$$
\Gamma_{m_i \to m_j} = \frac{2\pi}{\hbar} |\langle m_j | \hat{W} | m_i \rangle|^2 \rho(E = \hbar\omega_L) \quad (4.15)
$$

where $\hat{W}$ is the coupling between the ground states $|m_i\rangle$ and $|m_j\rangle$ with energies
$E_{m_j} = E_{m_i}$, and $\rho(E)$ the density of states of energy $E$. If the excited state connecting
$|m_i\rangle$ and $|m_j\rangle$ is weakly populated, then the two photon effective coupling $\hat{W}$ reads in
second order perturbation theory:

$$
\hat{W} = \frac{1}{2} |\boldsymbol{E}_L||\boldsymbol{E}_v| \frac{1}{\hbar} \sum_{|\phi'\rangle} \frac{\boldsymbol{\epsilon}_v^* \cdot \hat{\boldsymbol{D}} |\phi'\rangle \langle \phi'| \boldsymbol{\epsilon}_L \cdot \hat{\boldsymbol{D}}}{\omega_{\phi'} - \omega_L} + \frac{\boldsymbol{\epsilon}_v \cdot \hat{\boldsymbol{D}} |\phi'\rangle \langle \phi'| \boldsymbol{\epsilon}_L^* \cdot \hat{\boldsymbol{D}}}{\omega_{\phi'} + \omega_L} \quad (4.16)
$$

Here, $\mathbf{E}_v(\mathbf{E}_L)$ stands for the electric field of the vacuum (laser) of polarization $\boldsymbol{\epsilon}_v$ ($\boldsymbol{\epsilon}_L$). Note that here we consider that the polarization of the excitation is $\pi$, so that the polarization of the spontaneous photon is well defined for a considered arrival state $m_j$. Injecting the effective two photon coupling into Fermi golden rule finally gives, after neglecting the anti-resonant term:

$$
\Gamma_{m_i \to m_j} = \frac{3\pi\epsilon_0\hbar c^3 |\mathbf{E}_L|^2 \Gamma_{nat}^2}{\omega^3 4 \hbar^2} \left| \sum_{|\phi'\rangle} \frac{(2F' + 1)}{\omega_{\phi'} - \omega_L} \begin{pmatrix} F & 1 & F' \\ m_j & m' - m_j & -m' \end{pmatrix} \begin{pmatrix} F & 1 & F' \\ m_i & 0 & -m' \end{pmatrix} \right|^2 \quad (4.17)
$$

which can be rewritten with the Clebsch-Gordan coefficients

$$
\Gamma_{m_i \to m_j} = \frac{3\pi\epsilon_0\hbar c^3 |\mathbf{E}_L|^2 \Gamma_{nat}^2}{\omega^3 4 \hbar^2} \left| \sum_{|\phi'\rangle} \frac{C_{Fm_j,1m'-m_j}^{F'm'} C_{Fm_i,10}^{F'm'}}{\omega_{\phi'} - \omega_L} \right|^2 \quad (4.18)
$$

This result for the scattering rate associated with small population of the excited
state is actually known as the Kramers-Heisenberg formula [RF99]. In our case, the
polarisation of the laser electric field is pure $\pi$, i.e. $\boldsymbol{\epsilon}_L = \mathbf{u}_0$, and $\Gamma_{nat}$ is the inverse life
time of the excited states $|\phi'\rangle$.

The decay to the final state depends on the polarisation of the spontaneous emitted
photon, such that the total scattering rate $\Gamma_i = \sum_j \Gamma_{i \to j}$ from the initial spin state $|m_i\rangle$
can be decomposed over the scattering to the three possible final states $|m_i\rangle$ ($|m_i \pm 1\rangle$)
for a $\pi$ ($\sigma_\mp$) emitted photon. Scattering rate of $m_F = +3/2$ for each polarization of the
<figure><img src="image_38.png" /></figure>

Figure 4.1: Calculated scattering rate as a function of the light frequency spanned within the hyperfine structure F=7/2,9/2,11/2 of the $^1S_0 \leftrightarrow ^3P_1$ transition. The scattering rate is calculated using experimental light parameters, which is a 6 mW gaussian laser beam with a waist of 180 µm and polarization π. Left: scattering rate from $m_F = +3/2$, for each polarization of the spontaneously emitted photon, in dashed lines. The total scattering rate is plotted in solid black line. Right: total scattering rate for each zeeman sub-level. Here, only positive spin states are represented because $\Gamma_{tot}$ is symmetric in $m_F$. On both figures, the red vertical solid lines are the hyperfine resonances.

spontaneous photon is plotted on figure 4.1, left, as well as the resulting total scattering
rate. In this figure, the calculation is made with a π polarized light excitation, and the
intercombination line $^1S_0 \leftrightarrow ^3P_1$, $F=9/2$ only is considered for the calculation.

As expected, the spin dependent scattering rate is maximum closer to resonance.
There exist detuning values δ such that the scattering process towards a specific final
state, i.e. emitted photon polarisation, is canceled, as it can be seen when the scattering
rate reaches zero. The sum over the hyperfine states F' = 7/2, F' = 9/2, and F' = 11/2,
accounts for the different paths through those excited states leading from the initial
state to the final state, and the probability for a scattering event to happen results from
the interference between these different possible paths. This can be seen in equation
(4.18) from the sum over the excited states of $C_jC_i/\delta$ which can be zero for specific $\delta$, depending on the coupling strengths $C_jC_i$, i.e. on $|m_j\rangle$ and $|m_i\rangle$. However, this doesn't mean that the light scattering is canceled. Indeed, the total light scattering accounts for all possible scattering paths.

Since the polarization of the excitation is well defined, the final states associated with scattering are distinguishable and there are no interferences involved into the total scattering from one $m_F$, i.e. $\Gamma_i = \sum_j \Gamma_{i \to j}$. The total light scattering from each spin state is plotted on figure 4.1, right, and it can be seen that indeed it is never zero for any frequency of the light excitation.

### 4.1.3 Tensor light shift associated with the intercombination line

Tensor light shifts can be easily computed numerically from equation 4.5, given that the transition strength defined in equation 4.14 is known. If the transition is not closed, e.g. if the lower state has fine or hyperfine structure, then the branching ratios of the transitions have to be taken into account [Lud08; Shi+15]. Regarding the intercombination line of $^{87}$Sr, the excited state $^3P_1$ can decay only to the ground state $^1S_0$. Hence, the radiative decay rate of the excited state $\Gamma_{^3P_1}$ gives the $L-L'$ transition strength:

$$ \Gamma_{^3P_1} = \frac{\omega_0^3}{3\pi\hbar\varepsilon_0 c^3} D_{L,L'}^2 \quad (4.19) $$

where $L=0$ and $L'=1$. Here, the hyperfine structure is not accounted for, and the branching ratios are needed to write the transition strength in the hyperfine coupled basis [MS99; Wal21]:

$$ D_{F,F'}^2 = (2L'+1)(2F+1)(2F'+1)(2J+1)(2J'+1) \begin{Bmatrix} J' & 1 & J \\ L & S & L' \end{Bmatrix} \begin{Bmatrix} F' & 1 & F \\ J & I & J' \end{Bmatrix} D_{L,L'}^2 \quad (4.20) $$

and it finally reads:

$$ |\langle F | |D| |F' \rangle|^2 = \Gamma_{^3P_1} \frac{3\pi\hbar\varepsilon_0 c^3}{\omega_0^3} (2L'+1)(2F+1)(2F'+1)(2J+1)(2J'+1) \begin{Bmatrix} J' & 1 & J \\ L & S & L' \end{Bmatrix} \begin{Bmatrix} F' & 1 & F \\ J & I & J' \end{Bmatrix} \quad (4.21) $$
<figure><img src="image_39.png" /></figure>

Figure 4.2: (a) Calculated light shifts as a function of the light frequency spanned within the hyperfine structure F=7/2,9/2,11/2 of the $ ^1S_0 \leftrightarrow ^3P_1 $ transition, for each spin state of the ground state manifold. The $m_F$ spin states are noted above the corresponding solid lines. The spin dependent light shift is calculated using experimental light parameters, which is a 6 mW gaussian laser beam with a waist of 180 µm and polarization $\pi$. (b) Degeneracy lift of the $m_F \leftrightarrow m_F \pm 1$ raman resonances within the ground state manifold, computed as $U_{dip}(m_F) - U_{dip}(m_F \pm 1)$. The corresponding spin states are noted above the curves. On both figures, the red vertical solid lines are the hyperfine resonances.

Now, the tensor polarizability associated with the intercombination line can be computed for a defined polarization of the electric field, from equations 4.5, 4.6 and 4.21. Figure 4.2 (a) shows a numerical computation of spin dependent light shifts within the ground state manifold $^1S_0$ with $\pi$ light, with respect to the detuning to the $^1S_0 \leftrightarrow ^3P_1$, $F=9/2$ transition, within second order perturbation theory. Because the approximation $\delta \gg \Gamma_{^3P_1}$ has been made, the calculated light shifts diverge close to the resonance with each hyperfine state. However, it is interesting to see that for every spin state, the dipole force goes from attractive to positive, when the light frequency goes from the $F=9/2$ to $F=11/2$ resonance. Moreover, a light detuned at approximately -700 MHz from $F=9/2$ highlights the spin dependence of the dipole potential, where half the spins are attracted and the other half expelled by the potential. Practically, figure 4.2 (a) shows that with those parameters, it is possible to expell all spin states
but +9/2 and -9/2 with $\omega_l - \omega_{9/2} \simeq -1.1$ GHz, effectively preparing a SU(2) $^{87}$Sr gas, similarly to [Son+20]. The opposite can also be realized by setting the light frequency at $\omega_l - \omega_{9/2} \simeq +1$ GHz, preparing a SU(8) gas. The resulting energy splits between the $m_F$ ground states are plotted with respect to the angular frequency of the light on figure 4.2 (b). The energy differences are rather uniform when the light frequency is tuned in the middle of two hyperfine states, while close to resonance the spectral resolution diverges for all the $|m_F\rangle \leftrightarrow |m_F + 1\rangle$ opened gaps.

Most importantly, we see here that because of the spin-dependent light shift, the ground state manifold acquires spectrally resolved $|m_F\rangle \leftrightarrow |m_F \pm 1\rangle$ transitions. With only a scalar or linear dependence on the Zeeman sub-levels, all the transitions would be resonant. This means that only thanks to the quadratic spin dependence of the tensor light shift, two photon Raman transitions can be used to selectively couple two ground spins states. For pure $\pi$ polarized light, this lifted degeneracy can be easily written as :

$$ \alpha(m_F + 1) - \alpha(m_F) = \frac{-3\alpha_T}{F(2F - 1)}(2m_F + 1) \quad (4.22) $$

according to equations 4.8 and 4.13, where $\alpha(m_F) = \langle m_F | \hat{\alpha} | m_F \rangle$. Hence, a quadratic spin dependent light shift results in linear spin dependent $|m_F\rangle \leftrightarrow |m_F \pm 1\rangle$ Raman resonances. Increasing the spectral resolution of the Raman resonances by tuning the light close to resonance also increases the scattering rate as shown in figure 4.1, not only heating the atoms but also enhancing spin relaxation and decoherence. It should be noticed that close to the $F = 9/2$ resonance, stretched states have highest polarizability, resulting in maximal light scattering and energy shifts for these spin states, while close to the $F = 11/2$ resonance, the minimal magnetic spins, i.e. minimal $|m_F|$, are the most shifted and scattering spin states. The competition between light shift and light scattering thus has non trivial spin dependence. This competition can be looked at by comparing the energy difference between two successive spin states, to the mean of the scattering rate of these two states. The ratio of these two quantities is plotted in figure 4.3, and it shows that optimal light frequency tuning might exist to maximize the degeneracy light while minimizing scattering rate, and this result highly depends
<figure><img src="image_40.png" /><figcaption>Figure 4.3: Calculated degeneracy lift of Raman resonances within the $^1S_0$ manifold, over spin dependent scattering rate, computed as the mean over the two connected spin states, as a function of the frequency of near resonant light field spanned within the hyperfine structure of the $^3P_1$ state. The red vertical solid lines are the hyperfine resonances. This calculation is made with our experimental parameters</figcaption></figure>

on the spin-spin transitions. This has non negligible consequences on the choices of experimental schemes to realize spin swaps by efficient adiabatic passages.

The predictions for the tensor light shift associated with the intercombination line of $^{87}$Sr are well characterized, as well as for the associated scattering rate. We are now interested in the experimental realization and measurements of the tensor light shifts associated with the intercombination line of $^{87}$Sr.
## 4.2 Engineering the degeneracy lift of $^{87}$Sr ground state manifold

### 4.2.1 Experimental design

To realize spin dependent light shifts on the experiment, we use a dedicated laser setup presented in figure 4.4. The laser head is made of a diode with extended cavity, and it is beat-locked onto our Master red laser. By setting the frequency for the beat signal, it is possible to tune the frequency of the laser beam within a 1 GHz wide frequency window centered on the $F = 9/2$ hyperfine resonance. For the experiment, we use two double pass accousto-optical modulators, such that we can tune independently the frequencies of two laser beams. The RF frequencies for the two AOMs derive from two direct digital synthetizers (DDS) which share the same clock and are therefore phase coherent. The first one is used to realize the spin dependent light shift, which I will call *dressing beam* for simplicity. The second beam is used to flip the spins with a two photon process together with the first beam, and I will call this laser beam *Raman beam*. Before interacting with the atoms, the dressing and Raman beams are superimposed with a beam-splitter, and the outgoing electric fields of the two beams are orthogonal. The quantization axis can be chosen collinear with the dressing beam wave vector, such that in this frame the electric field of the Raman laser is $\sigma_+ + \sigma_-$ polarized. To do this, we set the power of the Raman beam one thousand time weaker than the dressing beam. This way, it is possible to realize well defined $|m_F\rangle \leftrightarrow |m_F \pm 1\rangle$ two photon Raman transitions with one $\pi$ photon from the dressing beam, and one $\sigma_\pm$ photon from the Raman beam. The Raman beam has the same waist than the dressing beam, i.e. 180 $\mu$m, at the location of the atoms. In this case, the relative intensities are equal to the relative powers.

### 4.2.2 Measurement of the spin dependent light shifts

We characterize the spin-dependent light shift by measuring the energy difference between two consecutive spins of the ground state manifold. This is done with a Raman spectroscopy after the tensor light shift is turned on, and our measurement shows that the resonances are separated by a quadratic light shift.
<figure><img src="image_41.png" /><figcaption>Figure 4.4: Dedicated laser setup for the spin dependent light shift and spin selective spin flips. This scheme is simplified for easy reading, only the main components are presented. Two accousto-optical modulators (AOM) in double passe configuration are used to independently tune the frequencies of the dressing beam and the Raman beam. A third path is used for beat lock onto the Master laser. The Raman and dressing beam are finally recombined before the vacuum chamber.</figcaption></figure>

The measurement proceeds as follows. The gas is prepared in a spin polarized state, as presented in section 3.2.3 of chapter 3, after forced evaporation. Then, the homogeneous magnetic field is slowly rotated such that it is aligned with the linear polarization of the electric field of the dressing beam. This way, the polarization of the light is well defined and it is purely $\pi$. Then, the dressing beam is slowly turned to its maximum power $P = 6$ mW within 2 ms. Now, the degeneracy of the ground state is lifted by the quadratic light shift. We then rely on Raman spectroscopy to find the $|m_F\rangle \leftrightarrow |m_F \pm 1\rangle$ resonances by pulsing the Raman laser beam at different frequencies. The power of the Raman beam is set as low as possible, here 5 $\mu$W, in order to narrow the measurement of the resonances, and the time of pulse at fixed frequency long enough, here 10 ms, to maximize the frequency resolution of the Raman spectroscopy.

The result of the experiment is shown on figure 4.5. Atoms are initially in $m_F = -5/2$, and the Raman detuning is scanned to find the $|-5/2\rangle \leftrightarrow |-3/2\rangle$ and $|-5/2\rangle \leftrightarrow |3/2\rangle$.
<figure><img src="image_42.png" /><figcaption>Figure 4.5: Left: Raman spectroscopy of the $|-5/2\rangle \leftrightarrow |-5/2 \pm 1\rangle$ resonances, after degeneracy lift with quadratic light shift. Atoms are initially in $m_F = -5/2$, and the population transfer to $m_F = -7/2$ and $m_F = -3/2$ are measured with spin-dependent momentum transfer (see chapter 3). The solid lines are fits of each resonance with Lorentz functions (see equation 4.23). In this experiment, the dressing beam has measured power 6 mW and waist $185 \pm 5\mu$m at the location of the polarized fermi sea, and the Raman beam has measured power 5 $\mu$W and same waist. The filled regions correspond to the resonances prediction with experimental parameters and associated uncertainty on the waist $185 \pm 5\mu$m. Right: calculated degeneracy lift of the ground state manifold with experimental parameters, and the two photon Raman transitions correspond to the ones measured.</figcaption></figure>

$|-7/2\rangle$ resonances. The populations of the coupled spin states after the pulse are measured with the spin dependent momentum transfer technique presented on chapter 3. The populations are fitted with Lorentz functions:

$$L(f, A, f_0, \Gamma) = \frac{A}{1 + \left(\frac{f - f_0}{\Gamma/2}\right)^2} \qquad (4.23)$$

and their centers $f_0$ is found at $-11.5 \pm 0.3$ kHz for the $|-5/2\rangle \leftrightarrow |-3/2\rangle$ resonance and $-16.5 \pm 0.3$ kHz for the $|-5/2\rangle \leftrightarrow |-7/2\rangle$ resonance. This measurement confirms that the energy difference between two consecutive spin states is as expected for a quadratic light shift, see equation 4.22.
This result is compared to numerical prediction with experimental parameters, in-
cluding the uncertainty on the waist measurement 180 ± 5 µm, which is shown as the
filled region on the same figure, for each resonance. The computation predicts compati-
ble resonances, -11.2±0.6 kHz for the |−5/2⟩ ↔ |−3/2⟩ transition and -16.80±0.9 kHz
for the |−5/2⟩ ↔ |−7/2⟩ transition. The uncertainty comes only from the measurement
of waist of the dressing beam. The degeneracy lift of the ground state manifold, numeri-
cally calculated, is plotted on figure 4.5, right, where we can see both the quadratic light
shift, as well as the spin dependence of the energy gaps between two consecutive spin
states. The whole spectrum is numerically computed with experimental parameters, as
it can easily be inferred from the measurement of α<sub>T</sub>.

4.2.3 Measurement of the scattering rate

We are interested in the light scattering associated with the spin dependent light shift,
as it is a limitation to the conservative spin manipulations. On the experiment, the
scattering rate is characterized by measuring the heating rate associated with the spin
dependent light shift. After preparing a cold gas of strontium with ten spin compo-
nents, the dipole trap is recompressed to its maximum depth U<sub>dip</sub> ≃ 65 µK and either
the dressing beam or the Raman are pulsed, with detuning Δ = −400 MHz from the
<sup>3</sup>P<sub>1</sub>, F = 9/2 hyperfine state, and with respective powers P<sub>D</sub> = 6 mW and P<sub>R</sub> = 6 mW.
Atoms acquire kT of heating energy during the light pulse, and the deep trap with
U<sub>dip</sub> ≫ kT ensures that they don't leave the trap, so that all the energy acquired by
the uncoherent scattering events is maintained into the gas. For each duration of pulse,
the temperature of the gas is then measured after a time of flight, and the result of
the experiment is shown on figure 4.6, left. It is compared to a witness heating rate
measurement, which is done by realizing the same sequence without ever turning on the
dressing nore the Raman beam, and that I call here *residual heating*. We suppose that
this heating is due to mechanical vibrations of the optical dipole trap in which atoms
are trapped during the measurement. Right figure shows the heating slope after remov-
ing the residual heating signal, and the heating rate associated with the spin dependent
light shift is measured between 220 ± 10 nK.s<sup>-1</sup>.mW<sup>-1</sup> and 255 ± 12 nK.s<sup>-1</sup>.mW<sup>-1</sup> from
the dressing beam and Raman beam respectively, assuming the same waist. Note that
<figure><img src="image_43.png" /><figcaption>Figure 4.6: Measurement of the heating rate associated with the spin dependent light shift. For this measure, the detuning is set at $\Delta = -400$ MHz from the $^3P_1, F = 9/2$ hyperfine state. The heating rate of the Raman (empty squares) and dressing (empty dots) beams are compared to the residual heating (black dots), and to the same measurement after spectral mode filtering of the laser (blue filled dots). Left figure shows the straight measure of the gas temperature, with $P_R = 6$ mW, $P_D = 6$ mW and $P_{cav} = 2.5$ mW. Right figure shows the same data normalized to the respective beam powers, and the slope of the residual heating has been removed. The prediction (hatched region) is computed according to equation 4.26 with waist $185 \pm 5$ µm.</figcaption></figure>

the uncertainty doesn't take into account the power and waist measurements of the laser beams, only the fit covariance is given. Since the trap is recompressed, atoms are hot and dense enough such that the collision rate is much bigger than the scattering rate, so we can assume that the gas is always at equilibrium, hence the scattering rate can be inferred from heating rate as follows.

When an atom scatters photons, a two step process happens. The atom absorbs the photon of momentum $\hbar\vec{k}_1$ from the coherent laser beam, then randomly emits another photon of momentum $\hbar\vec{k}_2$ in any direction. The exchange of momenta between the atoms and the photons after a scattering event can be written as $\delta\vec{p} = \hbar\vec{k}_2 - \hbar\vec{k}_1$.
Assuming that the trapped 3D gas is at equilibrium $<\vec{\delta p}>$= 0 with temperature $T$, and no other process can be responsible for energy variations, then the time evolution of the kinetic energy of the gas $E_K = 3k_B T/2$, where $k_B$ is the Boltzmann constant, is only due to the exchange of momenta $\vec{\delta p}$ happening at rate $\gamma_{scatt}$, and it reads

$$ \frac{dE_K}{dt} = \gamma_{scatt} \frac{\langle \delta p^2 \rangle}{2m} \Leftrightarrow \frac{3}{2} \frac{dk_B T}{dt} = \gamma_{scatt} \frac{\langle \delta p^2 \rangle}{2m} \quad (4.24) $$

where $m$ is the mass of an atom. The spontaneous emission of a photon with random momentum $<\vec{k}_{se}>$= 0 implies that the absorbtion-emission processes are independent $<\vec{k}_1\vec{k}_2>=$ $<\vec{k}_1>\langle\vec{k}_2>=$ 0, such that $<\vec{\delta p}^2> = 2(\hbar k_R)^2$ where $\hbar k_R$ is the recoil momentum of a 689 nm photon. Therefore, the heating rate associated with the light scattering reads:

$$ \frac{dk_B T}{dt} = \frac{2}{3} \gamma_{scatt} E_R \quad (4.25) $$

where $E_R = \hbar^2 k^2/2m$ is the recoil energy of a photon absorbed or emitted by a strontium atom. Finally the scattering rate can be deduced from the heating rate that is experimentally measured:

$$ \gamma_{scat} = \frac{3}{2} \frac{d}{dt} \left( \frac{k_B T}{E_R} \right) \quad (4.26) $$

From the measured heating, we infer a scattering rate $1.4 \pm 0.06 \text{ s}^{-1}$ for the Raman beam and $1.7 \pm 0.08 \text{ s}^{-1}$ for the dressing beam, per 1 mW, i.e. approximately $2.9 \text{ W/cm}^2$. With the experimental parameters, the prediction of the scattering rate from equation 4.18, averaged over the 10 spin states, gives $<\gamma_{scatt}>_{m_F} \simeq 0.54 \text{ s}^{-1}$, which is three times lower than our measurement. We explain our measurement of enlarged scattering by spectral imperfections of the diode generating the laser beam, resulting in amplified spontaneous emission (ASE) [SGS13].

To circumvent this issue, we added a Fabry-Perot cavity to our setup. After filtering the spectral mode of the laser diode, we measure a heating rate lowered to $60 \pm 20 \text{ nK.s}^{-1}\text{mW}^{-1}$, as shown on 4.6, corresponding to scattering rate $0.39 \pm 0.12 \text{ s}^{-1}\text{mW}^{-1}$. This result is in agreement with the prediction at the upper limit of
the uncertainty, which embeds only the fit covariance. This measurement is enough
to show that the spectral mode of the laser is broadened, and after cavity filtering,
the scattering rate associated with the tensor light shift is properly understood and
predicted. Note that the cavity was not further used because of instability issues, and
is presently being upgraded for persistent use.

4.2.4 Spin depolarization

Our setup to engineer the spin dependent light shifts involves spin dependent couplings, notably the dressing beam and the quantization field. Hence, attention has to be paid to possible second order spin-spin couplings, which would allow spin rotations, and in the case of a polarized Fermi sea, result in depolarization. For instance, the electric-dipole operator resulting from the interaction of atoms with a dressing beam with not pure π polarization allows $|m_F\rangle \leftrightarrow |m_F \pm 1\rangle$ couplings, as shown in equations 4.11 and 4.12. In this case, the spin eigenbasis is SU(N) rotated and the definition of the new eigenstates is not trivial, which is unwanted for proper understanding of our schemes.

To avoid these complications, it is possible to physically rotate the magnetic field
such that the polarization of the electric field of dressing beam is well defined and
purely π, σ+ or σ−. Is it particularly true if the dressing beam is linearly polarized
and the magnetic field aligned with the direction of the polarization and in this case,
the eigenstates are the same with or without dressing beam. In our experiment, we
optimized the alignement of the magnetic field with the π polarized electric field as
follows: we prepare a polarized Fermi sea, and for each rotation of the quantization
field the dressing beam is pulsed at maximum power, then we measure the number of
atoms remaining in the initial spin state.

4.3 Selective global manipulation of the spin states

In this section, I present our scheme to selectively and coherently flip the ground spin states of strontium 87. After lifting the degeneracy of the ground state manifold with a quadratic light shift, we take advantage of the resolved $|m_F\rangle \leftrightarrow |m_F \pm 1\rangle$ transitions
to engineer selective adiabatic passages. With minimal spontaneous emission, we flip
the spin of the atoms from their initial state to a well determined final spin state. With
a measurement of the spin populations we infer 80% efficiency for the preliminary
experiments. We finally compare our results with simulations.

**4.3.1 Scheme overview**

In our setup, the spin flips are induced by two photon adiabatic Raman passages. One photon of the Raman process comes from the dressing beam, with $\pi$ polarization, while the second photon comes from the Raman beam, with either $\sigma_+$ or $\sigma_-$ polarization. Raman transitions can be engineered either with $\pi + \sigma_-$ photons, or with $\pi + \sigma_+$ photons, as presented on figure 4.7. Because of the quadratic degeneracy lift and selection rules, only one of the two polarizations of the Raman beam crosses a Raman resonance for one Raman detuning $\delta = \omega_D - \omega_R$. Hence, for each Raman transition, it is possible to choose either $\pi + \sigma_+$ or $\pi + \sigma_-$ two photon transition. Since we usually work with negative spin states because of our optical pumping scheme (see chapter 3), and since Clebsch-Gordan coefficients associated with $\sigma_-$ transitions are stronger for negative spin states, we focus on the $\pi + \sigma_-$ case only. Indeed, stronger Rabi couplings allow faster passages, which means that the time needed to engineer spin flips is lowered compared to typical time for light scattering, which is strongly favorable in presence of noise. In this case, the Raman resonances are crossed for $\delta > 0$ for negative spin states and $\delta < 0$ for positive spin states, as shown on figure 4.7.

**4.3.2 Spectrum of the Raman transitions in the $^1S_0$ manifold**

Now, I will discuss adiabatic passages in $\Lambda$ scheme in the large spin states manifold. Let's consider the ground state manifold of $^87\text{Sr}$ and its 10 spin states, coupled to two photon fields $\mathbf{E}_\pi, \mathbf{E}_{\sigma_-}$ with $\pi$ and $\sigma_-$ polarizations. To write the associated Hamiltonian, it is conveniant to define the dressed state basis $|m_F + n\rangle \equiv |m_F + n, N_\pi - n, N_\sigma + n\rangle$, where $N_\pi$ ($N_\sigma$) is the number of photons in the field $\mathbf{E}_\pi$ ($\mathbf{E}_{\sigma_-}$), and $n$ is integer. Each spin state $|m_F\rangle$ is connected to $|m_F \pm 1\rangle$ states by the $\pi + \sigma_-$ two photon Raman process, with associated Raman detuning $\delta = \omega_\pi - \omega_{\sigma_-}$. Hence, in the absence of the quadratic degeneracy lift, each dressed state $|m_F + n\rangle$ has energy $n\delta$.
<figure><img src="image_44.png" /><figcaption>Figure 4.7: Scheme of Raman two photon couplings within the spin states manifold after quadratic degeneracy lift, within $m_F < 0$ spin states. R and D letters above Rabi frequencies stand for Raman beam, $\sigma_+ + \sigma_-$ polarized, and Dressing beam, $\pi$ polarized. Left: positive detuning $\delta = \omega_D - \omega_R > 0$, such that negative spin states are coupled with $\pi + \sigma_-$ photons. Right: negative detuning $\delta = \omega_D - \omega_R < 0$, such that negative spin states are coupled with $\pi + \sigma_+$ photons.</figcaption></figure>

Then, the off-diagonal couplings of the dressed Hamiltonian are effective two photon Raman couplings which can be computed from second order perturbation theory:

$$ \frac{\hbar \Omega_{m_F 1}^{m_F 2}}{2} = |\mathbf{E}_\pi| |\mathbf{E}_{\sigma_-}| \sum_{|e\rangle} \frac{\langle m_F 2 | \boldsymbol{\epsilon}_\pi^* \cdot \hat{\mathbf{D}} | e \rangle \langle e | \boldsymbol{\epsilon}_{\sigma_-} \cdot \hat{\mathbf{D}} | m_F 1 \rangle}{\Delta_e} \quad (4.27) $$

where $\Delta_e \simeq \omega_e - \omega_\pi \simeq \omega_e - \omega_{\sigma_-}$. This approximation holds since the Raman detuning is in the order of the kHz, while the detuning from the excited state is in the MHz range. After applying the Wigner-Eckart theorem, the summed terms $\mu_{m_F,e} = \langle e | \varepsilon \cdot \hat{\mathbf{D}} | m_F \rangle$ can be written as:

$$ \mu_{m_F,e} = (-1)^{1+L'+S+J'+J'+I-m'_F} \begin{pmatrix} F & 1 & F' \\ m_F & q & -m'_F \end{pmatrix} D_{F,F'} \quad (4.28) $$

where ' symbol stands for the excited state $|e\rangle$, and $D_{F,F'}$ has been defined in
equation 4.21 with branching ratios. The effective two photon Raman couplings are then completely defined for any $|m_F\rangle \leftrightarrow |m_F \pm 1\rangle$ transition.

Taking into account the quadratic degeneracy lift of the ground state manifold by $E_\pi$ as a diagonal operator $\langle m_F | \hat{H}_q | m_F \rangle = -1/4 \langle m_F | \hat{\alpha} | m_F \rangle |E_\pi|^2$, where $\langle m_F | \hat{\alpha} | m_F \rangle$ is defined in equation 4.8, the effective tridiagonal hamiltonian in the dressed state basis finally reads:

$$ \hat{H}(\delta) = \frac{\hbar}{2} \begin{pmatrix} -9\delta & \Omega_{-9/2}^{-7/2} & 0 & \dots & 0 & 0 & 0 \\ \Omega_{-9/2}^{-7/2*} & -7\delta & \Omega_{-7/2}^{-5/2} & \dots & 0 & 0 & 0 \\ 0 & \Omega_{-7/2}^{-5/2*} & -5\delta & \dots & 0 & 0 & 0 \\ \vdots & \vdots & \ddots & \ddots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & \dots & 5\delta & \Omega_{5/2}^{7/2} & 0 \\ 0 & 0 & 0 & \dots & \Omega_{5/2}^{7/2*} & 7\delta & \Omega_{7/2}^{9/2} \\ 0 & 0 & 0 & \dots & 0 & \Omega_{7/2}^{9/2*} & 9\delta \end{pmatrix} + \hat{H}_q \quad (4.29) $$

This approach to write the effective hamiltonian gives the same result as obtained with the standard components of the tensor polarizability in the case of an electric field with polarizations $\pi + \sigma_-$ treated in section 4.1.1, with amplitudes $|E_\pi|$ and $|E_{\sigma_-}|$. The spectrum of hamiltonian 4.29, giving the energy crossings and hence light induced spin-spin transitions, can be obtained by direct diagonalization. The result of the diagonalization, is plotted in figure 4.8, and it is calculated with experimental parameters (see caption).

The spectrum within $m_F < 0$ manifold only is shown here for better visibility. This spectrum shows that the energy crossings between $|m_F\rangle$ and $|m_F \pm 1\rangle$ states are avoided, authorizing two photon transitions, and those are the transitions that we use to selectively flip the spins. The insets of figure 4.8 show that crossings between $|m_F\rangle$ and $|m_F \pm 2\rangle$ states are also avoided, authorizing transitions involving the exchange of four photons. To make sure that the four photon transitions do not damage the selectivity of the spin flips that we realize with the two photon transitions, it is relevant to investigate on the relative strength of the two process.
<figure><img src="image_45.png" /></figure>

Figure 4.8: Spectrum of Hamiltonian defined in equation [4.29] with $\pi + \sigma_{-}$ two photon couplings, with respect to Raman detuning $\delta$. For this computation, experimental parameters are used: detuning of the dressing beam $\Delta/2\pi = -700$ MHz from the hyperfine state $^{3}P_{1}$, $F = 9/2$ and intensity $I_{D} \simeq 9.3$ W/cm$^{2}$, and much lower intensity of the Raman beam $I_{R} \simeq 5.1$ mW/cm$^{2}$. Top: eigenergies with respect to Raman detuning $\delta = \omega_{\pi} - \omega_{\sigma_{-}}$, where the solid lines are the eigenenergies, while the dashed lines indicate the energy of the dressed states. Two photon transitions are annotated in red, and four photon transitions are annotated in grey. Upper insets zoom in respective four photon transitions. Bottom: states mixing, defined in [4.30] as a function of the two photon Raman detuning $\delta$. The solid (dashed) line indicates two (four) photon transitions, and the respective Rabi couplings, defined in equation [4.27] ([4.32]) are given by the half width at $1/\sqrt{2}$ of the states mixing.

The strength of the transitions, given by the Rabi frequency $\Omega$, can be graphically evaluated by the size of the gap at the avoided crossings, and figure 4.8, top, indicate
the two photon transitions are stronger. It is also convenient to look at the states mixing [MS99]:

$$ \sin(2\theta) = \frac{\Omega^2}{\sqrt{\Omega^2 + (\delta - \omega_R)^2}} \qquad (4.30) $$

as a function of the detuning $\delta - \omega_R$ with respect to the resonance $\omega_R$ of the considered transition. The states mixing is shown on figure 4.8, bottom, for every resonances. Here, $\theta$ is the mixing angle [MS99, FIM05], which defines the coherent superposition of the atom $|\Psi\rangle$ over the two spin states $|m_F\rangle$ and $|m'_F\rangle$ involved into the transition, which can read:

$$ |\Psi\rangle = \cos(\theta) |m_F\rangle + \sin(\theta) |m'_F\rangle \qquad (4.31) $$

At resonance $\delta - \omega_R = 0$, the states mixing is maximal $\sin(2\theta) = 1$, i.e. $|\Psi\rangle = (|m_F\rangle + |m'_F\rangle)/\sqrt{2}$. The half-width of the resonances at $\sin(2\theta)/\sqrt{2}$, for both two and four photons transitions, is then given by the Rabi frequency $\Omega$, and 4.8, bottom, illustrates again that with the experimental parameters, the two photon transitions are much stronger than the four photon transitions.

From the dressed states point of view, which is schemed on figure 4.9, the four photon transitions are very similar to standard two photon Raman transitions. At resonance, the effective coupling between $|m_F\rangle$ and $|m_F \pm 2\rangle$ is simply $\Omega_4 = \Omega_2^*\Omega_2/2\Delta$, where $\Delta = \delta - (\varepsilon_{m_{F+1}} - \varepsilon_{m_F})$ is the detuning with respect to the intermediate state, which is actually the detuning to the two photon transition. Hence, the effective four photon couplings read:

$$ \Omega_{m_F}^{m_F+2} = \frac{\Omega_{m_F}^{m_F+1} \Omega_{m_F+1}^{m_F+2}}{2\Delta} \qquad (4.32) $$

At resonance with the four photon transition, $\delta = (\varepsilon_{m_{F+2}} - \varepsilon_{m_F})/2$. Hence, $\Delta$ is simply the difference between with the four photon and two photon resonances. It is very interesting to note here that both resonances depend on the quadratic light shift which is proportional to the intensity of the dressing beam, and then $\Delta \propto I_D$. Since the two photon Rabi couplings are proportional to $\sqrt{I_D I_R}$, then $\Omega_{m_F}^{m_F+2} \propto I_R$, which
<figure><img src="image_46.png" /><figcaption>Figure 4.9: Four photon transitions in the dressed states point of view. Left: energy levels with quadratic degeneracy lift, and example of a four photon transition with associated resonance ($\epsilon_{-5/2} - \epsilon_{-9/2}$)/2 and Raman detuning $\delta$. Right: dressed state picture of 2 and 4 photons couplings, where $\Delta_{2ph} = \delta - (\epsilon_{-7/2} - \epsilon_{-9/2})$ is the detuning with respect to the 2 photons transition, and $\Delta_{4ph} = 2\delta - (\epsilon_{-5/2} - \epsilon_{-9/2})$ to the 4 photons transition. $N_\pi$ and $N_{\sigma_-}$ are the number of photons in respective photon fields $E_\pi$ and $E_{\sigma_-}$.</figcaption></figure>

first shows that the four photon transitions are much weaker than the two photon. And most importantly, the relative strength of the effective couplings finally scale as:

$$ \frac{\Omega_4}{\Omega_2} \propto \sqrt{\frac{I_R}{I_D}} \qquad (4.33) $$

This shows that the relative strength of the four photon couplings is strongly reduced with an appropriate setting of the Raman and dressing beams intensities. In this example shown on figure 4.8, the laser intensity into the Raman beam is 500 times smaller than that of the dressing beam. It is sufficient to strongly displace the eigenenergies, enabling the avoided crossing that we use for adiabatic passages, while the relative strength of the four photon couplings is as low as $\Omega_4/\Omega_2 \simeq 5\%$.

Note that all this derivation is done in the $|E_\pi| \gg |E_{\sigma_-}|$ regime. In the opposite case, where the dressing beam is $\sigma_-$ polarized and the Raman beam $\pi$ polarized, the
degeneracy lift of the ground state manyfold should be derived for a $\sigma_-$ excitation, as presented in 4.1.1, with the main difference that the polarizability acquires a linear dependence in $m_F$. Regarding the Raman transitions, it has no consequences, and hence the previous estimation of the relative strength of the two photon and four photon couplings remains valid.

In the intermediate regime $I_D \approx I_R$, the tensor polarizability accounts for the combination of the two laser and associated polarizations, and the ground state manifold is rotated from the $|m_F\rangle \leftrightarrow |m_F \pm 1\rangle$ couplings defined in equations 4.11 and 4.12 as presented in section 4.1.1. This means that spin flips can be better realized with a STIRAP sequence [BVS15; Vit+01; MSB96], i.e. a sequence of pulses sequence by varying the laser intensities with respect to time rather than the Raman detuning.

In our experiment, we manipulate the spins in the $|E_\pi| \gg |E_{\sigma_-}|$ regime that we considered here, and this is the case that I will also consider in the following discussions.

### 4.3.3 Engineering selective spin flips

In the frame presented on figure 4.8, a selected spin can be flipped to a target spin with adiabatic passage by ramping the Raman detuning through the corresponding Raman resonance. With proper parameters to center the ramp on the resonance and with initial and final detunings such that no other resonances are crossed, the success of the adiabatic passage presents no difficulty. However, it is possible to take advantage of the conditions for adiabatic passage to facilitate experimental engineering of the spin flips. Indeed, I have shown in the previous section that, accordingly to equation 4.33, the relative strength of the four photon couplings can be dramatically minimized by lowering the power of the Raman beam or increasing the power of the Dressing beam. Regarding adiabatic passages, a four photon transition has much lower probability of success than a two photon transition considering a fixed duration of ramp of the Raman detuning. It is then possible to engineer a spin flip with a ramp crossing multiple four photon transitions.

To realize selective spin flips on the experiment, we first prepare a spin polarized gas in spin state $m_F = -7/2$, as presented in section 3.2.3 of chapter 3. Then, the
magnetic field is slowly rotated such that it is aligned with the linear polarization of the electric field of the dressing beam. This way, the polarization of the light is well defined and it is purely linear. Then, the dressing beam, $\pi$ polarized, is slowly turned to its maximum intensity $I_D \approx 9.3 \text{ W/cm}^2$ within 2 ms, with detuning $\Delta = -700 \text{ MHz}$ from the $^3P_1$, $F=9/2$ hyperfine state. Now, the degeneracy of the ground state $^1S_0$ is lifted by the quadratic light shift, and the spin-spin transitions are well resolved. At this step, the experiment is ready for adiabatic passage.

After slowly turning on the Raman beam up to $I_R \approx 5.1 \text{ mW/cm}^2$ within 2.5 ms, ensuring smooth connection to the eigenstates (see annexe C), the frequency of the Raman beam is ramped such that the Raman detuning $\delta = \omega_\pi - \omega_{\sigma_-}$ is spanned from 5 KHz to 13 kHz, for different durations of the frequency ramp. Then, the Raman beam is turned off within 2.5 ms, and the populations in the spin states are measured with the spin dependent momentum transfer scheme (see chapter 3). The result of this spin flip experiment is presented on figure 4.10, left. The top figure shows the frequency ramp through the Raman resonances, and the bottom figures shows the measurement of the spin populations, which are shown as round dots with associated errorbars and rescaled by the 80% efficiency of the populations measurement, for different durations of the detuning ramp. Note that not all atoms are detected during the populations measurement. This is shown by the total percentage of detected atoms in gray dots in bottom plots of figure 4.10, which is not constantly 100% and even goes as low as 80%.

Initially, approximately 90% atoms are in $-7/2$ and 10% in $-5/2$. For fast ramps, atoms remain in the initial state $-7/2$ and all transitions fail. For slow enough ramp with duration $\approx 20 \text{ ms}$, 80% spins have flipped to $-5/2$, while the population in $-3/2$ has raised to 10%, and approximately 10% atoms remain in $-7/2$. To evaluate the efficiency of the adiabatic passage, it is convenient to realize a back passage. From the final state of the forth passage, the same procedure is done with a reversed ramp from $\delta = 13 \text{ kHz}$ to $\delta = 5 \text{ kHz}$, and the Raman beam is turned off between the two passages. The back passage is shown on right figures. For slow enough ramp with duration $\approx 20 \text{ ms}$, we detect 60% atoms in $-7/2$, and nearly 10% atoms are in both $-5/2$ and $-3/2$. With this experiment, we infer that approximately 65% out of the 90% atoms initially in $-7/2$ have successfully realized the back and forth adiabatic passage, which
<figure><img src="image_47.png" /><figcaption>Figure 4.10: Experimental back and forth adiabatic passages from −7/2 to −5/2, with Δ = −700 MHz, $I_D \simeq 9.3$ W/cm$^2$, and $I_R \simeq 5.1$ mW/cm$^2$. Left: forth passage from −7/2 to −5/2 with ramp of the Raman detuning $\delta = \omega_\pi - \omega_{\sigma_-}$ from 5 kHz to 13 kHz. Right: consecutive back passage from −5/2 to −7/2 with ramp of the Raman detuning from 13 kHz to 5 kHz. The Raman beam is turned off between the two passages. Ramps through the spectrum of the two and four photon transitions are shown on top figures. The measured populations, with associated errorbars and rescaled by the 20% efficiency of the measurement, in the different spin states are shown in bottom figures, with respect to the duration of the ramp. The gray large points correspond to the total percentage of measured atoms. On both figures, solid and dashed lines are results from time integration of the Lindblad equation, including respectively predicted scattering rate and four times the predicted scattering rate.</figcaption></figure>
results in 80% efficiency for a one way adiabatic spin transfer. This experiment verifies the applicability, and robustness, of this adiabatic spin transfer scheme. Even if the detuning ramp crosses multiple two and four photon resonances, there is an optimal duration of the frequency ramp such that the four photon transitions are avoided while the adiabatic followings are ensured for two photon transitions. With proper choice of the direction of the frequency ramp, it is then possible to ensure a reliable selectivity on the spin transfer $|m_F\rangle \leftrightarrow |m_F \pm 1\rangle$, depending only on the initial spin state.

The result of the experiment is compared to numerical time integration of Lindblad equation [Man20]:

$$ \frac{d\hat{\rho}}{dt} = \frac{1}{i\hbar}[\hat{H}, \hat{\rho}] + \sum_k \gamma_k \left( \hat{L}_{ik}\hat{\rho}\hat{L}_{ik}^{\dagger} - \frac{1}{2}\{\hat{L}_{ik}\hat{L}_{ik}^{\dagger}, \hat{\rho}\} \right) \quad (4.34) $$

where the hamiltonian is defined in equation 4.29, $\hat{\rho}$ is the density matrix, and the $\hat{L}_{ik}$'s are the jump operators, i.e. $|m_{Ff}\rangle\langle m_{Fi}|$ projectors [DCM92] for $m_{Fi} \rightarrow m_{Ff}$ transitions, associated with rate $\gamma_k$. Here, $\{A, B\} = AB + BA$ is the anticommutator operator for A and B. The sum runs over all the possible spin projections, in our case set by Raman scattering defined in equation 4.18. Note that the elastic Rayleigh scattering $\Gamma_{m_F \rightarrow m_F}$ is taken into account, even if it doesn't change the spin state, as it is a source of spin decoherence and hence it can decrease the efficiency of the coherent adiabatic passage. Also, no atom loss is considered here, since the measured heating rate is low enough so that the atoms cannot be blasted from the trap, according to our measurement presented on figure 4.6.

The prediction with experimental parameters (intensities and detuning) and no other adjustable parameter is plotted on figure 4.10, bottom left and right, as thick solid line, and it shows that the efficiency of the adiabatic passage should be significantly higher than measured, as high as 95%. Because the measurement of the scattering rate showed that it is much higher than predicted (see section 4.2.3), it is relevant to artificially enlarge the scattering rate as high as measured to compare prediction with the spin flip measurement. This is shown by the thin dashed line on 4.10, bottom left and right, where the scattering rate is four times higher than predicted, according
to the measurement. While the expected populations in −7/2 and −3/2 are higher
by a few percents to what is measured, this second simulation seems to be in good
agreement with the experimental results, accounting for the non 100% total percentage
of detected atoms. Hence, this simulation shows that our experiment is only limited
by spontaneous emission due to the enlarged light scattering, that can be corrected by
filtering the spectral mode of the laser with a Fabry-Perot cavity. Then, we expect that
the adiabatic passage scheme to selectively flip spin state should reach 90% efficiency.
Note that a cavity has been set and used to measure the scattering rate (see section
4.2.3), however its sensitivity to vibrations has prevented us from using it for a spin flip
experiment. The cavity is presently being upgraded for future use.

## 4.4 Site selective adiabatic passage to prepare spin textures - Outlook

Previous experiments have demonstrated experimental realizations of patterns of atoms selectively placed at the sites of optical lattices, using an optical super lattice [Pei+03], or site selective light shifts [Gri+06]. More recently, experiments have shown that patterns can be assembled atom by atom using optical tweezers [Bar+16; Kum+18]. In this section, I present our scheme to prepare alternate spin textures [Li+16]. Using a spin dependent optical super-lattice [Man+03; Hei+20] and a homogeneous Raman laser, we will selectively flip half the spins of a single component band insulator.

4.4.1 General Idea

Starting from a polarized Fermi gas loaded in the ground band of a 2D lattice, half the
spins are selectively flipped to prepare a Neel order [Sun+21], i.e. an alternate pattern
of up and down spins, as shown on figure 4.11. This is done with a space dependent
tensor light shift [Hei+20] which varies at the micrometer scale, and it is engineered
by retro-reflecting the laser beam responsible for the spin dependent light shifts. Since
the resonances of the lifted spin spin Raman transitions are proportionnal to the local
laser intensity, it is possible to realize a chirp of Raman detuning that crosses the
resonances at sites with maximal depth of the spin dependent potential only, using
<figure><img src="image_48.png" /></figure>

Figure 4.11: Scheme for writing spin textures. Left: the spin dependent lattice (red background) is superimposed to a 2D spin polarized lattice. A cut of the potential (horizontal green solide line) is presented on the middle figure. Middle: spin dependent potential (red solide line) along an eigenaxis of the 2D lattice. Each spin (blue dots) localized at the sites of the 2D lattice (x axis) feels a different light shift which is proportional to the light intensity (yaxis). The chirp of the raman detuning is shown by the black arrow and crosses the Raman resonances (see figure 4.10) of the maximally shifted sites only, which are flipped from blue spin to red spin. The resulting spin texture is shown on right figure.

an homogeneous Raman beam. Then, our demonstrated scheme to coherently flip the
nuclear spin states acquires a well defined site dependence, enabling the possibility to
handwrite a predictable spin texture.

4.4.2 Geometry of the lattices

The geometry of the laser setup used to engineer the spin dependent lattice is presented on figure 4.12. The dressing beam at 689 nm is retro-reflected to shape a 1D lattice, that I will call for simplicity *dressing* lattice, and the azimuthal angle between the dressing beam and the 532 nm beams is 45°, such that the dressing lattice is along the diagonals of the 2D trapping lattice. With elevation angle 16°, the projection of the dressing lattice on the 2D layers has site spacing $a_{689} \approx 360$ nm, which is smaller by less than 10% than the diagonal of the 2D lattice $a_{532}\sqrt{2} \approx 390$ nm (i.e. the distance between neighbours at 45° of the main lattice axis). Neglecting the 10% difference in first
<figure><img src="image_49.png" /><figcaption>Figure 4.12: Experimental design of the setup for lattices. The retro-reflected 1D spin dependent lattice is nearly colinear with one arm of the 1064 nm lattice, with azimuthal angle 45° with each arm of the 2D lattice at 532 nm and 16° elevation angle.</figcaption></figure>

approximation (see figure 4.13, (a)), it means that the spin dependent light shift is maximum one out of two diagonals of the 2D lattice, and hence the depth of the spin dependent potential would alternate along an eigen-axis of the 2D lattice. In this case, a chirp of a Raman detuning crossing only the maximally shifted resonances to flip half the spins would realize exactly a Neel order. If considering the 10% difference, Figure 4.13 (b.1) shows that trapped atoms actually see 11 sites periodic pattern of alternate maximum and minimum depth of the spin dependent potential. This pattern has maximum contrast at the center over about 7 sites and is blured at the edges over about 4 sites, and this results in alternate pattern for the Raman resonances between higher and lower frequencies, see figure 4.13 (b.2), over 11 sites.
<figure><img src="image_50.png" /><figcaption>Figure 4.13: Spatial and spin dependence of the tensor light shift associated with the intercombination line. (a) The eigenaxis of the 689 nm 1D lattice is along the diagonal of the trapping 2D lattice, considering $a_{689} = a_{532}\sqrt{2}$. (b.1) Considering the experimental situation with sites spacing $a_{689} \approx 360$ nm, which is 90% of the diagonal of the 2D lattice, this is the projection along one eigenaxis of the 2D square lattice at 532 nm of the trapping (dashed lines) and spin dependent (solid lines) potentials, for each spin state. (b.2) Projection along one eigenaxis of the 2D square lattice of the raman resonances defined as $U_{689}(m_F) - U_{689}(m_F + 1)$, where U is the dipole potential.</figcaption></figure>
<figure><img src="image_51.png" /></figure>

Figure 4.14: Site dependent Raman resonances. The 532 nm lattice is sketched as the green solid line, and the target two photon $|−7/2⟩ \leftrightarrow |−5/2⟩$ transition (Red) is sketched for each site, as well as the other Raman resonances subtitled at the central site. The ramp of Raman detuning from $\delta_i$ to $\delta_f$, sketched by the black arrow, crosses the resonance one over two sites only, as shown by the blue filled region. The initial frequency is chosen as 90% of the less shifted $|−7/2⟩ \leftrightarrow |−5/2⟩$ resonance among selected sites, and the final frequency as 110% of the most shifted resonance, which corresponds to the lattice depth.

### 4.4.3 Site dependent Raman resonances

The Raman resonances of the spin spin transitions, with acquired site dependence, are presented on figure 4.14 along 11 sites of an eigen-axis of the 2D lattice at 532 nm. The Raman resonances, proportional to the local intensity of the dressing lattice, are then site resolved.

One every two sites, say odd sites, the resonances are maximally shifted, so that chirp of the Raman detuning with appropriate initial and final frequencies can cross the resonance of a target two photon resonance on these sites only, while it is not crossed on the other half sites, say even sites. This way, it is possible to realize an adiabatic spin
flip on the odd sites, while atoms at the even sites are not affected by the frequency chirp. It is pictured by the black arrow between an initial $\delta_i$ and final $\delta_f$ detunings, and $\delta_i$ is chosen such that the $|−9/2⟩ \leftrightarrow |−7/2⟩$ two photon resonances of the avoided sites are not crossed by the ramp, for instance on sites -5 and 5 on figure 4.14.

### 4.4.4 Predicted texture writing

The frequency chirp presented on figure 4.14 is simulated by numerical integration of the Lindblad equation defined in equation 4.34 for each one of the 11 sites, and the result is shown on figure 4.15. For this simulation, the intensities are $I_D \approx 11$ W/cm$^2$ and $I_R \approx 6$ mW/cm$^2$, where $I_D$ ($I_R$) is the intensity of the dressing (Raman) beam, with detuning $\Delta = -700$ MHz from the excited hyperfine state $|^3P_1, F = 9/2⟩$. The Raman detuning $\delta = \omega_D - \omega_R$ is ramped from 63% to 110% of the maximally shifted $|−7/2⟩ \leftrightarrow |−5/2⟩$ target resonance (site 0 of figure 4.14) during 5 ms. The amplitude of the Raman beam is ramped up (down) during 2.5 ms before (after) the frequency ramp, as shown by the red filled background, to improve the connection to the followed eigenstates. The left column, considering predicted spontaneous emission, shows the spins populations dynamics during a transfer from the initial spin state $−7/2$ to the final spin state $−5/2$. Sites with deeper tensor light shift ($U_{dip}/U_{max} > 0.5$) are flipped with 90% chance of success, while the avoided sites ($U_{dip}/U_{max} < 0.5$) have less than 3% chance of success, and on this 11 sites sample, the fidelity of the spin texture writing, here defined as the average of the final $−5/2$ populations in target sites and final $−7/2$ in avoided sites, can be estimated as high as 97%. If taking into account the measured spontaneous emission which is four times higher than the prediction (see section 4.2.3), the result of the simulation is shown on the right figure, and the estimated fidelity of the spin texture writing is lowered to 85%.

The conditions for adiabatic passage have to be carefully looked at for every site. Indeed, the spin dependent light shift is not the same for all selected sites, and hence the Landau-Zener (LZ) condition for adiabatic following (see appendix C) varies accordingly. It is therefore important to focus on the site dependence of the adiabatic following. First, let's recall that the Landau-Zener probability of success for an adia-
<figure><img src="image_52.png" /><figcaption>Figure 4.15: Simulation of the frequency chirp presented on figure 4.14 with numerical integration of the Lindblad equation (4.34) for each one of the 11 sites, to flip spins from -7/2 to -5/2. The success of passage is indicated in big black font, and the final populations in -9/2 and -3/2 are indicated in small red font. The simulation parameters are detailed in section 4.4.4. In left column, the simulation is realized with predicted scattering rate, while right column takes into account measured scattering which is four times larger.</figcaption></figure>
batic following is defined as:

$$P_{LZ}(\Omega^2, \dot{\delta}) = 1 - \exp\left(-2\pi \frac{\Omega^2}{4\dot{\delta}}\right) \qquad (4.35)$$

where $\Omega^2$ is the coupling strength and $\dot{\delta}$ is the speed of the detuning ramp. Considering that the intensity of the Raman beam is homogeneous over the whole lattice, and that the detuning ramp is linear in time, then the chance of success is lowered at sites with lower spin dependent light shift, since $\Omega^2 \propto I_D$. However, the scattering rate associated with the tensor light shift is obviously higher at sites with stronger intensity of the dressing lattice. Finally, the probability of success for four photon transitions does not depend on the depth of the tensor light shift. This means that there exists optimal conditions for adiabatic passage, which corresponds to a chirp duration slow enough to maximize the relative chance of success of the targeted two photon transitions on the less shifted sites with respect to the avoided four photon transitions, and fast enough to minimize spontaneous emission on the most shifted sites.
# Chapter 5

## 2D Fermi gases in the lowest band
of a low recoil 1D lattice

The objective of our experiment is to realize a Fermi Hubbard model with 2D lattices [BDZ08; GB17]. Such experiments require the preparation of ultra cold Fermi gases, that are trapped into the lowest Bloch band of optical lattices with minimal entropy [MD11]. Practically, it requires to circumvent the difficulties associated with the manipulation of ultra cold gases of fermions [Ess10]. For instance, atoms are typically cooled to quantum degeneracy with forced evaporation assisted by collisions [GWO00; LRW96], that are inhibited for ultra-cold fermions [DeS+10]. Yet, active research on the Fermi Hubbard Model led to the realization of numerous quantum simulators, typically using alkali fermions such as potassium 40 [Sch+08; Che+15; Dre+17] or litium 6 [Gre+16; Bol+16; Par+16]. Regarding alkali atoms, Feshbach resonances [Chi+10] can be used to tune the scattering length $a_s$, which helps prepare ultra-cold gases, and at the realization of isentropic transformations assisted by collisions, such as lattice loading. Inside lattices, entropy can be further reduced, for instance with Raman side band cooling [Ham+98; GWO00; Ker+00; Tho+13], or by filtering of atoms populating the excited bands [Arn+19].

In our experiment, ultra cold strontium 87 shall be trapped in a 3D optical lattice, that is made of the combination of a 1D lattice with large site spacing [Huc+09] and negligible tunneling, with a 2D lattice where the spin dynamics takes place (see chapter
1). The 1D lattice, that is first loaded, has site spacing $2\mu m$, so that prior to the
loading the gas extends over approximately 8 sites. The large spacing of the lattice
permits that the inter site tunneling is inhibited with moderate lattice depths of a few
tens of $kHz$. From this geometry, we prepare several independent 2D Fermi gases that
are vertically stacked. The 2D gases shall next be adiabatically loaded in a 2D square
lattice to prepare several band insulators [Leb+18] that do not interact with each other.
This chapter is dedicated to the loading of the lowest band of the 1D lattice.

*Chapter summary:* first, I will discuss a critical limit set by the Pauli exclusion principle at the loading of the lowest band only of the 1D lattice with large site spacing. The Fermi energy of the 2D gases cannot exceed the lattice band gap, which limits the number of atoms that can be loaded. However, in practice, the Fermi momentum of our evaporated gases exceed the recoil momentum of the lattice, so that atoms connect to excited bands when turning on the lattice. In this case, loading the lowest band of the lattice requires a quasi-static transformation assisted by s-wave collisions [Ess10].

Then, I will present our loading experiment. I will describe two attempts to do so. In the first, the gas is initially prepared at very low temperature, by strong evaporation in the dimple. I show that we are able to load the 1D lattice with 93(2)% atoms into the lowest band, using a Boltzmann approximation and a global analysis over the entire cloud distributed in a few layers. It is even possible to extract information that depends on the layers of the 2D lattice. From the Fermi statistics, I estimate the chemical potential of each layer, as well as the temperature, based on an independent measurement of the distribution of atoms between the layers.

Using this method, I find that the 2D gases have negative chemical potentials, in-
dicating that they are not degenerate, and the infered probability of occupation of the
lowest band is the same than the classical analysis. A comparison with the initial ther-
mal state of the gas prior to the loading shows that the high probability of occupation
of the lowest band is enabled by a deep evaporation prior to the lattice loading, until
our lower limit yet of measurable temperature $T = 5 \pm 5\,nK$ into the dipole trap with
approximately 5000 atoms. However, the degeneracy of the loaded gases are higher
than expected for an isentropic loading. Hence, this measurement shows the quasi-
static behavior failed, and it suggests that the collisions were not sufficient. This study
serves as a guideline for a preliminary result.

In a second attempt, improved adiabaticity is shown by holding sufficient dipole trap compression prior to the loading into the lattice, ensuring sufficiently high collision rate. This way, we are able to produce 4 independent 2D degenerate SU(10) Fermi gases with $\mu/\epsilon_F \ge 0.4$ and approximately 99% atoms into the lowest band.

## 5.1 Limit to the atom number for fermions in the lowest band

In our experiment, the site spacing of our 1D lattice is $a \approx 2 \mu m$ with associated recoil momentum $k_R = \pi/a$. In this case, the recoil energy $E_R = \hbar^2 k_R^2/2m$ is as low as $E_R \approx k_B \times 7 \text{nK} \approx h \times 150 \text{Hz}$, which is much lower than in typical retroreflected optical lattices. Consequently, the site to site tunneling $t \sim \exp(-2\sqrt{V_0/E_R})$ [BDZ08] is negligible for a lattice depth $V_0$ of a few tens of kHz, and in our experiment we measure $V_0 \approx 1800 E_R$ (see chapter 1), so that the motional degree of freedom along the lattice eigen axis is practically frozen. The phase space is then restrained to the on-site 2D motion, with associated phase-space cell $d^2rd^2p/h^2$, and to the band excitation $n$. Assuming that only the lowest band $n=0$ is populated, every site of the 1D lattice is reduced to 2D dimensionnality $(r,p) = (r_x, r_y, p_x, p_y)$.

In the $T=0$ K limit, fermions occupy all energy states below the Fermi energy, and for non zero temperature, higher energy states are accessible. This sets a critical limit to the number of atoms that can be loaded into the lowest band of a 1D lattice. Assuming that the transverse confinement of the lattice sites can be approximated to harmonic traps of frequency $\omega_\perp$, and that the tunneling is negligible, the limit reads

$$ \epsilon_F - \epsilon_0 = \hbar\omega_\perp(2N)^{1/2} < \hbar\omega_\parallel \quad (5.1) $$

where $\hbar\omega_\parallel = 2\sqrt{V_0E_R}$ is the lattice band gap, and $\epsilon_0$ the energy of the lowest band. This result can be shown by derivating the Fermi energy $\epsilon_F$ of a gas trapped into a 3D harmonic trap, tightly confined along one direction only and with zero band excitation (see appendix A). It is simply the Fermi energy of a 2D gas shifted by the energy of
the lowest band. In our experiment, $\hbar\omega_\parallel \simeq 12$ kHz and $\hbar\omega_\perp \simeq 80$ Hz, so that the Fermi energy is lower than the energy of the first excited band only if $N < 10\,000$ atoms per spin state. If the transverse harmonic confinement can be adjusted without changing the lattice depth, for instance by using a combination of a red and blue detuned dipole traps, the maximum number of atoms in the lowest band can be practically adjusted [Sch+08].

## 5.2 Adiabatic loading of the lowest band : collision requirements

Typically, lattice loading requires adiabaticity in the single particle sense. It relies on the adiabatic theorem that implies that atoms follow the eigenstates of the time dependent Hamiltonian if it is modified sufficiently slowly [Den+02]. This way, an atom in a plane wave state $k_{\text{atom}}$ evolves into a Bloch state, that will be the ground state if $k_{\text{atom}} < k_R$. The momentum recoil $k_R = \pi/a$ of the lattice defines the limit of the first Brillouin zone (FBZ), and momenta exceeding the FBZ connect to the excited bands of the lattice. In the case of a Fermi gas, the loading of the lowest band only thus implies, in the single particle limit, that $k_F < \pi/a$. In our case, the recoil energy $E_R \simeq k_B \times 7$ nK $\simeq h \times 150$ Hz is much lower than in typical retroreflected optical lattices, and this is cause for concern. First, cooling a thermal gas of strontium 87 fermions in this limit is difficult as collisions are inhibited by Pauli blocking at deep evaporation stages [DeS+10], and closed shell strontium 87 atoms do not benefit from Feschbach resonances to favorably tune the collision rate. Second, even if we did, the average distance between atoms would scale as the site spacing $a$. In the prospect of next loading a 2D lattice with site spacing $a/8$ (see chapter 1) to realize a band insulator, such low densities are not favorable. Therefore, the idea of adiabatic following at the single particle level is not fruitful. Fortunately, there is also the possibility to be quasi-static in the thermodynamic sens, if collisions are allowed, and this approach is typically used to prepare Mott insulators [CHU09; Fuk+09; Jör+08].

This second possibility for loading the ground band of the lattice relies on collisions, which means that the adiabaticity of the process is not an eigenstate following which
relies on single particle physics, but an adiabatic process in the sense of remaining at
thermal equilibrium relying on s-wave collisions [Ess10]. Furthermore, this many-body
transformation requires a mixture of fermionic species. When the lattice is turned on,
the initial thermal state of the transformation is a 3D gas in a 3D harmonic trap with
Fermi energy $\epsilon_F^{3D} \propto \hbar(\omega_\perp^2\omega_\parallel)^{1/3} \gg \hbar\omega_\parallel$, where $\hbar\omega_\parallel$ is the band gap of the lattice. Hence,
atoms populate excited bands. When the depth of the lattice is increased, the band
gap increases faster than the Fermi energy, and collision assisted relaxation permits
to empty the band excitations. In the final thermal state, the gas can be 2D only if
$\epsilon_F^{2D} < \epsilon_1$, as discussed in the previous section, where $\epsilon_F^{2D} = \hbar\omega_\perp(2N)^{1/2}$ is the 2D Fermi
energy and $\epsilon_1 = \hbar\omega_\parallel$ the excitation energy of the first vibrationnal state. Therefore, the
lattice loading can be seen as an adiabatic trap compression assisted by collisions along
the 1D lattice eigen-axis so that $\omega_\parallel \gg \omega_\perp$, which permits a dimensionality cross-over
from 3D to 2D.

While the single-particle approach is limited by $\epsilon_F < E_R$, the collision assisted
loading of the lowest band is limited by $\epsilon_F < \epsilon_1$, where $\epsilon_1 \gg E_R$. Hence, loading
the lowest band of the lattice with a mixture of fermions allowing for s-wave collisions
permits to produce much more dense 2D gases. However, the adiabaticity criterion
is very different between the two schemes. In the single-particle approach, which can
be realized with only one fermionic species, the timescale associated with the lattice
depth variations is in the order of a ms for band gaps in the order of 1 kHz, which
is more favorable than the timescale associated with the inter atomic collisions, that
require a mixture of fermionic species, and that is in the order of the hundreds of ms
for degenerate Fermi gases of strontium.

5.3 Loading experiments and measurement of the band populations with a Boltzmann approximation

In this section, I present our protocole to produce a 2D Fermi gas of $^{87}$Sr. From time
of flight absorption images of the loaded gases, the kinetic energy of the gas expanding
transversaly to the lattice eigen axis is used as a thermometer, and the probability of
occupation of the lowest band $r_0$, that I will refer to as loading efficiency, is infered
from a Boltzmann approximation. While the kinetic energy along the lattice eigen
axis converges to that one of the lowest Wannier state and the temperature of the gas
decreases, the aspect ratio measured on the time of flight images increases, indicating
that the Fermi gas reaches the 2D regime. Our measurement shows that we are able to
produce 2D gases with 93(2) % atoms into the lowest band of the 1D lattice.

*Experimental procedure:* we produce a spin mixture $m_F = +9/2, -5/2, -9/2$ of $^{87}$Sr with a sequence of spin selective optical pumping, as presented in chapter 3, and the relative populations are respectively nearly 10%, 70% and 20% due to the sequence of the mixture preparation (see section 3.2.2). The gas is cooled by forced evaporation, with final number of atoms $N_{at}$ and degeneracy $T/T_{FODT}$ tunable with respect to the final depth of the 3D optical dipole trap. Then, the 1D lattice is slowly ramped up in two steps. First, while the collision rate per atom is as low as approximately $\gamma_{col} \simeq 3 \text{ s}^{-1}$, inhibited by the Pauli exclusion principle after reaching low $T/T_{FODT}$, the lattice is slowly turned on from zero depth to $V_0 \simeq 270 E_R$, i.e. $h \times 40$ kHz, in 600 ms. During this stage, the gas is split in different layers of the 1D lattice as pictured on figure 5.1, and for such depth, the tunneling is negligible. Here we assume that the gases in every layer remain at thermal equilibrium, and it is discussed later in section 5.4.2. Then in a second stage, the lattice is ramped up to its maximal depth $V_0 \simeq 1780 E_R$, i.e. 255 kHz, in 600 ms. The measurement of the depth of the low recoil lattice is presented in section 1.2.1, and at full depth, the band gap $\hbar\omega_\parallel = 2\sqrt{V_0E_R}$ is 12 kHz. During this stage, the optical dipole trap is slowly turned off such that the geometry of the trap is completely determined by the lattice laser. After loading the lattice, there is 250 ms hold time. Then, the lattice is abruptly shut off and the atoms are imaged with absorption imaging after a time of flight of 10 ms. The result of the experiment is shown on figure 5.2.

After time of flight of the loaded gas, the imaged densities, shown on figure 5.2, (a.1)
and (a.2) for different number of atoms and respective final depth of the dipole trap, are
analysed in the classical approximation, and fitted with Gaussian distributions. The
<figure><img src="image_53.png" /></figure>

Figure 5.1: Depth of the lattice with respect to the time of ramp. The loading is done in two stages, split by the dashed line on the plot at $V_0 \simeq 40 \text{ kHz}$. The insets are illustrations of the loading at different stages, the black curve is the density and the red curve is the 1D lattice potential. At 0 ms, the lattice is off and the gas is bulk, at 40 kHz the lattice depth is deep enough such that the thermal gas is split into different 2D layers, at 255 kHz the depth is maximal and the atoms shall occupy the ground band only, i.e. the lowest Wannier state, with a Gaussian density distribution.

kinetic energy $<E_k> = <p^2>/2m = k_B T/2$ measured along $O_r$, i.e. along the 2D layers, is presented on figure 5.2, (b), empty squares, as a function of the number of atoms. Assuming that the gas is at thermal equilibrium, we use the measurement of the momentum distribution $<p_r^2>$ as a thermometer for this experiment, the temperature drops linearly with respect to the number of atoms, from approximately 450 nK to 200 nK in this data. The momentum distribution measured along $O_z$, i.e. along the lattice eigenaxis, (filled circles on figure 5.2), converges towards that one of Wannier state of the lowest band with momentum spread $\Delta p^2/2m = \hbar\omega_\parallel/2$ (dashed line). Atoms in the lowest Wannier state of the lattice have their momentum distribution defined by the harmonic oscillator length only: considering the Heisenberg uncertainty principle $\Delta x\Delta p = \hbar/2$, where $\Delta x = a_{ho}/\sqrt{2}$ for the lowest Wannier state, then $\Delta p^2 = m\hbar\omega_\parallel/4$. It shows that obviously, fewer atoms occupy excited bands for deeper evaporation prior
<figure><img src="image_54.png" /><figcaption>Figure 5.2: Classical analysis from tof images, with 10 ms time of flight. (a.1) and (a.2): Absorption images of gases with respectively 93(2) % and 73(1) % atoms in the lowest band, with indicated RMS radius in red dashed line. (b) Kinetic energy, along Or, i.e. along the 2D layers (empty squares), and along Oz, i.e. along the lattice eigenaxis (filled circle), as a function of the number of atoms. The horizontal dashed line indicates the kinetic energy of the lowest Wannier state. (c), probability of occupation of the lowest band p₀/pₜₜ, as a function of the number of atoms. (d) probability of occupation of the lowest band as a function of the aspect ratio σz/σr. On (b) and (c), the grey filled slope are linear fits, filled at ±1σ fit uncertainty.</figcaption></figure>

to the loading.

From the measurement of the temperature of the gas loaded into the 1D lattice, we infer the bands populations from the Boltzmann factor:

$$p_n(T) = e^{-\epsilon/k_B T} \qquad (5.2)$$

and the probability of occupation of the band n is defined as p_n/Z where Z is the partition function. For a given temperature, it is defined in the canonical ensemble as Z(T) = Σ_n p_n(T) = pₜₜ. The probability of occupation of the lowest band in this
classical approximation $r_0|_C$, that I here define as the efficiency of the loading:

$$r_0|_C = p_0(T)/p_{tot} \qquad (5.3)$$

is shown on figure 5.2, (c), as a function of the number of loaded atoms $N_{at}$. We are able to load up to 93(2)% atoms into the lowest band of the 1D lattice, with approximately 5000 atoms.

The aspect ratio of the loaded gas is compared to the inferred $p_0/p_{tot}$ on figure 5.2, (d), which shows the strong correlation, $C = 0.85$, between the two quantities. Indeed, the lower bound of the momentum distribution $<p_z^2>$ along the eigen axis of the lattice is the momentum spread of the lowest Wannier state. Considering a gas cold enough so that all excited bands are empty, which reads with the Botzmann factors

$$e^{-(\epsilon_1-\epsilon_0)/k_B T} \ll 1 \Leftrightarrow k_B T \ll \epsilon_1 - \epsilon_0 = \hbar\omega_{\parallel} \qquad (5.4)$$

the kinetic energy within the 2D layers decreases as much as the temperature while it remains constant along the lattice eigen axis. Hence, the increase of the aspect ratio is a good signature that the band excitation degree of freedom is vanishing and the gas reaches a 2D regime.

This measurement shows that we are able to produce 2D gases of strontium 87 with 93(2)% atoms into the lowest band. However, this classical approach doesn't account for the Fermi statistics. Differently populated sites have different chemical potentials, and hence different probabilities of occupation of the lowest band. Indeed, with sites spacing $a = 2 \mu m$, the atomic cloud extends over a few sites only, where $a_{ho} \simeq 1 \mu m$ prior to the loading, and the tunneling rapidly decays when the lattice depth is increased so that the populations are not redistributed between the sites. An accurate description of the loaded gases then requires to consider different chemical potentials at every site, and it cannot be accounted for in the classical analysis of the time of flight images since the density of a classical gas does not depend on the chemical potential. Hence, I will now discuss a semi-classical approach that I use to accurately describe the thermal state of degenerate fermions loaded in several layers of a low recoil 1D lattice with a
non uniform distribution of the number of atoms.

## 5.4 Band populations in the semi classical approximation for a 2D gas of Fermions

In this section, I first derive the analytic equations from the Fermi statistics that permit to infer the probability of occupation of the lowest band of a deep 1D lattice. Accounting for the non uniform population in every site of the 1D lattice, I derive a formula that relates the chemical potentials of every site, to the relative populations. The fitting function used to analyse absorption images after time of flight of the Fermi gas trapped into the 1D lattice is then derived, accounting for the several layers with different number of atoms.

### 5.4.1 Derivation of the number of atoms in each band

In the following derivations, I consider that the sites of the 1D lattice are 2D harmonic oscillators with quasi-classical phase space cells $d^2rd^2p/h^2$. In the deep 1D lattice approximation, with $V_0 \simeq 1800E_R \gg E_R$, the bands of the lattice are considered as vibrationnal states of a quantized harmonic oscillator [BDZ08] with degree of freedom $n$, which I will next consider equivalently. In this case, the Hamiltonian with no interactions reads

$$H_n(r, p) = \frac{p^2}{2m} + \frac{1}{2}m\omega_{\perp}^2 r^2 + (n + \frac{1}{2})\hbar\omega_{\parallel} \quad (5.5)$$

This hamiltonian is separable, such that the in plane physics and axial quantized physics can be treated separately, and $\epsilon_n(r, p) = \epsilon_{2D}(r, p) + \epsilon_n$, where $\epsilon_n = (n+1/2)\hbar\omega_{\parallel}$, and the associated Fermi Dirac occupation number reads

$$f_n(r, p) = \frac{1}{\exp(\beta(\epsilon_{2D}(r, p) + \epsilon_n - \mu)) + 1} \quad (5.6)$$

The band energies $\epsilon_n$ can be injected into the chemical potential $\mu_n = \mu - \epsilon_n$, which is a similar approach to the Thomas-Fermi approximation [GPS08], here associated to the vibrationnal degree of freedom of the quantized harmonic oscillator. In this case,
the phase space density of each band $n$ is the same except that the chemical potential is shifted by exactly the energy of the band. It is then possibly to directly derive the density $n(r)$ as the sum of 2D densities over all populated bands:

$$n(r) = \sum_{n} n_{2D}(r, \mu_{n}) \qquad (5.7)$$

where $n_{2D}(r, \mu_n)$ is the local density of each vibrationnal state $n$:

$$n_{2D}(r, \mu_n) = \frac{-1}{\Lambda_{DB}^2} Li_1(-e^{\beta(\mu_n - V(r))}) \qquad (5.8)$$

It is calculated by integrating the phase space density over the 2D momentum (see appendix A), and using the definition of polylogarithm for the integral of the Fermi Dirac distribution at $T \neq 0$, according to equation A.18. The number of atoms $N_n$ in the band $n$ is then computed by integrating the density according to the integration rule for the polylogarithm:

$$N_n(T, \mu_n) = - \left( \frac{k_B T}{\hbar \omega_{\perp}} \right)^2 Li_2(-e^{\beta \mu_n}) \qquad (5.9)$$

The probability of occupation of the lowest band $n_0$ is then:

$$r_0(T, \mu) = \frac{N_0}{\sum_n N_n} \qquad (5.10)$$

This is consistent with the fact that occupation of energy states is defined by the Fermi Dirac distribution, as a function of the temperature and chemical potential. The numerical result of equation 5.10 is presented on figure 5.3. Obviously, the probability of occupation of the lowest band increases at lower temperature, as the occupation number of high energy states is lower than at high temperature. Moreover, for any fixed temperature, a lower chemical potential implies a higher fraction of atoms in the ground band. Let's focus on the $T = 0$ K regime where the chemical potential is the Fermi energy, which is pictured as the thick dashed line on figure 5.3. Until Fermi energy $\epsilon_F$ reaches the energy of the first excited band, pictured as the filled region at $\epsilon = \frac{3}{2}\hbar\omega_\parallel$, all atoms occupy energy states lower than $\epsilon_F$, which is the ground band
<figure><img src="image_55.png" /><figcaption>Figure 5.3: Ratio of atoms in the ground band of the 1D lattice, with respect to the temperature T and the chemical potential $\mu$. The red fills indicate the energy of the first two vibrationnal states at $\epsilon_0 = \hbar\omega_\parallel/2$ and $\epsilon_1 = 3\hbar\omega_\parallel/2$. The chemical potential read at $T = 0$ is the Fermi energy, and the ratio of atoms in the first band at $T = 0$ with respect to the Fermi energy is plotted in dashed line. The ratio has been computed with equation 5.10 by taking into account the populations into the first 10 vibrationnal states. Here, $\mu > 0$, assuming that the gas is degenerate with $T/T_F < 0.6$.</figcaption></figure>

only, and $r_0 = 1$. As soon as $\epsilon_F \ge \epsilon_1$ atoms can populate the first excited band. As introduced in section 5.1, this sets a limit in the maximum number of atoms $N_{max}$ that can populate the ground band only, such that $\epsilon_F \le \epsilon_1$. Using the definition of $\epsilon_F$ from equation 5.1, this limit reads

$$N_{max} = \frac{1}{2} \left( \frac{\hbar\omega_{\parallel}}{\hbar\omega_{\perp}} \right)^2 \qquad (5.11)$$
Regarding the experiment this limit is very important. Considering there are 25% atoms in the most populated 2D layer, where $\omega_{\perp}/2\pi \simeq 80$ Hz and $\omega_{\parallel}/2\pi \simeq 12$ kHz, then the 1D lattice could not be loaded with a gas which has initially more than approximately 10 000 atoms per spin state and per lattice site at $T = 0$ K if we want the first band only to be populated in order to prepare a 2D gas. Obviously, the $T = 0$ K limit can't be reached. At $T \neq 0$ K, the wings of the Fermi Dirac distribution are enlarged, such that probability of occupation of the first excited states is enlarged and the chemical potential has to be lowered, hence the number of atoms, in order to maintain a high $r_0$.

### 5.4.2 Accounting for multiple layers in the lattice

The site spacing $a$ of the 1D lattice is approximately $2\mu m$, which means that the gas can extend over several sites just before loading the 1D lattice. In our experiment, we measure the number of atoms in each site with a magnification of the loaded lattice (see section 1.2.1 of chapter 1), and the result is recalled on figure 5.4. Our measurement shows that approximately 8 sites are significantly loaded with different populations. To account for this feature to measure the number of atoms in each band, I assume the following assertions.

First, the recoil momentum associated with the lattice $k_R = \pi/a$ is very low, as a consequence of the large site spacing, and therefore the tunneling $t \propto \exp(-2\sqrt{V_0/E_R})$ [BDZ08] rapidly vanishes when the lattice depth $V_0$ is increased, up to $V_0 \simeq 1800 E_R$ in our experiment. Hence, the inter layer coupling is negligible and we do not assume inter layer equilibrium. It is then possible to assign a different chemical potential to each site $i$, according to a local chemical potential of the gas binned around site $i$, making use of the local density approximation (LDA). This means that each layer of the 1D lattice must have a different degeneracy, chemical potential, and entropy per particle, depending on the number of atoms $N_i$ which have been initially loaded in each lattice site $i$.

Then, before loading the lattice, the initial gas is at thermal equilibrium, with a
<figure><img src="image_56.png" /><figcaption>Figure 5.4: 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. The color bar represents the optical depth. Right: 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</figcaption></figure>

well defined temperature. Temperature is an intensive thermodynamic quantity, such that when the gas is separated into the different sites, each fraction of the gas must have the same temperature, since the potential is the same for each fraction of the gas. Then the lattice depth is slowly increased, and assuming that the trap curvature is negligible over a few sites of the lattice (8 in our case), the adiabatic theorem for dipole trap compression $T/\omega = T'\omega'$ holds, and the compression is the same at each site. Therefore, the temperature is the same in every site of the deep 1D lattice.

Now I use these assertions to define the chemical potential for each site. Each site, or 2D layer, is independent and at thermal equilibrium, so that the relation between the number of atoms and chemical potential defined in equation 5.9 can be used independently for each layer. Since the temperature and confinement frequency are the same in each layer, it is possible to link the chemical potential of two layers i and j with respect to the ratio of the number of atoms only:
$$Li_2(-e^{\beta\mu_i}) = \frac{N_i}{N_j} Li_2(-e^{\beta\mu_j}) \qquad (5.12)$$

With the distribution of atoms into the different layers known, that can be measured with the magnification of the lattice sites, all the chemical potentials are then inferred from a single free parameter $\mu_0$, and it is then possible to determine all of the thermodynamic and statistical quantities of the independent gases loaded in several 2D layers of the 1D lattice.

### 5.4.3 Fit function with time of flight imaging

Let's now derive the fitting function to measure the thermodynamics quantities from the expanded density after time of flight, assuming that the mean-field energy can be neglected and that ballistic expansion is a good approximation. From this measurement, it is then possible to infer the bands populations in every site of the 1D lattice. It is an application of the fitting functions for the imaged densities of expanding Fermi gases [KZ08; DeM01] to the Hamiltonian defined in 5.5, accounting for the vibrationnal degree of freedom $n$ and the different 2D layers $i$.

The Hamiltonian [5.5] is separable, and the expansion during the time of flight can be treated separately along the layers and along the axis of the lattice.

First, along the lattice eigen-axis, atoms occupy the localized Wannier states, which expand as the harmonic oscillator eigenstates in the deep-lattice approximation. Assuming that most atoms are in the lowest Wannier state, the momentum distribution along the lattice eigenaxis can be considered Gaussian as a first-order approximation.

Then, along the 2D layers, we want to derive the expansion of the density of atoms defined in equation [5.8]. Let's consider a single layer first. The expanded density $n(r, t)$ after time of flight of duration $t$ is derived from the initial phase space density distribution $f_n(r_0, p_0)$ in the trap, respecting energy conservation so that $r(t) = r_0 + p_0t/m$:
$$ 
\begin{aligned}
n_{\text{tof}}(r,t) &= \sum_n \int d^2r_0 \int \frac{d^2p_0}{h^2} f_n(r_0, p_0) \delta(r - (r_0 + p_0t/m)) \\
&= \frac{2mk_B T}{h^2 b^2} \sum_n \int d^2q \frac{-1}{e^{q^2/z} - 1}
\end{aligned} 
\quad (5.13) $$

where $b^2(t) = 1+\bar{\omega}^2t^2$ is the scaling factor [KSS96; CD96], and the change of variable $q^2 = \beta b^2 p_0^2/2m$ has been made. Here, $z = \beta(\mu_n - V(r)/b^2(t))$, where $\epsilon_n$ is the excitation energy of band $n$. Using the formula for the integration of the polylogarithm, the 2D density after time of flight reads:

$$ n_{\text{tof}}(r,t) = \frac{-1}{\Lambda_{\text{DB}}^2 b^2(t)} \sum_n Li_1(-e^{\beta(\mu - \epsilon_n)} e^{-\beta V(r)/b^2(t)}) \quad (5.14) $$

In our experiment, we image by the side of the 1D lattice, i.e. along one of the axis of the 2D confinement. The integrated density profile along the imaging axis is then:

$$ n_{\text{tof}}^{\text{integrated}}(x,t) = \frac{-2\pi a_{ho}^2}{\Lambda_{\text{DB}}^3 b(t)} \sum_n Li_{3/2}(-e^{\beta(\mu - \epsilon_n)} e^{-\beta V(x)/b^2(t)}) \quad (5.15) $$

which depends on the free parameters $(\omega_\perp, \omega_\parallel, T, \mu)$. The depth of the lattice can be measured independently (see section 1.2.1 of chapter 1), so that the lattice band gap $\hbar\omega_\parallel$ is known. Moreover, the harmonic confinement of frequency $\omega_\perp/2\pi$ within the 2D layers can be inferred from a measurement of the size of the gaussian laser beams. Accounting for the different 2D layers $i$, the fit function used on time of flight images of gas loaded in the 1D lattice is then:

$$ n_{\text{tof}}^{\text{integrated}}(x, T, \mu_0) = \sum_i n_{\text{tof},i}^{\text{integrated}}(x, T, \mu_i) \quad (5.16) $$

where

$$ n_{\text{tof},i}^{\text{integrated}}(x, T, \mu_i) = \frac{-2\pi a_{ho}^2(\omega_\perp)}{\Lambda_{\text{DB}}^3(T, \omega_\perp)b(\omega_\perp)} \sum_n Li_{3/2}(-e^{\beta(\mu_i - \epsilon_n(\omega_\parallel))} e^{-\beta V(x, \omega_\perp)/b^2(\omega_\perp)}) \quad (5.17) $$

with free parameters $T$ and $\mu_0$, and the other $\mu_i$'s are inferred from $\mu_0$ using equation 5.12 and the relative number of atoms in every layer, that is measured independently.
This concludes the derivation of the fit function with the semi-classical approach, and
now, it can be applied to our data.

5.5 Semi-classical measurement of the bands popu-
lations of several independent 2D layers

In this section, I present the application of the semi-classical approach to analyse the time of flight images of the gases loaded in the 1D lattice. The procedure of loading is detailed in section 5.3. The gases are loaded after evaporation, and the initial temperature, number of atoms, and degeneracy, before loading the lattice, are set by tuning the depth of dipole trap prior to the loading. The gas are released 250 ms after loading, and imaged with absorption imaging after 10 ms time of flight. Using the semi-classical analysis, I show that we are able to prepare several independent 2D Fermi gases with 93(2) % atoms in the lowest band of the 1D lattice, which is the same result than the classical analysis, and the measurement of the chemical potentials further shows that the 2D gases are not degenerate with µ < 0. Then I will discuss the efficiency of the loading as a function of the evaporation depth. With deep evaporation, our experiment reaches its lower limit of measurable temperatures 5 ± 5 nK, and the efficiency of the loading of the lowest band is further improved by pursuing the evaporation.

5.5.1 Analysis of a single image

To apply the semi-classical analysis, I assume that the temperature is well defined and equal in all the sites, and that the inter-site tunneling is negligible. Moreover, the distribution of atoms in the sites of the lattice is well known, with 8 populated sites, from the magnification measurement (see figure 5.4). The fit function 5.16 can then be used to measure the temperature *T* and a reference chemical potential *μ*₀, that I attribute to the most populated site. The chemical potentials of all other sites can then be inferred from equation 5.12. Note that the three spins mixture is accounted for by considering three independent population distributions in every layer, that are weighted by the relative populations in every spin state. The chemical population of each spin state in every layer is then inferred from equation 5.12. The fitted chemical potential is
<figure><img src="image_57.png" /><figcaption>Figure 5.5: Semi classical analysis of time of flight absorption images of a gas loaded in several sites of a deep 1D lattice. (a) Absorption image (b) Integrated optical depth along the lattice eigenaxis Oz (dots) and result of the fit using equation 5.16 (solid line). (c) Fit residuals, and measured temperature T and 2D chemical potential $\mu_0 = \mu - \epsilon_0$ of the most populated layer for this image. (d) Artificial split of the optical depths of every site, and associated local chemical potential inferred from $\mu_0$. (e) Fermi Dirac occupation number $f_{FD}(\epsilon, \mu, T)$ for every site. The energy axis is in bang gap $\hbar\omega_\parallel$ units, and the population of the lowest band is indicated above each site.</figcaption></figure>

that of the most populated spin in the most populated layer, from which the chemical potentials for the different spin states in the different layers are deduced.

The fit function 5.16 can be used with two free parameters only, $T$ and $\mu_0$. The parameters $T$, $\mu_0$, $\omega_\perp$, and $\omega_\parallel$, are highly correlated, so that the fit uncertainty is strongly increased if all parameters are free. It is then convenient to infer $\omega_\parallel$ and $\omega_\perp$
from independent measurements, so that they are fixed and the uncertainty of the fits is drastically reduced.

The band gap $\hbar\omega_\parallel = 2\sqrt{V_0E_R}$ is directly inferred from the measurement of the depth $V_0$ of the 1D lattice (see chapter 1). In this run, $V_0 \simeq h \times 255$ kHz, which sets the associated band gap frequency to $\omega_\parallel/2\pi \simeq 12$ kHz. The radial confinement within each layer of the 1D lattice can be approximated as a harmonic potential in the bottom of the sites:

$$V_0 e^{-2r^2/w^2} \bigg|_{r \to 0} \simeq V_0(1 - 2\frac{r^2}{w^2}) \simeq V_0 + \frac{1}{2}m\omega_r^2 r^2 \quad (5.18)$$

where $w$ is the waist of the laser beam. Hence, the radial frequency $\omega_r$ can be directly inferred from the depth $V_0$ and waist $w$ as:

$$\omega_{x,y}^2 \simeq \frac{4|V_0|}{mw_{x,y}^2} \quad (5.19)$$

In our experiment, $|V_0| \simeq 255$ kHz, and the measured waists are $w_x \simeq 170$ µm and $w_y \simeq 105$ µm. The geometric mean of the radial frequency $\omega_\perp = \omega_x\omega_y$ is then:

$$\frac{\omega_\perp}{2\pi} \simeq 80 \text{ Hz} \quad (5.20)$$

For the analysis, $\omega_\perp$ and $\omega_\parallel$ are then fixed to the respective values $2\pi \times 80$ Hz and $2\pi \times 12$ kHz.

The result of the semi-classical analysis of a time of flight absorption image is presented on figure 5.5. The optical depth, shown on (a), is integrated along the eigen axis of the lattice (Oz), and it is fit using the fit function 5.16. For this gas, the measured temperature is $T = 210 \pm 14$ nK, and the measurement of the chemical potential of the most populated layer $\mu_0/k_B T = -0.40(5)$ is then used to infer the chemical potential of the other layers using relation 5.12. Figure 5.5 (d), shows an artificial split of the sites and their respective chemical potentials. The thermodynamics properties are known in every site, and the Fermi Dirac occupation number $f_{FD}$ is well determined, and it is plotted on figure (e). The band populations can be inferred from equation 5.9, and the probability of occupation of the lowest band $r_0$, defined in equation 5.10, here measured $r_0 = 93 \pm 1\%$, is shown on figure (e) for every site.
<figure><img src="image_58.png" /><figcaption>Figure 5.6: (a) Population in the lowest band as a function of the Fermi energy $\epsilon_F$ in units of the energy of the first excited band $\epsilon_1 = 3/2\hbar\omega_\parallel$, into the most populated layer. (b) Population in the lowest band as a function of the measured temperature into the lattice. (c) Number of atoms into the most populated layer as a function of the temperature.</figcaption></figure>

This result shows that it is possible to infer the thermodynamic properties of independent sites of the 1D lattice with the distribution of atoms known, even if they are not distinguishable on a single absorption image, assuming a uniform adiabatic compression in every site and negligible tunneling. With this method, we measure that we are able to prepare a set of independent 2D Fermi gases with $r_0 > 90\%$ that can be analysed simultaneously. The semi-classical analysis is made easier with independent measurements of the lattice depth, and the harmonic confinement within each site. Note that here the signal to noise ratio (SNR) of the integrated density profile is slightly lower than 10. With a better SNR, the independent measurements on the lattice geometry should not be required to have significant measurements.

The joint measurement of the chemical potential of the most populated layer $\mu_0/k_B T = -0.40(5)$ and temperature $T \simeq 210$ nK gives a measurement of the Fermi energy, which is lower than the energy of the first excited vibrational state $\epsilon_F/\epsilon_1 \simeq 0.3$, where $\epsilon_1/k_B \simeq 870$ nK. Hence, the population of the lowest band is not saturated by the number of atoms, and assuming thermal equilibrium, the residual band excitations are then due to the non zero temperature. Figure 5.6, (a) shows the increase of $r_0 = N_0/N_{tot}$ as a function of the Fermi energy $\epsilon_F$, which approximately amounts to the number of atoms, and (b) as a function of the temperature $T$ into the lattice. This measurement shows
that our production of multiple 2D degenerate gases with best $r_0 \approx 93(1)\%$ is due as much to a reduced number of atoms, i.e. chemical potential, as to a low temperature.

*Comparison with the classical analysis:* both the classical and semi-classical provide the same measurement of the probability of occupation of the lowest band, and this shows that the degeneracy of the 2D gas is not sufficient for the density to significantly deviate from a Gaussian distribution. This is further confirmed by the measurement of negative chemical potentials. However, the semi-classical approach provides an accurate description of the loading of several independent 2D Fermi gases which are not uniformly populated, and this approach can be applied on further degenerate gases. Moreover, the chemical potentials and degeneracies of every layer can be measured from a single absorption image, with the distribution of atoms known.

### 5.5.2 Systematic study as a function of the evaporation depth, for SU(3) and SU(10) gases

The efficiency of the lattice loading with a three spins mixture with populations 10%, 20% and 70%, and an equipopulated SU(10) gas, are here compared as a function of the evaporation depth prior to the loading, read as the geometric mean frequency of the dipole trap $\bar{\omega}$. It is inferred from an independent calibration of the trap geometry. Note that in this study, the evaporation approaches spilling, so that the estimation of $\bar{\omega}$ is uncertain. Hence, the value of $\bar{\omega}$ has to be read with caution, and its use only intends to give a reading of the depth of the evaporation, else than the numeric command that we use on our experiment.

The probability of occupation of the lowest band is plotted on figure 5.7, (a), as a function of $\bar{\omega}$ at the end of the evaporation, and the tendency is a regular increase towards 100% for both mixtures, however a slight increase of the slope can be observed for $\bar{\omega}/2\pi < 50$ Hz. In this measurement, the efficiency of loading with a SU(10) gas is larger by approximately a multiplicative factor 1.05. Figure 5.7, (b), and (c), respectively show the number of atoms in a single spin state, here the most populated of the SU(3) preparation, and temperature of the gases prior to the loading as a function of
<figure><img src="image_59.png" /><figcaption>Figure 5.7: SU(3) (empty squares) and SU(10) (filled circles) loading of the lowest band as a function of the evaporation depth. (a) Probability of occupation of the lowest band $r_0$ as a function of the geometric mean frequency $\bar{\omega}$ of the dipole trap, which is inferred from independent measurements. (b) Number of atoms $N_{at}$ in $m_F = -5/2$, into the dipole trap before loading as a function of $\bar{\omega}$ It represents 70% of the total number of atoms into the SU(3) gas, and 10% into the equipopulated SU(10) gas. (c) Temperature of the gas before loading as a function of $\bar{\omega}$.</figcaption></figure>

$\bar{\omega}$. Ultra-cold temperatures $T \approx 10$ nK are reached from $\bar{\omega}/2\pi \approx 50$ Hz, and further evaporation results in atom depletion only. These measurements show that deep evaporation prior to the lattice loading increases the probability of occupation of the lowest band after loading, and this is associated with very low initial temperatures as well as low chemical potentials.

On figure 5.8, the degeneracy of the 2D gas $T/T_F$ after loading is compared to the depth of evaporation, for both SU(3) and SU(10) gases. In both cases, the measurement shows that deeper evaporation prior to the loading doesn’t permit to produce more degenerate 2D gases. A possible explanation is that in this experiment, we reach very low temperatures at the end of the evaporation by reducing simultaneously the horizontal and vertical confinements of the dipole trap, which reduces significantly the collision rate. Consequently, when the lattice is turned on, adiabatic following of the lattice depth compression is made very difficult, which is already challenging for degenerate fermions due to Pauli blocking.
<figure><img src="image_60.png" /><figcaption>Figure 5.8: Degeneracy $T/T_F$ for SU(3) (left) and SU(10) (right) gases after loading of the lowest band as a function of the evaporation depth.</figcaption></figure>

### 5.5.3 Preparation of degenerate Fermi 2D gases with improved adiabaticity

To enhance the loading efficiency, we experienced two modifications to the previous protocol.

First, we compensated for the low collision rate when the lattice depth is increased. Closed shell $^{87}$Sr atoms do not benefit from Feschbach resonances to tune the interaction strength. Hence, we rather hold a strong transverse harmonic confinement compared to the previous experiment, with the vertical laser beam of the optical dipole trap (ODT), that propagates mostly along the lattice eigenaxis. In the previous experiment, its power was lowered to reach deep evaporation, until the transverse frequency $\omega_{\perp}^{ODT}$ reached approximately $2\pi \times 30$ Hz, and it is now held at approximately $2\pi \times 85$ Hz.

Second, the lattice depth is increased to only $h \times 70$ kHz in 500 ms, which reduces the band gap from $\omega_{\parallel} \approx 2\pi \times 12$ kHz to $\omega_{\parallel} \approx 2\pi \times 6.4$ kHz. Consequently, the transverse harmonic confinement of the lattice is also reduced. However, the vertical ODT beam is not turned off, and the total transverse frequency after loading is now approximately $\omega_{\perp} \approx 2\pi \times 95$ Hz.

The result of the lattice loading with a SU(10) gas of 12000 atoms initially at $T \lesssim 20$ nK and $T/T_F \gtrsim 0.15$ is shown on figure 5.9. With this preparation, the
<figure><img src="image_61.png" /><figcaption>Figure 5.9: Degenerate 2D SU(10) Fermi gas loaded in several sites of a deep 1D lattice. (a) Absorption image (b) Integrated optical depth along the lattice eigenaxis Oz (dots) and result of the fit using equation 5.16 (solid line). (c) Fit residuals, and measured temperature T and 2D chemical potential $\mu_0 = \mu - \epsilon_0$ of the most populated layer for a single spin state. (d) Artificial split of the optical depths of every site, and associated *local* chemical potential inferred from $\mu_0$. (e) Fermi Dirac occupation number $f_{FD}(\epsilon, \mu, T)$ for every site. The energy axis is in bang gap $\hbar\omega_\parallel$ units, and the population of the lowest band is indicated above each site. In every figure, unwritten errorbars are below 0.5% of the fit value.</figcaption></figure>

probability of occupation of the lowest band is increased to approximately 99% in every layer, and the 2D gases are degenerate with $\mu_0/\epsilon_F \ge 0.4$ in the four most populated layers, with approximately 300 atoms per spin state per layer.

This shows that while the collision rate of $^{87}$Sr suffers from a lack of tunability, it can be compensated for by increasing the trapping frequencies. Furthermore, these two ex-
periments indicate that it is much more favorable to hold sufficiently high harmonic con-
finement for isentropic transformations with degenerate alkaline-earth fermions, than
to reach ultra deep evaporation where collisions are strongly inhibited. Also, the final
depth of the lattice can be an important parameter that has to be investigated. Further
systematic study for this protocole is planned.

**5.6 Conclusion and outlook**

In this chapter, I have shown the production of several independent 2D Fermi gases
of $^{87}$Sr atoms with populations ranging from 1000 to 1700 atoms in a three spins
mixture with relative populations 10%, 70% and 20%. In a first set of experiments,
the probability of loading of the lowest band of the 1D lattice is 93(2)% with a band
gap $\omega_{\parallel}$ = $2\pi \times 12$ kHz and Fermi energy $\varepsilon_{F}$ one third of the energy of the first excited
band. Improved adiabaticity is shown in a second experiment by holding sufficient
dipole trap compression prior to the loading into less deep lattice. This way, we are
able to produce 4 independent 2D degenerate SU(10) Fermi gases with $\mu_{0}/\varepsilon_{F} \ge 0.4$
approximately 99% atoms into the lowest band. This preparation will be the starting
point for the loading of 2D lattices to realize a SU(1) band insulator. The 2D gases shall
be adiabatically loaded in a 2D square lattices to prepare several non interacting band
insulators [Leb+18]. The independent 2D Fermi gases can be later used as independent
realizations of a same experiment in a 2D geometry [Fel+11], similarly to wire arrays
experiments [Gre+01; Par+04].
# Conclusion

In this thesis, I reviewed my contributions to the various protocols required for the realization of the 2D Fermi-Hubbard model with enlarged SU(N) symmetry with strontium 87. These key protocols are now sufficient to reach the short term objectives of the experiment, i.e. an adiabatic following to the lowest energy states of a SU(2) quantum magnet, with a full control and monitoring over the spin dynamics in 2D lattices that will provide a first demonstration of super-exchange dynamics on our experiment.

## Summary

In chapter 2, I presented our method to measure local densities with absorption imaging of objects smaller than the resolution limit of our imaging setup. I have shown that the fast variations of the local density are averaged over the resolution limit, and this results in non-linear errors on the imaged density. This correction is demonstrated on elongated Fermi gases, from which we infer the unresolved transverse size, as small as one fourth of our imaging resolution limit. The density profile along the longitudinal axis, distorted by the non linear error, is also recovered with our method. 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. For this, we wanted to further enhance the mean-field signal with a 1D expansion, i.e. by allowing the gas to expand along direction axis only, so that the full mean-field energy is converted to kinetic energy into this axis only. Although this investigation was not conclusive due to thermalization failure, it yielded the establishment of the demonstrated method, which was required for the analysis of the tightly confined gas expanding in 1D.

In the last chapter, I have shown that we can prepare 2D Fermi seas with $\mu/\epsilon_F \gg$
0.4 and approximately 99% atoms in the lowest band of the 1D lattice. With spin selective blast, the Fermi gases will next be polarized, as demonstrated in chapter 3, and adiabatically loaded in the 2D square lattices to prepare several band insulators with a single spin component.

In chapter 4, I demonstrated a method to selectively flip the nuclear spins of strontium 87 with Raman adiabatic passages and a tensor light shift associated with the intercombination line, with actual approximately 80% efficiency. Next, using a spin dependent optical superlattice realized by simply retroreflecting the tensor light shift beam, a Néel spin order shall be written from the band insulators, with approximately 97% fidelity according to my simulations. It will then be possible to adiabatically approach the regime where many-body physics are driven by super-exchange interactions, by slowly reducing the depth of the spin dependent superlattice.

The alternate magnetization along the quantization axis will be measured by first reversing the texture writing protocol, then measuring the spin populations as demonstrated in chapter 3. The dynamics of the spin ordering, that shall deviate from the initial preparation, will provide a first measurement of the super-exchange dynamics.

## Short-term prospects

The spin correlations, that can result effectively in squeezing of the alternate magnetization, will be detected using the same spin dependent optical superlattice. However, this measurement requires supplementary implements, not discussed in this thesis, and yet to be investigated. The squeezing detection requires a measurement of the variance of the alternate magnetization associated with the SU(2) rotation generators $\hat{S}_x, \hat{S}_y$, and $\hat{S}_z$. Practically, these rotations will be engineered with $\pi/2$ light pulses, which notably require a stabilization of the intensity of the laser realizing the tensor light shift.

Moreover, the detection of quantum correlations require to reduce the detection noises and fluctuations as close as possible to the standard quantum limit (SQL) [Ala22]. For this, the implementation of a high resolution imaging system with diffraction limit 1.25 µm and reduced noise sensitivity is ongoing. Note that the ground state of strontium 87 with zero angular momentum is very weakly sensitive to magnetic fields, which
is a strong advantage for minimal systematic noise on the measurement of quantum
correlations.

Also, the current setup is planned to include a 2D spin independent superlattice at 1064 nm. It will permit to isolate 2x2 clusters on the 2D lattice at 532 nm, to prepare local singlet states. By reducing the depth of the superlattice, either abruptly or slowly, this setup will provide an original measurement of the dynamics from local to global correlations.

**Improvement suggestions**

I would like to propose some easy implements that could improve the detection efficiency of the spin populations and facilitate statistical measurements of the spin correlations, as last contribution.

In chapter 3, I have discussed the limitation of our method to measure the spin populations, which is almost entirely due to the laser intensity. In our experiment, we use the retroflected vertical MOT beam to realize the measurement, which has a very large waist of approximately 1 cm and approximately 8 mW power. I suggest to use a dedicated laser setup with a beam of much smaller waist. For instance, a waist of 100 µm with only 80 µW power would increase the light intensity by two orders of magnitude, which would significantly increase the efficiency of measurement of the spin populations to approximately 99%, and hence increase the reliability of the detection of quantum correlations. Furthermore, the simultaneous measurement of multiple spin populations, currently limited by the efficiency of a single adiabatic passage, would be facilitated.

Using a dedicated laser setup, it is possible to change the direction of the momentum
transfer, which is presently along the 1D lattice eigen axis. I suggest that the momentum
transfer should be orthogonal to the lattice eigen axis. In chapter 1, I demonstrated
that we are able to spatially separate the sites of the 1D lattice with the quantum
magnifier method. Combined with an orthogonal spin dependent momentum transfer,
it would be possible to simultaneously measure the spin populations of every layer in
a single shot. Furthmore, I demonstrated in chapter 5 that we are able to prepare at
least 4 independent degenerate 2D gases, that are good candidates to prepare Mott
insulators. Hence, it would be possible to realize at least 4 independent experiments,
simultaneously measurable.

Considering the required statistical measurements to detect quantum correlations
close to the SQL, such a setup would permit to both criticaly enhance the reliability
of the spin populations measurement, and dramaticaly reduce the average time per
experiment to approximately 5 s.

Outlook

This experiment is designed to study SU(N) quantum magnetism with strontium 87.
The ongoing demonstration of the method to prepare spin textures in SU(2) is easily
generalized to SU(N), N > 2, by realizing consecutive spin flips. Very interesting
investigations will then be lead on the growth of correlations, with tunable N. For
instance, frustrated spin ordering is expected [HGR09], and this experiment will offer
a unique platform for this study in square lattices with N as high as 10.

Taking advantage of enhanced large-spin Pomeranchuck cooling [Tai+12], this ex-
periment shall produce strongly correlated 2D spin ensembles. With a control on the
filling factor, a possible implementation of the fermionic t-J model [And87] can be con-
sidered, in SU(2) symmetry, and possibly generalized, in enlarged SU(N) symmetry,
with tunable N.

Remarkably, this experiment makes use of protocoles, notably the artificial magnetic
field, very similar to the ones used to implement topological lattices. Accounting for the
lattice geometry, and the spatially dependent spin orbit coupling associated with the
intercombination line, already implemented on the experiment, artificial gauge fields
could be engineered [Sch+20].
# Appendix A

## Basic theory for d-dimensional Fermi gases in harmonic traps

### A.1 Density of states and Fermi energy

Let's consider a shallow d-dimensional (1D, 2D or 3D) harmonic trap. In the semi-classical limit $k_B T \ll \hbar\bar{\omega}$, the Hamiltonian reads

$$H(r,p) = \sum_{i=1}^{d} \frac{1}{2} m \omega_i^2 r_i^2 + \frac{p_i^2}{2m} \qquad (A.1)$$

The density of states is then derived from Thomas-Fermi rule, which reads in the semi-classical limit

$$\rho(\varepsilon) = \int \frac{d^d r d^d p}{\hbar^d} \delta(\varepsilon - H(r,p)) \qquad (A.2)$$

$$\rho(\varepsilon) = \frac{1}{(d-1)!} \frac{\varepsilon^{d-1}}{(\hbar\bar{\omega})^d} \qquad (A.3)$$

From the density of states, it is then possible to derive the number of atoms with integration over the density of states and Fermi-Dirac occupation number $f_{FD}$

$$N = \int_{0}^{\infty} d\varepsilon \rho(\varepsilon) f_{FD}(\varepsilon, \mu, T) \qquad (A.4)$$
at $T = 0$ K, fermions condensate in the energy states lower than the Fermi energy which is defined as $\epsilon_F = \mu|_{T=0}$, such that the Fermi-Dirac distribution reduces to the heaviside distribution and the number of atoms becomes

$$N = \int_{0}^{\epsilon_F} d\epsilon \rho(\epsilon) \qquad (A.5)$$

Using the definition of the density of states A.3, the d-dimension Fermi energy can be computed directly, with respect to the number of atoms and mean trap frequency $\bar{\omega} = (\prod_i \omega_i)^{1/d}$

$$\epsilon_F = \hbar\bar{\omega}(d!N)^{1/d} \qquad (A.6)$$

On lower dimensions, the density of states is lowered, such that for a fixed number of atoms, fermions have to occupy higher energy states, and thus the Fermi energy is increased, and this is embedded in the $N^{1/d}$ scaling of the Fermi energy.

### A.1.1 3D harmonic trap tightly confined along one direction only

In the following derivation, I consider a gas trapped into a 3D harmonic trap tightly confined along one direction only. The transverse shallow confinement is a shallow 2D harmonic trap with associated phase-space cell $d^2rd^2p/h^2$, and the tight confinement is a quantized harmonic trap with associated degree of freedom $n$, the band excitations. In this case, the Hamiltonian with no interactions reads

$$H_n(r, p) = \frac{p^2}{2m} + \frac{1}{2}m\omega_{\perp}^2 r^2 + (n + \frac{1}{2})\hbar\omega_{\parallel} \qquad (A.7)$$

This hamiltonian is separable, such that the in plane physics and axial quantized physics can be treated separately. In this geometry, the density of states $\rho(\epsilon)$ of the 2D harmonic confinements within each site of the lattice reads:

$$\rho(\epsilon) = \sum_n \int \frac{d^2r d^2p}{h^2} \delta(\epsilon - (\epsilon_{2D}(r,p) + \epsilon_n)) \qquad (A.8)$$

Picturing $\epsilon - \epsilon_n$ as a shift of the 0 energy state, where $\epsilon_n = (n+1/2)\hbar\omega_{\parallel}$, it amounts to assign an independent 2D density of state $\rho_n$ for each energy band $\epsilon_n$:
$$ \rho_n(\varepsilon) = \frac{\varepsilon - \varepsilon_n}{(\hbar\omega_{\perp})^2} \qquad (A.9) $$

and then the total density of states is the sum over all bands:

$$ \rho(\varepsilon) = \sum_n \frac{\varepsilon - \varepsilon_n}{(\hbar\omega_{\perp})^2} \qquad (A.10) $$

One can consider the *local* density of states as defined in equation A.9 for each band, so that the number of atoms in each band *n* can be computed as follows:

$$ N_n = \int_{\varepsilon_n}^{\varepsilon_{n+1}-\delta\varepsilon} d\varepsilon \rho_n(\varepsilon) f_n(\varepsilon, \mu, T) \qquad (A.11) $$

where the Fermi-Dirac occupation number $f_n(\varepsilon, \mu, T)$ is defined as

$$ f_n(\varepsilon, \mu, T) = \frac{1}{\exp(\beta(\varepsilon_{2D}(r,p) + \varepsilon_n - \mu)) + 1} \qquad (A.12) $$

In the $T = 0$ limit of a 2D Fermi gas with $\varepsilon_F < \varepsilon_1$, the number of atoms into the lowest band is then inferred from

$$ N|_{T=0} = \int_{\varepsilon_0}^{\varepsilon_F} d\varepsilon \rho_0(\varepsilon) f_{FD}(\varepsilon, \mu, T) \qquad (A.13) $$

and by replacing $\rho_0$ by its expression given in equation A.9 and $f_{FD}|_{T=0}$ by the heaviside distribution, the Fermi energy reads

$$ \varepsilon_F = \hbar\omega_{\perp}(2N)^{1/2} + \varepsilon_0 \qquad (A.14) $$

This result shows that by taking into account the vertical confinement which allows to engineer a 2D Fermi gas by populating the ground band only of a 1D lattice, the 3D Fermi energy is simply the Fermi energy of the gas living in a 2D layer, c.f. equation A.6, shifted by the energy of the ground band.
## A.2 Density and chemical potential, exact derivation in the semi-classical regime

The in-situ density of a Fermi gas is obtained from momentum integration the Fermi-Dirac phase space density

$$n(r) = \int \frac{d^d p}{h^d} f(r, p) \qquad (A.15)$$

where d is the dimensionality. To derive this integral with the Fermi-Dirac distribution, it requires to use the polylogarithm functions, which can be defined as follows [KZ08]:

$$ \begin{gathered} Li_n(z) = \frac{1}{\pi^n} \int d^{2n}q \frac{1}{e^{q^2/z} - 1} \qquad (A.16) \\ \int_{-\infty}^{+\infty} dx \, Li_n(ze^{-x^2}) = \sqrt{\pi} Li_{n+\frac{1}{2}}(z) \end{gathered} $$

Equation A.15 has to be rewritten in order to apply the definition of the polylogarithm integral. This is easily done by isolating the kinetic energy $p^2/2m$ and multiplying by $-1$:

$$n(r) = - \int \frac{d^n p}{h^n} \left[ \frac{e^{\beta p^2 / 2m}}{-e^{\beta(\mu - V(r))}} - 1 \right]^{-1} \qquad (A.17)$$

with change of variable $y^2 = \beta p^2 / 2m$ and replacing $n = d/2$, the identification with polylogarithm is straightforward and the density reads:

$$n(r) = \frac{-1}{\Lambda_{DB}^d} Li_{d/2}(-e^{\beta(\mu - V(r))}) \qquad (A.18)$$

where $\Lambda_{DB} = h/\sqrt{2\pi mk_B T}$ is the DeBroglie wavelength. It is interesting to see here that the fugacity $q = \beta\mu$ defines the shape of the cloud, while the potential $\beta V(r) = \frac{1}{2}m\omega r^2$ defines the size of the cloud. Indeed, the fugacity is an homogeneous prefactor to the potential term in the highly non linear polylogarithm function, which distorts the density profile defined by $\beta V(r) = \frac{1}{2}m\omega r^2$, for high positive values when
$\mu/\epsilon_F \to 1$ and $T/T_F \to 0$. On the other hand, the phase space density tends to a gaussian distribution for a Boltzmann gas if $\mu/k_B T \to -\infty$.

The number of atoms of atoms can then be computed through integration successive integrations of the density and use of the polylogarithm integration defined in equation A.16:

$$N_d = - \left( \frac{k_B T}{\hbar \bar{\omega}} \right)^d L i_d(-e^{\beta \mu_d}) \quad (A.19)$$

This result is important because it directly shows how the number of atoms is linked to the temperature, the trap, and the chemical potential. This makes sense because the chemical potential is the energy cost to add a new particle to the degenerate gas, whose energy and degeneracy is defined by the trap, the temperature, and the number of atoms. Finally, the chemical potential can be derived from equation A.3 by injecting the definition of Fermi energy A.6.

$$L i_d(-e^{\beta \mu_d}) = d! \left( \frac{T}{T_F} \right)^d \quad (A.20)$$

This derivation requires to invert the polylogarithm, which can easily be done numerically with an interpolation. Note that this result shows that the fugacity only depends on the degeneracy of the Fermi gas and on the dimensionality.

## A.3 Thermodynamics and entropy, derivation in the semi-classical regime

In a statistical ensemble at equilibrium, thermodynamics are defined from the first principle of thermodynamics:

$$dU = \partial_S U dS + \partial_V U dV + \partial_N U dN \quad (A.21)$$

where the thermodynamic potential $U$ is the internal energy for an isolated system. Conjugated thermodynamic quantities ($\mu, N$), ($S, T$) and ($p, V$) can be seen as canonical variables connected by the thermodynamic potential $U$. In this picture, one quantity
is a generalized force $\vec{F}$ deriving from the thermodynamic potential $U$, through its derivative from a generalized displacement $\vec{X}$, i.e. $\vec{F} = -\nabla_{\vec{X}}U$. This is embedded in the well known definitions

$$ 
\begin{aligned}
T &= \partial_S U \\
p &= -\partial_V U \\
\mu &= \partial_N U
\end{aligned} 
\qquad (A.22) $$

The choice of the thermodynamic ensemble doesn't actually change the average values of the thermodynamic quantities [Hua87], but working in the proper ensemble is mandatory to derive the fluctuations. For instance, a gas of ultra-cold atoms in a dipole trap is an isolated system, so it requires that the fluctuations are derived in the micro-canonical ensemble. In our case, we are mainly interested at the average values. For this reason, I will next derive entropy and chemical potential in the grand-canonical ensemble, which is more convenient regarding the study of degenerate fermions. (the chemical potential is properly defined by the degeneracy of the gas, the temperature is properly defined by the cooling procedure (MOT then evaporative cooling (open system here), need to explain it briefly). In the grand-canonical ensemble, the thermodynamic potential is the free energy of Landau $J = U - TS - \mu N$. In this ensemble, the first principle of thermodynamics becomes:

$$ dJ = \partial_T J dT + \partial_V J dV + \partial_\mu J d\mu \qquad (A.23) $$

The chemical potential $\mu$ and entropy $S$ can then defined from the free energy of Landau as

$$ 
\begin{aligned}
S &= -\partial_T J|_{\mu,V} \\
N &= -\partial_\mu J|_{T,V}
\end{aligned} 
\qquad (A.24) $$

Those relations are convenient for fermions in the grand canonical ensemble since $J$ is well defined for degenerate gases, with respect to the number of occupations $f_{FD}$. In the semi-classical limit for fermions, the grand potential reads:
$$J = -k_B T \int d\epsilon \rho(\epsilon) \log(1 - f_{FD}(\epsilon, \mu, T)) \quad (A.25)$$

The density of states can be replaced by equation A.3, and after injecting the Fermi energy derived in equation A.6, one gets

$$J_d = -N k_B T d \int \frac{d\epsilon}{\epsilon_F} \left(\frac{\epsilon}{\epsilon_F}\right)^{d-1} \log(1 - f_{FD}(\epsilon, \mu, T)). \quad (A.26)$$

It is then convenient to rewrite all energies in Fermi energy units $\epsilon/\epsilon_F$, as well as temperature in Fermi temperature units $T/T_F$. This permits to have a universal reading of the results with respect to a single variable $T/T_F$ which can be read as the degeneracy of the Fermi gas. Also, the chemical potential expressed in Fermi energy units gives in a direct reading of the degeneracy, since $\epsilon_F \equiv \mu|_{T=0}$, such that $\mu/\epsilon_F$ has to saturate to 1 at low temperature. This makes sens because all low energy states are occupied and the energy cost to add a new particle converges to the Fermi energy. Hence, reading the saturation of $\mu/\epsilon_F$ gives a direct reading on the energy states occupation. This can be convenient for instance when looking at the sites occupation of the ground band of a lattice. If the Fermi energy is lower than the first excited band, a saturating chemical potential indicates that the occupation number tends to 1 per site. For those reasons, I strongly recommend students to get used to be working on normalized units, such as $T/T_F$ and $\epsilon/\epsilon_F$ in this case. And more practicaly, it makes the derivations easier.

In normalized units, the Fermi-Dirac distribution is simply rewritten as follows:

$$ \begin{aligned} f_{FD}(\epsilon, \mu, T) &= \frac{1}{e^{\beta(\epsilon - \mu)} + 1} \\ &= \left[ \exp\left(\frac{1}{T/T_F}\left(\frac{\epsilon}{\epsilon_F} - \frac{\mu}{\epsilon_F}\right)\right) + 1 \right]^{-1} \end{aligned} \quad (A.27) $$

and then with a change of variable $x = \epsilon/\epsilon_F$, the free energy of Landau can be defined in Fermi energy units, with respect to the degeneracy $T/T_F$:

$$\frac{J}{\epsilon_F}(T/T_F) = -N \frac{T}{T_F} d \int dx \, x^{d-1} \log\left(1 + \exp\left(-\frac{1}{T/T_F}\left(x - \frac{\mu_d}{\epsilon_F}\right)\right)\right) \quad (A.28)$$
This equation might look complicated but it is in fact very simple since it depends only on the number of atoms N and the degeneracy $T/T_F$, such that $J$ can be simply written as

$$ \frac{J}{\varepsilon_F}(T/T_F) = -NI(T/T_F) \qquad (A.29) $$

where the integral $I$ depends only on $T/T_F$.

$$ I(T/T_F) = \frac{T}{T_F} d \int dx \, x^{d-1} \log \left( 1 + \exp \left( -\frac{1}{T/T_F} \left( x - \frac{\mu_d}{\varepsilon_F} \right) \right) \right) \qquad (A.30) $$

Using definition A.24, the entropy per particle can be finally calculated by derivating $I(T/T_F)$ with respect to $T/T_F$:

$$ \frac{S}{N} = k_B \left[ \frac{\partial I(T/T_F)}{\partial (T/T_F)} \right]_{\mu} \qquad (A.31) $$

The entropy per particle can then be computed numerically. In order to have a numerical result in the semi-classical regime with no further approximation, the chemical potential can be computed by using equation A.20. Note that $\mu/\varepsilon_F = \beta\mu\frac{T}{T_F}$.

## A.4 Thermodynamics, entropy and chemical potential, degenerate approximations

Previous derivations allow to compute the chemical potential and entropy per particle numerically, but it can be interesting to derive analytic formulas for better insight. In the degenerate regime, the so called Sommerfield expansion or low temperature expansion is commonly used to approximate integrals dependent on the Fermi Dirac distribution function.

$$ \begin{aligned} I &= \int_{0}^{\infty} d\varepsilon h(\varepsilon) f_{FD}(\varepsilon, \mu, T) \\ &= H(\mu) + \frac{\pi^2}{6} (k_B T)^2 h'(\mu) + \mathcal{O}(k_B T^2) \end{aligned} \qquad (A.32) $$
where $H$ is a primitive of $h$ and $h'$ its derivative. To take advantage of the low temperature expansion, it is convenient to used the grand potential definition in equation A.25 without injecting the Fermi energy expression:

$$J = -\frac{k_B T}{(d-1)!(\hbar\bar{\omega})^d} \int d\epsilon \epsilon^{d-1} \log(1 - f_{FD}(\epsilon, \mu, T)) \quad (A.33)$$

First integrating by part $\epsilon^{d-1}$ and $\ln(1 + e^{-\beta(\epsilon-\mu)})$, the Sommerfield expansion for the free energy of Landau reads in a d-dimensional harmonic trap reads:

$$J(\mu, T) = -\frac{\mu^{d+1}}{(d+1)!(\hbar\bar{\omega})^d} \left(1 + \frac{\pi^2}{6} d(d+1) \left(\frac{k_B T}{\mu}\right)^2 + \mathcal{O}(k_B T^2)\right) \quad (A.34)$$

The derivation of entropy $S$ is then straightforward by directly derivating $J$ with respect to $T$:

$$\frac{S}{N} = d \frac{\pi^2}{3} k_B \frac{T}{T_F} \quad (A.35)$$

The chemical potential is obtained by first computing the number of atoms by derivating $J$ with respect to $\mu$

$$N = \frac{\mu^d}{d!(\hbar\bar{\omega})^d} \left( 1 + d(d-1) \frac{\pi^2}{6} \left( \frac{k_B T}{\mu} \right)^2 \right) \quad (A.36)$$

Then by injecting the d-dimensional Fermi energy equation A.6, and by using a low temperature approximation $k_B T/\mu \simeq T/T_F \ll 1$, the chemical potential finally reads [BR97]:

$$\frac{\mu}{\epsilon_F} \approx \left( 1 - (d-1) \frac{\pi^2}{6} \left( \frac{T}{T_F} \right)^2 \right) \quad (A.37)$$

The analytic formulas obtained for entropy in equation A.35 and chemical potential in equation A.37 are interesting because they show that for stronger degeneracy, i.e. when $T/T_F \to 0$, Fermi gas in low dimension tend to condensate faster than in a higher dimension. This can be seen from the chemical potential that saturates faster to 1 for fixed $T/T_F$ in low dimension, which means that the low energy states, which have indeed lower density in lower dimension as shown by equation A.3, are faster saturated. The resulting entropy is then obviously lower. The numerical results shown in figures
<figure><img src="image_62.png" /><figcaption>Figure A.1: 2D and 3D Chemical potential for degenerate fermi gases. Comparaison between low and high temperature analytic approximations with numerical resolution.</figcaption></figure>

A.1 and A.2 compare the analytical results for low temperature expansion with the numerical computation of the exact solution in the semi-classical regime. First, they show that the low temperature approximation reach about 10% error from $T/T_F \approx 1$.

<figure><img src="image_63.png" /><figcaption>Figure A.2: 2D and 3D entropy for degenerate fermi gases. Comparaison between low and high temperature analytic approximations with numerical resolution.</figcaption></figure>
for 2D gases while the same error is already reached for $T/T_F \approx 0.5$ for 3D gases. This is consistent with the fact that in low dimension, Fermi gases condense faster because of the lower density of states, and hence the degeneracy in the sense of $\mu/\epsilon_F \to 1$ is reached faster for low dimension gases, such that the low temperature expansion holds better at higher $T/T_F$ in 2D than in 3D.

## A.5 Thermodynamics, entropy and chemical potential, classical gas approximations

Finally, it can be interesting to have a glance at the high temperature approximations to bring more insight into the intermediate state from classical to degenerate gas for fermions. Indeed, the condensation of fermions is not a phase transition contrary to bosons, and there is no critical point to define that fermions reach a degenerate state. Hence, it can be convenient to compare the classical gas approximation with the fermi regime in order to better characterize the emergence of degeneracy, when the Boltzmann description doesn't hold anymore.

To derive the average thermodynamic quantities for a Boltzmann gas, it is convenient to work in the canonical ensemble. The relevant thermodynamic potential is the free energy $F$ defined as:

$$F = E - TS \qquad (A.38)$$

where the constraint over the entropy is released with respect to an isolated system. In this case, the first principle of thermodynamic reads:

$$dF = \partial_T FdT + \partial_V FdV + \partial_N FdN \qquad (A.39)$$

The averages of $S$ and $\mu$ are then derived from the free energy with respect to the conjugated variables $T$ and $N$:

$$ \begin{aligned} S &= -\partial_T F \\ \mu &= -\partial_N F \end{aligned} \qquad (A.40) $$
The free energy is computed from the total number of configurations of the system of N particles $Z_N$, also called the partition function.

$$F = -k_B T \log(Z_N) \qquad (A.41)$$

The partition function for N indiscernable particles is directly related to the partition function for 1 particle $Z$ by $Z_N = Z^N/N!$, where

$$ \begin{aligned} Z &= \int \frac{d^d r d^d p}{h^d} e^{-\beta H(r,p)} \\ &= \left( \frac{k_B T}{\hbar \bar{\omega}} \right)^d \end{aligned} \qquad (A.42) $$

The entropy $S$ and chemical potential $\mu$ can then be derived from the free energy, and in the classical approximation they read [BR97]:

$$ \begin{aligned} \frac{S}{N} &= k_B(d + 1 + \log(d! \left(\frac{T}{T_F}\right)^d)) \\ \frac{\mu}{\varepsilon_F} &= -\frac{T}{T_F} \log(d! \left(\frac{T}{T_F}\right)^d) \end{aligned} \qquad (A.43) $$

The analytical result for the Boltzmann gas in d-dimension are plotted in figures A.1 and A.2 and compared to the derivation in the degenerate regime. For low dimensional gas, it is clear that the Boltzmann description doesn't hold for $T/T_F < 1$. For 3D gases, it is possible to identify a smooth transition from the Boltzmann to the Fermi description at $T/T_F \simeq 0.5$, which corresponds to a transition from negative to positive values for $\mu/\varepsilon_F$, and $S/N \simeq 4$.
# Appendix B

## Basic theory for optical lattices

### B.1 Eigenstates and energy bands

Let's consider a periodic 1D dipole trap $V_a(x) = V_0 \sin^2(k_L x)$, of period $a$ such that $V_a(x + a) = V_a(x)$, where $k_L = \pi/a$ is the recoil momentum of the lattice and $V_0$ the depth of the trap, resulting from the interference pattern of a retroflected far off detuned laser beam. The hamiltonian reads:

$$H(x, p) = \frac{p^2}{2m} + V_a(x) \qquad (B.1)$$

Since this hamiltonian is invariant by spatial translation, the Bloch theorem can be applied, such that the eigenstates of this hamiltonian are bloch states

$$\psi_q(x) = u_q(x)e^{iqx} \qquad (B.2)$$

where the quasi-momentum is a quantum number in the reciprocal space used to define each bloch state, and $u_q$ are defined as periodic functions such that $u_q(x + a) = u_q(x)$. It is convenient to rewrite the hamiltonian according to the bloch states definition for diagonalization:

$$e^{iqx} \left[ \frac{(p + \hbar q)^2}{2m} + V_a(x) \right] u_{nq}(x) = \varepsilon_n(q) u_{nq}(x) e^{iqx} \\ \left[ \frac{(p + \hbar q)^2}{2m} + V_a(x) \right] u_{nq}(x) = \varepsilon_n(q) u_{nq}(x) \qquad (B.3)$$
where the relation $p(e^{iqx}u_q(x)) = e^{iqx}(p + \hbar q)u_q(x)$ has been used. The discrete index $n$ stands for each eigenvector of the eigen sub space corresponding to a single value of the real valued quasi-momentum $q$. We can see here that the spectrum of the hamiltonian is completely determined by the functions $u_q$. It is then relevant to decompose the a-periodic $u_q$ functions on the plane wave basis as a Fourier sum:

$$u_q(x) = \sum_{j=-\infty}^{\infty} C_j(q)e^{2i\pi jx/a} \qquad (B.4)$$

where $j$ indicates the Fourier component along the plane wave of momentum $|2jk_L\rangle$. The periodic potential can be rewritten as follows

$$V_a(x) = V_0/2 - V_0/4(e^{2ik_L x} + e^{-2ik_L x}) \qquad (B.5)$$

By injecting the fourrier decomposition of the $u_q$ functions and the linearised potential $V_a(x)$ into equation B.3, we obtain the equation used to compute the spectrum of the Hamiltonian:

$$\left(\left(2j - \frac{q}{k_L}\right)^2 + \frac{V_0}{2E_R}\right)C_j(q) - \frac{V_0}{4E_R}\left(C_{j-1}(q) + C_{j+1}(q)\right) = \frac{\varepsilon_n(q)}{E_R}C_j \qquad (B.6)$$

where $E_R = \hbar^2 k_L^2 / 2m$ is the recoil energy of the lattice. The spectrum is defined with respect to two variables, discrete $n$ standing for the energy levels and continuous $q$ standing for the quasi-momentum. We can clearly see here that lattice potential couples plane waves $|2jk_L\rangle$ to $|2((j \pm 1)k_L)\rangle$, which corresponds to an absorption of one photon and emission of one counter-propagating photon into the lattice photons modes. It is important to note here that the bloch states are periodic with respect to the quasi-momentum $q$, which periodicity which is exactly twice the recoil momentum $k_L$ in the case of simple lattices such as this case. Hence, a $2k_L$ translation in the Bravais lattice doesn't change the state of the particle. For this reason, the spectrum of the hamiltonian has to be defined over one period of the Bravais lattice, which has a width $\delta q = 2k_L$. It is then convenient to define a momentum space $[-k_L; k_L]$, which is the so-called the First Brillouin zone, where the eigen states and eigen energies are properly defined. Equation B.6 can be numerically computed. The result is plotted on
<figure><img src="image_64.png" /><figcaption>Figure B.1: Band structure computed from equation B.6 in a 1D lattice for different depth $V_0$. Energies are in $E_R$ recoil units. The hatched region represent the gap between bands n=0 and n=1.</figcaption></figure>

figure B.1. Energy bands are flattened in deep lattices, and this is mostly visible on the ground band $n = 0$. The energy difference between the ground band and the first excited band $\epsilon_{bg} = min(\epsilon_1(q)) - max(\epsilon_0(q))$ where bg stands for band gap increases with the depth of the lattice, as shown by the hatched region. The band gap opens in the border of the FBZ where the two bands are degenerate at $V_0 = 0$, i.e. $q = \pm k_L$, such that actually $\epsilon_{bg} = \epsilon_1(\pm k_L) - \epsilon_0(\pm k_L)$. In the shallow lattice regime, the coupling between plane waves $j \leftrightarrow j \pm 1$ which is ensured by the lattice potential with strength $-V_0/4$, as shown in equation B.6 can be reduced to first order, such that the hamiltonian developed in equation B.6 reduces to:

$$H = \begin{pmatrix} E_R + V_0/2 & -V_0/4 \\ -V_0/4 & E_R + V_0/2 \end{pmatrix} \qquad (B.7)$$

The diagonalisation of this hamiltonian allows to compute the band gap in the shallow lattice regime:

$$\epsilon_{bg} = \frac{V_0}{2} \qquad (B.8)$$
## B.2 Adiabatic lattice loading with fermions

On ultra-cold atoms experiment with lattices, we typically want to study the lowest energy states of the system by loading only the ground band. This is done by first cooling a gas to degeneracy, until the temperature is close to $T = 0$ K. In this regime, the momentum distribution peaks around $\hbar k = 0$, and the atoms lay in the lowest energy states. In this low energy states, if the momentum distribution doesn't exceed the lattice recoil, then the atomic wave function maps on the ground Bloch states only. By ramping up the lattice depth slower than the coupling between the bands, which can be considered as the band gap, atoms follow the ground Bloch states and remain on the ground band. The condition on the initial momentum distribution, can be quickly shown as follows.

Let's first consider the condition on initial momentum distribution for loading the ground band only. The atomic wave function expands over the eigenstates of the lattice as:

$$|\Phi\rangle = \sum_n \int_{-k_L}^{k_L} dq \langle n, q | \Phi \rangle |n, q\rangle \qquad (B.9)$$

where $|n, q\rangle$ are the Bloch states $\psi_{n,q}(x) = u_{n,q}(x)e^{iqx}$ in the momentum basis. From equation B.4, the decomposition of the Bloch states over the plane wave basis reads:

$$|n, q\rangle = \sum_j C_j(q) |q + 2jk_L\rangle \qquad (B.10)$$

The atomic wave function can then be rewritten in the plane wave basis with a relation de fermeture

$$|\Phi\rangle = \sum_n \int_{-k_L}^{k_L} dq \sum_j C_j^*(q) \langle q + 2jk_L | \Phi \rangle |n, q\rangle \qquad (B.11)$$

If we consider the $V_0 = 0$ case, then the Bloch states are simply plane waves, such that $|n, q\rangle = |q + 2nk_L\rangle$ and then the atomic wave function reduces to:
# Appendix C

## Adiabatic passages in Λ scheme

Optical or radio-frequency (RF) pulses are commonly used to swap the populations of the two coupled quantum states. These schemes rely on the control of Rabi oscillations, requiring a perfect stabilization of the strength of the couplings as well as the duration of the pulse. The advantage of adiabatic passages over pulses is their robustness against experimental fluctuations, e.g. of laser frequencies and amplitudes. While a Raman pulse would require a timed pulse to swap the populations between two spin states, an adiabatic passage requires a time dependent transformation fulfilling the so-called Landau-Zener condition. These conditions can be accurately described and quantified in the dressed-states basis of atom-light interaction. First I will summarize the conditions for adiabatic passage in a three level scheme with 2 photons coupling. Then I will extend the scheme for a 3 spin states manifold.

### C.1 Two ground states

Let's first consider two degenerate ground states $|g_1\rangle$, $|g_2\rangle$, and an excited state $|e\rangle$. Two photon fields $E_1, E_2$ of angular frequency $\omega_1, \omega_2$ couple the ground and excited states as following :

$$ \begin{aligned} \hbar\Omega_1 &= -\langle e | \hat{D} \cdot \mathbf{E}_1 | g_1 \rangle \\ \hbar\Omega_2 &= -\langle e | \hat{D} \cdot \mathbf{E}_2 | g_2 \rangle \end{aligned} \qquad (C.1) $$

where $\Omega_1$ and $\Omega_2$ are the Rabi frequencies, and $\hbar = \frac{\hbar}{2\pi}$ the reduced planck constant.
The scheme for this three level system is shown on figure C.1, left. If both photons are sufficiently detuned from the $|g_1\rangle \leftrightarrow |e\rangle$ and $|g_2\rangle \leftrightarrow |e\rangle$ resonance $\omega_0$, the excited state can be ruled out using second order perturbation. In the dressed states basis $\{|\tilde{g}_1\rangle, |\tilde{g}_2\rangle\}$, the effective hamiltonian can then be written as :

$$ \hat{H}_{eff}(\delta) = \frac{\hbar}{2} \begin{pmatrix} \delta & \Omega \\ \Omega^* & -\delta \end{pmatrix} \qquad (C.2) $$

where $\Omega = \Omega_1\Omega_2^*/\Delta_e$ is the effective two-photons coupling between $|\tilde{g}_1\rangle$ and $|\tilde{g}_2\rangle$, while $\delta = \omega_2 - \omega_1$ is the detuning between the two angular frequencies of two photons from the fields $E_1$ and $E_2$. $\Delta_e$ stands for the detuning $\omega_0 - \omega_1 \simeq \omega_0 - \omega_2$. The spectrum of the effective hamiltonian is obtained by direct diagonalization. Figure C.1, middle, shows the energies of eigenstates $|+\rangle, |-\rangle$, which read:

$$ \begin{aligned} E_{|-\rangle}(\delta) &= -\frac{\hbar}{2}\sqrt{\Omega^2 + \delta^2} \\ E_{|+\rangle}(\delta) &= +\frac{\hbar}{2}\sqrt{\Omega^2 + \delta^2} \end{aligned} \qquad (C.3) $$

with respect to the Raman detuning $\delta$. Far of the Raman resonance $\delta = 0$, i.e. when $\delta \ll \Omega$, the eigenstate $|-\rangle$ ($|+\rangle$) is colinear to the dressed state $|\tilde{g}_1\rangle$ ($|\tilde{g}_2\rangle$), which corresponds to the bare atomic state $|g_1\rangle$ ($|g_2\rangle$), as shown on figure C.1, right. When ramping up the frequency of $\omega_2$, the detuning $\delta$ between the two photons gets closer to zero and so to the Raman resonance, where the effective coupling $\Omega$ lifts the degeneracy by exactly $\Omega$ at resonance.

To realize an adiabatic passage, atoms in state $|\psi\rangle$ can be prepared in $|g_1\rangle$, which is colinear to $|-\rangle$ far of Raman resonance, $\delta \ll \Omega$. If the rate of the detuning ramp is slow enough with respect to the avoided crossing, i.e. $\delta \ll \Omega^2$, then the state $|\psi\rangle$ adiabatically follows the eigenstate $|-\rangle$ all along the ramp. Far of the other side of the Raman resonance, toward the right of figure C.1, right, the dressed state $|-\rangle$ is now colinear to $|g_2\rangle$, so is $|\psi\rangle$. The population has then been transfer from $|g_1\rangle$ to $|g_2\rangle$. The occupation probabilities, initially at $|\langle g_1|\psi\rangle|^2 = 1$, swap at the Raman resonance such that $|\langle g_2|\psi\rangle|^2 = 1$ at the end of the passage.
<figure><img src="image_65.png" /><figcaption>Figure C.1: Two photons coupling with respect to Raman detuning $\delta$. Left: atomic states $|g_1\rangle$ and $|g_2\rangle$ coupled to excited state *kete* with off resonant electric dipole interaction with respective Rabi frequencies $\Omega_1$ and $\Omega_2$. The detuning between the frequencies of the two electric fields is noted $\delta$. Middle: energy spectrum of the dressed states and eigenstates with respect to $\delta$, where the excited state has been rule out. At Raman resonance $\delta = 0$, the degeneracy if lifted by exactly twice the effective rabi coupling $2\Omega$. Right: projection of the eigenstate $|-\rangle$ along dressed states $|\tilde{g}_1\rangle$ and $|\tilde{g}_2\rangle$ with respect to $\delta$.</figcaption></figure>

Adiabatic passage can be seen in the frame of nuclear magnetic resonance. A spin in precession will stay colinear to a rotating magnetic field if the rotation 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. In this discussion, the two ground states are considered degenerate. If they are not, it is all the same but that the Raman resonance happens at $\delta = \epsilon_2 - \epsilon_1$ where $\epsilon_1, \epsilon_2$ are the eigenergies of the bare atomic states $|g_1\rangle, |g_2\rangle$.

In order to maximize the probability of success of the adiabatic passage, three conditions must be respected. First, the initial conditions of the passage must be so that the initial atomic state is colinear to the followed eigenstate, i.e. $\delta \ll \Omega \Rightarrow |g_1\rangle \simeq |-\rangle$. Second, the final conditions of the passage must be so that the final atomic state is colinear to the same followed eigenstate, i.e. $\delta \gg \Omega \Rightarrow |g_2\rangle \simeq |-\rangle$. These two first similar conditions ensure that the connexion between the atomic states and eigenstates
is smooth, preventing Rabi oscillations. The third condition is that the rate of the adiabatic transfer must be slow with respect to the effective coupling between the atomic states, i.e. $\dot{\delta} \ll \Omega^2$. This last condition is embedded in the Landau Zener condition for adiabatic passage, which is that the probability of success of the adiabatic passage is given by:

$$P_{L.Z.}(\Omega, \dot{\delta}) = 1 - \exp\left(-2\pi \frac{\Omega^2}{4\dot{\delta}}\right) \qquad (C.4)$$

and this equation highlights the high sensitivity in $\Omega^2/\dot{\delta}$ as it is exponential.

## C.2 Three states manifold

Let's now consider a more complicated case: three degenerate $|m_F\rangle$ spin states $|-1\rangle$, $|0\rangle$, $|1\rangle$, two photon fields $E_\pi$, $E_{\sigma_+}$ with respective angular frequencies $\omega_\pi$, $\omega_{\sigma_+}$ and polarizations $\pi, \sigma_+$, and an excited states manifold. Obviously, $|m_F\rangle \leftrightarrow |m_F \pm 1\rangle$ are coupled by absorption of a $\pi$ ($\sigma_+$) photon and spontaneous emission of a $\sigma_+$ ($\pi$) photon. In the dressed state basis, the energy difference between two consecutive dressed spin states, i.e. $|-1\rangle$, $|0\rangle$, and $|1\rangle$ is the energy difference between the absorbed and emitted photons $\hbar\delta = \hbar(\omega_\pi - \omega_{\sigma_+})$. Hence, dressed states $|-1\rangle$ and $|1\rangle$ have energy difference $2\hbar\delta$, which corresponds to a four photons transition. This is consistent with conservation of energy which imposes that the energy difference between any two dressed states is equal to the remaining energy of the corresponding photon exchanges. This principle can be extended to more than three states and is discussed in chapter 4. To fully determine the effective hamiltonian in the dressed state basis, we need the effective couplings between the spin states. Similarly to the previous case in C.1, the excited state manifold can be ruled out using second order perturbation if the photon fields are sufficiently detuned from resonance with the excited states. In this case, the effective couplings $|m_F\rangle \leftrightarrow |m_F + 1\rangle$ couplings read:

$$ \begin{aligned} \Omega_{-1,0} &= \langle -1 | \hat{D} \cdot \mathbf{E}_{\sigma_+}^* | e \rangle \langle e | \hat{D} \cdot \mathbf{E}_\pi | 0 \rangle / \Delta_e \\ \Omega_{0,1} &= \langle 0 | \hat{D} \cdot \mathbf{E}_{\sigma_+}^* | e \rangle \langle e | \hat{D} \cdot \mathbf{E}_\pi | 1 \rangle / \Delta_e \end{aligned} \qquad (C.5) $$
where $\Delta_e$ is the detuning with the transition $\omega_0 - \omega_\pi \approx \omega_0 - \omega_{\sigma_+}$. Now, all the matrix elements of the effective hamiltonian in the dressed state basis are known, and it reads:

$$ \hat{H}_{eff}(\delta) = \hbar \begin{pmatrix} \delta & \Omega_{-1,0}/2 & 0 \\ \Omega_{-1,0}^*/2 & 0 & \Omega_{0,1}/2 \\ 0 & \Omega_{0,1}^*/2 & -\delta \end{pmatrix} \qquad (C.6) $$

where $\delta = \omega_\pi - \omega_{\sigma_+}$ is the Raman detuning. In this derivation, the three states are degenerate. Actually, even if they are not degenerate, the physical arguments hold if the energy difference between the ground states is much lower than $\Delta_e = \omega_0 - \omega$, where $\omega \approx \omega_\pi \approx \omega_{\sigma_+}$. In this case, the effective hamiltonian obviously reads:

$$ \hat{H}_{eff}(\delta) = \hbar \begin{pmatrix} \delta & \Omega_{-1,0}/2 & 0 \\ \Omega_{-1,0}^*/2 & 0 & \Omega_{0,1}/2 \\ 0 & \Omega_{0,1}^*/2 & -\delta \end{pmatrix} + \hbar \begin{pmatrix} \varepsilon_{-1} & 0 & 0 \\ 0 & \varepsilon_0 & 0 \\ 0 & 0 & \varepsilon_1 \end{pmatrix} \qquad (C.7) $$

where $\varepsilon_{-1}$, $\varepsilon_0$, $\varepsilon_1$ are the eigenergies of respective bare atomic states $|-1\rangle$, $|0\rangle$, $|1\rangle$. Hence, the degeneracy lift is simply propagated into the energies of the associated respective dressed states. In this case, the 2 photon Raman resonances happen at $\delta = \varepsilon_{-1} - \varepsilon_0$ and $\delta = \varepsilon_0 - \varepsilon_1$, which can be non-zero.

The spectrum of this hamiltonian, obtained after diagonalization, is presented on figure C.2, (a). The crossing between dressed states $|-1\rangle$, $|0\rangle$ at the Raman resonance $\delta = -6$ kHz is avoided by the two photon effective coupling $\Omega = 400$ Hz, as well as the $|0\rangle$, $|1\rangle$ crossing at $\delta = +2$ kHz. The four photons cCaption coupling between $|-1\rangle$ and $|1\rangle$ at $\delta = -2$ kHz is zoomed in the bottom right inset, highlighting that the four photons coupling, scaling as $\Omega^2$, is much lower than the two photons coupling, and hence is the degeneracy lift.

Adiabatic state following is simulated in this manifold and the result for different simulations is presented on figure C.2, (b). The atomic state $|\psi\rangle$ is initially along $|0\rangle$ at $\delta = -14$ kHz, and the detuning is ramped up to $\delta = 11$ kHz for a total chirp of 23 kHz. In this simulation, we follow the projection of $|\psi\rangle$ onto each of the three dressed states all along the detuning ramp, for different rates of the ramp. Figure (b.1) shows a
<figure><img src="image_66.png" /></figure>

Figure C.2: Adiabatic state following in a three level manifold. For this computation, the energies of the bare atomic states are $\epsilon_{-1} = -6 \text{ kHz}$, $\epsilon_0 = 0 \text{ kHz}$, and $\epsilon_1 = -2 \text{ kHz}$, and $\Omega = 400 \text{ Hz}$. The Clebsch-Gordan coefficients are neglected. (a) Colored dashed lines: energies of the dressed states, and solid black lines: eigenenergies of hamiltonian defined in equation C.7, both with respect to the Raman detuning $\delta$. Bottom right inset shows the avoided crossing between $|\tilde{-1}\rangle$ and $|\tilde{1}\rangle$ by four photons coupling at $\delta = -2 \text{ kHz}$. (b) Simulations for adiabatic following of the yellow eigenstate in (a), from initial state $|\psi\rangle$ along $|\tilde{0}\rangle$ at $\delta = -14 \text{ kHz}$, up to $\delta = 11 \text{ kHz}$. Solid lines represent the projection of the atomic state along the dressed states $|\langle\tilde{d}|\psi\rangle|^2$, $d = -1, 0, 1$, with respect to the detuning $\delta$, with colors respective to (a). The detuning is ramped up to $\delta = 11 \text{ kHz}$, and each simulation corresponds to a different chirp rate. (b.1) Frequency ramp in 30 ms. (b.2) Frequency ramp in 200 ms. (b.3) Frequency ramp in 10000 ms.

fast ramp during 30 ms which corresponds to $\Omega^2/\dot{\delta} \simeq 1$. Even though the atomic state seems to flip from $|\tilde{0}\rangle$ to $|\tilde{-1}\rangle$ at the avoided crossing at $\delta = -6 \text{ kHz}$, there remains
strong rabi oscillations and the final projection onto $|\tilde{-1}\rangle$ doesn't converge to 1, which illustrates that the rate of the ramp is too high. The four photon transition is crossed without following, and hence the final population in $|\tilde{1}\rangle$ is negligible, even though a slight increase of the population in $|\tilde{1}\rangle$ is visible. With such a fast ramp, it is then possible to cross the whole spectrum with the frequency ramp and target the $|\tilde{0}\rangle \Rightarrow |\tilde{-1}\rangle$ transition with 95% efficiency. Figure (b.2) shows a slower ramp of duration 200 ms, which corresponds to $\Omega^2/\delta \simeq 8$. Obviously, the two photons transition at $\delta = 6$ kHz is now properly followed, and the four photons transition at $\delta = -2$ kHz is also initiated, such that the population in $|\tilde{1}\rangle$ initially at zero grows to nearly 15%. The last two photon transition at $\delta = 2$ kHz is also properly followed with same conditions than the first one, and populations in $|\tilde{0}\rangle$ and $|\tilde{1}\rangle$ are properly exchanged. This simulation illustrates that with $\Omega^2/\delta \simeq 8$, it is enough to maximize the efficiency of the two photon adiabatic following, while the efficiency of the four photon transition is as low as 15%. Figure (b.3) shows a simulation with frequency ramp duration of ten seconds. This corresponds to a much slower rate such that $\Omega^2/\delta \simeq 400$. This is the maximum ramp rate such that the probability of success of the four photons transition reaches almost 100%, and the eigenstate is properly followed all along the ramp.

This eigenstate following, although useless since the initial and final states are the same, illustrates the different situations that can be encountered in a three levels manifold. The efficiency of a targeted two photon transition for a $|\tilde{0}\rangle \Rightarrow |\tilde{-1}\rangle$ transition can be easily reduced by the near four photons transition as shown by the simulation (b.2), while maximizing the efficiency of the four photon transition requires very low chirp rates compared to the two photons transitions, as shown by simulation (b.3). Actually, even with multiple crossing of the dressed states energies along the frequency ramp, it is possible to achieve maximal efficiency for targeted transitions while avoiding non-wanted transitions.

## C.3 Experimental optimization of adiabatic passage

On experiments, multiple crossing of the dressed states energies are often met. In this case, it is might be possible to tune the rate of the frequency ramp to minimize
the probability of the success of the non-wanted transitions, such as four photons,
while keeping the targeted transition at maximal efficiency. In section C.2, such a
situation is presented, as well as its limits: with a chirp rate fast enough to suppress
the four photons transition, the following of the targeted two photons transition is not
perfect, and a slight decrease of the rate is enough to start the four photons transition.
Actually, the four photons transition can be easily avoided by choosing initial and final
frequencies of the ramp such that it is not crossed. Such a case is presented in figure
C.3, left, where the spectrum is computed from hamiltonian defined in equation C.7.
The black arrow shows the narrow frequency ramp centered around the $|-1\rangle \Rightarrow |0\rangle$
two photons transitions at $\delta = -6 \text{ kHz}$, for an adiabatic state following to flip atomic
state from initial $|0\rangle$ to final $|-1\rangle$. On figure C.3, center, the result of the simulation
for adiabatic following with a ramp duration of 20 ms shows that by narrowing the
frequency ramp, the smooth connexions between dressed states and eigenstates is not
ensured. Indeed, at $\delta_i$, dressed state and eigenstate are not colinear, and hence quickly
turning on the coupling light results in Rabi oscillations. Hence, even at the end of the
ramp, oscillations remain and the final state is a superposition of $|0\rangle$ and $|-1\rangle$, the
adiabatic following has failed. To circumvent this issue, it is possible to smoothly turn
on and off the intensity of the coupling light simultaneously to the frequency ramp.
This way, atoms smoothly connect to the eigenstate at the beginning of the ramp, and
then smoothly reconnect to the atomic state at the end of the ramp. Figure C.3, right,
shows such a simulation, where the intensity ramp is plotted in the inset. The Rabi
oscillations are completely suppressed and atoms end in $|-1\rangle$ with probability 1, the
adiabatic following is successfully achieved.
<figure><img src="image_67.png" /><figcaption>Figure C.3: Left: diagonalization of hamiltonian defined in equation [C.7] with respect to Raman detuning $\delta$. Green(red) dashed line: dressed state $|0\rangle$($|-1\rangle$). Dressed state $|1\rangle$ is not involved in this scheme and greyd. Black vertical lines: initial and final frequencies of the ramp, indicated by the large black arrow. Blue solid line: followed eigenstate. Center and right: simulation of adiabatic state following with $\delta_i = -8 \text{ kHz}$ and $\delta_f = -4 \text{ kHz}$, with ramp duration 20 ms. Center: square intensity window. Right: gaussian intensity window with standard deviation $\sigma = 5$ ms. The intensity windows are shown in respective insets.</figcaption></figure>
# Appendix D

## Guided expansion - parallaxe - anticonfinement

This appendix is dedicated to the following problem: we let a thermal Boltzmann gas expand while channeled in a far red detuned laser beam. The gas is tightly confined in two directions, and is allowed to expand in the axial direction of the laser beam. The motion of each individual atom is determined by gravity and the atom-light dipole force. Along the longitudinal extension of the channeling atomic gas, the dipole force can be locally approximated as a mean force acting on the center of mass of the atoms, and a local curvature acting on the spread of the atomic gas. The red-detuned trapping beam at $\lambda = 1070$ nm is focused at 4 Rayleigh lengths above the initial position of the gas, with a $25~\mu m$ waist. Therefore, the atoms lay after the inflexion point of the dipole potential, such that the atoms feel an anharmonic trapping. The position dependant local curvature can then be associated with an imaginary frequency $\omega$. Furthermore, the laser beam has angle $\theta$ with respect to the vertical direction. Gravity is then affected by parallax $\theta$, and the sizes and center of mass positions of the atomic gas during its channeled expansion are imaged with the same parallax, $\theta$. The goal is to measure the frequency associated with the anharmonic trapping, and the parallax.
<figure><img src="image_68.png" /><figcaption>Figure D.1: Geometry of the problem. Atoms (black ball) expand while falling along the far red detuning laser beam whose direction, here noted Oz makes angle $\theta$ with the gravity axis. The waist of the beam is located above the atoms, resulting in an axial attractive force $F_{dip}$. Absorption images are realized with parallaxe $\theta$.</figcaption></figure>

# D.1 Derivation

## D.1.1 Effective gravity

Let's consider a single atom trapped by an **isotropic and non-astigmatic** laser beam, and falling along the axis of the laser beam. The dipole force associated with the atom-light interaction along the axis of the beam can be written as

$$ 
\begin{aligned}
U_{dip}(z) &= \frac{U_0}{w(z)^2} \\
w(z) &= w_0 \sqrt{1 + \left(\frac{z - z_0}{z_R}\right)^2}
\end{aligned} 
\qquad (D.1) $$

where $U_0$ is depth of the dipole potential, $z_R$ the Rayleigh length, and $w(z)$ the width of the laser beam at $I_0 e^{-2}$. The mean force $F_{dip} = -\partial_z U_{dip}$ can be derived from the potential.

$$ F_{dip}(z) = -2 \frac{U_0 (z - z_0) / z_R^2}{w_0^2 (1 + (z - z_0)^2 / z_R^2)^2} \qquad (D.2) $$
Since the atoms start at $4z_R$ from the focus of the beam, we can approximate $z - z_0 \gg z_R$ such that

$$F_{dip}(z) \approx -2 \frac{U_0}{w_0^2 \bar{z}^3}. \qquad (D.3)$$

We write $z - z_0 = \bar{z}$ for simplicity. For a short time of channeling into the laser beam with low initial velocity, the dipole force can be approximated as a uniform force field along the trajectory of the atoms, and then associated with an effective gravity $g_{eff}$

$$F_{dip}(z) \approx -mg_{eff} \qquad (D.4)$$

The local curvature can then be associated with a frequency $\omega$ such that

$$\partial_z F_{dip} = -m\omega^2 \qquad (D.5)$$

Using eq. D.3, D.4 and D.5, we can link the effective gravity $g_{eff}$ with the frequency $\omega$ such that $-m\omega^2 = \frac{-3}{\bar{z}}(-mg_{eff})$ and

$$g_{eff} = -\frac{1}{3}\omega^2\bar{z}. \qquad (D.6)$$

The atoms lay after the inflexion point of the dipole potential, such that the atoms feel an anharmonic trapping. The position dependant local curvature can then be associated with an imaginary frequency $\omega$. Thus we define $\omega = i\sqrt{\alpha} \Leftrightarrow \alpha = -\omega^2 > 0$. Finally

$$g_{eff} = \frac{1}{3}\alpha\bar{z}. \qquad (D.7)$$

## D.1.2 Center of mass

The atoms fall along the axis of the beam with a slope $\theta$, see fig. D.1. Furthermore, the channelled fall of the atoms is slowed down by an effective anti-gravity coming from the dipole attraction of the focus of the beam. The motion of the atoms is then given by

$$\ddot{z} = g \cos\theta - g_{eff} \qquad (D.8)$$
The positions of the atoms are imaged by absorption imaging at different times of channelled fall. The camera chip, supposed to be orthogonal to the imaging beam and parallel to gravity, the atoms are imaged with parallax $\theta$. Thus in the camera frame, the motion $z_M$ is imaged with parallax $\theta$ such that

$$z_M(t) = \cos(\theta)\left(\frac{1}{2}g \cos(\theta)t^2 - \frac{1}{2}g_{eff}t^2\right) \qquad (D.9)$$

The integration constants $v_0$ and $z_0$ can be neglected since the measurement is made on the center of mass of the thermal gas. A direct measurement of fall acceleration $g_M$ without taking into account the effective gravity and parallax would be $z_M(t) = \frac{1}{2}g_M t^2$, such that the measurement can be linked to the parallax $\theta$ and effective gravity $g_{eff}$. Using eq. D.7

$$g_M = g \cos(\theta)^2 - \frac{1}{3} \alpha \bar{z} \cos(\theta) \qquad (D.10)$$

We now need a second measurement to solve the system.

### D.1.3 Cloud width

The gas is initially trapped at thermal equilibrium with distribution

$$f(z, v) \propto \exp\left(-\frac{1}{2} \frac{m\omega_0^2 z^2}{k_B T}\right) \exp\left(-\frac{1}{2} \frac{mv^2}{k_B T}\right) \qquad (D.11)$$

with $\omega_0$ the frequency of the harmonic trapping potential, and $T$ the temperature of the gas. Assuming that an atom is isolated into the laser beam, its motion along the axis of the beam depends only on the dipole force. Using the anharmonic trap local approximation and $\omega = i\sqrt{\alpha}$, the equation of motion can be written as $\ddot{z} = \alpha(t)z$, and then integrated to

$$z(t) = z_0 \cosh(\sqrt{\alpha}t) + \frac{v_0}{\sqrt{\alpha}} \sinh(\sqrt{\alpha}t) \qquad (D.12)$$

The position probability distribution $P(x_0, t) = \int dx dv f(x, v) \delta(x(t) = x_0)$ can then be computed from eq. D.11 and eq. D.12. The width of the probability distribution $\sigma_z$ is

$$\sigma_z^2(t) = \frac{k_B T}{m} \left( \frac{\sinh^2(\sqrt{\alpha}t)}{\alpha} + \frac{\cosh^2(\sqrt{\alpha}t)}{\omega_0^2} \right) \qquad (D.13)$$
The measurement of the size $\sigma_z$ with parallax $\theta$ gives $\sigma_M = \cos(\theta)\sigma_z$. Furthermore, the temperature is inferred from the measurement of the size $\sigma_{3D}^2 = k_B T/m$ after a time of flight of a gas totally released from the trap. For a long enough time of flight, $\sigma_{3D}^2(t) \simeq \frac{k_B T}{m} t^2$, such that

$$ \frac{\sigma_M^2}{\sigma_{3D}^2}(t) \approx \frac{\cos^2(\theta)}{t^2} \left( \frac{\sinh^2(\sqrt{\alpha}t)}{\alpha} + \frac{\cosh^2(\sqrt{\alpha}t)}{\omega_0^2} \right) \quad (D.14) $$

With the two equations D.10 and D.14, we define a set of two equations with two unknowns $\alpha$ and $\theta$, and dependent on two experimentally measurable quantities, the fall of the center of mass and the expansion of the channelled thermal gas. Hence, it is now possible to infer the anticonfinement parametrized by $\alpha$ and the parallaxe $\theta$.

## D.2 Measurements

The measurements of the centers of mass and sizes, shown on figure D.2, are made on hot thermal gas $T > 2 T_F$, and for time of flights $t_{tof} \ge 14$ ms long enough such that the approximation $\sigma_{3D}^2(t) \simeq \frac{k_B T}{m} t^2$ holds. The channelling gas is imaged with absorption imaging, while inside the beam, and this in-situ absorption imaging requires to correct the imaged densities such that the size $\sigma_z$ is correctly measured, see chapter 2.

On figure D.2, (a), the effective gravity $g_{eff} \simeq 7.2 m/s^2$ is inferred from the measurement of the channeled fall, and the gravity is calibrated from time of flight measurement. The agreement between the measured effective gravity, and its expected value from equation D.10, for all values of $(\alpha, \theta)$, is shown on figure D.2, (b). All sets of $(\alpha, \theta)$ values which verify equation D.10, so that the expected value match the measured value, are obtained and plotted on the white dotted line. The sizes of the channelled gas and the freely expanding gas are measured and plotted on figure D.2, (c), with respect to time. The agreement between the measured effective gravity, and its expected value from equation D.10, for all values of $(\alpha, \theta)$, is shown on figure D.2, (d). All sets of $(\alpha, \theta)$ values which verify equation D.14 and so that the expected value is in agreement with the size measurement (c), are obtained and plotted on the white dotted line.

The two sets of $(\alpha, \theta)$ values obtained from the center of fall measurements and size
<figure><img src="image_69.png" /><figcaption>Figure D.2: (a) Measurement of the position of the center of mass, for the channeled expansion (squares) and time of flight experiment (dots), with respect to the time. (b) 2D resolution of equation D.10 with respect to $\alpha$ and $\theta$. The white line (closest to zero) indicates the values with best agreement with the data shown on (a). (c) Measurement of the size of the gas, for the channeled expansion (squares) and time of flight experiment (dots), with respect to the time. (d) 2D resolution of equation D.14 with respect to $\alpha$ and $\theta$. The white line (closest to zero) indicates the values with best agreement with the data shown on (c). For (a) and (c), the size of a pixel is 6.45µm.</figcaption></figure>

expansion measurements are put together, see figure D.3. The 2D system is then solved as only one tuple $(\alpha, \theta)$ is solution, here ($2^2 \text{Hz}^2$, 30.21°). On figure D.4, we compare
<figure><img src="image_70.png" /><figcaption>Figure D.3: Combination of the resolutions of equations D.14 and D.10 shown on figures D.2 (b) and (d). The crossing reveals the unique solution $(\alpha, \theta)=(2^2, 30.21^\circ)$.</figcaption></figure>

<figure><img src="image_71.png" /><figcaption>Figure D.4: Curvature of the axial dipole potential in (Hz). The atoms start the channeled fall at position 0 and are imaged at 15 mm away from initial position after 20 ms of channeled fall. This expectation is obtained from the known focus, size and depth of our optical dipole trap, which were calibrated from center of mass oscillations and beating of the cloud size.</figcaption></figure>
the obtained anti-confinement with expected value (geometry optimization of the trap
with oscillation frequencies). After 20 ms of channeled fall, the atoms are 1.5 mm away
from the initial position. At this position, the anti-confinement is expected to be a
little below 2 *Hz*, which is in good agreement with the measurement measurement.
Furthermore, the parallax has been measured directly on the experiment with a ruler,
$\theta_{ruler} = 29.5 \pm 1.5^{\circ}$, also in good agreement with this method.
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# Manipulation of the nuclear spin states of $^{87}$Sr in degenerate SU(N)-symmetric Fermi gases

## Abstract

This thesis reviews our implementation of a quantum simulator for the Fermi-Hubbard model with strontium 87. Closed-shell $^{87}$Sr with large nuclear spin I=9/2 benefits from a 7.4 kHz narrow intercombination line that played a central role in this thesis. It offers ideal conditions for the realization of spin orbit coupling schemes to coherently manipulate the 10 nuclear spin states with minimal spontaneous emission. First, we demonstrated a method to measure local densities smaller than the resolution limit of our imaging system. We overcome the limits due to aberrations and fast variations of the density profiles. Derived from the Beer-Lambert law, the non-linear correction lets us infer sizes as small as 1/4 pixel. Then, we prepare polarized Fermi seas of $^{87}$Sr at $T/T_F \simeq 0.25$, with a spin purification by optical pumping. The spin populations are measured with a spin orbit coupling scheme associated with the narrow line, realizing a spin dependent momentum transfer. We then demonstrate a method to coherently and selectively flip the spin states. The degeneracy of the ground state is lifted with a quadratic light shift. With a Raman adiabatic passage, the spins are flipped with 80% efficiency that will be improved up to 95%. Designed as an optical super lattice, this scheme will permit to write deterministic spin textures that will be the initial state for the quantum simulator. Finally, we study the loading of Fermi gases into the ground band of a 1D lattice with large sites spacing 2 µm. Our experiment highlights the critical importance of maintening a sufficient collision rate to adiabatically follow the 3D to 2D dimensionnality cross-over, and we simultaneously produce four 2D Fermi seas.

**Keywords:** nuclear spins, strontium 87, spin orbit coupling, Raman adiabatic passage, degenerate Fermi gas, SU(N) symmetry, optical lattices.

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

## Résumé

Cette thèse présente notre implémentation d'un simulateur quantique pour le modèle de Fermi-Hubbard avec des atomes de strontium 87. Le $^{87}$Sr de grand spin nucléaire I=9/2 bénéficie d'une raie d'intercombinaison étroite de 7.4 kHz jouant un rôle central dans cette thèse. Elle offre un cadre idéal pour manipuler les 10 états de spin nucléaires par couplage spin-orbite avec une émission spontanée minimale. D'abord, nous démontrons une méthode pour mesurer des densités locales non résolues par notre système d'imagerie. Nous surpassons les limites dues aux abbérations et variations rapides du profile de densité, par une correction non-linéaire établie à partir de la loi de Beer-Lambert. Nous déduisons des tailles jusqu'à 0.25 pixel. Ensuite, nous préparons des mers de Fermi de $^{87}$Sr polarisées à $T/T_F \simeq 0.25$ en utilisant une purification de spin par pompage optique. Les populations de spin sont mesurées par des transferts d'impulsion sélectifs en spin grâce au couplage spin orbit. Nous démontrons de plus une méthode pour retourner les spins de manière cohérente et sélective. La dégénérance de l'état fondamental est levée par un déplacement lumineux quadratique. Par passage Raman adiabatique, les spins sont retournés avec une efficacité de 80% qui sera améliorée à 95%. Conçu comme un réseau optique, cette méthode permettra l'écriture deterministe d'une texture de spin qui sera l'état initial du simulateur quantique. Enfin, nous étudions le chargement d'un gaz de Fermi dans la bande fondamentale d'un réseau 1D de grande période 2 µm. Nous montrons l'importance de maintenir un taux de collision suffisant pour suivre le cross-over de géométrie de 3D à 2D, et nous produisons simultanément 4 mers de Fermi 2D.

**Mots clefs:** spins nucléaires, strontium 87, couplage spin-orbit, passage adiabatique Raman, gas de Fermi dégénérés, symmétrie SU(N), réseaux optiques.