ \delta \vec{k} = n \vec{k}_L + \sum_{i=1}^{n} \vec{k}_{se,i} \qquad (3.23) $$

Regarding this energetic consideration, the relevant quantity is the momentum spread, which reads as the quadratic momentum variation averaged over the realizations:

$$ \begin{aligned} \langle \delta \vec{k}^2 \rangle &= \langle n^2 k_L^2 + \left( \sum_{i=1}^n \vec{k}_{se,i} \right)^2 + 2n \sum_{i=1}^n \vec{k}_{se,i} \cdot \vec{k}_L \rangle \\ &= n^2 \langle k_L^2 \rangle + n \langle k_{se}^2 \rangle \end{aligned} \qquad (3.24) $$

This results from the independence between the absorption and emission process, so that the momentum of the absorbed photon is not correlated with the momentum of the spontaneous photon, and $\langle \vec{k}_L \cdot \vec{k}_{se} \rangle = \langle \vec{k}_L \rangle \langle \vec{k}_{se} \rangle$. Similarly, every spontaneous emission is inpendant so that $\langle \vec{k}_{se,i} \cdot \vec{k}_{se,j} \rangle = \langle \vec{k}_{se,i} \rangle \langle \vec{k}_{se,j} \rangle$. Since the momentum of spontaneously emitted photons is random, $\langle \vec{k}_{se} \rangle = \vec{0}$, which cancels both products. Now, the recoil energy $E_R = \hbar^2 k_R^2 / 2m$ associated with the intercombination line can be inserted to define the average energy acquired:

$$ \langle \delta E \rangle = n^2 E_R + n E_R \qquad (3.25) $$
which shows the quadratic dependence on $n$ of the energy acquired from the 1D acceleration and the linear dependence on $n$ of the 3D momentum diffusion resulting from the spontaneous emission.

In our experiment, the decay time associated with optical pumping of a spin state is approximately between 1 and 2 ms (see figure 3.11), while the collision rate is slightly lower than 1 collision per atom per millisecond. Considering that the gas necessitates several collisions to thermalize, and the pumping duration is smaller than the dipole trap period, it is then reasonable to consider that an atom is approximately isolated when it is pumped into another spin state. The estimated average energy acquired by the atom is:

$$ \langle \delta E \rangle = n^2 E_R + n E_R \qquad (3.26) $$

The associated heating for a 3D thermal gas in a harmonic trap with total energy $E_{tot} = 3k_B T$ finally reads:

$$ \delta T = \frac{1}{3}n(n + 1)T_R \qquad (3.27) $$

Where the recoil temperature associated with the intercombination line of $^{87}$Sr is $T_R = E_R/k_B = 230$ nK. Considering that the heating associated with one spin flip is redistributed to the whole gas after thermalization, the heating associated with the total depletion of a spin population can finally be predicted from:

$$ \delta T = \frac{N_i}{N_{tot}} \frac{1}{3} n(n + 1)T_R \qquad (3.28) $$

where is the initial number of atoms in the pumped spin state and $N_{tot}$ the total number of atoms in the gas, and $n = 1/p_{success}$ the average number of spontaneous emissions needed to change spin population, where $p_{success}$ is defined in equation 3.22. For instance, considering that the gas is initially unpolarized with 10% atoms per spin state, atoms in -3/2 have 0.5 probability to change spin state, so that emptying the population of -3/2 heats the gas by approximately 115 nK. Note that the populations of the low negative $m_F$ spin states, with strong CGs associated with $\sigma_-$ relaxation, see
figure 3.6, need to experience more spontaneous emission than the positive spin states
to be transfered, for instance -3/2 spins are heated twice as much as 9/2.

Optical pumping is best realized on a hot gas in a deep trap, rather than on an
ultra-cold gas in a shallow trap which would consequently strongly heat the atoms and
might blast them out of the trap. For this reason, the spin populations are emptied
just after loading the dipole trap which is 90µK deep, while the gas is unpolarized
with approximately 10% atoms per spin state (see section 3.1) at 7.2 µK (see section
1.1.6 of chapter 1). In this case, the heating associated with the pumping of one spin
population is hardly measurable.

3.2.2 Controlling the number of spin components from 1 to 10

In the previous section, I presented our method to empty the population of a $m_F$ spin state and transfer it to $m_F - 1$ and $m_F - 2$. I will now explain how we can tune the number of spin components in the Fermi gas of $^{87}$Sr from 10 to 1. To prepare a selective spin mixture, we rely on the pure polarization of the laser, such that consecutive spin states can be emptied in descending $m_F$ order, while never bringing back atoms in previously emptied spin states. Considering the preparation of a Fermi gas with 2 spin components with initially all 10 spin states populated, we sequentially empty all $m_F$ states with decreasing $m_F$, but one spin state that we voluntary skip, say -5/2, and -9/2 that cannot be pumped with $\sigma_-$ excitation, so that all spin states are emptied, but -5/2 and -9/2.

Figure 3.12 shows absorption image after time of flight of a SU(2) $^{87}$Sr thermal gas (right) after a pumping sequence to prepare a {−9/2, −1/2} mixture, and a SU(10) gas (left) with no prior optical pumping. We compare the temperature of the gas after the pumping sequence and after allowing for 500 ms thermalization, to the initial temperature of the gas prior to pumping, as well as the number of atoms. The increase of temperature is reported on figure 3.12, (a), for the {−9/2, −1/2} preparation, as well as other preparations, {−9/2, 3/2}, {−9/2, −3/2}, and {−9/2, −7/2}. We measure an increase of 1 µK after thermalization of the {−9/2, −7/2} preparation, and smaller
<figure><img src="image_33.png" /></figure>

Figure 3.12: Measurement of the heating and atom loss associated with the preparation of a 2 component Fermi gas of $^{87}$Sr with a pumping sequence. Top figures, absorption images after time of flight of a SU(10) gas (left) and SU(2) gas with remaining -9/2 and -1/2 spin states. The number of atoms and temperatures and inferred from a Gaussian fit (white dashed line) of the integrated optical depths (black solid lines). (a): heating of $\delta T = T_{SU(10)} - T_{SU(2)}$, inferred from a simulated pumping sequence (black dots), neglecting evaporation, and measurement with associated fit uncertainty. (b) density of the remaining spin component, here defined as $\propto N_{m_F}/T^3$, inferred from a simulation (black dots) and measurement (empty squares), where the population in $m_F$ is inferred from the measured number of atoms and the predicted population balance after the pumping sequence. (c) Prediction of the ratio of atoms in $m_F$ after preparation of a -9/2, $m_F$ two component gas, using Clebsch-Gordan coefficients.

increase of approximately 800 nK for the other preparations, and it is compared to simulations of the associated pumping sequences using only the Clebsch-Gordan coefficients, and equations 3.22 and 3.28. Our measurement shows that heating is less spin dependent contrary to the prediction and moreover it is generally lower. A possible explanation is that during thermalization, the two component gas continues evapora-
tion, attenuating the heating from the pumping sequence, and our measurement of the number of atoms $N_{at}^{SU(2)} \simeq 10^6$ for all preparations confirms that one third of the atoms are lost after thermalization. We assume that thermalization is homogeneous for all spin states, and that there is no spin exchange due to the SU(N) symmetry of the collision properties [Gor+10]. Then, the population in the remaining $m_F$ component can be inferred from the total number of atoms measured and the predicted population imbalance after the pumping sequence, shown on figure 3.12, (c). This way, we deduce the density of atoms in the remaining $m_F$ component, as shown on figure 3.12, (b).

On figure 3.13, we compare the final state of several gases after evaporation, depending on spins mixture preparation prior to evaporation. The initial mixtures are either a SU(10) gas with no optical pumping prior to evaporation, or a SU(2) gas prepared in $\{-9/2, -7/2\}$, $\{-9/2, -3/2\}$, $\{-9/2, -1/2\}$, or $\{-9/2, +3/2\}$ (these preparations are the ones discussed on figure 3.12). After the evaporation, all realized from initial trap frequency 330 Hz ramped down to 85 Hz during 5 seconds, we compare the densities and degeneracies of each preparation, accounting for the predicted balance of the population, see figure 3.12.

This measurement first shows that a SU(10) gas of $^{87}$Sr has four times more atoms than a SU(2), and thermalizes at a temperature lower by 20%, for this evaporation depth. Accounting for the fact that initially SU(10) gas has approximately 30% more atoms, this shows that relatively more atoms are evaporated in SU(2) preparations. This is consistent with previous observations that the collision rate of a SU(2) gas is lowered compared to that one of a SU(10) gas [Son+20]. However, accounting for the number of atoms per spin state, the phase space density $N_{mf}/T^3$ of the most populated spin state of the SU(2) preparations $\{-9/2, -7/2\}$ and $\{-9/2, -5/2\}$ is equal to that one of the SU(10) preparation, as well as the degeneracy $T/T_F$. The $\{-9/2, -1/2\}$ and $\{-9/2, +3/2\}$ preparations, with less atoms in the selected spin states, are consequently less degenerate. This shows that the manipulation of the spins populations is viable for the preparation of ultra cold gases of $^{87}$Sr with tunable N. Moreover, we are not interested into the choice of the spin mixture regarding the latter study of the Heisenberg model in enlarged SU(N) symmetry, since the collision properties of $^{87}$Sr do not depend on the spin states. It is then reasonable to select the most favorable spin mixture
<figure><img src="image_34.png" /><figcaption>Figure 3.13: Final state of different spin mixtures after evaporation. Top: absorption images after 12 milliseconds of time of flight of the different mixtures, SU(10) on left, and SU(2) preparations. From the measured number of atoms and temperatures, we infer the density per spin state $N_{m_F}/T^3$ for each preparation, and degeneracy $T/T_F$, using predicted populations per spin states. The final trap frequency of evaporation is indicated at top right corner.</figcaption></figure>

regarding only the production of degenerate gases.
Collision rate for unbalanced mixtures. The preparation of the SU(2) mixtures in-
duces populations imbalances between the remaining spin states. It is then interesting
to investigate the consequences regarding the efficiency of evaporation, that depends
on collisions. Symmetry rules forbid s-wave collisions between two identical fermions,
so that collisions are inhibited in the degenerate regime and the efficiency of the evap-
oration decreases when $T \to 0$ [DeS+10]. In case of population imbalance between the
colliding fermions, the collision rate per atom will be lowered for the most populated
spin state, $m_F$ in our case. Indeed, the less populated has more partners to collide with,
and Pauli exclusion principle forbids collisions for which the arrival state is already oc-
cupied by a fermion of the same specie, so that at thermal equilibrium between the two
spin states, the most populated state has more occupied states in the arrival state for
the collision, inhibiting the probability for this collision. It is possible to numerically
predict the inhibition of collisions due to a population imbalance in a two components
fermi mixture. The collision rate of two colliding species $\alpha$ and $\beta$ can be computed
thanks to the collisional integral [LRW96; Geh+03]:

$$
\begin{align}
\rho(\epsilon_4) \dot{f}_{\alpha}(\epsilon_4) &= \frac{M\sigma}{\pi^2 \hbar^3} \int_0^{\infty} d\epsilon_1 d\epsilon_2 d\epsilon_3 \rho(\epsilon_{min}) \delta(d\epsilon_1 + d\epsilon_2 - d\epsilon_3 - d\epsilon_4) \nonumber \\
&\quad (1 - f_{\alpha}(\epsilon_1)(1 - f_{\beta}(\epsilon_2)f_{\alpha}(\epsilon_3)f_{\beta}(\epsilon_4)) \tag{3.29} \\
\Gamma_{coll}^{\alpha\beta} N_{\alpha} &= \int_0^{\infty} d\epsilon_4 \rho(\epsilon_4) \dot{f}_{\alpha}(\epsilon_4) \nonumber
\end{align}
$$

where M is the mass, $\sigma = 4\pi a_s^2$ the low energy s-wave collisions cross-section with $a_s$ the scattering length, and $\hbar$ the reduced Planck constant. $\rho(\epsilon) = \epsilon^2/2(\hbar\bar{\omega})^3$ stands for the density of states, in a harmonic trap of geometric mean frequency $\omega$. The Pauli exclusion principle is taken into account by the $(1 - f(\epsilon))$ non-occupation number, where $f(\epsilon)$ is the Fermi-Dirac occupation number:

$$
f_{FD}(\epsilon, T, \mu) = \frac{1}{e^{\beta(\epsilon - \mu)} + 1} \qquad (3.30)
$$

Here, $\beta = 1/k_B T$ and the chemical potential $\mu$ is defined as the Fermi energy at zero temperature $\mu(T = 0) = \epsilon_F = k_B T_F$, $T_F$ the Fermi temperature. When solving equation 3.29, it is important to take into account the fact that the two unbalanced
<figure><img src="image_35.png" /><figcaption>Figure 3.14: Inhibition of the collision rate per atom for unbalanced populations in the two colliding Fermi components $\alpha$ and $\beta$. The numbers above each curve indicates the ratio of atoms $p_{\alpha}$ in the considered state $\alpha$ with respect to the total number of atoms $N_{tot} = N_{\alpha} + N_{\beta}$, i.e. $N_{\alpha} = p_{\alpha}N_{tot}$. The calculation has been made for trap frequency $\bar{\omega} = 120$ Hz and $N_{tot} = 10^{5}$, representative of our trap after evaporation.</figcaption></figure>

populations have different Fermi energy $\varepsilon_{F\alpha} = \hbar\bar{\omega}(6N_{\alpha})^{1/3}$ and hence Fermi temperature and chemical potential. So the occupation numbers $f_{\alpha}$ and $f_{\beta}$ are not the same for the two species. The details for the computation can be found in appendix A. In figure 3.14 are presented different collision rates per atom in state $m_{Fsel}$ with respect to $T/T_{F\alpha}$ and populations imbalance, for fixed $N_{tot}$ and $\bar{\omega}$. If the two spin states are equally populated, $N_{\alpha} = N_{\beta} = N_{tot}/2$, then collision rate per atom is maximal, and the forced evaporation efficiency optimal. Then if the state $m_{Fsel}$ starts to be more populated, i.e. $p_{\alpha} = N_{\alpha}/N_{tot} > p_{\beta}$, the collision rate drops quickly, from -20% for $p_{\alpha} = 0.6$ with respect to equally populated states ($p_{\alpha} = p_{\beta} = 0.5$) to approximately -80% for $p_{\alpha} = 0.9$.

This shows that unbalanced mixtures suffer from inhibition of the collisions, which results in a less efficient evaporation. This is consistent with our measurement of the
number of atoms after evaporation presented on figure 3.13, which indicates that the
most unbalanced mixture {−9/2, −7/2} is slightly lower than the other SU(2) prepa-
rations. However, it is not significant, as T/T<sub>F</sub> is smaller. Regarding the preparation
of SU(2) degenerate Fermi gases with forced evaporation, the best mixtures are hence
−9/2 with either −7/2 or −5/2, according to our measurement.

A spin mixture with more than two components can be prepared exactly the same
way, except that another $m_F$ spin has to be skipped during the pumping sequence. Note
that the preparation of the Fermi gas with tunable number of spin components is real-
ized in the deep dipole trap prior to evaporation because of the heating associated with
optical pumping, so it is mandatory to keep at least two spin components to authorize
forced-evaporation assisted by s-wave collisions. For this reason, our preparations of
polarized Fermi seas have to be done after evaporation, which I will now discuss.

3.2.3 Spin purification to prepare a polarized Fermi sea

To prepare a polarized Fermi sea from a two components degenerate Fermi gas, we selectively blast one spin state out of the trap after evaporation. This is done by realizing our optical pumping scheme on spin state -9/2, i.e. with a long pulse of saturating $\sigma_-$ light resonant with the $|^1S_0, -9/2\rangle \leftrightarrow |^3P_1, F = 11/2, -11/2\rangle$ transition. This transitions is cyclic, and atoms absorb photons from the laser and emit spontaneous photons until they acquire enough energy to leave trap. Then, there remains only the other spin state and the Fermi gas is spin polarized.

Figure 3.15 shows our experimental realization of the blast to prepare a polarized Fermi gas in $m_F = -1/2$, from an evaporated gas previously prepared in a $-9/2, -1/2$ spins mixture with optical pumping. Right figure shows absorption images of the $|-1/2\rangle \leftrightarrow |+3/2\rangle$ diffraction line, with respect to the duration of the $\sigma_-$ pulse, and the corresponding longitudinal integrated optical depths are superimposed for each blast duration. The population in the non diffracted cloud, which corresponds to atoms in $-9/2$, and centered on pixel 30, is progressively reduced with longer pulses until it reaches a minimum while the population in $-1/2$ seems not affected, and the absence of a second
<figure><img src="image_36.png" /><figcaption>Figure 3.15: Preparation of a polarized Fermi gas in $m_F = -1/2$ by blasting atoms in $m_F = -9/2$ out of a two components trapped Fermi gas. (a) Depletion of the -9/2 population, and relative increase of the -1/2 population, with respect to the frequency of the blast with fixed duration 5 ms. (b) Ratio of atoms in $m_F = -1/2$ with respect to the blast duration. It is fit with an exponential decay of the -9/2 population $e^{-t/\tau}$, with fitted decay time $\tau = 97 \pm 14\mu s$, and the ratio of atoms in $-1/2$ saturates at approximately 66% in this measurement. (c) Measured number of atoms and temperature of the diffracted cloud associated with -1/2 population, with respect to the duration of the blast. Right figure: absorption images of the $|-1/2\rangle \leftrightarrow |+3/2\rangle$ diffraction line, with respect to the duration of the resonant blast with saturating $\sigma_-$ excitation, $I \approx 70 I_{sat}$, used for the measurements on figures (b) and (c).</figcaption></figure>
diffracted cloud confirms that the population in +3/2 is zero. From this measurement, we infer the ratio of atoms in -1/2 with respect to the blast duration, shown on figure (b), which raises consequently to the exponential decay of the population in -9/2, with fitted decay time $\tau = 97 \pm 14$ µs. We observe here that the population in -1/2 saturates close to 70%, and this is also observed on a scan of the optimal frequency of the blast with 5 ms of pulse, shown on figure (a), with a plateau of the depletion in -9/2 over 150 kHz of scan. This poor efficiency of the blast was due to a hardware failure damaging the efficiency of the spin-dependent momentum transfer, so that atoms in -1/2 were not diffracted for this specific data. This failure was later corrected.

Nevertheless, our measurement shows that during the blast of atoms in -9/2, atoms in -1/2 remain unaffected, as the diffracted cloud has an approximately constant number of atoms, and the temperature remains stable compared to the temperature of the initial gas $T_i \approx 45$ nK, as shown on figure 3.15, (c). This is consistent with the fitted decay rate $\gamma = 1/\tau \approx 2\pi \times 1.5$ kHz of the -9/2 population, which is much higher than the collision rate and the trap frequency. Atoms in -9/2 leave the trap before colliding with atoms in -1/2, and the Zeeman degeneracy lift of the excited state ensures that there is no possible light scattering from atoms in -1/2 on those timescales.

### 3.2.4 Polarized Fermi sea

We measure the fiability of our preparation of a polarized Fermi sea of $^{87}$Sr in $m_F = -7/2$, by measuring the number of atoms in every spin state with the spin-dependent momentum transfer scheme. Figure 3.16 shows the results of our preparation with absorption images of each diffraction line (see section 3.1.4). From the diffraction line $|-7/2\rangle \leftrightarrow |-3/2\rangle$ (see (a)), we measure $75 \pm 3\%$ diffracted atoms from $-7/2$.

The signal in the non diffracted cloud, in both the absorbtion image (top) and the integrated optical depth (a), shows that atoms at rest prior to the momentum transfer, at the center of the gas, are better diffracted than atoms with non zero velocity away from the center. This is a systematic effect that leads to an underestimate of the number of atoms from $-7/2$. The feature is taken into account in figure (b). The inset shows the velocity dependence of the efficiency of the momentum transfer, which is infered from
<figure><img src="image_37.png" /><figcaption>Figure 3.16: Measurement of the spin polarization after preparation of a polarized Fermi sea in $m_F = -7/2$. Top: absorption images of each diffraction line, the labels correspond to the spins expected at location indicated by the dashed circles. (a) Fit of the $|-7/2\rangle \leftrightarrow |-3/2\rangle$ diffraction line (solid line) with the sum of gaussian distributions. Insets: residuals of thermal (left) and degenerate (right) fits of a non diffracted polarized Fermi sea. (b) Expected density profile (dashed-line), accounting for the velocity $v$ dependent efficiency $P(v)$ of the spin-dependent momentum transfer (see inset). (c) Integrated optical depths of the other diffraction lines with indicated fraction of diffracted atoms. (d) Measured number of atoms in $-7/2$ (empty square), accounting for the velocity dependent efficiency (empty circle), and calibrated by the total fraction of diffracted atoms (filled square and circle).</figcaption></figure>
numerical integration of hamiltonian 3.1 accounting for an opposite doppler shift on
the $\sigma_+$ and $\sigma_-$ photons. The number of atoms in $-7/2$ is then measured by accounting
only for atoms at rest, i.e. the peak density of the diffracted cloud, and the density
at $v = 0$ for the non diffracted cloud. On figure (b), the dashed line indicates the
expected density profile by neglecting the velocity dependence. With this correction,
we infer $81 \pm 3\%$ diffracted atoms from $-7/2$. These two results are finally calibrated
by taking into accounts the atoms detected in the other spin states (see figure (c)), and
we find $89.5 \pm 10\%$. Figure (d) reports the result of our measurements: accounting
only for the detected atoms in $-7/2$ (empty square), with the correction of the velocity
sensitivity (empty circle), and calibrated by the efficiency of the diffraction measurement
(respectively filled square and circle).

Finally, the degeneracy of the gas is measured from a non diffracted gas prepared
in the same conditions, with a fit of the absorption images with a degenerate density
distribution [KZ08] (see residuals and fit result in inset of figure 3.16, (a)), and we
measure $T \simeq 0.25 T_F$, $T_F \simeq 250$ nK with approximately 20 000 atoms.

This finally shows that we are able to prepare a degenerate polarized Fermi gas of
$^{87}$Sr at $T/T_F \simeq 0.25$ with $20 \times 10^3$ atoms, and we estimate that $89.5 \pm 10\%$ atoms are
in the same spin state $|-7/2\rangle$. Therefore our measurement is compatible with a 99%
SU(1) degenerate Fermi gas.
Chapter 4

Coherent manipulation of the
nuclear spin states of $^{87}$Sr

The ground state $^1S_0$ of fermionic $^{87}$Sr has an electronic closed-shell. As mentionned
in chapter 3, it is not practical to use magnetic fields to manipulate the 10 degenerate
ground spin states. In the prospect of studying the Heisenberg model in enlarged
SU(N) symmetry, we rely on spin-orbit coupling and spin resolved Raman transitions
to coherently probe and manipulate the spins with light. To do this, we take advantage
of spin dependent light shifts.

The bosonic isotopes of strontium have zero nuclear spin and J=0 in the ground
state, and hence no hyperfine structure. However, one of the most interesting properties
of fermionic $^{87}$Sr is the hyperfine coupling between total momentum J and the large
nuclear spin I = 9/2 in excited states. For instance, the excited state $^3P_1$ with non-zero
momentum J is split into 3 hyperfine states F = 7/2, 9/2, 11/2, which are all separated
by more than 1 GHz. These three hyperfine states are highly resolved compared to the
narrow linewidth Γ/2π = 7.4 kHz of the optical transitions. Therefore, the dynamic
polarizability associated with the intercombination line can acquire a significant and
tunable spin dependence [Shi+15], so called tensor polarizability [MOR86], with low
spontaneous emission. While experiments for optical lattice clocks take specific care
of minimizing spin dependent light shifts [Boy07; Lud08] on alkaline earth species to
prevent perturbations on the clock frequencies, degenerate gases experiments can take
advantage of the tensor polarizability. It has been demonstrated on both strontium
[SGS11] and ytterbium [Tai+10] that spin components can be separated using conser-
vative spin dependent forces, enhancing the study potential of many-body physics in
enlarged SU(N) symmetry [CHU09; Gor+10; Tai+12].

In our experiment, we use a *dressing* laser beam that realizes spin dependent light
shifts associated with the intercombination line of $^{87}$Sr, so that the degeneracy of Raman
transitions within the ground state spin manifold is lifted. Then, we selectively and
coherently flip chosen nuclear spin states with two photon Raman process, with one
photon from the *dressing beam*, and a one photon from another laser, that I call *Raman
beam* in this chapter, and whose frequency is adiabatically swept through the selected
Raman resonance. Designed as a dipolar optical lattices, the spin dependent light
shifts are then associated with a spatial selectivity that can be taylored at the optical
wavelength scale. Therefore, the Raman process acquire a site dependence, that can be
used to prepare, and detect, spin textures into lattices. In the prospect of the realization
of a quantum simulator for Heisenberg model, this scheme enables the possibility to
probe long range spin ordering, and to detect spin entanglement [Tai+20; Sun+21].

The basics of the tensor polarizability and the derivation of the quadratic light shift, as well as the associated scattering rate, are well known results that are first recalled in this chapter. We measured the quadratic light shift with Raman spectroscopy, and results are compared to predictions. Then, I show how we manipulate the ground spin states, taking advantage of the tensor polarizability to engineer spin selective adiabatic passages within the 10 spins ground state manifold of $^{87}$Sr. Finally I discuss the short-term prospect of associating spatial selectivity to the spin dependent light shift, to prepare spin textures into lattices with site dependent adiabatic passages.
# 4.1 Degeneracy lift associated with spin dependent light shift

## 4.1.1 Tensor dynamic polarizability

The rotational invariance of AC Stark shift can be broken for atoms which have fine and hyperfine structures, and in this case, the polarizability can acquire a dependence on the zeeman sub-levels $m_F$. Here, I will focus on the main steps to summarize the derivation of the irreducible tensor components of the atomic polarizability $\alpha$ in order to expand the spin-dependant light shifts in terms of a scalar component, independent on $m_F$, a vector component, linearly dependent on $m_F$, and a tensor component, quadratically dependent on $m_F$. This well-known result can already be found in several publications and pedagogical contents. [MOR86; Ovs+06; Shi+15]. A clear and interesting introduction to the angular momentum algebra and spherical tensors can be found in [Wal21], which are explicitly used for the following derivations.

The interaction between the atom electric dipole $\hat{D}$ and the classic monochromatic light field $\mathbf{E}$ of angular frequency $\omega_l$, $\mathbf{E} = \frac{1}{2}\epsilon|\mathbf{E}|e^{i(\omega_l t - \phi)} + c.c.$, is described by the electric dipole potential operator $\hat{U}_{dip}$:

$$ \hat{U}_{dip} = -\frac{1}{4}\hat{\alpha}|\mathbf{E}|^2 \qquad (4.1) $$

where $\alpha$ is the polarizability operator. We are interested in the magnetic dependence of the polarizability $\langle m_1 | \hat{\alpha} | m_2 \rangle$, where $|m_1\rangle$ and $|m_2\rangle$ are Zeeman sub-levels of the ground state $^1S_0$. In second order perturbation theory, $\alpha$ reads:

$$ \hat{\alpha} = \frac{1}{\hbar} \sum_{|\phi'\rangle} \frac{\epsilon^* \cdot \hat{D} |\phi'\rangle \langle\phi'| \epsilon \cdot \hat{D}}{\omega_{\phi'} - \omega_l} + \frac{\epsilon \cdot \hat{D} |\phi'\rangle \langle\phi'| \epsilon^* \cdot \hat{D}}{\omega_{\phi'} + \omega_l} \qquad (4.2) $$

where $\hbar$ is the reduced Planck constant. Here, $|\phi'\rangle = |n'L'S'J'F'm'\rangle$ are the excited states, and $\epsilon$ is the polarization of the electric field. In the spherical basis of unit vectors $\mathbf{u}_q$, defined as $\mathbf{u}_{\pm 1} = \mp(\mathbf{x} \pm i\mathbf{y})/\sqrt{2}$, $\mathbf{u}_0 = \mathbf{z}$, with $\mathbf{u}_q = (-1)^q \mathbf{u}_{-q}$, the electric dipole operator $\hat{D} = -e\hat{R}$ can be expended as a linear combination of spherical harmonics
$\mathcal{Y}_l^m$ [Wal21]:

$$ \hat{\mathbf{D}} = \sum_q -e |\mathbf{r}| \sqrt{\frac{4\pi}{3}} \mathcal{Y}_1^q \mathbf{u}_q^* \quad (4.3) $$

This is enough to show that $\hat{\mathbf{D}}$ is a spherical tensor $T^k$ of rank $k=1$, with three standard components $T_{kq}$, $q=0, \pm 1$. It is then possible to apply the Wigner-Eckart theorem on the electric dipoles $D_1(|m_1\rangle, |\phi'\rangle) = \langle m_1|\epsilon^* \cdot \hat{\mathbf{D}}|\phi'\rangle$ and $D_2(|\phi'\rangle, |m_2\rangle) = \langle \phi'|\epsilon \cdot \hat{\mathbf{D}}|m_2\rangle$:

$$ \begin{aligned} D_1(m_1, \phi') &= \sum_{q_1} (\epsilon^* \cdot \mathbf{u}_{q_1}^*)(-1)^{F_1-m_1} \langle F_1 | |D| |F' \rangle \begin{pmatrix} F_1 & 1 & F' \\ -m_1 & q_1 & m' \end{pmatrix} \\ D_2(\phi', m_2) &= \sum_{q_2} (\epsilon \cdot \mathbf{u}_{q_2}^*)(-1)^{F_2-1+m'} \langle F' | |D| |F_2 \rangle \begin{pmatrix} F_2 & 1 & F' \\ m_2 & q_2 & -m' \end{pmatrix} \end{aligned} \quad (4.4) $$

where the angular dependence is contained into the Wigner-3j symbols. Here, $F_i$ and $F'$ are the hyperfine states associated with the $|m_i\rangle$, $i=1, 2$, Zeeman sub-levels and the excited state $|\phi'\rangle$. The double bar matrix elements $\langle F_1| |D| |F'\rangle$ and $\langle F' | |D| |F_2\rangle$ are the reduced dipole matrix elements which do not depend on $m_1$, $m_2$ or $m'$.

The product of the two spherical tensors $D_1$ and $D_2$ of rank $k_1$ and $k_2$ can be expanded as the sum of irreducible tensor operators of integer ranks $|k_1 - k_2| \le k \le k_1 + k_2$. This can be done either by expanding the product of Wigner-3j symbols over $m'$ using sum rules for angular momentum, or by identically expanding the product of the two uncoupled spherical tensors in a direct sum of irreducible tensors [Wal21]. Practically, these irreducible tensor operators have rank $k=0, 1, 2$ in the case of the combination of two electric dipole operators. The result of the expansion takes the simple form [MOR86; Ovs+06; Shi+15]:

$$ \langle m_1 | \hat{\alpha} | m_2 \rangle = \sum_{k=0}^{2} \alpha_k(\omega_l) \sum_{q=-k}^{k} (-1)^q C_{Fm_2,kq}^{Fm_1} \{\epsilon \otimes \epsilon^*\}_{kq} \quad (4.5) $$

where $\{\epsilon \otimes \epsilon^*\}_{kq} = \sum C_{1q_1,1q_2}^{kq} \epsilon_{q_1} \epsilon_{q_2}^*$ [MOR86], and $\epsilon_q = \mathbf{u}_q^* \cdot \epsilon$. Here, $q_1$ and $q_2$ are the projections in the spherical basis of the standard components of the electric dipole operator, as defined in equation 4.3. The $C$ symbols are the Clebsch-Gordan coefficients
defined as $C_{j_1q_1,j_2q_2}^{kq} = \langle j_1q_1, j_2q_2|kq \rangle$. This approach allows to isolate the contributions from the different physical components. Indeed, the polarization dependence is entirely contained in the tensor $\{\epsilon \otimes \epsilon^*\}_{kq}$, the magnetic dependence in the Clebsch-Gordan coefficient $C_{Fm_2,kq}^{Fm_1}$, and the light angular frequency in the tensor component $\alpha_k(\omega_l)$ of the dynamic polarizability, which reads [MOR86; Ovs+06; Shi+15]:

$$ \alpha_k(\omega_l) = \frac{1}{\hbar} \sqrt{\frac{2k+1}{2F+1}} \sum_{n'} \sum_{F'=F-1}^{F+1} (-1)^{k-F-F'} \left( \frac{1}{\omega_{\phi'} - \omega_l} + \frac{(-1)^k}{\omega_{\phi'} + \omega_l} \right) \begin{Bmatrix} F & F & k \\ 1 & 1 & F' \end{Bmatrix} |\langle nF | D | n'F' \rangle|^2 \quad (4.6) $$

where the bracket symbols are the Wigner-6j symbols. Two useful examples with purely polarized light illustrate the efficiency of this approach.

If the light is $\pi$ (linearly) polarized, such that $\epsilon = u_0^* = \epsilon^*$, then $\{\epsilon \otimes \epsilon^*\}_{kq}$ is non zero only if $q = 0$. In this case the momentum conservation rules impose $m_1 = m_2 = m$, which makes sense for purely polarized light, and parity rules impose that $k$ is even, else $C_{10,10}^{k0} = 0$. Thus, the polarizability matrix elements can be written as following [MOR86; Shi+15]:

$$ \langle m | \hat{\alpha} | m \rangle \Big|_{\pi} = C_{Fm00}^{Fm} C_{1010}^{00} \alpha_0(\omega_l) + C_{Fm20}^{Fm} C_{1010}^{20} \alpha_2(\omega_l) \quad (4.7) $$

$$ \langle m | \hat{\alpha} | m \rangle \Big|_{\pi} = -\frac{1}{\sqrt{3}} \alpha_0(\omega_l) - \sqrt{\frac{2}{3}} \frac{3m^2 - F(F+1)}{\sqrt{F(F+1)(2F-1)(2F+3)}} \alpha_2(\omega_l) \quad (4.8) $$

The polarizability matrix elements are now reduced to a sum of two terms, as seen on the right hand side. The first term is independent of the magnetic sub-levels, depending only on the scalar component of the dynamic polarizability, while the second term is quadratically dependent on the magnetic sublevels. This first result shows that if the tensor component of rank 2 $\alpha_2(\omega_l)$ is non zero, then the dynamic polarizability acquires a dependence in the magnetic sub-levels, and for pure $\pi$ polarized electric field, this dependence is quadratic.

If the light is $\sigma_{\pm}$ (circularly) polarized, such that $\epsilon = u_{\pm 1}^*$, i.e. $\epsilon_{\pm 1}^* = -\epsilon_{\mp 1}$, then again $\{\epsilon \otimes \epsilon^*\}_{kq}$ is non zero only if $q = 0$, and momentum conservation rules impose
$m_1 = m_2 = m$. But $k$ only needs to respect triangle inequality in $C_{1\pm1,1\mp1}^{k0}$, such that rank 1 irreducible tensor of polarizability expansion is not obviousy zero. In this case, the polarizability matrix elements read:

$$ \langle m | \hat{\alpha} | m \rangle |_{\sigma_{\pm}} = C_{Fm00}^{Fm} C_{1\pm1,1\mp1}^{00} \alpha_0(\omega_l) + C_{Fm10}^{Fm} C_{1\pm1,1\mp1}^{10} \alpha_0(\omega_l) + C_{Fm20}^{Fm} C_{1\pm1,1\mp1}^{20} \alpha_2(\omega_l) \quad (4.9) $$

$$ \langle m | \hat{\alpha} | m \rangle |_{\sigma_{\pm}} = \frac{1}{\sqrt{3}} \alpha_0(\omega_l) \mp \alpha_1(\omega_l) m \frac{1}{\sqrt{2F(F+1)}} + \frac{1}{2} \sqrt{\frac{2}{3}} \frac{3m^2 - F(F+1)}{\sqrt{F(F+1)(2F-1)(2F+3)}} \alpha_2(\omega_l) \quad (4.10) $$

Finally, let's consider an electric field with mixed polarizations, for instance $\pi + \sigma_+$, i.e. $\epsilon = (\mathbf{u}_0^* + \mathbf{u}_+^*)\sqrt{2}$ and $\epsilon^* = (\mathbf{u}_0 - \mathbf{u}_-)^\sqrt{2}$. In this case, $\{\epsilon \otimes \epsilon^*\}_{kq}$ can be non zero for all four possible 2 photon process involving $\pi$ and $\sigma_+$ photons, which corresponds to $(q_1, q_2)$ pairs $(0, 0)$, $(1, -1)$, $(1, 0)$ and $(0, -1)$. The two first possibilities correspond to absorbtion and reemission of a photon with same polarization, which is exactly the two first introduced examples with pure polarized light. However, the $(1, 0)$ and $(0, -1)$ pairs respectively correspond to absorption of a $\pi(\sigma_+)$ photon and reemission of a $\sigma_+(\pi)$ photon. This coupling is actually a Raman two photon coupling, which is non diagonal in the Zeeman sub-levels basis and hence involves a basis rotation. The tensor polarizability acquires non diagonal components $\langle m - 1 | \hat{\alpha} | m \rangle$ for $(q_1 = 1, q_2 = 0)$ and $\langle m + 1 | \hat{\alpha} | m \rangle$ for $(q_1 = 0, q_2 = -1)$, and selection rules respectively impose that $q = 1$ or $q = -1$ and hence $k > 0$. Finally, the new non-diagonal matrix elements of the tensor polarizability read:

$$ \langle m - 1 | \hat{\alpha} | m \rangle |_{\pi + \sigma_+} = (C_{Fm11}^{Fm-1} C_{11,10}^{11} \alpha_1(\omega_l) + C_{Fm21}^{Fm-1} C_{11,10}^{21} \alpha_2(\omega_l))/2 \quad (4.11) $$

$$ \langle m + 1 | \hat{\alpha} | m \rangle |_{\pi + \sigma_+} = (C_{Fm1-1}^{Fm+1} C_{10,1-1}^{1-1} \alpha_1(\omega_l) + C_{Fm2-1}^{Fm+1} C_{10,1-1}^{2-1} \alpha_2(\omega_l))/2 \quad (4.12) $$

Tensor polarizability is then completely defined from equations 4.7, 4.9, 4.11, and 4.12. Note that a factor 1/2 has to be added to the diagonal elements when summing
the $\pi, \pi$ and $\sigma_+, \sigma_+$ contributions, which comes from the $1/\sqrt{2}$ normalization of the unit polarization of the electric field.

This shows that the polarizability matrix elements can be completely defined by Clebsch-Gordan coefficients and three parameters, which are commonly defined as the scalar $\alpha_S$, vector $\alpha_V$ and tensor $\alpha_T$ irreducible components of the polarizability, and usually obtained by rescaling $\alpha_0, \alpha_1, \alpha_2$ [MOR86; Shi+15]:

$$ 
\begin{align} 
\alpha_S &= \alpha_0 \frac{1}{\sqrt{3}} \nonumber \\ 
\alpha_V &= \alpha_1 \sqrt{\frac{2F}{F+1}} \tag{4.13} \\ 
\alpha_T &= \alpha_2 \sqrt{\frac{2F(2F-1)}{3(F+1)(2F+3)}} \nonumber 
\end{align} 
$$

One has to know the transition strength $D_{\phi\phi'}^2 = |\langle nF | |D| | n'F' \rangle|^2$ to fully compute the dynamic polarizability. Extensive research on the characterization of the transition rates for atomic spectra has yielded an affluent database on atomic spectroscopy, for instance [SN10] and more recently [HS22] regarding strontium 87. It is then possible to determine the transition strength element from the transition angular frequencies $\omega_{\phi\phi'}$, oscillator strengths $f_{\phi\phi'}$ [Lud08] and transition rates $\Gamma_{\phi\phi'}$:

$$ 
\begin{align} 
\Gamma_{\phi\phi'} &= \frac{\omega_{\phi\phi'}^3}{3\pi\hbar\varepsilon_0 c^3} D_{\phi\phi'}^2 \nonumber \\ 
f_{\phi\phi'} &= \frac{2}{3} \omega_{\phi\phi'}^3 \frac{m}{\hbar e^2} D_{\phi\phi'}^2 \tag{4.14} 
\end{align} 
$$

where $m$ and $e$ are the electron mass and charge.

### 4.1.2 Scattering rate

The scattering of photons can be modeled by the absorption of a photon from the laser field with angular frequency $\omega_L$, and the spontaneous emission of another photon. This two photon process connects to the continuum energy many-fold, which can be described by the Fermi golden rule to derive the scattering rate: