assumption is made about the state to be reconstructed (as opposed to the evolution fits shown in Sec. 3.3.1). The only details needed by the algorithm are “contextual”, namely the lattice depth $s_0$ and the data time stamps corresponding to the evolution data fed to the algorithm. We start with reminders on measurement in quantum mechanics [150], and we then present the reconstruction algorithm that we use in the following section. Let $\hat{\rho}$ be the density matrix<fnref n="10" /> associated to the state of a system that we want to reconstruct from a set of measurements.

**Projective measurement.** We consider the projective measurement of the physical quantity $a$ associated with the observable operator $\hat{A}$ with eigenvalues and eigenvectors $\{a_i, |a_i\rangle\}$. When measuring $\hat{A}$ on $\hat{\rho}$, one gets the result $a_i$ with probability

$$ \pi_i = \mathrm{tr}\{\hat{P}_i \hat{\rho}\} \quad \text{and the state becomes} \quad \hat{\rho} \rightarrow \hat{\rho}' = \frac{\hat{P}_i \hat{\rho} \hat{P}_i}{\pi_i}, \qquad (3.41) $$

where $\hat{P}_i = |a_i\rangle\langle a_i|$ is the projector on the corresponding eigenstate, with $\sum_i \hat{P}_i = 1$. This ideal projective measurement scheme is also refered to as von Neumann measurement [150].

**Generalized measurement.** We can describe a more general measurement that is not based on projectors anymore. We consider a set of operator $\hat{M}_i$ (named Kraus operators) acting on the system, the only condition being $\sum_i \hat{M}_i^\dagger \hat{M}_i = 1$. When measuring the state of the system, we get the result $i$ with probability

$$ \pi_i = \mathrm{tr}\{\hat{\rho}\hat{M}_i^\dagger \hat{M}_i\} \quad \text{and the state becomes} \quad \hat{\rho} \rightarrow \hat{\rho}' = \frac{\hat{M}_i \hat{\rho} \hat{M}_i^\dagger}{\pi}. \qquad (3.42) $$

Projective measurement is a special case of generalized measurement where the Kraus operators $\hat{M}_i$ are projectors. We define the positive operator-valued measure (POVM) formed by the set of operators $\hat{E}_i = \hat{M}_i^\dagger \hat{M}_i$ (which are not necessarily projectors).

We now consider an unknown state $\hat{\rho}$, prepared a large number of times, on which have been performed a set of measurements of the POVM $\{E_i\}$. We are given the frequencies $f_i$ (with $\sum_i f_i = 1$) with which the results $i$ were obtained. Reconstructing $\hat{\rho}$ consists in finding the theoretical state most likely to have produced these results. This state $\hat{\rho}_{\mathrm{ML}}$ is the one that maximizes the likelihood function $\mathcal{L}$ defined as

$$ \mathcal{L}[\hat{\rho}] = \prod_i \mathrm{tr}\{\hat{\rho}\hat{E}_i\}^{f_i} \quad \text{and} \quad \hat{\rho}_{\mathrm{ML}} = \arg\max_{\hat{\rho}} \{\mathcal{L}[\hat{\rho}]\} \qquad (3.43) $$

However, finding $\hat{\rho}$ is a tedious endeavor, as one needs to set all the matrix elements of $\hat{\rho}_{\mathrm{ML}}$. Precisely, in a Hilbert space of dimension $d_H$, symmetric density matrices have $d_H$ real numbers

<footnote n="10">¹⁰For a one-body pure state $|\psi\rangle$, one has $\hat{\rho} = |\psi\rangle\langle\psi|$. As will be explained below, we will however not restrict our analysis to pure states. In general, one rather has $\hat{\rho} = \sum_i p_i |\psi_i\rangle\langle\psi_i|$, where $p_i$ is a classical probability for the state defined by $\hat{\rho}$ to be measured in the state $|\psi_i\rangle$, with $\sum_i p_i = 1$.</footnote>
on the diagonal, and $d_H(d_H - 1)/2$ complex numbers as unique off-diagonal coefficients, which amounts to $(d_H)^2$ real number to determine. We therefore turn to a numerical iterative process to reconstruct $\hat{\rho}_{ML}$.

**Iterative maximum likelihood algorithm and application to our case.** The algorithm that we implement to reconstruct $\hat{\rho}_{ML}$ is inspired from quantum optics [138, 140, 141] with recent implementations on matter wave systems [151, 152]. We consider a POVM $\{E_i\}$, with the experimental frequency $f_i$ for the measure $i$, and a guess density matrix $\hat{\rho}$. The algorithm is based on the definition of an operator $\hat{R}[\hat{\rho}]$:

$$ \hat{R}[\hat{\rho}] = \sum_i \frac{f_i}{\text{tr}\{\hat{\rho}\hat{E}_i\}} \hat{E}_i, \qquad (3.44) $$

and the transformation

$$ \hat{\rho} \rightarrow \hat{\rho}' = \frac{\hat{R}[\hat{\rho}] \hat{\rho} \hat{R}[\hat{\rho}]}{\text{tr}\{\hat{R}[\hat{\rho}] \hat{\rho} \hat{R}[\hat{\rho}]\}}. \qquad (3.45) $$

We remark that if $\hat{\rho}$ is the maximum likelihood density matrix $\hat{\rho}_{ML}$, we have $\pi_i = \text{tr}\{\hat{\rho}_{ML}\hat{E}_i\} \approx f_i$, implying that $\hat{R}[\hat{\rho}_{ML}] \approx \sum_i \hat{E}_i = 1$. Thus $\hat{\rho}_{ML}$ is left unchanged by the transformation (3.45), with $\mathcal{L}[\hat{\rho}'] \ge \mathcal{L}[\hat{\rho}]$ [140]. The iterative application of the transformation (3.45) thus yields a series of density matrices that have increasing likelihood with respect to the experimental data.

Concretely, we measure in our case the relative plane wave atomic populations $|c_\ell(t)|^2$ of the prepared state evolving in the static lattice during a time interval $t \in [t_c, t_c + t_{\text{rec}}]$ with a number $N_t$ of measurement times (unless specified otherwise, we use $t_{\text{rec}} = 100$ µs and $N_t = 21$ time steps). Considering all the evolution data as our measurement ensemble, an atom is measured at time $t$ in the plane wave $\ell$ with frequency

$$ f_i = f_{\ell,t} = \frac{1}{N_t} |c_\ell(t)|^2. \qquad (3.46) $$

As we intend to use these measurements to reconstruct the state prepared at $t_c$, the elements of the POVM are therefore

$$ \hat{E}_i = \hat{E}_{\ell,t} = \frac{1}{N_t} \hat{U}^\dagger(t, t_c) |\chi_\ell\rangle \langle\chi_\ell| \hat{U}(t, t_c), \qquad (3.47) $$

with $\hat{U}(t, t_c)$ the evolution operator in the static lattice potential from $t$ to $t_c$. This allows to define the operator $\hat{R}$ (3.44) for a given density matrix. With these expressions, we implement the algorithm summarized in the following box.
## Summary of the iterative maximum likelihood algorithm

1. Set an initial guess state $\hat{\rho}^{(i=0)}$.

2. At iteration $i$, construct $R[\hat{\rho}^{(i)}] = \sum_j f_j E_j / \text{tr}\{\hat{\rho}^{(i)} E_j\}$.

3. Transform $\hat{\rho}^{(i)} \rightarrow \hat{\rho}^{(i+1)} = R[\hat{\rho}^{(i)}] \hat{\rho}^{(i)} R[\hat{\rho}^{(i)}] / \text{tr}\{R[\hat{\rho}^{(i)}] \hat{\rho}^{(i)} R[\hat{\rho}^{(i)}]\}$.

4. Iterate from step 2 until $\mathcal{L}[\hat{\rho}^{(i+1)}] - \mathcal{L}[\hat{\rho}^{(i)}] \approx 0 \Leftrightarrow \hat{\rho}^{(i+1)}$ converged to $\hat{\rho}_{\text{ML}}$.

<figure><img src="image_40.png" /></figure>

As our initial guess for the one-body density matrix, we set $\hat{\rho}^{(0)} = 1_{d_H}/d_H$, with $d_H = 2\ell_{max} + 3$ chosen as the cut-off dimension of the Hilbert space in order to avoid boundary effects, $\ell_{max}$ being the highest diffraction order at which some signal is experimentally detected (for the experiments presented in the following section, we have $2 \le \ell_{max} \le 6$). Our choice for $\hat{\rho}^{(0)}$ is the state that maximizes the von Neumann entropy$^{11}$, which corresponds to the most unbiased guess with no assumption on the state to be reconstructed. Finally, two indicators are computed to certify the preparation: the fidelity of $\hat{\rho}_{\text{ML}}$ to the numerically propagated state $F_{\text{exp}} = \langle\psi_{\text{QOC}}|\hat{\rho}_{\text{ML}}|\psi_{\text{QOC}}\rangle$ and the purity $\gamma = \text{tr}\{\hat{\rho}_{\text{ML}}^2\}$ which is an indicator of our preparation reproducibility over the realizations used for the reconstruction (see next paragraph). We illustrate the quantum state reconstruction process in Fig.3.7 with an experimental example taken from the next section.

**Note on the framework shift and the introduction of density matrices.** Beside the reconstruction formalism, one can apprehend the shift from working with kets to working with one-body density matrices as the need to introduce mixed states in order for the reconstructed evolution to have a chance to match the experimental data. Indeed, we recall that our experimental data along time are the preparation of several states $|\psi_j(t_c)\rangle$ identical to the extent of our experimental abilities and evolved for increasing times (before being destructed for the momentum distributions measurement, see Sec. 2.3). Thus, one can expect the theoretical state whose evolution maximizes the likelihood with the experimental data to be a state resulting of some statistical mix of the states $|\psi_j\rangle$, with a purity $\gamma$ quantifying the reproducibility of the preparation.

In the following, we apply this state reconstruction scheme to evaluate the preparation of translated Gaussian states, superpositions of Gaussian states and squeezed Gaussian states using our QOC protocol.

### 3.3.4 Non-squeezed Gaussian states

In this section, we prepare and reconstruct non-squeezed Gaussian states centered at arbitrary positions in phase space. As already introduced in Sec. 1.4.1 where we introduced the Husimi representation of a state in the $(x,p)$ phase space, we define non-squeezed Gaussian states in our Bloch system as the Gaussian states that have the same $(x,p)$ aspect ratio as the

<footnote>$^{11}$The von Neumann entropy reads $S = -\text{tr}\{\hat{\rho}\ln\hat{\rho}\}$.</footnote>
<figure><img src="image_41.png" /><figcaption>**Figure 3.7** | Quantum state reconstruction in a Bloch system by likelihood maximization. (a-b) Density matrices $\hat{\rho}$ with $\arg\{\hat{\rho}_{i,j}\}$ color coded and $|\hat{\rho}_{i,j}|$ size coded (not to scale between panels). (a) Identity matrix $1_{d_H}/d_H$ ($d_H = 11$) as the initial guess. (b) Density matrix of maximum likelihood $\hat{\rho}_{ML}$. (c) Stack of experimental integrated absorption images taken during the evolution of the prepared state in the static lattice at $s = 5.5 \pm 0.5$. (d) Diagonal terms of the numerical propagation of $\hat{\rho}_{ML}$, whose evolution maximizes the likelihood of results in (c). These data correspond to the experiment of Fig. 3.8(d) (see below).</figcaption></figure>

ground state of the harmonic oscillator that approximates the bottom of each lattice well. We here denote as $|g(u, v)\rangle$ this state displaced in phase space by $(u, v) = (k_L\langle\hat{x}\rangle_{g(u,v)}, \langle\hat{p}\rangle_{g(u,v)}/\hbar k_L)$. In this system of coordinates, Eq. (1.50) for the plane wave coefficients of this state becomes

$$c_{\ell}(u, v) = \left( \frac{2}{\pi \sqrt{s_0}} \right)^{1/4} e^{iuv/2} e^{-ilu} e^{-(l-v)^2/\sqrt{s_0}}. \qquad (3.48)$$

We write $\Delta x_0$ and $\Delta p_0$ the position and momentum standard deviation for this non-squeezed state. We have $k_L\Delta x_0 = s^{-1/4}$ and $\Delta p_0/\hbar k_L = s^{1/4}/2$.

To present the results for the preparations of such phase space distributions, we compare the Husimi function $Q_{\hat{\rho}}(u, v) = \langle g(u, v)|\hat{\rho}|g(u, v)\rangle/2\pi$ (see Sec. 1.4.1) of the numerically prepared states $\hat{\rho}_{QOC} = |\psi_{QOC}\rangle\langle\psi_{QOC}|$ and corresponding density matrices $\hat{\rho}_{ML}$ reconstructed from experimental data. We show results of non-squeezed state preparations in Fig. 3.8(a-c) with quantitative details in Table 3.4. We see that we are able to prepare non-squeezed Gaussian states with high fidelity to numerical simulations ($F_{\text{exp}} \ge 0.86$) and great purity ($\gamma \ge 0.93$). As discussed in the previous section, our experimental reconstruction data comes from several
<figure><img src="image_42.png" /><figcaption>**Figure 3.8** | Experimental preparation and measurement of non-squeezed Gaussian states. (a-e) Husimi representations in the phase space of the static lattice. Top (red): states $|\psi_{QOC}\rangle$ numerically prepared by optimal control. Bottom (green): density matrices $\hat{\rho}_{ML}$ reconstructed from experimental data by likelihood maximization. The relative phases in the superpositions (d,e) are respectively 0 and $\pi$ (see text). The colormap value for each Husimi function extends from 0 to its maximum value. See Table 3.4 for associated experimental parameters and figures of merit.</figcaption></figure>

<table><caption>**Table 3.4** | Parameters used for the preparation of non-squeezed Gaussian states and figures of merit obtained from their reconstruction. For all experiments $F_{num} > 0.995$ and $t_c = 1.75 T_0$. This table refers to Fig. 3.8.</caption><thead><tr><th>Fig. 3.8</th><th>a</th><th>b</th><th>c</th><th>d</th><th>e</th></tr></thead><tbody><tr><td>u</td><td>π/2</td><td>0</td><td>π/2</td><td>±π/2</td><td>±π/2</td></tr><tr><td>v</td><td>0</td><td>√s</td><td>√s/2</td><td>±√s</td><td>±√s</td></tr><tr><td>F<sub>exp</sub></td><td>0.95</td><td>0.86</td><td>0.93</td><td>0.89</td><td>0.91</td></tr><tr><td>γ</td><td>0.95</td><td>0.96</td><td>0.93</td><td>0.82</td><td>0.91</td></tr><tr><td>s</td><td>5.50±0.25</td><td>5.49±0.20</td><td>5.57±0.20</td><td>5.5±0.5</td><td>5.30±0.25</td></tr></tbody></table>

independent initial states evolved for different durations before measurement. We thus attribute the eventual decrease in purity to experimental fluctuations. In Fig. 3.8(d,e), we realize even and odd superpositions of non-squeezed Gaussian states, that is

$$|\psi_T^{(\Delta\phi)}\rangle = \frac{1}{\sqrt{2}} \left( |g(u, v)\rangle + e^{i\Delta\phi} |g(-u, -v)\rangle \right), \qquad (3.49)$$

with $\Delta\phi = 0, \pi$ for (d,e) respectively. While very little difference is observed between the Husimi functions<fnref n="12" /> of the two superpositions realized (both for $|\psi_{QOC}\rangle$ and $\hat{\rho}_{ML}$), the differences between their momentum evolutions (shown in Fig. 3.9) allow to unambiguously identify the prepared states. This allows us to iteratively reconstruct density matrices $\hat{\rho}_{ML}^{(d)}$ and $\hat{\rho}_{ML}^{(e)}$ with good purity

<footnote n="12">¹²The apparent similitude between the Husimi functions of the Gaussian state superpositions realized is a known feature of the Husimi quasi-distribution of probability [153].</footnote>
<figure><img src="image_43.png" /><figcaption>Figure 3.9 | Evolution of the momentum distribution of superpositions of Gaussian states in the static lattice. Top panels (a) correspond to the even superposition of Fig. 3.8(d). Bottom panels (b) correspond to the odd superposition of Fig. 3.8(e). (a₁-b₁) Stacks of experimental integrated absorption images taken during the evolution of the prepared states in the static lattice. (a₂-b₂) Numerical evolution of the diagonal terms of the density matrices reconstructed from (a₁-b₁). (a₃-b₃) Numerical evolution of the numerically prepared states |ψ<sub>QOC</sub>⟩. See Table 3.4 for details.</figcaption></figure>

(γ ≥ 0.82) and consistency with numerical simulations: $\mathcal{F}_{\text{exp}}^{(d)} = 0.89$ and $\mathcal{F}_{\text{exp}}^{(d)} = 0.91$ (see Table 3.4). This is further confirmed by the very low cross fidelities $\mathcal{F}(|\psi_{\text{QOC}}^{(d)}\rangle, \hat{\rho}_{\text{ML}}^{(e)}) = 0.026$ and $\mathcal{F}(|\psi_{\text{QOC}}^{(e)}\rangle, \hat{\rho}_{\text{ML}}^{(d)}) = 0.042$ (as the targets for Fig. 3.8(d,e) are orthogonal). To our knowledge, there is no adiabatic method for preparing such superpositions of translated Gaussian states in the lattice.

### 3.3.5 Squeezed Gaussian states

In this last section of experimental results about wave function control, we apply our preparation and reconstruction procedures to the squeezing of Gaussian states. We define the x-squeezing parameter as the ratio of standard deviations $\xi = \Delta x/\Delta x_0 = (\Delta p/\Delta p_0)^{-1}$, with $\Delta x_0$ and $\Delta p_0$ the non-squeezed standard deviations (see previous section). Including $\xi$ in the definition of our Gaussian states, Eq. (3.48) becomes

$$c_{\ell}^{(\xi)}(u, v) = \left( \frac{2\xi^2}{\pi\sqrt{s_0}} \right)^{1/4} e^{iuv/2} e^{-ilu} e^{-\xi^2(l-v)^2/\sqrt{s_0}}. \qquad (3.50)$$

For the squeezed Gaussian state $|g^{(\xi)}\rangle$ at lattice depth $s_0$, position and momentum standard deviations are given by $k_L \Delta x = \xi s_0^{(-1/4)}$ and $\Delta p/\hbar k_L = s_0^{1/4}/2\xi$. In our periodic system, the upper bound on $\xi$ is infinite and is reached when only a single diffraction order $\ell$ is populated (we have shown results up to $|\ell| = 10$ in Sec. 3.2.2). Figure 3.10 and Table 3.5 display results of squeezed Gaussian states positioned at the center of the phase space $((u, v) = (0, 0))$ for $1/\xi$
<figure><img src="image_44.png" /><figcaption>**Figure 3.10** | Experimental preparation and measurement of squeezed Gaussian states. (a-e) Husimi representations in the phase space of the static lattice. Top (red): states $|\psi_{QOC}\rangle$ numerically prepared by optimal control. Bottom (green): density matrices $\hat{\rho}_{ML}$ reconstructed from experimental data by likelihood maximization. The colormap value for each Husimi function extends from 0 to its maximum value. See Table 3.5 for details.</figcaption></figure>

<table><caption>**Table 3.5** | Parameters used for the preparation of squeezed Gaussian states and figures of merit obtained from their reconstruction. For all experiments $(u, v) = (0, 0)$.</caption><thead><tr><th>Fig. 3.10</th><th>a</th><th>b</th><th>c</th><th>d</th><th>e</th></tr></thead><tbody><tr><td>1/ξ</td><td>0.44</td><td>0.62</td><td>1.65</td><td>2.75</td><td>4.34</td></tr><tr><td>F<sub>num</sub></td><td></td><td>&gt; 0.995</td><td></td><td>0.980</td><td>0.965</td></tr><tr><td>F<sub>exp</sub></td><td>0.99</td><td>0.96</td><td>0.98</td><td>0.93</td><td>0.75</td></tr><tr><td>γ</td><td>1.00</td><td>1.00</td><td>1.00</td><td>0.92</td><td>0.72</td></tr><tr><td>s</td><td>5.49±0.20</td><td>5.49±0.20</td><td>5.45±0.40</td><td>5.57±0.20</td><td>5.62±0.25</td></tr><tr><td>t<sub>c</sub>/T<sub>0</sub></td><td></td><td></td><td>1.75</td><td></td><td>2</td></tr><tr><td>t<sub>rec</sub> (μs)</td><td></td><td></td><td>100</td><td></td><td>125</td></tr></tbody></table>

ranging from 0.44 to 4.34. Up to $1/\xi = 2.75$, we prepare and reconstruct states with good fidelities and purities ($F_{\text{exp}} \ge 0.93$ and $\gamma \ge 0.92$, see Fig. 3.10(a-d)). For the highly squeezed state $1/\xi = 4.34$ of Fig. 3.10(e), we increase $t_c$ to $2T_0$ in order for the QOC algorithm to converge to a control field giving a reasonable numerical fidelity $F_{\text{num}}$. This is due to the complexity of the target state which consists in the superposition of 13 significantly populated momentum components ($|c_{|l|<7}|^2 > 0.025$) with as many complex coefficients to control. This large number of populated momentum components also makes the reconstruction more challenging, requiring a Hilbert space of dimension $d_H = 15$, with $15^2 = 225$ real parameters to determine in order to find $\hat{\rho}_{\text{ML}}$ (Sec. 3.3.3). To help the algorithm set this many parameters, we increase the amount of information we give it by increasing the duration of the evolution after the QOC preparation to $t_{\text{rec}} = 125$ µs with $N_t = 26$ time steps. The simultaneous population of many momentum components also affects the reconstruction in that it reduces the signal-to-noise
<figure><img src="image_45.png" /><figcaption>**Figure 3.11 | Evolution of the momentum distribution of a highly squeezed state in the static lattice.** (a) Stack of experimental integrated absorption images taken during the evolution of the prepared state in the static lattice. (b) Numerical evolution of the diagonal terms of the density matrix reconstructed from (a). (c) Numerical evolution of the state $|\psi_{QOC}\rangle$ obtained by optimal control. (b,c) correspond to the evolutions of the states presented in Fig. 3.10(e). See Table 3.5 for details.</figcaption></figure>

ratio due to the lower number of atoms per diffraction peak. We illustrate this by drawing in Fig. 3.11 the momentum evolutions of the experimental, reconstructed and numerical states during their evolution in the lattice, where one sees an important loss of signal-to-noise ratio in experimental data as the magnitude of the considered momentum increases. Nevertheless, we achieve a fidelity $F_{\text{exp}} = 0.75$ even in that extreme case, and all the Husimi functions of Fig. 3.10 show qualitatively very good agreement between $\hat{\rho}_{\text{ML}}$ and $|\psi_{\text{QOC}}\rangle$ for the squeezing of Gaussian states.

Interestingly, one can consider the effective lattice depth $s_{\text{eff}}$ at which the ground state matches the squeezed Gaussian state (3.50) generated at depth $s_0$ with squeezing parameter $\xi$. Comparing Eqs. (3.48) and (3.50), one finds $s_{\text{eff}} = s_0/\xi^4$. In that sense, Fig.3.10(e) shows the realization of the ground state of a lattice of effective depth $s_{\text{eff}} \approx 2000$ in our lattice of depth $s_0 = 5.62$. This is, to our knowledge, the first realization of such a state, the production of which is technically impossible with adiabatic methods. To give an idea, the laser for the lattice in our experiment (see Sec. 2.4 and App. C) has a maximum power of 15 W, which corresponds for our setup to a maximum lattice depth of $s_0 \approx 40$. With the same setup, we thus would require a laser of about 750 W to produce a lattice depth of $s_0 \approx 2000$.

We also targeted Gaussian states both squeezed and rotated in the $(x, p)$ plane. Target state definition and results for those experiments are presented in App. E.

## 3.4 An application to quantum simulation

We conclude this chapter with an example of application of our QOC protocol to quantum simulation, namely the production of the initial state for the study of dynamical tunneling in
an amplitude-modulated one-dimensional optical lattice.

### 3.4.1 Introduction to dynamical tunneling

As we discussed in the first chapter of this manuscript (Sec. 1.1.2), the mixed stroboscopic phase portrait of a periodically driven dynamical system features classical trajectories that are either quasi-periodic (regular) or chaotic (resulting respectively in continuous lines or spread points in the phase portrait, see e.g. Fig. 3.12(a2)). In such a system, quantum particles can undergo dynamical tunneling, which consists in oscillating from one region of regular trajectories to another, crossing classically impassable Kolmogorov-Arnold-Moser surfaces (see Sec. 1.1.2 and [89]). Dynamical tunneling occurs when two non-degenerate Floquet states (Sec. 1.3.2) span the same regular regions of phase space, with a tunneling oscillation frequency proportional to the quasi-energy difference between the two states in the quasi-energy spectrum [142, 89].

In previous experiments with cold atoms in optical lattices, dynamical tunneling was studied with an initial sudden shift of the lattice to bring the ground state of the system in one of the tunnel-coupled regular regions (a representation of this approach is shown in Fig. 3.12(a) for the parameters of the dynamical tunneling experiment performed here; see below). Although this method provides evidence of the phenomenon [154, 155, 61], more than one frequency is observed in the tunneling signal as the initial states are only partially projected in the subspace of the two relevant Floquet states that carries the dynamical tunneling of interest. Moreover, the visibility of the oscillations is limited by the unequal-weight projection onto these states. We here apply QOC as a way to optimize the initial state for the observation of dynamical tunneling. Our goal is to compare the tunneling signal between the two methods.

Following the dynamical scaling for periodically modulated potentials summarized p. 34, the Hamiltonian that we study is that of the amplitude-modulated potential<fnref n="13" />:

$$ \tilde{H}(\tilde{x}, \tilde{p}, \tilde{t}) = \frac{\tilde{p}^2}{2} - \gamma [1 + \varepsilon_0 \cos(\tilde{t})] \cos(\tilde{x}), \quad (3.51) $$

which generates the mixed phase space of Fig. 3.12(a1,b) for the parameters $(\gamma, \varepsilon_0) = (0.25, 0.1)$. We focus on the center of the stroboscopic phase portrait, where a classical particle, stroboscopically observed every two periods of modulation, is bound to the lateral harmonic-oscillator-like region it started in (see Fig. 3.13(a)). We fix $\hbar_{\text{eff}} = 0.355$ (which corresponds to the experimental parameters $s_0 = 7.93$ and $\nu = 45.697$ kHz, see p. 34) for the dynamical tunneling timescale to be compatible with the two-period stroboscopic sampling<fnref n="14" />. Our optimal-control target is the state that maximizes the visibility of the tunneling oscillation, that is the equal-weight superposition of the two main Floquet states $|\varphi_A\rangle$ and $|\varphi_B\rangle$ in the central regular region of the stroboscopic phase portrait. Once the Floquet states are obtained through diagonalization of the Floquet

<footnote n="13">¹³Note that $\gamma$ is the scaled depth of the potential, and no longer the purity of a density matrix.</footnote>

<footnote n="14">¹⁴For $\hbar_{\text{eff}} = 0.355$, numerical simulations give a dynamical tunneling period of approximately $20T$ where $T = 1/\nu$ is the period of modulation of the potential. As the system is observed every 2 periods, we thus expect to sample the tunneling signal with approximately 10 measures per tunneling period, which is satisfactory.</footnote>
<figure><img src="image_46.png" /><figcaption>**Figure 3.12 | Initial states for dynamical tunneling.** Husimi representations and underlying classical phase spaces for (a) the states involved in the translation method and (b) the relevant Floquet states (see text). (a₁) Lattice ground state |φn=0⟩. (a₂) Translation of the ground state T̂Δ̃x|φn=0⟩ just after the start of the modulation. (b₁) One of the relevant Floquet state |φA⟩. (b₂) The other relevant Floquet state |φB⟩. (b₃) The ideal superposition of Floquet states |ψ(θR)⟩. See text for the definitions of these states. The colormap for each Husimi function extends from 0 to its maximum value. Parameters are (γ, ε₀, ℏeff) = (0.25, 0.1, 0.355).</figcaption></figure>

operator (i.e. the evolution operator over one modulation period, see 1.3.2), we identify |φA,B⟩ from their overlap with a non-squeezed Gaussian state (Eq. (3.48)) translated in the center of one of the lateral harmonic-oscillator-like region of the stroboscopic phase portrait. We show the Husimi functions of |φA,B⟩ in Fig. 3.12(b₁, b₂). The theoretical state that maximizes the visibility of the tunneling oscillation reads

$$|\psi(\theta)\rangle = \frac{1}{\sqrt{2}} (|\varphi_A\rangle + e^{i\theta} |\varphi_B\rangle) \qquad (3.52)$$

where the relative phase θ between the Floquet states |φA⟩ and |φB⟩ fixes the initial condition for the oscillation between the two lateral regular regions. We arbitrarily fix a starting point in the right region, so we set θ = θR defined as

$$\theta_R = \arg\max_{\theta} \left\{ \langle \hat{x} \rangle_{\psi(\theta)} \right\} \qquad (3.53)$$

Our optimal-control target is thus |ψ(θR)⟩. We draw the Husimi distribution of this state in Fig. 3.12(b₃). Finally, in order to compare the QOC approach with the signal obtained from the translated<fnref n="15" /> ground state T̂Δ̃x|φn=0⟩, we determine the optimal translation distance Δ̃x as the one that maximizes the overlap between T̂Δ̃x|φn=0⟩ and the ideal state |ψ(θR)⟩:

$$\Delta \tilde{x}_R = \arg \max_{\Delta \tilde{x}} \left\{ \left| \langle \psi(\theta_R) | \hat{T}_{\Delta \tilde{x}} | \phi_{n=0} \rangle \right|^2 \right\}. \qquad (3.54)$$

For our modulation parameters, we find a maximum overlap |⟨ψ(θR)|T̂Δ̃x|φn=0⟩|^2 ≈ 0.91 for the translation method. The Husimi distributions of the states |φn=0⟩ and T̂Δx|φn=0⟩ are

<footnote n="15">¹⁵See Sec. 1.2.1 for an expression of the translation operator.</footnote>
shown in Fig. 3.12(a).

### 3.4.2 Experimental results

For the experiment without QOC, the ground state is translated by a distance $\Delta \tilde{x}_R$ by applying a sudden phase shift $\varphi_0 = \Delta \tilde{x}_R$ (which spatially translates the lattice by a quantity $-\Delta \tilde{x}_R/k_L$, see Eq. (3.11)) right between the lattice loading and the beginning of the amplitude modulation. For the QOC experiment, we choose the ground state of the static lattice as our initial state for the QOC preparation. The amplitude modulation begins immediately after the QOC stage. Our control parameter remains the phase of the lattice $\varphi(t)$, with a control duration set to $t_c = 1.75 T_0$. The QOC algorithm (p. 72) converges to a control field reaching numerical fidelity $F_{\text{num}} > 0.995$.

In this dynamical tunneling experiment, the tunneling takes place between two region of the $x$ axis, separated by a fraction of the lattice step $d = 532$ nm. Our imagery setup (Sec. 2.3) cannot spatially resolve the phenomenon in-situ. The trick is to transfer the spatial information along the momentum axis using the sub-stroboscopic dynamics (see Fig. 1.4). To do so, after evolution in the modulated potential, and before TOF measurement, we modulate the potential during an additional half-period to perform a $\pi/2$-rotation around the center of the phase space and convert the population in the right (resp. left) regular region of the lattice well into negative (resp. positive) momentum components accessible through TOF [61]. This phase portrait rotation technique is illustrated in Fig. 3.13(a-b).

Figure 3.13(c-j) compares the results of dynamical tunneling experiments when the initial state is either approximated by a translation of the ground state (Fig.3.13(c-f)) or targeted by our optimal control method (Fig.3.13(g-j)). The spectral content of the oscillations is clearly refined when the two-Floquet state superposition is prepared, resulting in a greater signal-to-noise ratio for the measurement of the atomic tunneling frequency.

**Chaos-assisted tunneling.** Our application of QOC to dynamical tunneling is inspired from the study [61] of chaos-assisted tunneling (CAT) [88, 89] that we performed in 2020 in collaboration with M. Martinez, G. Lemarié and B. Georgeot from the Laboratoire de Physique Théorique (Toulouse) and O. Giraud and D. Ullmo from the Laboratoire de Physique Théorique et Modèles Statistiques (Paris). CAT is a type of dynamical tunneling where the tunneling between the two regular region is carried by two Floquet states mainly located on these regions, as well as by a third Floquet state delocalized over the chaotic sea (see Fig. 1.9 for an example of such a state). Under the variation of an external parameter of the system (e.g. the frequency of modulation), CAT resonances were predicted [88], associated with a non-monotonic variation of the tunneling frequencies in that three-level system. In our publication [61], we were the first to unambiguously observe such a resonance of CAT with matter wave. I recall that, while I chose not to present in my manuscript this research axis in which I participated, details of this study can be found in the thesis of M. Arnal [67], G. Chatelain [68] and M. Martinez [69].
<figure><img src="image_47.png" /><figcaption>**Figure 3.13** | **Quantum-optimal control applied to dynamical tunneling experiments.**
(a) Stroboscopic phase portrait for the amplitude-modulated potential (3.51) with the initially populated region (gray area with black solid line border) and the coupled regular region that gets populated through dynamical tunneling (gray area with black dashed line border). (b) Same as (a) after the $\pi/2$-phase space rotation (see text). (c-f) Results for the initially translated ground state. (g-j) Results with the initial state obtained from optimal control. (c,g) Numerical evolution of the momentum distribution. (d,h) Corresponding stack of experimental integrated absorption images. (e,i) Numerical and (f,j) experimental evolutions of the negative (solid line) and positive (dashed line) momentum populations, originating from the regions identified in (a,b) with the matching contouring line-style.</figcaption></figure>

Importantly, in our experiments on CAT [61], we observed that a large number of atoms in the BECs happens to have a deleterious effect on the time during which dynamical tunneling signals could be observed<fnref n="16" />. In [61], we indeed have had to measure signals over up to hundreds of modulation periods of the potential, and reducing the number of atoms in the BECs by a factor of 10 had a salutary effect on this experiment. Since then, the modification of the dipole trap geometry on our experiment (Sec. 2.2.3) has lead to the setup regularly producing BECs with up to five times more atoms. We thus point out that similar reduction of the number of atoms in the BECs were performed for the experiment of this section, where we worked with BECs of $N \approx 5 \cdot 10^4$ atoms, i.e. 10 times less that what the experiment produces. The protocol to reduce the number of atoms in the BECs in discussed in Sec. 2.2.4.

Conclusion

In this chapter, we discussed the successful implementation of quantum-optimal control

<footnote n="16">¹⁶We attribute this phenomenon to the interactions between atoms, but the exact mechanism is still unclear.</footnote>
(QOC) in our experiment to control the motional state of BECs in a one-dimensional optical lattice. QOC allows us to target states either unreachable by adiabatical means or in comparatively much shorter time.

In a first set of experiments, we established our ability to prepare given momentum distributions in the lattice. In a second step, we demonstrated full and reliable control of the external quantum state in which we prepare the atoms. We applied this control for the preparation of specific states in the phase space of the system. Such state control is a great tool for quantum simulation, notably in the semi-classical regime (Sec. 1.4.2) where quantum dynamics is strongly related to the underlying structures in the classical phase space. We concluded this chapter with an example of application of QOC for quantum simulation in a study of dynamical tunneling in the depth-modulated optical lattice. In this last example, we demonstrated our ability to target Floquet states of a given time-periodic Hamiltonian.

Quantum state control raises the question: How does one certifies proper state preparation? In our first experiments of momentum distribution preparation, the information was readily accessible through TOF imaging. We however required dedicated protocols when we targeted a broader range of quantum states (associated with sets of *complex* coefficients). Our general approach was to use data from the subsequent evolution of the prepared state in the static lattice as information to infer this prepared state. For simple momentum superposition targets, this was achieved through least-square analysis. To reconstruct the quantum state in the general case, we implemented a full quantum state tomography using a maximum-likelihood iterative approach, which allowed us to characterize our experimental states and confirm the success of the optimal-control preparation.
# Chapter 4 - Non-diffusive Hamiltonian ratchet

[...] because we pull up the shades and let the light out, because we cool off on the earth and get heat from the sun, the ratchets and pawls that we make can turn one way. This one-wayness is interrelated with the fact that the ratchet is part of the universe. It is part of the universe not only in the sense that it obeys the physical laws of the universe, but its one-way behavior is tied to the one-way behavior of the entire universe. It cannot be completely understood until the mystery of the beginnings of the history of the universe are reduced still further from speculation to scientific understanding.

Richard Feynman (The Feynman Lectures on Physics, Vol. 1, Chap. 46, 1963)

## Contents

<table><tr><td>Introduction</td><td>99</td></tr><tr><td>4.1 Designing a non-diffusive Hamiltonian ratchet in classical mechanics</td><td>102</td></tr><tr><td>4.1.1 Hamiltonian system considered and statement of the problem</td><td>102</td></tr><tr><td>4.1.2 Search for modulation parameters</td><td>104</td></tr><tr><td>4.1.3 Results of classical mechanics</td><td>106</td></tr><tr><td>4.2 Quantum ratchet transport along regular classical trajectories</td><td>108</td></tr><tr><td>4.2.1 A Floquet system</td><td>108</td></tr><tr><td>4.2.2 Influence of the effective Planck constant and Floquet state mixing</td><td>110</td></tr><tr><td>4.3 Experimental ratchet transport of matter waves</td><td>112</td></tr><tr><td>4.3.1 Transport from the ground state</td><td>113</td></tr><tr><td>4.3.2 Preparation of the ratcheting Floquet state using quantum-optimal control</td><td>116</td></tr><tr><td>Conclusion</td><td>118</td></tr></table>
<figure><img src="image_48.png" /></figure>
Introduction

A ratchet is a mechanical device that constrains the movement of a system in a defined direction. Its mechanism is schematically depicted in its rotating configuration in Fig. 4.1: if we apply torque on the gear, the combined effect of the pawl and the axial asymmetry of the toothing prevents the gear from rotating counterclockwise. Ratchets are widespread mechanisms that are part of many common devices, such as pendulum clocks, ratchet wrenches or bicycle freewheels.

<figure><img src="image_49.png" /><figcaption>**Figure 4.1 | Diagram of a mechanical rotating ratchet.** Both the red pawl and the blue gear can rotate around their fixed black pivots.</figcaption></figure>

In physics, ratchet became a subject of interest with G. Lippmann around 1900 [156, 157]. In a thought experiment, he considers a molecular size device that produces work from thermal noise using a ratchet mechanism to rectify the isotropy of Brownian motion. The first arguments against this setup apparently violating the second law of thermodynamics came from M. von Smoluchowski in 1912 [158], saying that as the ratchet part of the system also undergoes Brownian motion, it cannot consistently direct movement. In 1962, R. Feynman revisits the idea in the 46th chapter of his Lectures in Physics [159, 160], showing that for such a system to work, the ratchet must be held at a lower temperature than the rest of the machine, with heat flowing irreversibly between the parts (this ratchet system being *in fine* an intricate Carnot heat engine).

More abstractly, one defines the ratchet effect as the emergence of a directed current<fnref n="1" />

$$J = \lim_{T \to \infty} \frac{1}{T} \int_{0}^{T} \dot{x}(t) \, dt \neq 0, \qquad (4.1)$$

for a particle in a potential that is periodic in space and time:

$$V(x + d, t) = V(x, t + T) = V(x, t). \qquad (4.2)$$

<footnote n="1">¹We restrict ourselves to the one dimensional case.</footnote>
The ratchet effect is an intriguing phenomenon<fnref n="2" />, as it directly follows from the spatial periodicity of $V(x, t)$ that no net force derives globally from such potentials:

$$ \langle F(x,t) \rangle_{d,T} = \frac{1}{dT} \iint_{d,T} -\frac{\partial V(x,t)}{\partial x} dx dt = 0. \qquad (4.3) $$

The operating mechanism of the ratchet effect is well understood, and relies minimally on the fact that systems featuring this effect are out of equilibrium, with time-dependent potentials that break specific space and time symmetries [161, 162, 163, 164]. Among such systems, we can distinguish two families of ratchets:

(i) On one hand, Brownian ratchets [165, 166] consist in systems submitted to stochastic forces with a potential that rectifies the isotropy of Brownian motion [167] into a directed transport (in the fashion of the Lippmann-von Smoluchowski-Feynman ratchet). Interestingly, Brownian ratchets are thought to be the principle of operation of molecular motors [168, 169, 170, 171], as for instance for the kinesin [172], a motor protein found in eukaryotic cells. They are usually studied in the overdamped regime to account for the strong dissipation in biological media [166, 173]

(ii) On the other hand, one has deterministic ratchets, for which the classical dynamics is captured by the phase space of the systems (Sec. 1.1.2). Deterministic ratchets are either dissipative<fnref n="3" /> or Hamiltonian [76, 182, 163].

We focus in this chapter on Hamiltonian ratchets.

As we have seen in Chap. 1, Hamiltonian systems under moderate temporal driving exhibit a mixed dynamics, with a phase space displaying islands of regular trajectories embedded in a chaotic sea of non-integrable ones. A fundamental property of chaotic systems briefly addressed in Sec. 1.1.2 is ergodicity, which corresponds to the fact that a classical trajectory $(q_c(t), p_c(t))$ (Sec. 1.1.2) initialized in a chaotic region of phase space (index “c”) will eventually span all this chaotic region uniformly [69]. Considering a physical quantity $f(q, p)$, ergodicity formally means

$$ \lim_{T \to \infty} \frac{1}{T} \int_0^T f(q_c(t), p_c(t)) dt = \frac{1}{V_{\Pi_c}} \int_{\Pi_c} f(q, p) \, dq \, dp \qquad (4.4) \\ \langle f(q_c(t), p_c(t)) \rangle_t = \langle f(q, p) \rangle_{\Pi_c}, $$

where $\Pi_c$ stands for the considered chaotic region of phase space and $V_{\Pi_c}$ its hyper-volume. Put in words, ergodicity means that a trajectory from a given chaotic region of phase space is, in

<footnote n="2">²We add that particles are usually considered starting at rest in the reference frame of the potential, and the emerging asymptotic current $J$ (Eq. (4.1)) is also measured with respect to that reference frame. In other words, neither particles launched at high speed in unbounded phase space trajectories (Sec. 1.1.2) nor particles bound to a potential of the form (4.2) that travels linearly at $v = d/T$ are experiencing ratchet effect.</footnote>

<footnote n="3">³A few references on research about dissipative deterministic ratchet: [174, 175, 176, 177, 178, 179, 180, 181]</footnote>
time, representative of all trajectories from that chaotic region. Going back to one-dimensional Hamiltonian ratchets, the physical quantity $f(x(t), p(t))$ that goes into the definition of the asymptotic current $J$ (Eq. (4.1)) is the speed $\dot{x}(t) = p(t)/m$. From the ergodicity principle (4.4), we have, for a trajectory initialized in a chaotic region $\Pi_c$,

$$J = \frac{1}{m} \langle p_c(t) \rangle_t = \frac{1}{m} \langle p \rangle_{\Pi_c} \qquad (4.5)$$

From the right-hand side of this last equation, we see that the emergence of a non-zero asymptotic current (i.e. the ratchet effect) can result from the contribution of trajectories belonging to chaotic regions having a non-zero momentum barycentre. As a matter of fact, studies on Hamiltonian ratchets have so far mainly focused on so-called “diffusive Hamiltonian ratchets”, where the directed transport results from trajectories ergodically diffusing<fnref n="4" /> into such momentum-asymmetric chaotic seas [76, 182, 163].

We here take a different route, and study a system where the ratchet effect is based on integrable trajectories. The integrable ratchet effect has only been studied in the kicked rotor, where the ratchet effect occurs along the momentum axis (one speaks of accelerator ratchet [183, 184, 185]). The system that we study is a sine potential modulated in amplitude and phase in the fashion of the gating ratchet [186, 180]. In the simplest case of a single frequency of modulation, we determine modulation parameters such that the phase space of the system presents a regular island that travels between the sites of the periodic potential while periodically stopping in each site. This is to our knowledge the first study of such a non-diffusive ratchet effect along position space.

This chapter presents our study of this effect. The first section (Sec. 4.1) concerns the classical dynamics in this system. We then detail our method to determine modulations parameters for the integrable ratchet effect to emerge, and we then characterize the system. In the second section (Sec. 4.2), we consider quantum transport of wave functions in this Floquet system. Finally, we present experiments where we observed this non-diffusive ratchet effect with BECs in the modulated optical lattice.

The study presented in this chapter is the result of a quite personal work that I have been doing on the ratchet effect since my pre-thesis internship at LCAR in 2019. A publication is in preparation:

[65] N. Dupont, L. Gabardos, F. Arrouas, B. Peaudecerf, J. Billy and D. Guéry-Odelin, Matter wave transport from non-diffusive Hamiltonian ratchet effect, In preparation, (2022)

<footnote n="4">⁴We stress that, for Hamiltonian systems, this chaotic diffusion is deterministic.</footnote>
## 4.1 Designing a non-diffusive Hamiltonian ratchet in classical mechanics

In this section, we detail our method to determine, in a classical Hamiltonian system, modulation parameters of the sine potential in order for the system to feature a non-diffusive ratchet effect in position space.

### 4.1.1 Hamiltonian system considered and statement of the problem

**Our system.** We consider a classical particle of mass $m$ in a one-dimensional sine potential that is modulated in amplitude and phase with one harmonic at the same frequency. The corresponding time-dependent Hamiltonian can be written

$$H(x, p, t) = \frac{p^2}{2m} - [1 + \varepsilon_0 \cos(2\pi\nu t)] \frac{V_0}{2} \cos \left[ \frac{2\pi x}{d} + \varphi_0 \cos(2\pi\nu t + \Delta\phi) + \Delta\varphi \right], \quad (4.6)$$

with the fixed spatial periodicity $d$, and the adjustable modulation parameters,

* $V_0$, the average amplitude of the potential,

* $\varepsilon_0$, the amplitude of amplitude modulation,

* $\varphi_0$, the amplitude of phase modulation,

* $\nu$, the frequency of modulation,

* and $\Delta\phi$, the relative phase between the phase and amplitude modulations.

The parameter $\Delta\varphi$ is a constant spatial phase that we set to

$$\Delta\varphi = -\varphi_0 \cos(\Delta\phi), \quad (4.7)$$

in order for the spatial origin $x = 0$ to correspond to a minimum of the potential at time $t = 0$. As we have already seen in Chap. 1, one can look for a scaling of the Hamiltonian (4.6) such that the classical dynamics depends on fewer parameters. Following the procedure summarized Sec. 1.4.1 (p. 34), we get

$$\tilde{H}(\tilde{x}, \tilde{p}, \tilde{t}) = \frac{\tilde{p}^2}{2} - \gamma [1 + \varepsilon_0 \cos(\tilde{t})] \cos[\tilde{x} + \varphi_0 \cos(\tilde{t} + \Delta\phi) + \Delta\varphi], \quad (4.8)$$

with

$$\tilde{x} = \frac{2\pi}{d} x, \quad \tilde{t} = 2\pi\nu t, \quad \tilde{p} = \frac{d\tilde{x}}{d\tilde{t}} = \frac{1}{md\nu} p, \quad \text{and} \quad \tilde{H} = \frac{1}{md^2\nu^2} H, \quad (4.9)$$
and the scaled amplitude of the potential $\gamma = V_0/2md^2\nu^2$. The scaled dynamics effectively depends on the four modulation parameters $\{\gamma, \varepsilon_0, \varphi_0, \Delta\phi\}$ (with $\Delta\varphi$ still fixed by Eq. (4.7)).

**Correspondence with the gating ratchet.** Our Hamiltonian (4.8) is a known system from the ratchet community, called the *gating ratchet*. To understand this designation, we perform the Galilean transformation to switch from the inertial reference frame to the frame of the phase-modulated potential:

$$ \tilde{x}' = \tilde{x} + \varphi_0 \cos(\tilde{t} + \Delta\phi) + \Delta\varphi. \qquad (4.10) $$

In this non-inertial frame, the lattice is stationary, and the particle experiences the pseudo force

$$ \tilde{F}'(\tilde{t}) = -m\varphi_0 \cos(\tilde{t} + \Delta\phi). \qquad (4.11) $$

In this system, the amplitude-modulated potential barriers between sites act as opening and closing gates whose period can be correlated with the modulated pseudo force (4.11) to induce ratchet transport. This system has been theoretically studied in the Brownian case [186], as well as experimentally in the deterministic case with dissipation [180, 164]. We here consider this system in the Hamiltonian regime as a candidate for the emergence of a non-diffusive ratchet effect.

**Statement of the problem.** In order to design such a non-diffusive Hamiltonian ratchet (i.e. with a transport based on quasi-periodic trajectories in the phase space), we ask the question: Are there modulation parameters $\{\gamma, \varepsilon_0, \varphi_0, \Delta\phi\}$ such that a classical particle starting at rest at the bottom of a site ends up in the bottom of the next site after one modulation period ($T = 1/\nu$)? Mathematically put, we are looking for modulation parameters such that the trajectory $(\tilde{x}(\tilde{t}), \tilde{p}(\tilde{t}))$ starting in

$$ \begin{pmatrix} \tilde{x}(0) \\ \tilde{p}(0) \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix} \quad \text{goes to} \quad \begin{pmatrix} \tilde{x}(2\pi) \\ \tilde{p}(2\pi) \end{pmatrix} = \begin{pmatrix} 2\pi \\ 0 \end{pmatrix}, \qquad (4.12) $$

where we arbitrarily aim for a positive ratchet displacement. From the periodicity of the potential, a direct consequence of this hypothetical trajectory is that $\tilde{x}(\tilde{t} = 2\pi n) = 2\pi n$ with $n \in \mathbb{N}$, resulting in a constant<fnref n="5" /> ratchet current $J = 1$ (Eq. (4.1)).

<footnote n="5">⁵At least stroboscopically constant, i.e. when evaluated after an integer number of modulation periods.</footnote>
### 4.1.2 Search for modulation parameters

We are looking for modulation parameters {$γ, ε_0, ϕ_0, Δφ$} such that a classical particle described by the Hamiltonian (4.8) follows a trajectory passing by the two phase space points of Eq. (4.12).

From the correspondence with the gating ratchet discussed above, we simplify the problem by reducing the set of parameters: we set the relative phase $Δφ$ between the amplitude and phase modulations to a value that optimizes the gating effect in this system. With the gating potential (4.8), the most efficient way [180] to induce a positive ratchet current is to have $Δφ = π/2$ (i.e. a phase quadrature between the amplitude and phase modulations). In this case, the pseudo force becomes $\tilde{F}'(\tilde{t}) = mϕ_0 \sin(\tilde{t})$, which is maximally positive (resp. negative) a quarter of a period before the instant at which the potential barrier between neighboring sites is minimum (resp. maximum), resulting in a configuration that globally favors a positive ratchet transport. Our Hamiltonian becomes

$$ \tilde{H}(\tilde{x}, \tilde{p}, \tilde{t}) = \frac{\tilde{p}^2}{2} - \gamma [1 + \varepsilon_0 \cos(\tilde{t})] \cos[\tilde{x} - \varphi_0 \sin(\tilde{t} + \Delta\phi)], \quad (4.13) $$

with a set of modulation parameters reduced to {$γ, ϕ_0, ε_0$}.

To determine values for these three parameters, we note that, for the periodic trajectory (4.12) that we are looking to induce, the initial and final conditions have the same mechanical energy (evaluated by the Hamiltonian (4.13)), which is the global minimum of mechanical energy for this system:

$$ \tilde{H}(2\pi n, 0, 2\pi n) = -\gamma(1 + \varepsilon_0), \quad (4.14) $$

with $n \in \mathbb{Z}$. Our search for modulation parameters can thus be expressed as a minimization problem, where we want to minimize, as a function of {$γ, ε_0, ϕ_0$}, the increase of mechanical energy over one modulation period for a trajectory starting in $(\tilde{x}(0), \tilde{p}(0)) = (0, 0)$ and ending in the neighboring site$^6$. We define the function to minimize:

$$ g(\gamma, \varepsilon_0, \varphi_0) = \begin{cases} \tilde{H}_{\gamma, \varepsilon_0, \varphi_0}[\tilde{x}(2\pi), \tilde{p}(2\pi), 2\pi] - \tilde{H}_{\gamma, \varepsilon_0, \varphi_0}[\tilde{x}(0), \tilde{p}(0), 0] & \text{if } \pi < \tilde{x}(2\pi) < 3\pi \\ \text{a large number} & \text{otherwise,} \end{cases} \quad (4.15) $$

where the condition is set to force the particle to change site. We proceed numerically to minimize $g(\gamma, ε_0, ϕ_0)$, with numerical integration$^7$ of the trajectories $(\tilde{x}(t), \tilde{p}(t))$.

<footnote>$^6$One must be careful with this condition for the restatement of our problem, as, for instance, an absence of modulation ($ε_0, ϕ_0$) = (0, 0) results in a constant trajectory $(\tilde{x}(\tilde{t}), \tilde{p}(\tilde{t})) = (0, 0)$ that also trivially minimizes the variation of mechanical energy.</footnote>

<footnote>$^7$We use the Runge-Kutta “RK4” method to integrate classical dynamics.</footnote>
**Nelder-Mead algorithm.** To tackle this minimization problem, we use Nelder-Mead [187] method<sup>8,9</sup>: given a function $f(\mathbf{u}) : \mathbb{R}^N \to \mathbb{R}^+$ to be minimized and a guess $\mathbf{u}^{(0)}$, this iterative method consists in considering a polytope of $N + 1$ vertices $\mathbf{u}_i$ initialized in the neighborhood of $\mathbf{u}^{(0)}$. The function $f$ is evaluated at each vertex, which are then iteratively moved from the one that maximizes $f$ to the one that minimizes $f$ until they all converge toward the same local minimum for $f$. The Nelder-Mead method is an efficient alternative to gradient-descent methods when the gradient of the function to be minimized cannot be written explicitly, as it is the case for us with $f(\mathbf{u}) = g(\gamma, \varepsilon_0, \varphi_0)$ (Eq. (4.15)). As is generally the case when employing iterative methods of optimization, we want our initial guess to be in the vicinity of a global optimum.

**Initial guess.** To be able to minimize Eq. (4.15), we minimally require an initial guess such that the particle is located in the next site after one modulation period. To find such initial guesses, we start by maximizing the energy transferred to the particle at the beginning of the modulation by making our initial guess correspond to a modulation frequency that is resonant with the harmonic frequency $\nu_0$ at the bottom of the potential wells. For the Hamiltonian (4.13), the frequency of modulation is hidden in the scaled amplitude $\gamma = V_0/2md^2\nu^2$. From the potential of the unscaled Hamiltonian (4.6), we find $\nu_0 = \sqrt{V_0/2md^2}$, and we see that $\gamma = 1$ is the scaled amplitude corresponding to a resonant modulation at the bottom of the wells. We therefore set $\gamma = 1$ as our initial guess for this parameter.

To determine initial guesses for the amplitudes of modulation ($\varepsilon_0, \varphi_0$), we compute maps showing, after one modulation period, the position and the momentum of a particle starting at in $(\tilde{x}(0), \tilde{p}(0)) = (0,0)$ as a function of $(\varepsilon_0, \varphi_0)$. Such maps are shown in Fig. 4.2, for $\varepsilon \in [0, 1]$ (bounded so that the amplitude of the potential does not get negative, as required by the optical lattice setup in our experiment; Sec. 2.4) and $\varphi_0 \in [0, \pi]$ (corresponding to a peak-to-peak modulation of up to one spatial period). We use these maps to initialize $(\varepsilon_0, \varphi_0)$ to values such that $\tilde{x}(2\pi) \approx 2\pi$ and $\tilde{p}(2\pi) \approx 0$.

Figure 4.2 features an interesting behavior identified with a dashed black line. This linear<sup>10</sup> region corresponds to values of $(\varepsilon_0, \varphi_0)$ such that $\tilde{x}(2\pi) \approx \pi$ and $\tilde{p}(2\pi) \approx 0$. In words, they are modulation parameters such that the particle starting in $(\tilde{x}(0), \tilde{p}(0)) = (0,0)$ is brought to the top of the potential barrier, at the border between two lattice sites, and finishes there at rest. For our problem, we want to overcome this barrier of potential, and stop at the bottom of the following well. In Fig. 4.2, we see that this can be achieved for amplitudes of phase modulation $\varphi_0$ larger than those bringing the particle to a stop at the top of the potential (i.e. larger amplitudes for the pseudo force (4.11)). To the right of the dashed line, one sees a rather vast region of $(\varepsilon_0, \varphi_0)$ values such that $\tilde{x}(2\pi) \approx 2\pi$ and $\tilde{p}(2\pi) \approx 0$. We identify with solid black lines an example of good initialization $(\varepsilon_0, \varphi_0)$ for the algorithm.

<footnote n="8"><sup>8</sup>Using the open-source Python library SciPy.</footnote>

<footnote n="9"><sup>9</sup>We note that this problem could have very well been addressed using the formalism of optimal-control theory presented in Chap. 3. However, we ourselves were introduced to this field only after having found working modulation parameters for our non-diffusive Hamiltonian ratchet.</footnote>

<footnote n="10"><sup>10</sup>That is actually not trivial.</footnote>
<figure><img src="image_50.png" /><figcaption>Figure 4.2 | Parameter maps for initial guess determination in the search for ratchet modulation parameters. (a) Position $\tilde{x}$ and (b) momentum $\tilde{p}$ after one modulation period for a particle starting in $(\tilde{x}(0), \tilde{p}(0)) = (0, 0)$ and described by the Hamiltonian (4.13) with the scaled potential amplitude $\gamma = 1$ (see text). The solid black lines in $(\epsilon_0, \varphi_0) = (0.3, 1.9)$ mark a parameter pair for an initial guess (see text), and the dashed black line marks a region of parameters for which the particle ends up resting at the top of the potential barrier between the central well and the next one.</figcaption></figure>

### 4.1.3 Results of classical mechanics

Using the method detailed in the previous section, we converge to the modulation parameters

$$ (\gamma, \epsilon_0, \varphi_0) = (1.2, 0.3, 1.7). \qquad (4.16) $$

This solution for the problem that we set ourselves in Sec. 4.1.1 (p. 103) is not unique, and others can be found using the same method. In the rest of this chapter, we however focus on parameters (4.16). For these modulation parameters, we draw in Fig. 4.3(a,b) the position $\tilde{x}(\tilde{t})$ and the momentum $\tilde{p}(\tilde{t})$ over three modulation periods for the particle starting in $(\tilde{x}(0), \tilde{p}(0)) = (0, 0)$. As required, we get a periodic evolution, with a particle that travels one site per modulation period: $\tilde{x}(2\pi n) = 2\pi n (n \in \mathbb{N})$. As explained in the introduction of this chapter, the general ratchet effect is intriguing as no global net force derives from a periodic potential (Eq. (4.3)). In the specific case of our periodic ratchet, we have the stronger fact that the average force exerted per modulation by the potential on the ratcheting particle is zero along its trajectory:

$$ \langle \tilde{F}(\tilde{x}(\tilde{t}), \tilde{t}) \rangle_{2\pi} = \frac{1}{2\pi} \int_{0}^{2\pi} \tilde{F}(\tilde{x}(\tilde{t}), \tilde{t}) d\tilde{t} = 0. \qquad (4.17) $$

This is illustrated numerically in Fig. 4.3(c) where we plot, as a function of time, the instantaneous force $\tilde{F}(\tilde{x}(\tilde{t}), \tilde{t})$ exerted by the potential on the particle in $\tilde{x}(\tilde{t})$, as well as the average of
<figure><img src="image_51.png" /><figcaption>**Figure 4.3** | **Classical non-diffusive ratchet trajectory.** (a) Position $\tilde{x}$ as a function of time $\tilde{t}$ for the particle starting at rest at the bottom of the central well. (b) Momentum $\tilde{p}$ as a function of time for that same particle. (c) Instantaneous (dashed black line) and integrated average (solid black line) force $\tilde{F}(\tilde{x}(\tilde{t}), \tilde{t})$ as a function of time. We show three modulation periods, with modulation parameters $(\gamma, \epsilon_0, \varphi_0) = (1.2, 0.3, 1.7)$.</figcaption></figure>

this force from $\tilde{t} = 0$ up to that time. We see that this accumulated average is zero for $\tilde{t} = 2\pi n$ ($n \in \mathbb{N}$). This observation is also directly deduced from the conditions (4.12) that we imposed on the generated trajectory. Indeed, having $\tilde{p}(0) = \tilde{p}(2\pi)$ implies, from Hamilton equations and the fundamental theorem of calculus,

$$ -\frac{\partial \tilde{H}}{\partial \tilde{x}} = \dot{\tilde{p}} \quad \Rightarrow \quad \int_{0}^{2\pi} -\frac{\partial \tilde{H}}{\partial \tilde{x}} d\tilde{t} = \tilde{p}(2\pi) - \tilde{p}(0) \qquad (4.18) $$

$$ \Leftrightarrow \quad \int_{0}^{2\pi} \tilde{F}(\tilde{x}(\tilde{t}), \tilde{t}) d\tilde{t} = 0 \qquad (4.19) $$

This is a key difference between our periodic system and diffusive systems<fnref n="11" />, as well as between our spatial ratchet and the accelerator ratchet, also based on periodic trajectories, but ratcheting along the momentum axis<fnref n="12" /> in a kicked-rotor system.

<footnote n="11">¹¹For diffusive Hamiltonian ratchet based on a chaotic sea bounded in momentum, one has $\langle F \rangle_t$ only going to zero asymptotically in time.</footnote>

<footnote n="12">¹²With $p(nT) - p((n-1)T) \neq 0 \Rightarrow \langle F \rangle_T \neq 0$.</footnote>
<figure><img src="image_52.png" /><figcaption>**Figure 4.4 | Subperiod evolution of the stroboscopic phase portrait and ratcheting island.**
(a-e) Stroboscopic phase portraits for the same Hamiltonian (4.13) with $(\gamma, \epsilon_0, \phi_0) = (1.2, 0.3, 1.7)$ and the sub-stroboscopic observation times $\tilde{t}_0 = 0, 0.25, 0.5, 0.75$ and 1. In each panel, we artificially draw in continuous time the phase space trajectory at the center of the ratcheting island (dashed black line). See Sec. 1.1.2 and Fig. 1.2 for details on the procedure.</figcaption></figure>

We also compute the stroboscopic phase portrait (Sec. 1.1.2) for our ratchet modulation parameters (4.16). In Fig. 4.4, we draw five portraits associated with the sub-stroboscopic<fnref n="13" /> observation times $\tilde{t} = 0, 0.25, 0.5, 0.75$ and 1. We see that the determination of modulation parameters such that the trajectory starting in $(\tilde{x}(0), \tilde{p}(0)) = (0,0)$ periodically ratchets to the next site amounts to a “regularization” of the center of the phase space for this ratchet system. Indeed, we see that we generated a regular island that goes through one site of the potential per modulation period<fnref n="14" />, with the highly non-trivial fact that this island finds itself resting at the center of the sites periodically [188]. This is to our knowledge the first study of such a ratcheting regular island along the *x* axis. We also note from Fig. 4.4 that the rest of the phase portrait is greatly chaotic, which will later allows us to distinguish between evolutions carried by the regular ratcheting island and diffusive behavior in the chaotic sea.

We have so far considered classical mechanics in this system. In the next section, we study the transport of a wave function along this ratchet island.

## 4.2 Quantum ratchet transport along regular classical trajectories

### 4.2.1 A Floquet system

Given the modulation parameters $(\gamma, \epsilon_0, \phi_0) = (1.2, 0.3, 1.7)$, the scaled classical dynamics governed by the Hamiltonian (4.13) is completely determined, and represented by the phase space depicted in Fig. 4.4. We now consider wave function transport in this ratchet system. As introduced in Chap. 1, quantum dynamics in such scaled systems depends on an effective reduced Planck constant $\hbar_{\text{eff}}$. For our scaling (p. 102), we have

<footnote n="13">¹³See Sec. 1.1.2 and Fig. 1.2 for details.</footnote>

<footnote n="14">¹⁴i.e. a regular island with a winding number $w = 1$ (Sec. 1.1.2).</footnote>
$$ \hbar_{\text{eff}} = 2 \frac{\nu_{\text{L}}}{\nu}, \qquad (4.20) $$

with $\nu$ the unscaled modulation frequency and $\nu_L = h/2md^2$ a characteristic quantum frequency in Bloch systems (Sec. 1.2). As discussed in Sec. 1.4.1, $\hbar_{\text{eff}}$ sets the typical extent of quantum states with respect to the scaled phase space of the system. As we are interested in studying transport along the ratchet island that we designed in the previous section, we place our study “at the onset” of the semi-classical regime (see Sec. 1.4.1 and below), for values of $\hbar_{\text{eff}}$ such that the minimal extent of the quantum states is of the order of the area of the engineered ratcheting island. We recall that in the semi-classical regime, Floquet states are, in general, either localized on regular phase space structures or spread over the chaotic sea (see Sec. 1.4.1 and [86, 87, 69]). By “at the onset of the semi-classical regime”, we mean that the values of $\hbar_{\text{eff}}$ that we will consider will be small enough for quantum dynamics to be strongly affected by underlying phase space structures, but not be small enough for more than one Floquet state to be localized on the ratcheting island.

The quantum analogue of the periodic trajectory at the center of the ratcheting island (see Figs. 4.3 and 4.4) is the Floquet state $|\varphi_r\rangle$ that can be associated with this island, and we refer to it as the ratcheting Floquet state. For a given value of $\hbar_{\text{eff}}$, we compute the Floquet states $|\varphi_m\rangle$ by diagonalizing the Floquet operator (Sec. 1.3.2) around the sub-stroboscopic observation time $\tilde{t}_0 = 0$, i.e. when the ratcheting island is centered in the phase space (Fig. 4.4). We then identify the ratcheting Floquet state $|\varphi_r\rangle$ as the one with the greatest overlap with the ground state $|\phi_0\rangle$ of the static potential, a state itself centered in the phase space:

$$ |\varphi_r\rangle = \arg\max_{|\varphi_m\rangle} \left\{ |\langle\phi_0|\varphi_m\rangle|^2 \right\}. \qquad (4.21) $$

To quantify transport in a quantum state, we also define the expected transport between times $\tilde{t}_0$ and $\tilde{t}_1$ as the integral over this time interval of the averaged momentum<fnref n="15" />. For an arbitrary state $|\psi\rangle$, the expected transport can be written as

$$ \Delta \tilde{x}_{\tilde{t}_0, \tilde{t}_1}(|\psi\rangle) = \int_{\tilde{t}_0}^{\tilde{t}_1} \langle \hat{\tilde{p}} | \psi(\tilde{t}) \rangle d\tilde{t}. \qquad (4.22) $$

with $\hat{\tilde{p}}$ the scaled momentum operator. In the semi-classical regime, one expects Floquet states associated with regular structures to evolve as their associated structure, and, in particular for the ratcheting Floquet state, $\Delta \tilde{x}_{0,2\pi n}(|\varphi_r\rangle) \sim 2\pi n$. While this is generally true (see below), one has to account for Floquet state mixing which can complexify the picture for certain values of $\hbar_{\text{eff}}$.

<footnote n="15">¹⁵This definition is, among other things, motivated by our TOF imaging setup (Sec. 2.3), through which we measure the momentum distribution of the quantum state evolving in our lattice.</footnote>
4.2.2 Influence of the effective Planck constant and Floquet state mixing

We show in Fig. 4.5 numerical results of quantum transport in the ratcheting island as a function of $1/\hbar_{\text{eff}}$ (i.e. as a function of a classical scaling action in the system; see Sec. 1.4.1). We plot in Fig. 4.5(a,b) two quantities as a function of $1/\hbar_{\text{eff}}$: first, the overlap $|\langle\phi_0|\varphi_r\rangle|^2$ between the ratcheting Floquet state and the ground state of the static potential. This overlap somehow quantifies the “centrality” of $|\varphi_r\rangle$ in the phase space (as well as our upcoming ability to experimentally load the ratcheting Floquet state from the ground state of the optical lattice in order to observe ratchet transport). Second, we plot the expected transport per period $\Delta\tilde{x}_{0,2\pi}(|\varphi_r\rangle)$ of the ratcheting Floquet state. Having fixed the classical scaled dynamics (and, in particular, fixing $\gamma = V_0/2md^2\nu^2$), the limit $1/\hbar_{\text{eff}} \to 0 \Leftrightarrow \nu \to 0$ (Eq. 4.20) corresponds to a vanishing amplitude of the potential. In this radically deep quantum limit, both $|\varphi_r\rangle$ and $|\phi_0\rangle$ tend to the resting plane wave$^{16}$: their overlap tends to one and the transport of $|\varphi_r\rangle$ in the vanishing potential tends to zero (Fig. 4.5(a,b)). As $1/\hbar_{\text{eff}}$ varies, both $|\langle\phi_0|\varphi_r\rangle|^2$ and $\Delta\tilde{x}_{0,2\pi}(|\varphi_r\rangle)$ display rather sharp non-monotonic fluctuations ascribed to the phenomenon of Floquet state mixing (which we now introduce; the description of Fig. 4.5 is continued shortly below).

**Floquet state mixing.** As a parameter of the Hamiltonian is varied, the energy levels in the quasi-energy spectrum vary at different rates giving rise to the generic phenomenon of avoided crossings. This effect results in the coupled state mixing into reshaped orthogonal Floquet states near the crossing. We illustrate this phenomenon in Fig. 4.6 for realistic parameters, in the interval of $1/\hbar_{\text{eff}}$ identified in Fig. 4.5(a,b) by the gray shaded area, around a sharp drop of both $|\langle\phi_0|\varphi_r\rangle|^2$ and $\Delta\tilde{x}_{0,2\pi}(|\varphi_r\rangle)$. In Fig. 4.6(a), we draw a zoom of the quasi-energy spectrum (Sec. 1.3.2) where relevant Floquet levels have been identified from their overlap with the Floquet states$^{17}$ shown in panels (c,d). In Fig. 4.6(b), we again plot $|\langle\phi_0|\varphi_r\rangle|^2$ in this small interval of $1/\hbar_{\text{eff}}$. This quantity displays two drops that were not resolved in Fig. 4.5. These drops correspond to the two consecutive avoided crossings observed in panel (a), at which the ratcheting Floquet state $|\varphi_r\rangle$ is strongly reshaped (as one can see from the Husimi representations of Fig. 4.6(c-e)). Surprisingly, we observe in Fig. 4.5(b) that there are even values of $\hbar_{\text{eff}}$ such that the periodic transport of the ratcheting Floquet state becomes negative (even though the scaled classical dynamics still remains unchanged, with the ratcheting island going in the positive direction). As a matter of fact, there are such values of $\hbar_{\text{eff}}$ in the zoom region of Fig. 4.6, and we see from the Husimi representations shown in panels (c-e) that negative transport in $|\varphi_r\rangle$ results from couplings with other Floquet states having weights in the negative momentum part of phase space.

Going back to Fig. 4.5, we draw on panels (c-e) the Husimi representations of $|\phi_0\rangle$ (blue) and $|\varphi_r\rangle$ (green) for the three values of $1/\hbar_{\text{eff}}$ identified on panel (a,b). At $\hbar_{\text{eff}} \approx 0.79 \approx 1/1.27$, Fig. 4.5(d) is an example of a semi-classical, phase-space-centered and island-shaped $|\varphi_r\rangle$, with

<footnote>$^{16}$See for instance Sec. 1.2.2.</footnote>

<footnote>$^{17}$The Floquet states shown in Fig. 4.6(c,d) have themselves been identified by their overlap with the ground state $|\phi_0\rangle$.</footnote>
<figure><img src="image_53.png" /><figcaption>**Figure 4.5** | Ratcheting Floquet state and influence of the effective Planck constant. (a) Overlap between the ratcheting Floquet state $|\varphi_r\rangle$ and the ground state $|\phi_0\rangle$ of the potential as a function of $1/\hbar_{\text{eff}}$. (b) Periodic expected transport $\Delta\tilde{x}_{0,2\pi}$ of $|\varphi_r\rangle$ as a function of $1/\hbar_{\text{eff}}$. (c-e) Stroboscopic phase portraits and Husimi representations (Sec. 1.4.1) of $|\phi_0\rangle$ (top, blue) and $|\varphi_r\rangle$ (bottom, green) for the values of $1/\hbar_{\text{eff}}$ identified by the vertical black lines on the panels (a,b) (with $1/\hbar_{\text{eff}} \approx 0.704, 1.27$ and $1.56$ for (c-e) resp.). The Husimi functions in the inset of panels (d,e) are truncated to a quarter of their respective overall maximum value to reveal details. The gray shaded area in panels (a,b) corresponds to the interval of $1/\hbar_{\text{eff}}$ studied in Fig. 4.6. The Hamiltonian of the system is given by Eq. (4.13) with parameters $(\gamma, \epsilon_0, \varphi_0) = (1.2, 0.3, 1.7)$.</figcaption></figure>

a ground state overlap of $|\langle\phi_0|\varphi_r\rangle|^2 \approx 0.86$ and a periodic transport of $\Delta\tilde{x}_{0,2\pi}$ ($|\varphi_r\rangle$) $\approx 0.93$. On the other hand, Fig. 4.5(c) and (e) correspond to values of $\hbar_{\text{eff}}$ where a wave function (such as the ground state of the static lattice) placed in the ratcheting island will evolve out of it (towards a mode of high momentum$^{18}$ for Fig. 4.5(c) and over the chaotic sea for Fig. 4.5(e)).

This concludes our numerical study (both classical and quantum mechanical) about this non-diffusive Hamiltonian ratchet. In the following section, we apply our analysis of the influence of $\hbar_{\text{eff}}$ and the observations of Fig. 4.5 to experimentally observe ratchet transport of matter

<footnote>$^{18}$Quantum ratchet effects based on such couplings with Floquet states associated with large momentum have e.g. been studied in [189, 190].</footnote>
<figure><img src="image_54.png" /><figcaption>**Figure 4.6** | Floquet state mixing phenomenon illustrated on the ratcheting Floquet state.
(a) Portion of the quasi-energy spectrum as a function of $1/\hbar_{eff}$ for the modulated Hamiltonian (4.13) with parameters $(\gamma, \epsilon_0, \phi_0) = (1.2, 0.3, 1.7)$. $\epsilon$ denotes the quasi-energy that we plot reduced by the characteristic quantum energy scale of the potential $E_L = h^2/2md^2$ (Secs. 1.2 and 1.3.2). The Floquet states associated with the quasi-energy marked by the letters (c) and (d) are identified from their overlap with the ground state of the potential (see text). The colors green and red are associated with these states, and the spectrum is in return colored depending on the overlaps with these states (see Sec. 1.3.2 for method details). For all values of $\hbar_{eff}$, the most green quasi-energy corresponds to the ratcheting Floquet state $|\phi_r\rangle$ according to Eq. (4.21). (b) Overlap between the ratcheting Floquet state and the ground state $|\phi_0\rangle$ of the potential. The black vertical lines in (a,b) mark the position of the two avoided crossings seen in (a) corresponding to the two drops in ground state overlap seen in (b) (see text). (c-e) Stroboscopic phase portraits and Husimi representations of the three states associated with the quasi-energies identified in (a).</figcaption></figure>

waves along the ratcheting island that we engineered in Sec. 4.1.

## 4.3 Experimental ratchet transport of matter waves

In this last section, we present experiments of non-diffusive ratchet transport of BECs in the optical lattice. We focus on the two values of the effective reduced Planck constant $\hbar_{eff} \approx 0.79 \approx 1/1.27$ and $0.64 \approx 1/1.56$, which respectively correspond to the numerical simulations of Fig. 4.5(d) and (e). For both values, we see in Fig. 4.5(b) that the periodic transport of the ratcheting Floquet state is approximately one. However, their projection on lattice ground state (Fig. 4.5(a)) is quite different, with $|\langle\phi_0|\phi_r\rangle|^2 \approx 0.86$ for $\hbar_{eff} \approx 0.79$, while we only have
$|\langle\phi_0|\varphi_r\rangle|^2 \approx 0.56$ for $\hbar_{\text{eff}} \approx 0.64$ (resulting from a mixing with a Floquet state from the chaotic sea in this case).

In Sec. 4.3.1, we first present, for these two values of $\hbar_{\text{eff}}$, experiments of ratchet transport from the lattice ground state. In a second set of experiments (Sec. 4.3.2), we use the quantum-optimal control protocol detailed in Chap. 3 to prepare the ratcheting Floquet states and enhance ratchet transport and the periodicity of the evolution.

### 4.3.1 Transport from the ground state

The time-dependent Hamiltonian (4.13) studied in the two previous sections is directly implementable on our experiment of BECs in a one-dimensional optical lattice introduced in Chap. 2. To determine the experimental parameters for the modulation, we follow this procedure:

(i) We determine a value of $\hbar_{\text{eff}}$ at which we want to perform the experiment of ratchet transport (Sec. 4.2). As we require a given scaled phase space associated with the modulation parameters $(\gamma, \varepsilon, \varphi_0) = (1.2, 0.3, 1.7)$, we broadly set the lattice beams intensity in order to have the lattice depth $s_0 = 4\gamma/\hbar_{\text{eff}}^2$ (as $\gamma = s\nu_L^2/\nu^2$ and $\hbar_{\text{eff}} = 2\nu_L/\nu$; see p. 34). We can quickly verify the consistency of our settings by performing a “Kapitza-Dirac calibration” of the lattice (Sec. 2.4.2).

(ii) We precisely calibrate the lattice depth $s_0$ using our usual method of ground state translation (Sec. 2.4.2 and [112, 113]). As $\gamma = s_0(\nu_L/\nu)^2$, we set the value of the modulation frequency to $\nu = \nu_L\sqrt{s_0/\gamma}$, which fixes in return the true value of the effective Planck constant $\hbar_{\text{eff}} = 2\nu_L/\nu$ at which the experiment will be performed.

(iii) We perform the ratchet transport experiment (see below).

After obtaining the BECs (Sec. 2.2.3), we adiabatically load them into the ground state of the optical lattice at the calibrated depth $s_0$. We then modulate the depth and phase of the lattice to realize the Hamiltonian (4.13) during a given amount of time, before proceeding to TOF imaging (Sec. 2.3.2) allowing us to measure the momentum distribution of the atomic state in the lattice right before the TOF. For this experiment, we sample the evolution with four observations per modulation period.

We start by showing results of ratchet transport from the ground state of the lattice at $\hbar_{\text{eff}} \approx 0.79$ (the value of Fig. 4.5(d) associated to a semi-classical behavior). In Fig. 4.7(a-d), we show next to each other the stroboscopic phase portraits and the absorption images at corresponding sub-stroboscopic times during the first modulation period. In order to compare the momentum of the ratcheting island to that of the momentum components of the diffraction images, the classical dynamics in the phase portraits of Fig. 4.7(a-d) has been unscaled using

$$p/\hbar k_L = \tilde{p}/\hbar_{\text{eff}}, \qquad (4.23)$$
<figure><img src="image_55.png" /><figcaption>**Figure 4.7** | Semi-classical non-diffusive ratchet transport from the ground state. (a-d) Stroboscopic phase portraits (left) and experimental TOF absorption images (right) at sub-period observation times $\bar{t} = (n+r) \times 2\pi$, with $n \in N$ and $r = 0, 0.25, 0.5$ and $0.75$ for (a) to (d) respectively. The period shown for the experimental images is $n=0$, implying that (a) is the momentum distribution of the lattice ground state. (e) Momentum as a function of time, showing the experimental (blue markers linked with dotted lines) and numerical (plain red line) average of the momentum distribution with the ground state as the initial state, as well as the momentum of the classical trajectory at the center of the ratcheting island (plain black line).</figcaption></figure>

an expression that can be deduced from Eqs. (4.9) and (4.20) (with $k_L = 2\pi/d$). The great qualitative agreement observed between the momentum of the island and that of the momentum peaks in the diffraction pattern illustrates the semi-classicality of the observed ratchet transport, as well as its non-diffusive nature over this first period (this is demonstrated for longer times below). In Fig. 4.7(e), we compare three momentum evolutions: the experimental average momentum measured from the diffraction patterns, the corresponding numerical simulation for the evolution from the ground state, and the momentum of the classical trajectory at the center of the ratcheting island (as in Fig. 4.3(b), although unscaled using Eq. (4.23)). This plot further illustrates the semi-classicality of this ratchet transport.

The experimentally measured moment distributions (used to plot Fig. 4.7(e)) are shown in Fig. 4.8(a), as well as the corresponding numerically obtained momentum distributions. This
evolution is to be compared with the same experiment performed for the other value of the
effective Planck constant $\hbar \approx 0.64$. Numerical and experimental momentum evolutions for this
experiment are shown in Fig. 4.8(b). While the scaled phase space is fixed and the initial state
is in both cases the ground state of the static lattice, we observe a strong difference between the
momentum evolutions for these two experiments. For $\hbar \approx 0.79$ in panel (a), we observe a quasi-
periodic evolution with a rather constant momentum dispersion. This semi-classical evolution
results from the ground state $|\phi_0\rangle$ of the static lattice being well projected onto the ratcheting
Floquet state $|\phi_r\rangle$ for this values of $\hbar_{\text{eff}}$ (as seen in Fig. 4.5(d)). In the case of $\hbar \approx 0.64$
in Fig. 4.8(b), the momentum distribution is initially rather narrow before quickly widening,
never to return to a configuration close to that of the initial situation. This is emblematic of
a diffusion in the chaotic sea, which results from a situation where $|\phi_r\rangle$ is partially delocalized
over the chaotic sea (as seen in Fig. 4.5(e) and discussed in Sec. 4.2). This observation is of
purely quantum nature: although the atoms are initially at the center$^{19}$ of the regular island,
we here obtain a quantum evolution diverging from its classical counterpart.

From the sub-stroboscopic sampling of the momentum evolution, we compute the expected
transport (4.22). We plot this quantity in Fig. 4.8(c). In the case of experiment (a), we get
a transport of approximately 10 lattice sites in 10 modulation periods, as was expected from
Fig. 4.5. We also show this transport in the strongly aperiodic case of experiment (b). The
overall great agreement between experiments and numerical simulation shows our successful
implementation of the ratchet Hamiltonian (4.13), as well as our ability to precisely set and
determine modulation parameters in the experiment (notably thanks to our fine calibrations of
the lattice depths ; Sec. 2.4.2).

**Transport reversal.** A key feature of the ratchet effect is the ability to reverse the direction of the transport [163, 164]. For our Hamiltonian (4.13), this direction is imposed by the sign of the phase quadrature between the modulations of amplitude and phase. One expects the opposition of the phase modulation ($\varphi(t) = -\varphi_0 \sin(t) \rightarrow +\varphi_0 \sin(t)$) to result in a negative ratchet transport. In Fig. 4.8(c), we label (a') the integrated transport obtained experimentally (blue dots) and numerically (red line) for the parameters ($\gamma, \varepsilon, \varphi_0) = (1.2, 0.3, -1.7)$ and $\hbar_{\text{eff}} \approx 0.77$ (i.e. close to the experiment of panel(a)). We measure for this experiment a ratchet transport of approximately -10 sites in 10 modulation periods, a symmetric ratchet transport.

These results demonstrate the observation of quantum transport along a ratcheting island in
the phase space of our system. We have also observed how Floquet state mixing can complexify
the loading of the ratcheting phase space region, as this phenomenon reshapes the ratcheting
Floquet state. In the following section, we present a last experiment in which we use quantum-
optimal control to optimize the loading of this state.

<footnote n="19">¹⁹We also highlight that $\hbar_{\text{eff}}^{(a)} > \hbar_{\text{eff}}^{(b)}$ (corresponding to Fig. 4.5(d,e) resp.), meaning that the phase space extent of the initial state in panel (b) is *smaller* than that of panel (a), the latter yielding nevertheless a more semi-classical behavior.</footnote>
<figure><img src="image_56.png" /><figcaption>**Figure 4.8** | Experiments of ratchet transport from the ground state of the lattice. (a) Top: Numerical simulation of the momentum evolution during the ratchet modulation as a function of time for $\hbar_{eff} \approx 0.79$ (corresponding to Fig. 4.5(d)). Bottom: Corresponding experimental integrated absorption images. (b) Same as (a) for $\hbar_{eff} \approx 0.64$ (corresponding to Fig. 4.5(e)). (c) Expected numerical (solid red line) and experimental (blue markers) transport (see text) for data (a) and (b) as a function of time. Data (a') is approximately the reversed ratchet transport of (a), with $\phi_0 \to -\phi_0$ and $\hbar_{eff} \approx 0.77$ (see text).</figcaption></figure>

### 4.3.2 Preparation of the ratcheting Floquet state using quantum-optimal control

To improve the transport of a quantum state in a ratcheting island (as well as the periodicity of its evolution), the Floquet state $|\phi_r\rangle$ associated with the island must be prepared as the initial state for the modulation. To do so, we use the quantum-optimal control (QOC) protocol detailed in Chap. 3. As in that chapter, we use the lattice phase $\phi(t)$ as our control parameter, and we target $|\phi_r\rangle$ from the ground state $|\phi_0\rangle$ of the static lattice at depth of work. For this experiment, between the steps (ii) and (iii) of the experimental procedure presented in the previous section (p. 113), we add the following steps:

(ii.i) Having set all the modulation parameters, we numerically determine $|\phi_r\rangle$ by diagonalizing the Floquet operator, and selecting among its eigenstates the one that maximizes the overlap with the lattice ground state $|\phi_0\rangle$ (itself centered in the phase space).

(ii.ii) Using our QOC algorithm (Sec. 3.1.3), we numerically compute the control field $\phi(t)$ that drives $|\phi_0\rangle$ to $|\phi_r\rangle$. We set the control duration to $t_c = 1.75 T_0$ (see Sec. 3.1.4), and
<figure><img src="image_57.png" /><figcaption>**Figure 4.9 | Ratchet transport experiments of ratcheting Floquet states prepared by QOC.**
(a-c) Sketch of the QOC protocol to prepare the ratcheting Floquet state $|\phi_r\rangle$ from the ground state $|\phi_0\rangle$ at $\hbar_{eff} \approx 0.64$. (a) Husimi representation of $|\phi_0\rangle$ in the phase space of the static lattice. (b) Lattice phase evolution that drives $|\phi_0\rangle$ to $|\psi_{QOC}\rangle$ with the fidelity $F_{num} = |\langle\varphi_r|\psi_{QOC}\rangle|^2 \approx 0.996$. (c) Same as (a) for the prepared state $|\psi_{QOC}\rangle$. (d) Top: Numerical simulation of the momentum evolution of $|\psi_{QOC}\rangle$ during the ratchet modulation as a function of time for $\hbar_{eff} \approx 0.77$ (corresponding to Fig. 4.6(d) and Fig. 4.8(a)). Bottom: Corresponding experimental integrated absorption images. (e) Same as (d) for $\hbar_{eff} \approx 0.64$ (corresponding to Fig. 4.5(e) and Fig. 4.8(b)).</figcaption></figure>

the algorithm converges to control fields that numerically prepare the state $|\psi_{QOC}\rangle$ with numerical fidelity to the target $F_{num} = |\langle\varphi_r|\psi_{QOC}\rangle|^2 > 0.995$

The experimental sequence then consists in an adiabatical loading of $|\phi_0\rangle$ (Sec. 2.4.1), the QOC preparation with the computed control field, the ratchet modulation for a given amount of time and finally the TOF imaging. As in the previous section, we take four images of the momentum distribution of the atoms per modulation period.

We sketch in Fig. 4.9(a-c) the QOC procedure applied to this ratchet study for $\hbar_{eff} \approx 0.64$. This value corresponds to the case of the poor ratchet state loading of Fig. 4.5(e) and Fig. 4.8(b). In Fig. 4.9(d,e), we show numerical simulations and experimental results of optimized ratchet transport using QOC. The values of $\hbar_{eff}$ in Fig. 4.9(d,e) approximately correspond to the two values of this parameter previously studied in Fig. 4.8(a,b) (see the caption). While Fig. 4.8(a) already presented a clear regular ratchet transport, we see that the use of QOC enhances the periodicity of the evolution of the momentum distribution (as one expects from the evolution of a Floquet state). Comparing Fig. 4.9(e) with Fig. 4.8(b), one sees in this case an evident
improvement of periodicity. Interestingly, the momentum evolution of Fig. 4.9(e) display a broad momentum dispersion from the beginning of the modulation, corresponding to $|\phi_r\rangle$ being partially delocalized over the chaotic sea for this value of $\hbar_{eff}$ (Fig. 4.5(e)).

These results further demonstrate our experimental observation of matter waves ratcheting along a regular structure in the phase space of the modulated optical lattice. They also give another example of Floquet state preparations from QOC, in addition to the results of Sec. 3.4.

Conclusion

The ratchet effect is the intriguing emergence of a directed transport of particle in the global absence of net force. In this chapter, we presented the first study and experimental implementation of a Hamiltonian ratchet system that features a non-diffusive ratchet effect along space. Following an introduction on the ratchet effect, we presented the method that allowed to determine modulation parameters to observe this new kind of ratchet effect in a sine potential modulated in amplitude and phase. After having determined such modulation parameters, we characterized classical dynamics in this system. Interestingly, it follows from the quasi-periodicity of the ratcheting trajectories that, for this kind of ratchet, no net force is exerted on the particles locally, along their trajectories. In a second stage, we studied quantum mechanics in this ratchet system as a function of the effective reduced Planck constant $\hbar_{eff}$, and discussed the phenomenon of Floquet state mixing that can complexify the semi-classical picture for given values of $\hbar_{eff}$.

Following these numerical studies, we experimentally implemented this system using our setup of Bose-Einstein condensates in a controlled one-dimensional optical lattice (Chap. 2). We performed experiments of ratchet transport from the ground state of the optical lattice and experimentally observed the ratchet effect of matter waves. Finally, we optimized, using quantum-optimal control (QOC), the loading of the Floquet state associated with the ratcheting island even at values of $\hbar_{eff}$ for which Floquet state mixing couples the island with the chaotic sea. Following our results on QOC presented in the previous chapter, these experiments give another example of application of QOC to quantum simulation in a Floquet system, this time in a study of the ratchet effect.
# Part III

## Beyond the one-body wave function
<figure><img src="image_58.png" /></figure>
# Chapter 5 - Emergence of a tunable supercrystalline order in a Floquet-Bloch system

Spectacular. You appear to understand how a portal affects forward momentum, or to be more precise, how it does not. Momentum, a function of mass and velocity, is conserved between portals. In layman's terms, speedy thing goes in, speedy thing comes out.

GLaDOS (Portal, 2007)

## Contents

<table><tr><td>Introduction</td><td>123</td></tr><tr><td>5.1 Modulation regime and typical observations</td><td>125</td></tr><tr><td>5.2 An effective tight-binding model</td><td>126</td></tr><tr><td>5.2.1 Modeling the Floquet bands</td><td>128</td></tr><tr><td>5.2.2 Adding the interactions</td><td>131</td></tr><tr><td>5.2.3 Stability</td><td>135</td></tr><tr><td>5.2.4 Estimation of the interaction parameter</td><td>137</td></tr><tr><td>5.2.5 Results of the model and predictions</td><td>138</td></tr><tr><td>5.3 Experimental results</td><td>140</td></tr><tr><td>5.3.1 Band-mapping method and quasi-momentum of the decimal diffraction peaks</td><td>140</td></tr><tr><td>5.3.2 Tuning the momenta of the instabilities</td><td>142</td></tr><tr><td>5.3.3 Nucleation time</td><td>143</td></tr><tr><td>5.3.4 Survival of the instability</td><td>144</td></tr><tr><td>5.3.5 Amplitude modulation</td><td>145</td></tr><tr><td>5.4 Discussion on the realized state</td><td>146</td></tr><tr><td>5.4.1 Truncated-Wigner simulations</td><td>146</td></tr><tr><td>5.4.2 Shared features and differences between supersolids and the state emerging in our system</td><td>148</td></tr><tr><td>Conclusion</td><td>149</td></tr></table>
<figure><img src="image_59.png" /></figure>
Introduction

In the last two chapters, we presented results on the control and the transport of the
*one-body* wave function of BECs in a one-dimensional optical lattice. In both these studies,
we captured the dynamics of the BECs in the lattice with the Schrödinger equation. In this
linear framework, we designed efficient control fields for the preparation of non-trivial quantum
states (Chap. 3), and determined the right modulation parameters to observe ratchet transport
of matter waves along regular classical trajectories in the optical lattice (Chap. 4). In both
studies, we obtained very good agreements between the experiments and numerical simulations,
while completely neglecting the repulsive interatomic interactions in the system. This is made
possible by the weak interaction regime of our one-dimensional system combined with the short
timescales of the experiments presented so far (typically a few hundreds of microseconds). In this
final chapter, we present experiments performed over longer timescales that take us out of the
one-body formalism: the emergence of a supercrystalline order through spontaneous four-wave
mixing in the phase modulated optical lattice.

Four-wave mixing is an emblematic consequence of the introduction of interactions in a
wave theory. In non-linear optical media, it leads to the production of photon pairs, correlated
in momentum and energy according to conservation laws. As a direct consequence, the effect
strongly depends on the relation of dispersion of the media. Four-wave mixing is an essential
tool in quantum optics, notably for the generation of entangled states. In its formalism, one
speaks of *input* and *output* channels for the wave states before and after the process. In quantum
gases, the same phenomenon is at work through two-body interactions [191, 90]. At equilibrium,
it is at the root of the production of momentum-correlated atom pairs [192, 193, 194]. In direct
analogy with the optical parametric amplification, four-wave mixing of matter waves can also be
parametrized [195]. A way to achieve it in Bloch systems is by engineering the band structures
with a periodic modulation of an external parameter of the system<fnref n="1" />.

Using ultracold atoms in modulated optical lattices, parametric four-wave mixing has been
studied in effective one-band systems [196, 197], for frequencies of lattice modulation low in
comparison to the frequencies of interband transitions. The effect notably leads to the realization
of staggered states [198, 199, 200, 201] with output channels located at the edge of the Brillouin
zone.

In this chapter, we present experiments where we modulate our optical lattice at a frequency
that couples the ground band with an excited band, but not at the quasi-momentum where the
atoms are initially loaded. After tens of periods of modulation, we observe the emergence of new,
narrow and macroscopically populated diffraction peaks, between the momentum components
associated with the ground state. To understand these observations, we develop a two-level tight-
binding model with interactions that allows us to study the dynamical stability of our system. As
it turns out, we find sharp regions of dynamical instabilities located near the quasi-momenta at
which our modulation couples the bands. These instabilities indicate narrow four-wave mixing

<footnote n="1">¹One speaks of Floquet engineering.</footnote>
output channels in the Brillouin zone, where atoms colliding from $q = 0$ may accumulate. From the connection between the unstable modes and the avoided crossings, we deduce that we can tune the output channel quasi-momenta (and experimentally the momentum at which the structure emerges) by varying the position of the interband coupling in the Brillouin zone.

This last chapter is organized as follow: in the first section 5.1, we discuss the modulation regime of this study and present experiments with typical observations of the phenomenon. In the second section 5.2, we detail the development of an effective tight-binding model of two coupled bands that is able to reproduce the quasi-energy spectrum of our system. Following a perturbative Bogoliubov treatment of the interactions, we study the dynamical stability of our system for realistic parameters. We make a series of predictions from the behavior of the exponents of instability at the avoided-crossing. Section 5.3 presents experiments that test these predictions. The narrow diffraction patterns via which the new structures manifest hints at a preserved coherence in the emerging states. Moreover, the existence of closer diffraction peaks at adjustable momenta suggests, that a new, larger and controllable periodicity structures the BECs real space. We confirm these experimental hints in the last section 5.4 of this chapter by presenting results of Truncated-Wigner simulations, performed by Peter Schlagheck from the CESAM Research Unit in the University of Liège.

<footnote>The research work presented in this chapter results from a collaboration for the numerical studies with Prof. Peter Schlagheck from the CESAM Research Unit at the University of Liège. A publication is in preparation:</footnote>

<footnote>[66] N. Dupont, L. Gabardos, F. Arrouas, G. Chatelain, M. Arnal, J. Billy, P. Schlagheck, B. Peaudecerf, and D. Guéry-Odelin, *Emergence of a tunable supercrystalline order in a Floquet-Bloch system*, In preparation, (2022)</footnote>
## 5.1 Modulation regime and typical observations

In this chapter, we study the behavior of a weakly interacting BEC in a phase-modulated optical lattice over many modulation periods. The atoms experience the lattice potential

$$V(x) = -s_0 \frac{E_L}{2} \cos[k_L x + \varphi_0 \cos(2\pi\nu t)]. \quad (5.1)$$

At a given lattice depth $s_0$, we write $|\phi_{n,q}\rangle$ the $n^{th}$ eigenstate of the static lattice ($\varphi_0 = 0$) at quasi-momentum $q$. Before the modulation, the atoms are adiabatically loaded in the ground state $|\phi_{n,q}\rangle = |\phi_{0,0}\rangle$ (Sec. 2.4.1). We then modulate the lattice in phase, with a moderate modulation amplitude $\varphi_0$ and a frequency $\nu$ that couples the ground band $s$ to an excited band at $q \neq 0$. As the modulation does not resonantly couple $|\phi_{0,0}\rangle$ to another state of the static lattice$^2$, this situation results, in the Floquet picture, in a Floquet eigenstate $|\varphi_{m,q=0}\rangle$ that strongly resembles $|\phi_{0,0}\rangle$ for a given $m$. For instance at lattice depth $s_0 = 3.7$, the frequency of modulation$^3$ $\nu = 2.9 \nu_L$ couples the lattice bands $s$ and $d$ at $q \approx 0.28$ (Fig. 5.1(a)). In this situation, a phase modulation of amplitude $\varphi_0 = 15^\circ$ gives

$$\max_j \{|\langle\phi_{0,0}|\varphi_{j,0}\rangle|^2\} = |\langle\phi_{0,0}|\varphi_{m,0}\rangle|^2 \approx 0.995, \quad (5.2)$$

with Husimi representations$^4$ of $|\phi_{0,0}\rangle$ and $|\varphi_{m,0}\rangle$ displayed in Fig. 5.1. According to what we have studied so far in this thesis$^5$, one expects the stroboscopic evolution of $|\phi_{0,0}\rangle$ to be almost static, as it is mainly projected over one Floquet eigenstate. Experimentally (Fig. 5.2(c-d)), for the first tens of periods of modulation, we essentially observe the ground state of the static lattice and indeed see no evolution of the characteristic diffraction patterns, with momentum components separated by $\hbar k_L = h/d$. However, after $\sim 70$ periods of modulation, we distinctly see new diffraction peaks appear in between the original ones, with momenta $\hbar k^* \neq \ell \times h/d$ (with $\ell \in \mathbb{Z}$). In the following, we refer to these peaks as “decimal peaks”. We note that the decimal peaks are observed both for a coupling between the bands $s$ and $d$ (Fig. 5.2(c)), and a coupling between the bands $s$ and $f$ (Fig. 5.2(c)), i.e. regardless the curvature of the excited band. In both cases, the TOF images clearly display the emergence of a smaller periodicity in the reciprocal space. Through Fourier analysis, this would indicate a spontaneous in situ structuring with a new step $d^* > d$.

In Bloch systems, the drift of the whole diffraction pattern in the resting frame of the lattice results from a variation of quasi-momentum (Sec. 1.2.1). This well known effect is at the root of the Bloch oscillation phenomenon (see for example [148, 79]), where the quasi-momentum of the system is swept adiabatically by the mean of a constant force. In our case, only a part of

<footnote>$^2$At least not by a realistic number of phonons considering the low amplitude of modulation; see Sec. 1.3.2</footnote>

<footnote>$^3$We recall the characteristic angular frequency of the lattice $\nu_L = h/2md^2 \approx 8.1$ kHz in our system; see Sec. 2.4.</footnote>

<footnote>$^4$Drawn with respect to the stroboscopic phase portrait (Sec. 1.1.2) of the driven system to inform incidentally on the degree of chaoticity for our modulation parameters.</footnote>

<footnote>$^5$i.e. studying the evolution of the system with Schrödinger equation.</footnote>
<figure><img src="image_60.png" /><figcaption>**Figure 5.1** | Non-resonant modulation in the one-body wave function framework. (a) band structure of the static lattice (colored lines), with the point that corresponds to the initial state of the atoms (blue disk) and the energy of modulation (black arrows, only from the ground state) that leads to the spectrum (c). (b) Husimi representation of the ground state in the classical phase space of the static lattice. (c) Quasi-energy spectrum (colored lines) where the overlaps between the Floquet eigenstates and the eigenstates of the static lattice have been color coded (see Sec. 1.3.2) and projection of the atomic state (blue disk) at the beginning of the modulation. (d) Husimi representation of the Floquet state that maximizes the overlap with the ground state in the stroboscopic phase portrait for the modulated potential (5.1). Parameters are $s_0 = 3.7$ for all panels, and $\nu = 2.9\nu_L$ and $\phi_0 = 15^\circ$ for panels (c) and (d).</figcaption></figure>

the atoms are observed in shifted but symmetric diffraction peaks (Fig. 5.2(c-d)), behaving as if a local force was selectively and symmetrically applied on some of the atoms. According to this observation, and inspired by previous studies on dynamical instabilities in Floquet system [196, 201], we attribute the emergence of the decimal peaks to short-range interactions in the system. More precisely, we interpret the emergent momentum peaks as originating from two-body collisions that occur between atoms of the lattice ground state $q = 0$ and result in atoms accumulating in $q = \pm q^*$, i.e. four-wave mixing of atomic wave functions from the center of the Brillouin zone scattering into quasi-momenta $q = \pm q^*$. In this framework, an initial seed is needed in the output channels of the mixing. In our case, this role is provided by fluctuations of thermal or quantum origin as will be discussed in Sec 5.4.1.

To account for the interactions in the system, we develop an effective tight-binding model with interactions. As the real system is driven by a phase modulation that couples the *s* band with an excited state at $q \neq 0$, we implement a two-band model with a coupling at the quasi-momentum where the modulation is resonant. We detail the development of this model in the following section.

## 5.2 An effective tight-binding model

We consider *N* bosons placed into a periodic lattice potential of *L* sites. Labelling two energy bands 0 and 1, we define, in the second quantification formalism, the effective Hamiltonian
<figure><img src="image_61.png" /><figcaption>**Figure 5.2** | Experimental protocol and nucleation of decimals peaks. (a) Time-evolution of lattice depth and (b) phase, showing the adiabatic loading to the ground state of the lattice, the phase modulation for an integer number of periods, and the lattice release for TOF. (c,d) Stacks of experimental absorption images ($t_{TOF}$ = 35 ms) showing the stroboscopic evolutions of the momentum distribution as a function of the number $n$ of periods $T$. Parameters for (c) are: BECs of $N = 5 \cdot 10^5$ atoms, $s_0 = 3.7 \pm 0.10$, $\phi_0 = 15^\circ$, $\nu = 1/T = 25.5$ kHz. Parameters for (d) are: $N = 1 \cdot 10^5$ atoms, $s_0 = 3.4 \pm 0.10$, $\phi_0 = 20^\circ$, $\nu = 30.0$ kHz. The upper value of the colormaps for the ODs (optical densities; Sec. 2.3) are truncated to reveal details. (e) Lattice band structure for average $\bar{s}_0 = 3.55$ and transitions from the ground state for the modulation (c) and (d).</figcaption></figure>

$$ \hat{H}_{\text{eff}} = \hat{H}_0 + \hat{H}_{\text{int}} \qquad (5.3) $$

$$ \begin{aligned} \hat{H}_0 = & -J_0 \sum_{j=0}^{L-1} (\hat{a}_{j+1}^\dagger \hat{a}_j + \hat{a}_j^\dagger \hat{a}_{j+1}) \\ & -J_1 \sum_{j=0}^{L-1} (\hat{b}_{j+1}^\dagger \hat{b}_j + \hat{b}_j^\dagger \hat{b}_{j+1}) + \Delta E_1 \sum_{j=0}^{L-1} \hat{b}_j^\dagger \hat{b}_j \\ & +iW \sum_{j=0}^{L-1} (\hat{b}_j^\dagger \hat{a}_j - \hat{a}_j^\dagger \hat{b}_j) \end{aligned} $$

$$ \hat{H}_{\text{int}} = \frac{U}{2} \sum_{j=0}^{L-1} \hat{a}_j^{\dagger 2} \hat{a}_j^2, $$
where:

* our spatial coordinate is the lattice site $j$ and the periodicity is $d = 1$,

* $\hat{a}_j$ and $\hat{b}_j$ (respectively $\hat{a}_j^\dagger$ and $\hat{b}_j^\dagger$) are the annihilation (respectively creation) operators for the effective bands 0 and 1 on site $j$ of the lattice,

* $J_0$ and $J_1$ are the tunneling amplitudes with neighboring sites for the two bands (in the sine-potential system that we want to model, we have a ground band with $J_0 > 0$ coupled with an higher energy band with a tunneling amplitude $|J_1| > |J_0|$),

* $\Delta E_1$ offsets the energy of the band 1 with respect to the band 0, allowing us to tune the position of energy crossings in the Brillouin zone for the model,

* $W$ is a coupling amplitude between the bands on site $j$,

* $U$ is an effective on-site interaction energy in the lowest band ($U_0 = U$, see below).

The interaction Hamiltonian $\hat{H}_{\text{int}}$ here only takes into account the on-site two-body interactions for the band 0. One could include the interactions between an atom in band 0 and an atom in band 1 with a term of the form $U_{01} \sum_j \hat{a}_j^\dagger \hat{b}_j^\dagger \hat{b}_j \hat{a}_j/2$, or the interactions between two atoms in band 1 with $U_1 \sum_j \hat{b}_j^\dagger \hat{b}_j^2/2$. However, as we are interested in modeling the *beginning* of the decimal peaks nucleation from atoms initially *almost all* in band 0, we will see that $\hat{H}_{\text{int}}$ as written in Eq. (5.3) is sufficient to obtain our results (Sec. 5.2.5).

### 5.2.1 Modeling the Floquet bands

Setting aside the interactions for the moment, we show how $\hat{H}_0$ can model the two Floquet levels associated with the two lowest bands of the lattice that are coupled by the phase modulation (that is, for instance, the blue and green eigenvalues as a function of $q$ in the spectrum of Fig. 5.1(c)). Doing so, we will determine realistic parameters for the model. As we consider a finite number $L$ of lattice sites, the quasi-momentum $q$ becomes a discrete quantity:

$$q = \frac{2\pi k}{L} \quad \text{with the integer } k = \left\lfloor -\frac{L}{2} + 1, \dots, \frac{L}{2} \right\rfloor. \qquad (5.4)$$

In the limit $L \to \infty$, we retrieve a continuous quasi-momentum $q/k_L \in (-0.5, 0.5]$. We start by writing the annihilation operators in the reciprocal space, yielding Fourier modes $\hat{c}_k$ and $\hat{d}_k$ associated with the Bloch waves for our two bands:

$$\hat{c}_k = \sum_{j=0}^{L-1} \frac{e^{-i2\pi kj/L}}{\sqrt{L}} \hat{a}_j \quad \text{and} \quad \hat{d}_k = \sum_{j=0}^{L-1} \frac{e^{-i2\pi kj/L}}{\sqrt{L}} \hat{b}_j, \qquad (5.5)$$

along with the inverse transformations
$$ \hat{a}_j = \sum_k \frac{e^{i2\pi kj/L}}{\sqrt{L}} \hat{c}_k \quad \text{and} \quad \hat{b}_j = \sum_k \frac{e^{i2\pi kj/L}}{\sqrt{L}} \hat{d}_k, \qquad (5.6) $$

where the summation boundaries for $k$ have been defined in Eq. (5.4). The Hamiltonian $\hat{H}_0$ (Eq. (5.3)) becomes<fnref n="6" />

$$ \hat{H}_0 = \sum_k -2J_0 \cos\left(\frac{2\pi k}{L}\right) \hat{c}_k^\dagger \hat{c}_k + \sum_k \left(-2J_1 \cos\left(\frac{2\pi k}{L}\right) + \Delta E_1\right) \hat{d}_k^\dagger \hat{d}_k \quad (5.7) \\ + \sum_k iW \left(\hat{d}_k^\dagger \hat{c}_k - \hat{c}_k^\dagger \hat{d}_k\right). $$

We see that the eigenenergies of the uncoupled Bloch modes are cosine functions of the quasi-momentum, as expected for a tight-binding model:

$$ E_0(k) = -2J_0 \cos \left( \frac{2\pi k}{L} \right), \qquad (5.8) $$

$$ E_1(k) = -2J_1 \cos \left( \frac{2\pi k}{L} \right) + \Delta E_1. $$

The Hamiltonian (5.7) is now expressed as a sum over the Brillouin zone: $\hat{H}_0 = \sum_k \hat{H}_0(k)$, where the coupling Hamiltonian $\hat{H}_0(k)$ in the sub-space of quasi-momentum $q = 2\pi k/L$ reads

$$ \hat{H}_0(k) = E_0(k) \hat{c}_k^\dagger \hat{c}_k + E_1(k) \hat{d}_k^\dagger \hat{d}_k + iW (\hat{d}_k^\dagger \hat{c}_k - \hat{c}_k^\dagger \hat{d}_k). \qquad (5.9) $$

This Rabi Hamiltonian can be diagonalized by defining the mode $\hat{u}_k$ and $\hat{v}_k$:

$$ \hat{u}_k = \cos\left(\frac{\theta(k)}{2}\right) \hat{c}_k - i \sin\left(\frac{\theta(k)}{2}\right) \hat{d}_k, \qquad (5.10) $$

$$ \hat{v}_k = \sin\left(\frac{\theta(k)}{2}\right) \hat{c}_k + i \cos\left(\frac{\theta(k)}{2}\right) \hat{d}_k, $$

where the mixing angle $\theta(k) \in [0, \pi)$ is defined by

$$ \tan(\theta(k)) = \frac{2W}{E_0(k) - E_1(k)}. \qquad (5.11) $$

<footnote n="6">⁶We use the relation $\sum_{j=0}^{L-1} e^{i2\pi j(k-k')/L} = L \delta_{k,k'}$.</footnote>
<figure><img src="image_62.png" /><figcaption>Figure 5.3 | Partial fits of Floquet spectra with the two-level tight-biding model. (a,d) Typical Floquet spectra (colored lines) to be partially modeled (see text). Parameters are $s_0 = 3.7$, $\nu = 23.5$ kHz and $\phi_0 = 15^\circ$ for (a) and $s_0 = 3.4$, $\nu = 30.0$ kHz and $\phi_0 = 15^\circ$ for (d). (b,e) Zooms on the modeled parts of (a-e) resp., each only displaying the quasi-energies of the 2 Floquet states onto which the coupled static bands are maximally decomposed. (c,f) Spectra of $\hat{H}_0$, the Hamiltonian of the model (Eq. (5.3)) without interactions, with parameters adjusted to fit (b,e) The parameters for (c) are $(J_0, J_1, \Delta E_1, W) = (1.63 \cdot 10^{-3}, 1.73 \cdot 10^{-1}, -1.22 \cdot 10^{-1}, 3.11 \cdot 10^{-2}) \times E_L$ and the parameters for (f) are $(J_0, J_1, \Delta E_1, W) = (3.00 \cdot 10^{-3}, -2.75 \cdot 10^{-1}, 3.18 \cdot 10^{-1}, 3.81 \cdot 10^{-2}) \times E_L$. The gray areas in (b,c,e,f) indicate the intervals of quasi-momenta over which the fits are performed (see text).</figcaption></figure>

In the following, we write $\theta$ with an implicit dependence on $k$ to lighten the notation. We finally get

$$ \hat{H}_0 = \sum_k \left( E_u(k) \hat{u}_k^\dagger \hat{u}_k + E_v(k) \hat{v}_k^\dagger \hat{v}_k \right) \qquad (5.12) $$

with dressed energy levels

$$ E_u(k) = E_0(k) \cos^2\left(\frac{\theta}{2}\right) + E_1(k) \sin^2\left(\frac{\theta}{2}\right) + W \sin(\theta), \qquad (5.13) $$

$$ E_v(k) = E_0(k) \sin^2\left(\frac{\theta}{2}\right) + E_1(k) \cos^2\left(\frac{\theta}{2}\right) - W \sin(\theta). $$

With the effective dressed bands of Eq. (5.13), we can now fit the part of interest in the Floquet spectrum of the real system with the parameters $J_0, J_1, \Delta E_1$ and $W$. This is done in the continuous quasi-momentum limit ($L \to \infty$). We draw in Fig. 5.3 the fit of two spectra for modulation parameters similar to those of the experiments of Fig. 5.2. As the frequency of
modulation can also couple the excited state to even higher bands (or, slightly, the ground band to higher bands through multi-phonon processes, see Sec. 1.3.2), we see in Fig. 5.3(b,e) that more than two avoided crossings are seen in the part of interest of the Floquet spectra. Therefore, we only perform the fits in intervals of quasi-momenta around the main avoided crossings. For this adjustment, the quasi-energies of the real system are also offseted in order to have a zero eigenvalue in the center of the Brillouin zone for the Floquet state associated with the lower band of the model (as we only set an energy offset for band 1 in our model (5.3)). Figures 5.3(c,f) show the corresponding adjusted spectra of $H_0$ (with details presented in the figure caption). This method can be applied for other modulation parameters if they are moderate enough to identify effective static bands. Moreover, the results (5.9), (5.11) and (5.13) are general and can be used with other choices for the uncoupled energy levels $E_0(k)$ and $E_1(k)$, for instance with more paraboloid functions for the method to work at lower lattice depths, in a regime where the tight-binding approximation with only next-neighbor tunneling does not hold.

We can now add the interactions and study their effect in a model with two realistic coupled bands.

### 5.2.2 Adding the interactions

In Eq. (5.3), the presence of the interaction part $\hat{H}_{int}$, quartic in operators, complicates the diagonalization of the effective Hamiltonian $\hat{H}_{eff}$ in order to determine how the system evolves. To handle the quartic terms, we use a pertubative approach in the fashion of N. Bogoliubov [202, 90]. We choose to present a detailed derivation of the equations of evolution for the perturbation modes in the Brillouin zone. Results are summarized at the end of the derivation (p. 136).

In the experimental situation (Fig. 5.2), the vast majority of atoms (with the exception of perhaps some thermal excitations) is initially in the ground state. In the model, the corresponding mode is $\hat{c}_0$ from Eq. (5.5). This mode therefore contains a large number of atoms $N_0 = \langle \hat{c}_0^\dagger \hat{c}_0 \rangle$, and only a small number of atoms $N^* = N - N_0$ populates the other modes:

$$N^* = \sum_{k \neq 0} \left( \langle \hat{c}_k^\dagger \hat{c}_k \rangle + \langle \hat{d}_k^\dagger \hat{d}_k \rangle \right) + \langle \hat{d}_0^\dagger \hat{d}_0 \rangle \ll N_0 \approx N. \quad (5.14)$$

The perturbative treatment then consists in an expansion in powers of the small parameter $\epsilon = N^*/N$. Besides, the Bogoliubov prescription consists in replacing the macroscopically populated mode $\hat{c}_0$ by a scalar<fnref n="7" />: $\hat{c}_0 \approx \sqrt{N_0} \approx \sqrt{N}$. This prescription implies that we can decompose the mode $\hat{a}_j$ as (from Eq. (5.6))

$$\hat{a}_j = \sqrt{n_0} + \delta\hat{a}_j, \quad (5.15)$$

<footnote n="7">⁷A more rigorous method can be found in [203, 204].</footnote>
with $n_0 = N_0/L$ and

$$ \delta \hat{a}_j = \sum_{k \neq 0} \frac{e^{i2\pi kj/L}}{\sqrt{L}} \hat{c}_k. \qquad (5.16) $$

We note at this stage that, from these definitions, we have

$$ \sum_{j=0}^{L-1} \delta \hat{a}_j = \sum_{k \neq 0} \hat{c}_k \delta_{0,k} = 0, \qquad (5.17) $$

which is to say that the perturbation on band 0 is orthogonal to the mode of the condensate described by $\phi(j) = \sqrt{n_0}$:

$$ \sum_{j=0}^{L-1} \phi^*(j) \delta \hat{a}_j = 0. \qquad (5.18) $$

We also note that the number operator for the non-condensed modes can be written equivalently

$$ \hat{N}^* = \sum_{j=0}^{L-1} \delta \hat{a}_j^\dagger \delta \hat{a}_j + \sum_{j=0}^{L-1} \hat{b}_j^\dagger \hat{b}_j. \qquad (5.19) $$

We can now insert decomposition (5.15) into the $\hat{H}_{\text{eff}}$ (5.3) and consider terms in successive powers of the perturbations (i.e. containing 0, 1 or 2 quantum fields).

**Zeroth order.** At lowest order, $H_{\text{eff}}^{(0)}$ is a function of the atom number:

$$ H_{\text{eff}}^{(0)} = -2J_0 N_0 + \frac{N_0^2 U}{2L} \approx -2J_0 N + \frac{N^2 U}{2L}, \qquad (5.20) $$

where, in the second equation, we neglected second order corrections from perturbations $^8$. Equation (5.20) is the energy of the condensate of approximately $N$ atoms in the mode $\hat{c}_0$. In particular, we can identify the chemical potential of the condensate:

$$ \mu = \frac{\partial H_{\text{eff}}^{(0)}}{\partial N} = -2J_0 + nU. \qquad (5.21) $$

<footnote>$^8$As $N_0 \simeq N - \hat{N}^*$, see Eq. (5.19)</footnote>
**First order.** Injecting the decomposition (5.15) in $\hat{H}_{\text{eff}}$ (5.3) yields the following first-order terms:

$$ \hat{H}_{\text{eff}}^{(1)} = \hat{H}_{0}^{(1)} + \hat{H}_{\text{int}}^{(1)} \qquad (5.22) $$

$$ \hat{H}_{0}^{(1)} = -2J_{0}\sqrt{n_{0}}\sum_{j=0}^{L-1}(\delta\hat{a}_{j}^{\dagger} + \delta\hat{a}_{j}) + iW\sqrt{n_{0}}\sum_{j=0}^{L-1}(\hat{b}_{j}^{\dagger} - \hat{b}_{j}) $$

$$ \hat{H}_{\text{int}}^{(1)} = n_{0}^{3/2} U \sum_{j=0}^{L-1} (\delta \hat{a}_{j}^{\dagger} + \delta \hat{a}_{j}). $$

However, from Eq. (5.17), the only contribution that does not cancel out is

$$ H_{\text{eff}}^{(1)} \approx iW\sqrt{n}\sum_{j=0}^{L-1}(\hat{b}_{j}^{\dagger} - \hat{b}_{j}), \qquad (5.23) $$

where we have equated $n \simeq n_0$ (as it amounts to neglecting third-order terms) and neglected edge terms in rearranging the sums. We will make sense of this remaining term (5.23) shortly below.

**Second order.** The terms up to the second order in perturbations are

$$ \hat{H}_{\text{eff}}^{(2)} = \hat{H}_{0}^{(2)} + \hat{H}_{\text{int}}^{(2)} \qquad (5.24) $$

$$ \hat{H}_{0}^{(2)} = -J_{0}\sum_{j=0}^{L-1}(\delta\hat{a}_{j+1}^{\dagger}\delta\hat{a}_{j} + \delta\hat{a}_{j}^{\dagger}\delta\hat{a}_{j+1}) - J_{1}\sum_{j=0}^{L-1}(\hat{b}_{j+1}^{\dagger}\hat{b}_{j} + \hat{b}_{j}^{\dagger}\hat{b}_{j+1}) + \Delta E_{1}\sum_{j=0}^{L-1}\hat{b}_{j}^{\dagger}\hat{b}_{j} $$

$$ \hat{H}_{\text{int}}^{(2)} \approx \frac{nU}{2} \sum_{j=0}^{L-1} (\delta \hat{a}_{j}^{\dagger 2} + \delta \hat{a}_{j}^{2} + 4\delta \hat{a}_{j}^{\dagger} \delta \hat{a}_{j}). $$

Here, another contribution needs to be considered: at zeroth order in Eq. (5.20), we defined the energy of the condensate $H_{\text{eff}}^{(0)}(N_0) \simeq H_{\text{eff}}^{(0)}(N)$. However the corrections in this last equation are of order two. Therefore, we may write

$$ \hat{H}_{\text{eff}}^{(0)}(N_0) = \hat{H}_{\text{eff}}^{(0)}(N - \hat{N}^*) \approx H_{\text{eff}}^{(0)}(N) + 2J_0 \hat{N}^* - 2N \frac{\hat{N}^*U}{2L}, \qquad (5.25) $$

stopping the expansion at second order. This means that in order for our treatment to be consistent, we need to include in the second order Hamiltonian the extra term
$$ \hat{H}_{\mu}^{(2)} = (2J_0 - nU)\hat{N}^* = -\mu\hat{N}^*, \qquad (5.26) $$

where we recognized the chemical potential $\mu$ of the condensate (5.21). Equation (5.26) means that, in this approximation, the condensate is acting as a chemical potential for the perturbation modes.

**Dynamics in $k=0$.** From Eqs. (5.23), (5.24) and (5.26), we can isolate the terms pertaining to the dynamics of the mode $\hat{d}_0$ (noting that $\sum_j \hat{b}_j = \sqrt{L} \hat{d}_0$ by definition (5.5)):

$$ \hat{H}_{k=0}^{(2)} = iW\sqrt{N}(\hat{d}_0^\dagger - \hat{d}_0) + (-2J_1 + \Delta E_1 - \mu)\hat{d}_0^\dagger\hat{d}_0. \qquad (5.27) $$

This is, in fact, a Rabi Hamiltonian for the mode $\hat{d}_0$, driven by the coupling to the condensate mode $\hat{c}_0 \approx \sqrt{N}$. This coupling is non resonant provided the energy difference $(-2J_1 + \Delta E_1 - \mu)$ is large compared to the coupling. We will make sure to consider such situations, as in Figs. 5.1 and 5.3.

**Dynamics in $k \neq 0$.** The dynamics of the other modes is also best viewed in momentum space. Rewriting the corresponding terms of Eq. (5.24) with the use of Eqs. (5.16), (5.6) and (5.10), we get

$$ \hat{H}_{0,k \neq 0}^{(2)} = \sum_{k \neq 0} \left( E_u(k) \hat{u}_k^\dagger \hat{u}_k + E_v(k) \hat{v}_k^\dagger \hat{v}_k \right), \qquad (5.28) $$

and, likewise for the interaction terms<fnref n="9" />:

$$ \hat{H}_{\text{int},k \neq 0}^{(2)} = \frac{nU}{2} \sum_{k \neq 0} \left\{ \begin{aligned} & \left( \cos\left(\frac{\theta}{2}\right) \hat{u}_k + \sin\left(\frac{\theta}{2}\right) \hat{v}_k \right) \left( \cos\left(\frac{\theta}{2}\right) \hat{u}_{-k} + \sin\left(\frac{\theta}{2}\right) \hat{v}_{-k} \right) \\ & + \left( \cos\left(\frac{\theta}{2}\right) \hat{u}_k^\dagger + \sin\left(\frac{\theta}{2}\right) \hat{v}_k^\dagger \right) \left( \cos\left(\frac{\theta}{2}\right) \hat{u}_{-k}^\dagger + \sin\left(\frac{\theta}{2}\right) \hat{v}_{-k}^\dagger \right) \\ & + 4 \left( \cos\left(\frac{\theta}{2}\right) \hat{u}_k^\dagger + \sin\left(\frac{\theta}{2}\right) \hat{v}_k^\dagger \right) \left( \cos\left(\frac{\theta}{2}\right) \hat{u}_k + \sin\left(\frac{\theta}{2}\right) \hat{v}_k \right) \end{aligned} \right\}. \qquad (5.29) $$

Finally, the chemical potential term for the perturbation may be written

$$ \hat{H}_{\mu, k \neq 0}^{(2)} = -\mu \sum_{k \neq 0} \left( \hat{u}_k^\dagger \hat{u}_k + \hat{v}_k^\dagger \hat{v}_k \right). \qquad (5.30) $$

<footnote n="9">⁹We here use the fact that $\theta(k) = \theta(-k)$ for the symmetric band structure of $\hat{H}_0$, modeling our Floquet-Bloch system whose quasi-energy spectrum also presents that symmetry (e.g. Fig. 5.3) as it is a time-reversal symmetric system (see Eq. (5.1) and Sec. 1.2.2).</footnote>
With the quadratic expression for $\hat{H}_{\text{eff},k \neq 0}^{(2)}$ (i.e. the sum of Eqs. (5.28), (5.29) and (5.30)), we can now study the dynamics of the modes $(\hat{u}_k, \hat{v}_k)_{k \neq 0}$ in the system.

### 5.2.3 Stability

**Evolution of the perturbations.** To study the stability of the system, we write the Heisenberg equations for the evolution of the fields $\hat{u}_{k \neq 0}$ and $\hat{v}_{k \neq 0}$ from their commutation with $\hat{H}_{\text{eff},k \neq 0}^{(2)}$. In the following, we stop specifying that this study of dynamical stability is performed away from $k=0$. We remind that the operators $\hat{c}_k$ and $\hat{d}_k$, as well as $\hat{u}_k$ and $\hat{v}_k$, have bosonic commutation relations. We have $[\hat{u}_k, \hat{v}_{k'}] = [\hat{u}_k, \hat{v}_{k'}^\dagger] = 0$ ($\forall k$) and $[\hat{u}_k, \hat{u}_{k'}^\dagger] = [\hat{v}_k, \hat{v}_{k'}^\dagger] = 1$ ($\delta_{k,k'}$). Important consequences for our derivation are

$$[\hat{u}_k, \hat{u}_k^\dagger \hat{u}_k] = \hat{u}_k, \qquad \left[ \hat{u}_k, \sum_{k'} \hat{u}_{k'}^\dagger \hat{u}_{k'} \right] = \hat{u}_k \quad \text{and} \quad \left[ \hat{u}_k, \sum_{k'} \hat{u}_{k'}^\dagger \hat{u}_{-k'}^\dagger \right] = 2 \hat{u}_{-k}^\dagger.$$

With these relations, we find that the differential equations for the evolution of $\hat{u}_k$ and $\hat{v}_k$ are:

$$i\hbar \frac{\partial \hat{u}_k}{\partial t} \approx [\hat{u}_k, \hat{H}_{\text{eff}}^{(2)}] \qquad (5.31) \\ \approx A_{uu}(k)\hat{u}_k + A_{uv}(k)\hat{v}_k + B_{uu}(k)\hat{u}_{-k}^\dagger + B_{uv}(k)\hat{v}_{-k}^\dagger$$

$$i\hbar \frac{\partial \hat{v}_k}{\partial t} \approx [\hat{v}_k, \hat{H}_{\text{eff}}^{(2)}] \qquad (5.32) \\ \approx A_{uv}(k)\hat{u}_k + A_{vv}(k)\hat{v}_k + B_{uv}(k)\hat{u}_{-k}^\dagger + B_{vv}(k)\hat{v}_{-k}^\dagger.$$

The nomenclature for the coefficients is such that A couples the evolution of modes in the same quasi-momentum while B couples the evolution of modes at opposite quasi-momentum, with indices specifying the modes. From the parameters of the model, these coefficients read

$$A_{uu}(k) = \tilde{E}_u(k) + 2nU \cos^2\left(\frac{\theta}{2}\right) \qquad B_{uu}(k) = nU \cos^2\left(\frac{\theta}{2}\right) \qquad (5.32)$$

$$A_{vv}(k) = \tilde{E}_v(k) + 2nU \sin^2\left(\frac{\theta}{2}\right) \qquad B_{vv}(k) = nU \sin^2\left(\frac{\theta}{2}\right)$$

$$A_{uv}(k) = nU \sin(\theta) \qquad B_{uv}(k) = \frac{nU}{2} \sin(\theta),$$

where we defined the energy levels of the coupled bands offseted by the chemical potential $\tilde{E}_{u,v}(k) = E_{u,v}(k) - \mu$. In the following, we make implicit the dependence of $\tilde{E}_{u,v}$ (as well as that of the parameters of Eq. (5.32)) on the quasi-momentum $q = 2\pi k/L$. In Eq. (5.31), we see
that we get a set of four coupled differential equations. The last two can simply be obtained by complex conjugation of the first two and transforming<fnref n="10" /> $k \to -k$:

$$i\hbar \frac{\partial \hat{u}_{-k}^{\dagger}}{\partial t} \approx -B_{uu}\hat{u}_{k} - B_{uv}\hat{v}_{k} - A_{uu}\hat{u}_{-k}^{\dagger} - A_{uv}\hat{v}_{-k}^{\dagger} \quad (5.33)$$

$$i\hbar \frac{\partial \hat{v}_{-k}^{\dagger}}{\partial t} \approx -B_{uv}\hat{u}_{k} - B_{vv}\hat{v}_{k} - A_{uv}\hat{u}_{-k}^{\dagger} - A_{vv}\hat{v}_{-k}^{\dagger}.$$

We write this system in a matrix format:

$$i\hbar \frac{\partial}{\partial t} \begin{pmatrix} \hat{u}_k \\ \hat{v}_k \\ \hat{u}_{-k}^{\dagger} \\ \hat{v}_{-k}^{\dagger} \end{pmatrix} = \mathcal{L}(k) \begin{pmatrix} \hat{u}_k \\ \hat{v}_k \\ \hat{u}_{-k}^{\dagger} \\ \hat{v}_{-k}^{\dagger} \end{pmatrix}, \quad (5.34)$$

where the matrix $\mathcal{L}(k)$ that governs the evolution of the modes reads

$$\mathcal{L}(k) = \begin{pmatrix} A_{uu} & A_{uv} & B_{uu} & B_{uv} \\ A_{uv} & A_{vv} & B_{uv} & B_{vv} \\ -B_{uu} & -B_{uv} & -A_{uu} & -A_{uv} \\ -B_{uv} & -B_{vv} & -A_{uv} & -A_{vv} \end{pmatrix}. \quad (5.35)$$

The spectrum of $\mathcal{L}(k)$ contains the information about the stability of the system, as discussed in the following summary box.

## Result of the perturbative treatment of the interactions.

$\mathcal{L}(k)$ (5.35) is the coupling matrix for the differential equations (5.34) of the modes $\hat{u}_k$, $\hat{v}_k$, $\hat{u}_{-k}$, and $\hat{v}_{-k}$ (which are the coupled Fourier modes of our perturbations (5.10)). Its diagonalization yields eigenmodes whose angular frequencies of evolution are the four eigenvalues $\lambda_i(k)$ of $\mathcal{L}(k)$. These eigenvalues contain the information about the stability of the system. Indeed, any perturbation at quasi-momentum $q = 2\pi k/L$ will have a stable oscillating behavior if and only if all four eigenvalues $\lambda_i(q)$ are real. On the other hand, if either of the $\lambda_i(q)$ has non-zero imaginary part along the Brillouin zone, we get a “complex frequency” associated to exponential growth of unstable modes at that quasi-momentum. In the following, we will compute and display the instability parameter

$$w(q) = \max_i \{|\mathrm{Im}\{\lambda_i(q)\}|\}. \quad (5.36)$$

Evaluated for all quasi-momentum $-0.5 < q/k_L \le 0.5$ (taken in the continuous limit

<footnote n="10">¹⁰Using once again the central symmetry of the band structure (see footnote p. 134), implying here $A_{\alpha\beta}(-k) = A_{\alpha\beta}(k)$ and $B_{\alpha\beta}(-k) = B_{\alpha\beta}(k)$ for all modes $\hat{\alpha}, \hat{\beta}$.</footnote>
corresponding to $L \to \infty$), $w(q)$ informs us on the regions where to expect the growth of unstable modes in the Brillouin zone.

The matrix elements of $\mathcal{L}(k)$ (5.35) for the evaluation of $w(q)$ are computed using Eqs. (5.32), (5.13), (5.11) and (5.8).

To summarize, in Sec. 5.2.1, we developed a method to fit Floquet spectra of the real system and determine realistic parameters for the model. In this section, we explicitly obtained the elements of the evolution matrix for the perturbations in the Brillouin zone, allowing to determine regions of instability in the system through the parameter $w(q)$ (Eq. (5.36)). The only unknown quantity is the interaction parameter $nU$. In the next section, we discuss an estimation for this parameter.

### 5.2.4 Estimation of the interaction parameter

The interaction parameter $nU$ in the effective one-dimensional model has to take into account two features from the real system:

(i) the real system is a one-dimensional lattice of pancakes of atoms, highly confined in the lattice wells parallel to the lattice and loosely confined in perpendicular directions,

(ii) the number of atoms per site is not constant along the lattice, with more atoms in the center of the hybrid trap than further away from it.

A heuristic formula obtained from [205, 201] allows us to estimate the interaction parameter $n_0U_0$ in the central, most populated lattice site:

$$n_0U_0 \approx \hbar\Omega_{\perp}\sqrt{\frac{a_s}{\sqrt{2\pi}a_0}}\sqrt{n_0}, \quad (5.37)$$

where $\Omega_{\perp} \approx 2\pi \times 67$ Hz is the averaged angular frequency of the hybrid trap along the axes perpendicular to the lattice (see Sec. 2.2.4), $a_s \approx 5.23$ nm is the s-wave scattering length of $^{87}$Rb, $a_0 = \sqrt{\hbar/m\omega_0}$ is the characteristic size of the ground state in each lattice site in the harmonic approximation (with $\omega_0 = \sqrt{s_0} \times 2\pi\nu_L$, see Sec. 1.4.1) and $n_0$ is the number of atoms in this central lattice site. Working with BECs of $N \approx 5 \cdot 10^5$ atoms in a hybrid trap with angular frequencies ($\Omega_x, \Omega'_y, \Omega'_z$) $\approx 2\pi \times (10.4, 66, 68)$ Hz, we can estimate [201, 66] $n_0 \approx 4.4 \cdot 10^3$ atoms in this central lattice site. There, we thus get for the interaction parameter (in units of $E_L$): $n_0U_0/E_L \approx 0.86$. This estimate gives us an upper bound for $nU$ across the lattice, which permits to study the spectrum of $\mathcal{L}(q)$ (Eq. (5.35)) over a realistic range of interaction parameters.
<figure><img src="image_63.png" /><figcaption>**Figure 5.4** | Typical results of the two-band tight-binding model with interactions for realistic parameters. (a,d) Quasi-energy spectra. (b,e) Spectra of the effective Hamiltonian partially fitting (a,d) (see Sec. 5.2.1). (c,f) Corresponding instability parameter *w*(*q*) (5.36) as a function of *q* and *nU*. Parameters for (a-c) correspond to those of Fig. 5.3(a-c) and parameters for (d-f) correspond to those of Fig. 5.3(d-f). The vertical black dashed lines mark the quasi-momenta of the avoided crossings. A closer look of the crossings can be seen in Fig. 5.3.</figcaption></figure>

5.2.5 Results of the model and predictions

In the Secs. 5.2.1 and 5.2.4, we have determined realistic values for all the parameters of the
tight-binding model and we are now able to study the stability of the system by computing the
instability parameter *w*(*q*) along the Brillouin zone. According to our estimation of the upper
bound for the interaction parameter, we evaluate *w*(*q*) in the realistic interval *nU* ∈ [0, 0.1] *E*<sub>L</sub>.
Figure 5.4 shows a summary of our approach, from the fit of the Floquet bands to the study of
the spectrum of *L*(*q*). As *nU* is increased, we see emerging sharp stripes in quasi-momentum of
non-zero values of *w*(*q*), corresponding to region where *L*(*q*) has at least one complex eigenvalue
and where instabilities are expected. Strikingly, these sharp regions are observed in the vicinity
of the avoided crossings. As we will demonstrate in the next section of experimental results, the
decimal peaks observed in the experiments can be traced back to these unstable regions in the
Brillouin zone.
<figure><img src="image_64.png" /><figcaption>**Figure 5.5** | Predictions from the two-band tight-binding model with interactions. Instability regions versus $\Delta E_1$ and $W$ with the interaction parameter value fixed to $nU = 0.05 E_L$. (a) Instability parameter $w(q)$ as a function of $q$ and $\Delta E_1$ for $J_1 > 0$ (the parameters $J_0, J_1$ and $W$ are those of the fit of Fig. 5.3(c)). The variation of $\Delta E_1$ is such that the crossings position $q'$ (oblique dashed black lines) varies linearly, that is $\Delta E_1(q') = -2J_0 \cos(q') + 2J_1 \cos(q')$, with $0 \le q' \le \pi$ from the top to the bottom of the panel. (b) Same for $J_1 < 0$, with parameters corresponding to Fig. 5.3(f). (c) $w(q)$ as a function of $q$ and $W$ for $J_1 > 0$ (the parameters $J_0, J_1$ and $\Delta E_1$ are those of the fit of Fig. 5.3(c)). (d) Same for $J_1 < 0$, with parameters corresponding to Fig. 5.3(f). The vertical dashed black lines mark the crossings position and the horizontal plain black line mark the value of the varied parameter corresponding to Fig. 5.4.</figcaption></figure>

We see in Fig. 5.4 that the relative position of the unstable regions and the crossings depends on the curvature of the higher band 1 in the model. Indeed for $J_1 > 0$ (Fig. 5.4(c)), instabilities are observed outside the central region delimited by the crossings in the first Brillouin zone, while we observe them toward the inside this region for $J_1 < 0$ (Fig. 5.4(f)). This is a general trend of the model: for the regime of relative magnitude of $|J_0|$ and $|J_1|$ considered, instabilities are always observed away from the crossings, in the direction toward which the upper band grows (i.e. away from $p = 0$ in the unfolded band structure; see for instance Sec. 2.4.3). However, we will see in the next section (Sec. 5.3.1) that these small shifts are not resolved experimentally.

In Fig. 5.4(f), one can distinguish a broad region of low instability centered in $q = 0$. This may indicate in that case a less pronounced destabilization of modes near $q = 0$.

**Predictions.** We can make a series of experimental predictions from the model.

(i) A first prediction follows from the apparent connection between the avoided crossings and the instability region. We further investigate this connection in Fig. 5.5, where we fix the interaction parameter to a realistic intermediate value of $nU = 0.05 E_L$ and we vary the energy offset $\Delta E_1$ of the band 1. This changes the quasi-momenta at which the uncoupled levels of the model cross, and we see in Fig. 5.5(a,b) that the position of the instability regions follows this variation. Experimentally, the positions of the avoided crossings are mainly set by
the modulation frequency of the lattice (see Fig. 1.6). By tuning this parameter, we expect four-wave mixing to feed decimal peaks at momenta that will follow the variation of the quasi-momentum at which resonant coupling occurs. This experiment is presented in Sec. 5.3.2.

(ii) Another expected behavior directly stems from Fig. 5.4: we note that as $nU$ increases, so do the maxima of the instability parameter $w(q)$. In the model, this is associated with shorter instability timescales. In the experiment, we can *decrease* the interaction parameter $nU$ by reducing the number of atoms in the BECs (Sec. 2.2.4), which is expected to delay the decimal peaks nucleation. In Fig. 5.5(c,d), we also see that the variation of the coupling parameter $W$ results in a similar trend. Experimentally, the main parameter on which $W$ depends is the amplitude of modulation $\varphi_0$. Therefore, we should measure a strictly decreasing nucleation timescale as $\varphi_0$ increases. These experiments are presented in Sec. 5.3.3.

(iii) A last prediction concerns the generality of the model. Since the effective model is blind to the coupling mechanism between the bands, we can for instance expect similar instabilities with another type of modulation. With our experimental setup, we can very well generate similar looking Floquet spectra by modulating the *depth* instead of the phase of the lattice (see for instance Fig. 1.6). As a result, we expect to observe the nucleation of decimal peaks in depth-modulated optical lattices with similar modulation parameters.

We present experimental results in the next section. We start in 5.3.1 by measuring the quasi-momentum of the experimental decimal peaks to confirm the connection with the instability regions observed in the spectrum of $\mathcal{L}(q)$. In the rest of the section, we test our series of predictions.

# 5.3 Experimental results

## 5.3.1 Band-mapping method and quasi-momentum of the decimal diffraction peaks

In the previous section, we developed a tight-binding model that predicts the growth of instability modes in the neighborhood of the avoided crossings in the quasi-energy spectrum. Following this result, we perform a band-mapping experiment to access the quasi-momentum of the decimal peaks (observed in for example in Fig. 5.2) in order to confirm that it matches that of the instabilities predicted by the model. A reminder on the band-mapping technique is provided in Sec. 2.4.3.

Figure 5.6 shows a band-mapping experiment for the same modulation parameters as in Fig. 5.2(c). Comparing Fig. 5.6(c) and Fig. 5.2(c), we see how the TOF images are simplified through band-mapping. Indeed, as the atoms from the ground state only populate the plane wave of zero momentum after band-mapping, the rest of the signal stands out. In Fig. 5.6(d-f), we sketch the effect of the band-mapping process on the spectra of the system at different times. In panel (d), we place markers on the bands at the avoided crossings (in the model, these
<figure><img src="image_65.png" /><figcaption>**Figure 5.6** | Measurement of the quasi-momentum of the decimals peaks through band-mapping. (a) Depth of the lattice along time: adiabatical loading at $s_0$, held constant during the experiment, adiabatical unloading to band map (see text) and switch off for TOF imaging. (b) Phase of the lattice along time, sine-modulated with amplitude $\phi_0$ for an integer number $n$ of periods $T$. (c) Stack of experimental absorption images for increasing $n$, with $s_0 = 3.70 \pm 0.10$, $\phi_0 = 15^\circ$, $\nu = 1/T = 25.5$ kHz and $t_{\text{TOF}} = 35$ ms. (d) Corresponding quasi-energy spectrum (colored lines), with BEC (disk in $q=0$) and instability (disks in $q \neq 0$) modes. (e-f) Band structures of the lattices of depth $s_0 = 3.7$ for (e) and $s_0 = 0$ for (f) (solid colored lines) and follow-up of the modes (see text) with the same color code. (f-g) Borders of the Brillouin zone (black dotted lines). (g) Absorption image after $n = 80$ periods of data (c). The upper value of the colormaps for the ODs are truncated to reveal details.</figcaption></figure>

markers correspond to states in the modes $\hat{u}_q$ and $\hat{v}_q$ at the crossing) and a marker in $q=0$ that corresponds to the atoms in the ground state that initially macroscopically populate one Floquet state (see Fig. 5.1). When we stop the modulation (panel (e)), the populated states are projected on the Bloch eigenstates of the static lattice. Figures 5.6(f-g) correspond to the situation at the end of the band-mapping. We compare the position of the markers in the unfolded relation of dispersion with a TOF image from Fig. 5.6(c) after the nucleation of the decimal peaks. The agreement between the momentum of the markers and the position of the momentum peaks in the diffraction pattern confirms that, as predicted by the model, the quasi-momentum $q^*$ of the observed decimals peaks is related to the quasi-momentum of the avoided crossings in the quasi-energy spectrum. In the unfolded band structure, the model however predicts unstable modes with a slightly higher absolute momentum that the unfolded quasi-momentum of the crossings (Sec. 5.2.5). This trend has not been clearly observed in the experiments and we suspect that we cannot resolve this phenomenon due to the non-zero momentum dispersion of our atomic clouds, which translates in diffraction peaks of finite width.
<figure><img src="image_66.png" /><figcaption>**Figure 5.7** | Instability displacement with the modulation frequency. (a) Stack of experimental absorption images after $n = 100$ periods, averaged over 3 realizations, for an increasing modulation frequency $v$, with $s_0 = 3.57 \pm 0.10$, $\phi_0 = 15^\circ$ and $t_{\text{TOF}} = 35$ ms. The upper value of the colormaps for the ODs are truncated to reveal details. (b) Average instability position (in terms of the reduced quasi-momentum $k/k_L$ of the unfolded band structure) extracted from the fitted drift of the four $1 \le |p|/\hbar k_L \le 3$ orders of diffraction over all realizations (blue dots, errorbars correspond to the standard deviation of the 12-point sample) and calculated position of the resonant coupling as a function of the modulation frequency (solid black line). (c) Transition diagram from the lowest band $s$ over the first two Brillouin zones (solid black lines) and addressed transitions for data $b_1$, $b_2$ and $b_3$ (blue dots and solid lines). In (b-c) the gap between the transitions $s$-d and $s$-f (grey shaded area) and edge of the first Brillouin zone (black dotted line) are represented.</figcaption></figure>

### 5.3.2 Tuning the momenta of the instabilities

To further demonstrate that the quasi-momenta of the decimal peaks follows that of the avoided crossings, we tune the position of the crossings in the Floquet spectrum. This is simply achieved by varying the frequency $v$ of the phase modulation and taking TOF images after the nucleation of the decimal peaks. To this end, we fix the observation time at $n = 100$ periods for similar parameters as in Fig. 5.2. In Fig. 5.7, we compare, as a function of $v$, the measured reduced wave number $k/k_L$ of the emerging peaks with the unfolded quasi-momentum at which $v$ couples the bands. We vary $v$ from ~20.5 to ~40.5 kHz, resulting in a complete scan of the transitions $s$ to $d$ through $s$ to $f$ over the whole Brillouin zone (see Fig. 5.7(c)).

We observe that the decimal peaks in Fig. 5.7(a) follow the position of the avoided crossings in the quasi-momentum space (Fig. 5.7(b)), and even through the gap where the crossing transitions from the band pair ($s$, $d$) to the pair ($s$, $f$). Although this is globally in very good agreement with the predictions of the model, such a smooth correspondence over the whole Brillouin zone is not trivial as our interpretation of the model is valid when studying the instability at $q \neq 0$ (see Sec. 5.2).
This experiment demonstrates our ability to tune the position of the unstable modes in the quasi-momentum space, i.e. the quasi-momentum of the output channels for the four-wave mixing occurring at $q = 0$. Considering the experiment of Fig. 5.7, one can argue that the determination of the quasi-momenta of the decimal peaks as a function of $\nu$ would have been easier with a band-mapping before the measurements (as in Sec. 5.3.1). However, we here decided not to band-map in order to display the periodicity of both the ground state and the decimals peaks. Doing so, the results that we show in Fig. 5.7(a) suggest, from the momentum space, the coexistence of multiple spatial periodicities with a ratio of periods that can be tuned with the frequency of the phase modulation. At the end of this chapter (Sec. 5.4.1), we discuss numerical simulations that have been performed on this system, which corroborate the emergence of new spatial periodicities in the system.

### 5.3.3 Nucleation time

The two-band model (Sec. 5.2) can provide the position of unstable modes in the Brillouin zone and the magnitude of the exponents that characterize their growth, but it cannot provide the full kinetics of the mode growth nor their subsequent evolution. However, as hinted by Fig. 5.6(c), the full dynamics of the mode growth is readily accessible experimentally. In this section, we study the behavior of the nucleation time versus the amplitude $\phi_0$ of the phase modulation and the initial number $N$ of atoms in the BECs.

To easily extract the population of atoms in the decimal diffraction peaks, we once again perform band-mapping. The data analysis is presented in Fig. 5.8. For a given run (e.g. Fig. 5.8(a)), we count, as a function of the modulation time, the fraction of atoms in a set of lateral decimal peaks that are clearly visible. We do not take into account the decimal peaks that are too close to the central BEC, as the proximity with this highly populated region makes difficult to extract the populations in the emerging structures. We fit the fraction of atoms in the lateral decimal peaks in time with the sigmoid growth curve:

$$ \pi^*(t) = \frac{\pi_0^*}{1 + e^{-(t-t_n)/\tau}}, \qquad (5.38) $$

where:

* $\pi_0^*$ is the value of the plateau, i.e. the final fraction of atoms in the decimal peaks considered,

* $t_n$ is the time of half-maximum, which we choose to define the nucleation time,

* $\tau$ is the characteristic growth time of the instability.

For a frequency that couples the bands $s$ and $d$, we plot in Fig. 5.8(b) the growths of the atomic populations in the lateral order and the results of the fit for $\phi_0$ ranging from 10° to 20°. The extracted nucleation times $t_n$ are shown in Fig. 5.8(c) together with data corresponding to a resonant modulation between the bands $s$ and $f$. Fig. 5.8(d) shows a similar experiment, but
<figure><img src="image_67.png" /><figcaption>**Figure 5.8** | Measurement of the nucleation time of the instability. (a) Stack of absorption images after band-mapping for an increasing time of modulation, for $s_0 = 3.70 \pm 0.20$, $v = 25.5$ kHz (coupling bands *s*-d), $\phi_0 = 15^\circ$ and $N = (4.8 \pm 0.4) \cdot 10^5$ atoms in the BECs. The number of atoms in the grey stripes (here for $1 \lesssim |p|/(\hbar k_L) \lesssim 2$) is used to determine the nucleation time $t_n$ (blue vertical line, with a surrounding blue shaded area corresponding to one standard deviation on the fit). The upper value of the colormaps for the ODs are truncated to reveal details. (b) Growth curves extracted from data as in (a), with the purple, red, blue, orange, and green data corresponding to $\phi_0 = \{10^\circ, 12.5^\circ, 15^\circ, 17.5^\circ, 20^\circ\}$ respectively. The sigmoid fitting curves are shown and $t_n$ is displayed as in (a). (c) $t_n$ as a function of the modulation amplitude for data (b) (colored dots) and for $v = 30.5$ kHz and $s_0 = 3.56 \pm 0.20$ (black dots, coupling bands *s*-f). (d) Same as (c) as a function of the number of atoms $N$ in the BEC for $v = 25.5$ kHz and $s_0 = 3.58 \pm 0.30$, for a fixed modulation amplitude $\phi_0 = 15^\circ$. The blue marker/line in every panels corresponds to the same experiment. Error bars correspond vertically to one standard deviation on the fit and horizontally to one standard deviation over 4 independent measurements of the atom number.</figcaption></figure>

as a function of the initial number $N$ of atoms in the BECs for a frequency of modulation that couples the bands *s* and *d*. In the experiment, we reduce the number of atoms in the BECs during a holding time prior to the lattice loading (Sec. 2.2.4).

Results in Fig. 5.8 show that the nucleation time decreases with $\phi_0$ and $N$. These trends are in agreement with the predictions of the tight-binding model (Sec. 5.2.5): by increasing $\phi_0$, we mainly increase the coupling $W$ between the bands, which leads to larger instability exponents. The variation of the nucleation time with $N$ is qualitatively expected as well, as the initial interaction energy in the condensate increases with $N$, and is also associated with a stronger instability. We note that these results are similar to those obtained for the single-band Bogoliubov instability leading to staggered states [201].

### 5.3.4 Survival of the instability

Using the same experimental procedure as for the study of the nucleation time of the decimal peaks, we observe the behavior of the instability over a longer period of time. Figure 5.9 shows
<figure><img src="image_68.png" /><figcaption>**Figure 5.9** | Survival time of the nucleated pattern for $s_0 = 3.8 \pm 0.1$, $\nu = 25.5$ kHz, $\phi_0 = 15^\circ$ and $N \approx 5 \cdot 10^5$ atoms. (a) Stack of absorption images after band-mapping. The upper value of the colormaps for the ODs are truncated to reveal details. (b) Growth of the population of atoms in the instability regions. Same analysis as Fig. 5.8</figcaption></figure>

the result of the experiment for a large number of atoms and a coupling between the bands s and d. We see that the emerging patterns remain visible even ~2$t_n$ (~8 ms ~200 modulation periods) after their appearance ($t_n$ ~ 4 ms). Beyond that, we observe some heating, typical in modulated systems, that blurs the signal as a whole. We come back to the disappearance of the decimals peaks in Sec. 5.4.1.

### 5.3.5 Amplitude modulation

A last experiment suggested by the tight-binding model is the amplitude modulation of the lattice with similar lattice depth and frequency of modulation. Indeed, as the effective model is of static nature, it does not distinguish between phase and amplitude modulations. As the amplitude modulation of the real system can produce comparable Floquet spectra (see Fig. 5.10(a) below), four-wave mixing should trigger similar results in a depth-modulated lattice. Our amplitude-modulated potential is

$$V(x,t) = -s_0 \frac{E_L}{2} [1 + \varepsilon_0 \cos(2\pi\nu t)] \cos(x). \qquad (5.39)$$

Results are shown in Fig. 5.10. On panels (a,b), we plot the quasi-energy spectrum and its fit from the two-band tight-binding model. As we see, the pair ($s_0, \nu/\nu_L$) = (3.2, 2.84) couples the bands s and d close to the edge of the Brillouin zone. panel (c) shows the instability parameter $w(q)$, predicting unstable modes near the edge of the Brillouin zone for realistic interaction parameters $nU$ (Sec. 5.2.4). Experimentally (panel (e)), we indeed observe the emergence of diffraction peaks in between those associated to the lattice ground states (panel (d)). The emerging peaks have momentum slightly below half integer values of $\hbar k_L$, which is consistent with predictions made from $w(q)$ (panel (c)). This experiment highlights the generality of the phenomenon studied in this chapter, and the relevance of the two-level model that predicts
<figure><img src="image_69.png" /><figcaption>**Figure 5.10** | Instability nucleation through amplitude modulation of the lattice. (a) Quasi-energy spectrum (colored lines) for the amplitude modulation of Eq. (5.39) with parameters $s_0 = 3.2$, $\epsilon_0 = 0.325$ and $\nu = 2.84 \nu_L$. (b) Spectrum of the two-band tight-binding model partially fitting (a). (c) Instability parameter $w(q)$ over the Brillouin zone as a function of $nU$. (d) TOF absorption image of the ground state of the static lattice of depth $s_0 = 3.2 \pm 0.1$ at the beginning of the modulation with $N \approx 4 \cdot 10^5$ atoms. (e) TOF absorption image of the atom cloud after $n = 60$ periods of modulation with $\nu = 23.05$ kHz, corresponding to the spectrum (a). The upper value of the colormaps for the ODs are truncated to reveal details.</figcaption></figure>

the instabilities regardless the nature of the interband coupling (in this case the kind of lattice modulation employed).

## 5.4 Discussion on the realized state

### 5.4.1 Truncated-Wigner simulations

As discussed previously (Sec. 5.3.3), our two-level tight-binding model (Sec. 5.2) is able to predict trends on the timescale of the instability produced by four-wave mixing, but it is not suitable to study the kinetics of the phenomenon. To do so, one needs to make a simulation of the modulated system that takes into account both the interactions between the particles and fluctuations, either of thermal or quantum nature, to work as a seed for the spontaneous four-wave mixing process and feed the modes that eventually start to grow. To face these requirements, Truncated-Wigner (TW) simulations [201] were performed by our collaborator Peter
Schlagheck from the University of Liège. Through TW simulations, one is able to mimic quantum fluctuations by averaging over the integration of many independent trajectories with slightly different initializations. The simulations were performed with a realistic harmonic confinement (sec. 2.2.4) superimposed to the lattice potential.

These simulations are performed with $N = 1 \cdot 10^5$ atoms in the phase-modulated lattice (Eq.(5.1)), with parameters $s_0 = 3.4$, $\nu = 30$ kHz (which couples the bands $s$ and $f$ at $q = 0.36 k_L$) and $\phi_0 = 20^\circ$. These parameters are similar to those of the experiments featured in this chapter. These simulations reproduce what we observe experimentally, namely an absence of evolution for the first milliseconds of modulation, followed by the nucleation of momentum components at decimal multiples of $\hbar k_L$, and later a disappearance of the patterns (as experimentally shown in Fig. 5.9). In Figs. 5.11 and 5.12, we present typical results of TW simulations on the state of the system after a fixed $t = 5$ ms of lattice modulation.

First in Fig. 5.11, we present details on the reduced one-body density matrix $\hat{\rho}^{(1)}$ of atomic ensemble after the nucleation of the instability. panel (a) displays the eigenvalues of $\hat{\rho}^{(1)}$. The greatest eigenvalue corresponds to the initially populated state at $q = 0$ (closely alike the lattice ground state, see Sec. 5.1). We then see two eigenvalues emerging from the rest. This manifests the emergence of two condensed mode in the mixed state. panels (b-d) shows successively the momentum representation of the states associated with these three greatest eigenvalues. Focusing on the emerging modes (Fig. 5.11(d-e)), we see that the two secondary condensate wave functions are symmetric and antisymmetric combinations of plane waves $\langle x | \chi_{\pm q^*/k_L} \rangle \propto e^{\pm iq^*x}$ (Sec. 1.2.1), with $q^* \approx q = 0.36$, the quasi-momentum of the avoided crossings for our modulation parameter (see above). These modes are coherently populated by four-wave mixing during the evolution of the atoms in the modulated lattice.

With $\hat{\rho}^{(1)}$, we can also verify that the apparition of these modes translates to the emergence of a periodicity $d^*$ larger than the lattice step $d$. Figure 5.12 features a study of spatial correlations in the atomic ensemble. On panel (a), we show, as a function of the lattice site $j$, the atomic density distribution computed within the $s$ band. On the two individual TW trajectories displayed, we see evidence of a new structuration on a scale of about 3 sites. This is in agreement with the expected quasi-momentum at which the instability occurs, as a resonant modulation at $q/k_L = 0.36$ leads to a spacing $d^*/d \simeq 1/0.36 \simeq 2.8$. The phase reference of the density modulation appears random for each trajectory, so the modulation washes out on the spatial density averaged over many TW realizations. Nevertheless, we can recover evidence of the spatial density modulation in the averaged state by computing the normalized density-density correlation function $g^{(2)}(j) = \langle n_0 n_j \rangle / \langle n_0 \rangle \langle n_j \rangle$, between the central and the $j$th site of the lattice. This is plotted in Fig. 5.12(b). It reveals clear density oscillations across the lattice which signal the emergence of a new long-range order. Still within the ground band, we can also compute a normalized coherence $g^{(1)}(j) = \langle \hat{a}_0^\dagger \hat{a}_j \rangle / (\langle n_0 \rangle \langle n_j \rangle)^{1/2}$. We plot $g_j^{(1)}$ in Fig. 5.12(c). It displays a modulations at long-range with the same period. These last two panels show that coherent emergent states with a new period $d^*$ coexist with the initial condensate of period $d$. As we briefly discuss in the next section, there are shared features between supersolids and the exotic state that we realize in the modulated lattice.
<figure><img src="image_70.png" /><figcaption>**Figure 5.11** | Structure of the reduced one-body density matrix after nucleation of the instability. (a) Ordered eigenvalues of the reduced one-body density matrix normalized by the total number *N* of atoms. The eigenvalues identified by color and letter are shown in panel (b-d). (b-d) *s* band components of the eigenstates identified in (a) (scaled by the square roots of the associated eigenvalues) plotted in the momentum space, displayed by plotting their real parts (solid colored lines) and imaginary part (dashed colored line). Parameters are $s_0 = 3.4$, $\phi_0 = 20^\circ$, $\nu = 30$ (kHz) and $N = 1 \cdot 10^5$ atoms.</figcaption></figure>

### 5.4.2 Shared features and differences between supersolids and the state emerging in our system

In this last section, we discuss common features between supersolids and the emerging state in our system, as well as differences that make reluctant to use the work supersolid in our case. Briefly introduced, supersolidity is the emergence of a crystalline order within a coherent superfluid phase [206]. This mysterious state of matter was first introduce near 1960 (see [207, 208, 209]). Its existence was first discussed and debated in the context of $^4$He [210]. It was only in 2017, in the field of ultracold atoms, that conclusive observations of supersolidity were made. To bring out supersolid features, these pioneering experiments relied on different experimental techniques involving spin-orbit coupling [211], cavity-mediated long-range interactions [212], and later long-range dipolar interactions in ultracold erbium and dysprosium gases [213, 214]. Three key requirements [206] for a quantum state to be dubbed supersolid are

(i) a long range modulation of the spatial density accompanied by

(ii) a preserved coherence in the modes associated with the large density oscillations, and

(iii) the realization of a stable and stationary state (possibly the ground state of the system).

In our case, requirements (i) and (ii) are respectively demonstrated by the oscillations observed in the $g^{(2)}$ and $g^{(1)}$ correlations function (Fig. 5.12). However, as hinted by the disappearance of
<figure><img src="image_71.png" /><figcaption>**Figure 5.12** | Correlations and coherence after nucleation of the instability. (a) Spatial density distribution in the *s* band as a function of the site index *j*. Two individual TW trajectories (dashed and dotted lines) exhibit a clear periodic modulation of the density, which is washed out in the average of 1000 trajectories (thick line). (b) Average density-density correlation function *g*<sup>(2)</sup>(*j*) between the central site and the *j*<sup>th</sup> site in the *s* band. (c) Average amplitude correlation function *g*<sup>(1)</sup>(*j*) between the central site and the *j*<sup>th</sup> site in the *s* band. Parameters are *s*<sub>0</sub> = 3.4, *ϕ*<sub>0</sub> = 20°, *ν* = 30 (kHz) and *N* = 1 · 10<sup>5</sup> atoms.</figcaption></figure>

the emerging structure for long modulation times (observed both in the experiments (Fig. 5.9)
and in the TW simulations [66]), the state that emerges in our system is not a steady state
strictly speaking. This is the reason why we avoid to use the term of supersolidity, and prefer
to speak of the transient emergence of supercrystalline order.

**Conclusion**

In this last chapter, we presented results on the observation and control of a supercrystalline order emerging from short-range interatomic interactions in the phase-modulated optical lattice. By loading the ground state of the lattice and modulating its phase at a frequency that couples the ground band to an excited state in *q* ≠ 0, we make the peculiar observation of momentum components emerging between the diffraction peaks associated with the ground state.

To model this situation, we develop an effective two-level tight-binding model with inter-
actions to study the stability of the system. With the two-band model, we obtain instability
regions symmetric in quasi-momentum that are much narrower than what can be found with
one-band systems [199, 201, 197]. These instabilities are populated by pairs of atoms from the
BEC at q = 0 that coherently scatters into q = ±q* (the output channels of the four-wave
mixing process at stake). We find that these output channels follows the quasi-momentum of
the avoided-crossings in the Floquet spectrum, so their quasi-momenta can be tuned by the fre-
quency of lattice modulation. Through several experiments, we confirm the connection between
the instability regions of the model, the avoided crossings in the quasi-energy spectrum and
the emerging structures observed in TOF images. We demonstrate how the periodicity and the
nucleation timescale of the emerging momentum peaks can be tuned by engineering the Floquet
spectrum.

To quantitatively account for the kinetics of four-wave mixing, a numerical study of this
problem has to go beyond a mean-field treatment of the interactions. Such analysis were per-
formed by Peter Schlagheck with Truncated-Wigner simulations, typical results of which were
presented in the end of this chapter.

This work opens unexplored routes to generate exotic tunable states in higher dimensions
and/or other lattice geometries. A publication on this work is in preparation [66].
# General conclusion

"Audi panem quid meliora." Ça veut rien dire, mais je trouve que ça boucle bien.

Roi Loth (Kaamelott, Livre IV, Episode 1)

This manuscript features part of the results that I obtained during my PhD in the Cold Atoms group at *Laboratoire Collisions, Agrégats et Réactivité* (LCAR, Toulouse) between 2019 and 2022. Using Bose-Einstein condensates (BECs) of $^{87}$Rb in a controllable one-dimensional optical lattice, my PhD work concerned the development and implementation of quantum state control methods and their application to quantum simulation in a Floquet-Bloch system (i.e. a system periodic in space and time). In this conclusion, I begin with a brief summary of this work, and I then discuss perspectives opened by this thesis and the results of our group these last three years.

# Summary

**Quantum-optimal control of matter waves in a one-dimensional optical lattice.** In the first study presented in this manuscript (Chap. 3), we used the formalism of quantum-optimal control (QOC) to compute the way in which to continuously shift the optical lattice in order to arbitrarily control the motional state of the BECs in the lattice. Using this protocol, we first prepared specific momentum distributions, for which the success of the preparation could be assessed with a single time-of-flight (TOF) observation of the atoms, as shown in Fig. 6.1(a). We then targeted more involved quantum state, the preparation of which we had to certify by implementing a dedicated state tomography algorithm based on likelihood maximization. Using these tools, we demonstrated our ability to prepare Gaussian states that we squeezed in position up to more than a factor of four, an example of states unattainable by more conventional methods such as adiabatic approaches. I conclude this first chapter of results with the presentation of an experiment providing a proof-of-principle of QOC application to quantum simulation in a Floquet system. We demonstrated how the optimization on the initial state in a study of dynamical tunneling in the depth-modulated lattice resulted in clearer signals of tunneling oscillations. See [63, 64] for our publications on this subject.

**Non-diffusive Hamiltonian ratchet in space.** I detailed in Chap. 4 the design of an integrable ratchet effect in a sine potential whose amplitude and phase modulations are correlated. This
<figure><img src="image_72.png" /><figcaption>**Figure 6.1** | Summary figure of my PhD work. (a) Horizontal stack of independent TOF absorption images showing our control over the BEC momentum distribution in the optical lattice. This panel refers to [63, 64] and Chap. 3. (b) Horizontal stack of TOF absorption images showing the momentum evolution of the ratcheting Floquet state (here prepared by quantum-optimal control) over three ratchet periods. This panel refers to [65] and Chap. 4. (c) Horizontal stack of TOF absorption images showing the momenta at which emerge a transient supercrystalline structure as a function of the lattice phase modulation frequency. This panel refers to [66] and Chap. 5. (d) TOF absorption image featuring a scattering halo between the momentum components ±4 ħk_L (diffractions orders are not well resolved for this TOF shortened to 8 ms in order to reveal the halo). This panel refers to [62]. (e) Stroboscopic phase portrait and Husimi representation of a Floquet state (localized on regular islands) invovled in CAT in this system. This panel refers to [61].</figcaption></figure>

kind of ratchet effect consists in the emergence of a directed current of particles along quasi-
periodic trajectories, with the noteworthy detail that these integrable trajectories periodically
stop in each lattice site. After having presented our method to determine the modulation
parameters for this classical mechanics effect to emerge, I discussed wave function transport along
these classical trajectories: as a parameter of the system is varied (in our case the effective Planck
constant<fnref n="1" />), Floquet state mixing couples in a non-monotonous way the ratcheting region of phase
space with the chaotic sea. The identification of this effect allowed us to determine optimal
experimental parameters to observe the effect, and the chapter ends with the presentation of
experiments where we observed the transport of matter waves along the ratcheting structures
in the classical phase space (Fig. 6.1(b)). A publication on this study is in preparation [65].

**Emergence of a tunable supercrystalline order in a Floquet-Bloch system.** In the last part of this manuscript (Chap. 5), I presented the study of an effect that goes beyond one-body physics in our system: the transient emergence of a supercrystalline phase resulting from short-range interactions in the shaken optical lattice. In these experiments, we initially loaded the ground state (at quasi-momentum $q = 0$) of the lattice before applying a phase modulation that couples the ground band to an excited band away from the center of the Brillouin zone (at $|q| \neq 0$, resulting in avoided crossings between the two bands involved at $\pm q$). Experimentally, after tens of modulation periods, we observed the nucleation of additional momentum components in the TOF images. To understand this phenomenon, we developed a two-band tight-binding model with on-site interactions. Through a Bogoliubov treatment of the interactions, we found

<footnote n="1">¹Which is the only free parameter left once the classical phase space is fixed..</footnote>
the presence of instability regions near the quasi-momenta of the avoided crossings. Through Floquet engineering of the avoided crossings, we demonstrated a control on these unstable modes (namely their periodicity, as shown in Fig. 6.1(c), and nucleation timescale) resulting from spontaneous four-wave mixing. A publication on this study is in preparation [66].

**Research not featured in this manuscript.** I have contributed to two other research axes during my thesis, which I did not include in this manuscript for concision. The two following paragraphs briefly summarize these results:

**Observations of chaos-assisted tunneling (CAT) resonances.** CAT [88, 155, 154, 89] is a type of dynamical tunneling (Sec. 3.4) where the tunneling between the two regular regions is carried by two Floquet states mainly localized on these regions (Fig. 6.1(e)), as well as by a third Floquet state mainly delocalized over a chaotic sea. Under the variation of a parameter of the system, the phenomenon of Floquet-state mixing (described in Sec. 4.2) leads to CAT resonances [88], associated with a non-monotonic variation of the tunneling frequencies in that three-level system. In this sense, the mixing chaotic state can assist, or on the contrary freeze, dynamical tunneling in the system. In our work [61], we were the first to unambiguously observe such a resonance of CAT using BECs in the depth-modulated optical lattice. This work results from a collaboration with M. Martinez, G. Lemarié and B. Georgeot from the Laboratoire de Physique Théorique (Toulouse) and O. Giraud and D. Ullmo from the Laboratoire de Physique Théorique et Modèles Statistiques (Paris). For numerical and theoretical details on this work, see the thesis of M. Martinez [69]. For experimental details, see the theses of M. Arnal [67] and G. Chatelain [68].

**Observation and control of quantized scattering halos.** When BECs are released from an optical lattice to perform a TOF, atoms can undergo two-body collisions at the beginning of the TOF [62] resulting in well visible scattering halos in the images [107, 215, 216, 62]. In our experiment, the low relative kinetic energy between the colliding pair results in spherical s-wave scattering [90] (Fig. 6.1(d)), and the momentum quantization implied by the lattice periodicity (Sec. 1.2.1) results in a quantization of the possible relative momentum for the colliding atoms, thus leading to the observation of scattering halos of quantized diameter. In our work [62], we showed how one can control the momentum characteristics of these quantized halos: from a perturbative treatment of the collisions, one obtains that the number of atoms in the halos is proportional to the product of the colliding momentum component populations [62]. Thus, controlling halo momentum amounts to control the momentum probability distributions<fnref n="2" /> of the state in the lattice before the TOF. In our work [62], we demonstrated such control with a systematic technique that relies on a sudden lattice shift $\phi_0$ followed by a precisely timed evolution $\tau$ in the lattice of depth $s_0$. By determining optimal triplet $(\phi_0, \tau, s_0)$, we managed to control the atomic populations in the momentum components to the best of what is achievable from the evolution of a

<footnote n="2">²We point out that these results where obtained before our implementation of QOC in the experiment. This study on quantized scattering halos was actually our gateway to QOC.</footnote>
translated state in the static lattice, and thereby demonstrated our ability to control the scattering halos in the reciprocal space. This work results from a collaboration with P. Schlagheck from the CESAM research unit at the University of Liège. Details on this study [62] can be found in the thesis of G. Chatelain [68].

## Some prospects

Several lines of research are opened by these works. I highlight three of them here:

**QOC for qudit computation.** A qubit, the famous building block of quantum processors, can be defined as the superposition of two discrete quantum levels $|\psi\rangle = c_0|0\rangle + c_1|1\rangle$, with $c_0, c_1 \in \mathbb{C}$ and $|c_0|^2 = |c_1|^2 = 1$. A d-qudit [217, 218] is the generalization in d dimensions of the two-dimensional qubit: $|\psi_d\rangle = \sum_{n=1}^d c_n|n\rangle$, with $c_n \in \mathbb{C}$ and $\sum_{n=1}^d |c_n|^2 = 1$. In that regard, we remark that the external quantum states of BECs in Bloch systems (with their discretized Hilbert space) can be considered as qudits in momentum space. In Chap. 3 and [63, 64], we used QOC with the phase of the lattice as a control parameter $u(t)$, of which we computed the optimal evolution to drive an initial state $|\psi_a\rangle$ to a final state $|\psi_b\rangle$:

$$ |\psi_a\rangle \xrightarrow{u(t)} |\psi_b\rangle. \qquad (6.1) $$

One can try to go further and compute the optimized control field $u(t)$ in order to connect one-to-one the elements of two bases of the Hilbert space:

$$ \left\{ \begin{array}{c} |\psi_{a,1}\rangle \\ \vdots \\ |\psi_{a,d}\rangle \end{array} \right\} \xrightarrow{u(t)} \left\{ \begin{array}{c} |\psi_{b,1}\rangle \\ \vdots \\ |\psi_{b,d}\rangle \end{array} \right\}, \qquad (6.2) $$

with, for example in our system, a mapping between the plane waves $|\psi_{a,i}\rangle = |\chi_\ell\rangle$ and the Bloch eigenstates $|\psi_{b,i}\rangle = |\phi_n\rangle$ (see Sec. 1.2) for some relation $n = \sigma(\ell)$ between $\ell$ and $n$. Operations such as Eq. (6.2) correspond to unitary transformations, and allow one to design quantum gates acting on the qudits. Through this control approach, one does not have to compute the numerous matrix elements of the transformation (6.2), but only to optimize the control field $u(t)$. From here, one can implement basic quantum computing algorithms as for instance a version of the Deutsch algorithm [219] with a single qudit.

This application directly benefits from our implementation of QOC on the experiment. The connections with our work are here multiple, as, after the application of quantum processing operations, it will be necessary to measure the qudit. The maximum-likelihood state tomography technique implemented in Chap. 3 is well suited for this task.
**QOC for quantum sensing.** From our implementation of QOC, we can also design protocols for quantum sensing, i.e. the accurate measurement of an external parameter. A possible approach is to engineer a control field $u(t)$ in order to maximize, as a function of an external parameter to be measured, the rate at which diverge the different evolutions of a same initial state. Preliminary numerical results have been obtained for our setup in order to measure e.g. the lattice depth or a small magnetic force. Proof-of-concept experiments of quantum sensing will soon be realized by the group.

**Long-range chaos-assisted tunneling.** In our study of CAT [61], we have observed coupling resonances between pairs of regular islands of the same lattice site (Fig. 6.1). As a generalization of the effect, it has been predicted [220] that when the chaotic sea extends between lattice sites, there exists a harmonically decreasing coupling $t(n) \propto 1/n$ between the regular structures of a given site and the regular structures in the $n^{th}$ neighboring site. This slowly decreasing behavior of longer-range couplings can for instance be opposed with the exponentially decreasing coupling to neighboring sites in the case of the static lattice. Our observations of CAT resonances pave the way for a study of long-range chaos-assisted tunneling.

**New experimental setup.** Finally, our group is building a new experiment of $^{87}$Rb BECs in the newly built facility at LCAR. On this new setup, the implementation of optical lattices of higher dimensionality is planned. The developments presented in this manuscript are readily applicable to be utilized and further studied on systems of higher dimensionality.

These perspectives will be addressed during the theses of F. Arrouas and N. Ombredane, PhD students recently arrived in the group.
<figure><img src="image_73.png" /></figure>
# Part IV

## Appendices
<figure><img src="image_74.png" /></figure>
# Appendix A - Rubidium 87 D₂ line

This appendix refers to the Chap. 2. We present in Fig. A.1 the detail of the ⁸⁷Rb D₂ line, and we identify the transitions that we work with during the cooling sequence (Sec. 2.2). In a magnetic field gradient, the transition frequency between states $|F = 1, m_F = -1\rangle$ and $|F = 2, m_F = -1\rangle$ (on which microwave evaporation is based) decreases as the magnetic field increases, hence the negative detuning $\delta_{MW}(t)$ in Fig. A.1 (see Sec. 2.2.2). We remind that more information on the cooling sequence can be found in previous theses of our group [93, 94, 115, 67, 68].

<figure><img src="image_75.png" /><figcaption>**Figure A.1 | Rubidium 87 D₂ line.** During the MOT loading (Sec. 2.2.1), the cooling beam is red-detuned by a quantity $\delta_{MOT} \approx 2\Gamma$, with the spontaneous decay rate $\Gamma \approx 2\pi \times 6.07$ MHz (FWHM). The micro-wave frequency sweep $\delta_{MW}(t)$ is discussed in Sec. 2.2.2. The data in this figure are taken from [99].</figcaption></figure>
<figure><img src="image_76.png" /></figure>
# Appendix B - Dipole beams preparation and control

This appendix refers to the Chap. 2. We list the main elements that allow to prepare and control the dipole trap. We then present the optical diagram for the preparation of the dipole beams that constitute the trap in which Bose-Einstein condensation is achieved.

**Dipole beams.** The laser for the dipole trap is a monomode ALS-IR-1064-20-I-SF from Azurlight Systems, with wavelength $\lambda = 1064$ nm and maximum power $P = 20$ W. We operate it at $P = 17$ W. The two beams that make the dipole trap are focused at the same point in the cell. Their waist is $w_0 = 45$ µm with maximum power $P \approx 4$ W each. At the end of the evaporation in the dipole trap, their power is $P_f \approx 20$ mW each.

**AOMs.** The control of the dipole beams is the same for both beams. A radio frequency (RF) generator QMODP140,68-B-45-03 drives an acousto-optic modulator MQ40-B5A2-L1064-WSc (both components are from AA Opto-Electronic). This RF generator has an input channel that allows to vary the intensity of the driving RF and control the power of the diffracted beam. To prevent interference between the dipole beams, we work with opposite diffracted orders $\pm 1$ (see Fig. B.2).

<figure><img src="image_77.png" /><figcaption>Figure B.1 | Legend for the optical diagram.</figcaption></figure>
<figure><img src="image_78.png" /><figcaption>**Figure B.2** | Optical diagram for the dipole beams. This figure completes Fig. C.2 of App. C. See Fig. B.1 for the legend. Adapted from [67].</figcaption></figure>
# Appendix C - Lattice beams preparation and control

This appendix refers to the Chap. 2. We list the major components that constitute the optical lattice. We then present in Fig. C.2 the optical diagram for the preparation of the laser beams for the optical lattice. We remind the lattice dipole potential locally experienced by the atoms:

$$V(x, t) = -A(t) \times s_0 \frac{E_L}{2} \cos(k_L x + \varphi(t)). \quad (C.1)$$

with the lattice wavenumber $k_L = 2\pi/d$.

**Lattice beams.** The laser is a monomode YLR-15-1064-LP-SF from IPG Photonics. Its wavelength is $\lambda = 1064$ nm and it has maximum a power $P = 15$ W. At their focal point on the crossed dipole trap, the counterpropagating beams have maximum power $P = 2.3$ W and waist $w_0 = 150$ µm. The corresponding Rayleigh length is $x_R \approx 66.4$ mm, large compared to the lattice spacing $d = \lambda/2 = 532$ nm.

**Amplitude control.** The acousto-optic modulator (AOM) for the control of the lattice depth $A(t) \times s_0$ is placed before the beam splitting. It is a AA Opto-Electronic MQ80-A0.3-L1064-Z20-WLg driven by a radio frequency (RF) generator AA Opto-Electronic MODA80-B4-43. This latter component has a [0, 5] V modulation input acting on the amplitude of the RF that goes to the AOM, and its effect on the diffracted beam power is linear on the range [1.3, 3.3] V where we operate it [67]. We trigger an arbitrary waveform generator (AWG) (Keysight 33611A) during the sequence to control that input and therefore the lattice depth. We present in Fig. C.1 the Bode plot for the amplitude modulation of the lattice (measured using a photodiode with a sampling rate of 150 MHz placed before the cell on the lattice beam 2, see Fig. C.2). We see that we have approximately no loss in the amplitude of the depth modulation up to $\nu = 100$ kHz, a bound below which every experiments presented in this manuscript are performed.

**Phase control.** We control the lattice phase $\varphi(t)$ with one AOM per beam placed after the splitting. These AOMs are two AA Opto-Electronic MT200-A0.5-1064. By controlling the relative phase between their driving RFs, we control the relative phase between the diffracted beams. We generate their driving RF ourselves using AWGs, frequency doublers, filters and amplifiers (see [94, 115, 67]). We drive them at 160 MHz, i.e. slower than their nominal driving frequency of 200 MHz. The two AWGs are:

*Beam 1:* a Keysight 33612A with two channels. The first channel generates the AOM driving RF and the second channel modulates *internally* the phase of the first channel.

*Beam 2:* a Keysight 33611A to generate the AOM driving RF.
<figure><img src="image_79.png" /><figcaption>**Figure C.1** | Bode plot for the amplitude modulation of the lattice. The frequency at -3 dB is $\nu_0 \approx 10$ MHz.</figcaption></figure>

These two AWGs are synchronized with each other, as well as with the AWG that controls $A(t)$.

We are yet to implement an optical monitoring of the phase between the lattice beams from a beat signal. However, the result of two experiments allow us to estimate the cutoff frequency for the phase modulation of the lattice. On one hand, from electronical beat between the phase-modulated beam 1 RF and the unmodulated beam 2 RF, we measured an electronical cutoff frequency $\nu_{0,elec} \gtrsim 3$ MHz for the phase modulation of the beam 1 RF [68]. On the other hand, we measured the renormalization of the lattice depth in a fast phase-modulated optical lattice [198, 199, 113]. This effect, that depends on the amplitude of the phase-modulation, was measured up to $\nu = 500$ kHz [113, 67]. We therefore estimate the cutoff frequency for the phase modulation of the lattice to be somewhere in the range [0.5, 3] MHz.
<figure><img src="image_80.png" /><figcaption>**Figure C.2** | Optical diagram for the lattice beams that completes Fig. B.2 of App. B. See Fig. B.1 for the legend. Adapted from [67].</figcaption></figure>
<figure><img src="image_81.png" /></figure>
# Appendix D - Effect of the control duration in our QOC protocol

This appendix refers to the Chap. 3. We present a brief study of the experimental effect of the control duration $t_c$ in our QOC protocol. This study consists in targeting the state of Fig. 3.5(a):

$$|\psi\rangle = \frac{1}{\sqrt{2}} (|\chi_{-1}\rangle + |\chi_1\rangle), \qquad (D.1)$$

for different control times $t_c \in [0.75, 8] T_0$ and two lattice depth $s_0 = 2.55 \pm 0.10$ and $s_0 = 5.5 \pm 0.2$. Results are shown in Fig. D.1.

As discussed in Sec. 3.1.4, the lower bound on the control duration is of theoretical nature and consists in the lowest time needed for the target state to become reachable from the initial state. In the lattice, the dynamics of the atoms is dictated by the band structure. The initial state that we use for QOC protocols is generally the ground state of the lattice. The relevant timescale with respect to which the control duration needs to be sufficiently long is the largest inter-band transition period from the lattice ground state, i.e. $T_0$ as defined in Eq. (3.34).

For too long control times, experimental fluctuations have more time to accumulate. This is illustrated in Fig. D.1, as we observe a decreasing fidelity when $t_c$ increases. For the lattice depth $s_0 \approx 5.5$ (at which we performed most of the experiments in Chap. 3), we also observe larger fluctuations in the prepared state the longer the control time. Our rule of thumb is to set the control duration to the shortest time for which the numerical algorithm converged satisfactorily.

<figure><img src="image_82.png" /><figcaption>**Figure D.1 | Effect of the control duration** when targeting the state $(|\chi_1\rangle + |\chi_{-1}\rangle)/\sqrt{2}$. Fidelity $F^{(\beta)}$ (3.36) between the momentum distributions of the numerical and experimental states prepared by QOC as a function of the scaled control duration $t_c/T_0$ for two lattice depth ($s_0 = 2.55 \pm 0.10 \rightarrow T_0 \approx 87$ µs (blue dots) and $s_0 = 5.5 \pm 0.2 \rightarrow T_0 \approx 59$ µs (red triangles)). Error bars correspond to one standard deviation over a 10-realization statistics.</figcaption></figure>
<figure><img src="image_83.png" /></figure>
# Appendix E - Additional optimal control preparations

This appendix refers to the Chap. 3. We present three additional experiments that we have chosen to put in appendix in order to lighten the reading of that chapter. The three experiments featured in this appendix are (E.1) a preparation of the plane wave $|\chi_6\rangle$ from the resting plane wave $|\chi_0\rangle$, (E.2) a qualitative demonstration of our ability to control the relative phases between three momentum components and (E.3) the preparation and measurement of rotated squeezed Gaussian states.

## E.1 Momentum distribution control from the resting BEC

In Chap. 3, we demonstrate the efficiency of QOC for computing the manner in which to continuously shift the optical lattice in time in order to prepare arbitrary quantum states in our Bloch system. In the experiments presented in that chapter, the control stage always follows an adiabatical lattice loading (see Sec. 2.4.1), so the initial condition is the ground state of the lattice. We present in Fig. E.1 an experiment illustrating that this choice is somewhat arbitrary, and that control fields can as well be computed and experimentally applied to prepare a target starting from the resting BEC (i.e. the plain wave $|\chi_0\rangle$, see Secs. 1.2.1 and 2.4.1). Indeed, we manage to prepare the 6th momentum component with fidelity $F_{\text{exp}}^{(\beta)} = 0.85 \pm 0.06$ (averaged over 10 realizations with one standard deviation as uncertainty), i.e. as reliably as for the equivalent preparations of the Sec. 3.2.2 achieved from the ground state of the lattice.

## E.2 Control of the relative phases between three momentum components

We here extend the experiment of Sec. 3.3.1 and further display our ability to control the argument of the $c_{q,\ell}$ coefficients. We here target three-momentum component superpositions with specific relative phases between the plane wave coefficients:

$$|\psi_T^{(a)}\rangle = \frac{1}{\sqrt{3}} (|\chi_{-2}\rangle + |\chi_0\rangle + |\chi_2\rangle) \qquad (E.1)$$

$$|\psi_T^{(b)}\rangle = \frac{1}{\sqrt{3}} (|\chi_{-2}\rangle + e^{2i\pi/3} |\chi_0\rangle + e^{4i\pi/3} |\chi_2\rangle)$$

$$|\psi_T^{(c)}\rangle = \frac{1}{\sqrt{3}} (e^{2i\pi/3} |\chi_{-2}\rangle + |\chi_0\rangle + e^{4i\pi/3} |\chi_2\rangle)$$

Experimental results are shown in Fig. E.2. For each of these states, we show the experimentally measured evolution of the momentum distribution in the static lattice following the state preparation. We also present the theoretical evolutions of the numerically prepared states. We
<figure><img src="image_84.png" /><figcaption>Figure E.1 | Experimental preparation of the plane wave with momentum $p = 6 \hbar k_L$ from the resting BEC. (a) Depth of the lattice as a function of time. (b) Phase of the lattice as function of time, optimized for the preparation of the plane wave $|\chi_6\rangle$ from $|\chi_0\rangle$ using QOC in the interval $0 \le t \le t_c$. The numerical fidelity for this control field is $F^{(\beta)} \approx 0.977$ (see Sec. 3.2.1). (c) Numerical simulation of the momentum distribution in the lattice as function of time. (d) Experimental TOF absorption image after the preparation. The colormap for the OD is truncated to 0.8 of its maximum value (which saturates the center of the sixth momentum order) to see preparation defects. Parameters are $s_0 = 5.23 \pm 0.10$, $t_c = 3.25 T_0$ (see Sec. 3.1.2) and $t_{TOF} = 30$ ms.</figcaption></figure>

observe good qualitative agreement between the experimental and theoretical data, which allows us to identify the prepared superposition. Our general control on the $c_\ell$ coefficients is asserted when we target and reconstruct phase space distributions (see Secs. 3.3.4 and 3.3.5 as well as following section).

## E.3 Squeezed and rotated Gaussian states

This last section refers to Sec. 3.3. As a final example of optimal control preparation of Gaussian states, we present the rotation of squeezed Gaussian states. For a rotation angle $\theta$ (defined positive from the x axis to the p axis), the plane wave coefficients of a rotated squeezed Gaussian state can be written as:

$$c_{\ell}^{(\xi, \theta)}(u, v) = \left(\frac{\mathrm{Re}\{A\}}{\pi}\right)^{1/4} e^{iuv/2} e^{-ilu} e^{-A(l-v)^2/2}, \qquad (\mathrm{E.2})$$

with

$$A = \frac{\cosh(r) - \sinh(r) e^{2i\theta}}{\cosh(r) + \sinh(r) e^{2i\theta}} \quad \text{and} \quad r = \frac{1}{4} \ln\left(\frac{s}{4\xi^4}\right).$$

We focus on a rather large squeezing $\xi = 1/3$ (as compared to the experiments of Sec. 3.3.5),
<figure><img src="image_85.png" /><figcaption>**Figure E.2 | Preparation of three-momentum component states. (a,b,c)** Preparation of the superpositions $|\psi_T^{(a)}\rangle$, $|\psi_T^{(b)}\rangle$, and $|\psi_T^{(c)}\rangle$ (see Eq. (E.1)) respectively. Top: stack of integrated experimental images showing the evolution of the momentum distribution during a subsequent 110 µs holding time in a static lattice after applying the control field $\varphi(t)$ for the preparation of the target. Bottom: numerical propagation of the expected prepared state in a static lattice. For all three prepared states, the dimensionless lattice depth is $s_0 = 5.7 \pm 0.2$.</figcaption></figure>

phase space-centered states with $(u, v) = (0, 0)$ and we target the two rotation angles $\theta = \pm \pi/4$. As for most of our targets in Secs. 3.3.4 and 3.3.5, the control duration is fixed to $t_c = 1.75 T_0$ and we reconstruct the experimentally prepared states from the data of its evolution in the static lattice as detailed in 3.3.3. In Fig. E.3 we compare the Husimi representations of the numerically prepared and experimentally reconstructed states. In both cases, the QOC algorithm converged to a control field preparing the target with quantum fidelity $F_{\text{num}} > 0.995$. For $\theta = \pi/4$ (Fig. E.3(a)), the experimental fidelity to the numerically prepared state is $F_{\text{exp}} = 0.88$ with purity $\gamma = 0.89$ (see Sec. 3.3.3). For $\theta = -\pi/4$ (Fig. E.3(b)), we have $F_{\text{exp}} = 0.89$ and $\gamma = 0.88$.

<figure><img src="image_86.png" /><figcaption>**Figure E.3 | Preparation and measurement of rotated squeezed Gaussian states. (a-b)** Husimi representations in the phase space of the static lattice. Left (red): state $|\psi_{\text{QOC}}\rangle$ numerically prepared by optimal control. Right (blue): density matrix $\hat{\rho}_{\text{ML}}$ reconstructed from experimental data by likelihood maximization. The squeezing parameter for both targets is $\xi = 1/3$ and the rotation angles are $\theta = \pm \pi/4$ for (a,b) respectively. The lattice depth during the experiments is $s_0 = 5.45 \pm 0.30$. See text for fidelities and purities.</figcaption></figure>
<figure><img src="image_87.png" /></figure>
# Appendix F - Résumé du manuscrit en français

## Introduction

L'usage de la lumière et de champs magnétiques afin de contrôler les degrés de liberté externes des atomes (c'est-à-dire leur position et vitesse) a révolutionné le domaine de la physique atomique. Tirant parti du développement des lasers dans les années 1960, les techniques de piégeage et de refroidissement d'atomes [1, 2, 3] ont permis aux expérimentateurs d'atteindre un niveau de contrôle sans précédant sur les systèmes atomiques. Grâce à ces méthodes, il est possible de refroidir des ensembles atomiques proche du zéro absolu, jusqu'à des températures auxquelles une description quantique de la matière devient nécessaire. Un résultat emblématique de ce domaine d'étude est celui de la condensation de Bose-Einstein, une transition de phase quantique au travers de laquelle un gaz de bosons passe d'un ensemble de particules décrit classiquement par la statistique de Maxwell-Boltzmann à une fonction d'onde macroscopiquement peuplée par les atomes du gaz et dont l'évolution est gouvernée par la mécanique quantique: un condensat de Bose-Einstein (CBE).

La condensation de Bose-Einstein a été atteinte pour la première fois en 1995, dans les groupes de E. Cornell et C. Wieman [6] puis dans celui de W. Ketterle [7]. Cette prouesse expérimentale, récompensée par le prix Nobel de physique en 2001, marque la naissance du domaine de recherche sur les gaz quantiques, en constante croissance depuis lors. Ces gaz d'atomes ultrafroids, qui peuvent êtres de nature bosonique ou fermionique, constituent une plateforme hautement versatile avec laquelle étudier la mécanique quantique. En effet:

* L'utilisation de lasers et de champs magnétiques permet la réalisation d'une grande variété de potentiels [8, 9] dans lequels étudier l'évolution de ces systèmes quantiques. Il est par exemple possible de manipuler jusqu'à l'atome unique à l'aide de pinces optiques [10, 11] ou de réaliser des potentiels périodiques dans l'espace (dits de Bloch) finement contrôlables à l'aide e.g. de réseau optique [12] et de modulateurs spatiaux de lumière (spatial light modulators) [13, 14].

* En modulant périodiquement dans le temps ces potentiels artificiels (on parle alors de systèmes de Floquet), il devient possible de réaliser des hamiltoniens effectifs d'intérêt pour l'étude d'un système particulier [15, 16]. Un telle ingénierie de Floquet permet par exemple de créer des effets de magnétisme artificiel pour les atomes neutres [17, 18].

* Les effets des interactions entre particules quantiques peuvent être étudiés grâce aux résonances de Feshbach [19], qui permettent, pour les systèmes d'atomes ultrafroids, de contrôler la force et le signe des interactions à courtes portées entre les atomes. En outre,
d'autres systèmes sont développés afin d'étudier des effets d'interaction à plus longue portée, comme notamment les gaz dipolaires [20, 21, 22] et les systèmes d'atomes de Rydberg [23, 24, 25].

Le haut degré de contrôle qu'il est possible d'atteindre grâce aux expériences de gaz quantiques revêt un intérêt particulier pour les deux domaines que sont la simulation quantique et la métrologie quantique.

Le domaine de la simulation quantique [26, 27] vise à émuler des systèmes quantiques complexes sur des systèmes plus simples à observer et à contrôler. Cette idée a initialement été proposée par R. Feynman en 1982 [28] et a depuis été mise en œuvre sur de nombreux systèmes, comme par exemple les circuits supraconducteurs [29], les systèmes photoniques [30], les ions piégés [31] et les atomes neutres ultrafroids [32, 33, 34]. Nous nous intéressons dans ce manuscrit à cette dernière plateforme avec laquelle ont été réalisées, ces vingt dernières années, des expériences phare dans le domaine de la simulation quantique. On peut notamment citer la réalisation du modèle de Bose-Hubbard en trois dimensions, grâce à laquelle a été observée la transition entre une phase superfluide et un isolant de Mott [35], l'observation de la localisation d'Anderson en une [36, 37], deux [38, 39] et trois dimensions [40, 41, 42] dans des systèmes avec un désordre artificiel et la transition de Berezinskii-Kosterlitz-Thouless dans un gaz quantique confiné à deux dimensions [43, 44].

Un autre domaine fondamental qui bénéficie du haut niveau de contrôle offert par les gaz quantiques est celui de la métrologie quantique. Un exemple clé des avantages que recèle la manipulation d'un système quantique à des fins de métrologie est donné par les états comprimés, qui permettent d'atteindre des sensibilités de mesure de plus en plus proche des limites fondamentales dictées par le principe d'incertitude d'Heisenberg [45, 46]. C'est notamment le cas avec les états comprimés de photons pour la spectroscopie [47] et l'interférométrie [48, 49], et d'importants efforts sont fournis pour tirer parti de la compression d'états quantiques de matière dans le domaine de l'interférométrie avec des ondes de matières [50, 51, 52, 53, 54, 55].

Ces différents champs d'application encouragent le développement de nouvelles méthodes afin de contrôler et de manipuler les systèmes d'atomes froids. Plusieurs approches modernes sont proposées, parmi lesquelles les boucles de rétroaction [56], les raccourcis vers l'adiabaticité [57], l'apprentissage machine [58] et le contrôle optimal appliqué aux systèmes quantiques [59, 60].

Le groupe de recherche Atomes Froids dans lequel j'ai réalisé mon doctorat au Laboratoire Collisions, Agrégats et Réactivité (LCAR, Toulouse) travaille dans le domaine de la simulation quantique et du développement de techniques pour le contrôle de systèmes d'atomes ultrafroids. Le travail de thèse rapporté dans ce manuscrit porte à la fois sur le développement de telles méthodes et sur leur application pour la simulation quantique dans un système de Floquet-Bloch (c'est-à-dire un système périodique à la fois dans le temps et l'espace). Dans l'équipe Atomes Froids du LCAR, nous travaillons sur une expériences avec laquelle nous réalisons des CBE de $^{87}$Rb que nous plaçons ensuite dans un réseau optique unidimensionnel. Le long de l'axe du réseau, les atomes (qui peuvent être décrits par une fonction d'onde unique) font l'expérience d'un potentiel sinusoïdal dont nous pouvons contrôler la profondeur et la phase de manière arbitraire
au cours du temps à l'aide de modulateurs acousto-optiques. Ce système relativement simple constitue une plateforme idéale sur laquelle développer de nouvelles méthodes pour le contrôle d'états quantiques, ainsi que pour l'expérimentation sur la dynamique d'ondes de matière dans un potentiel sinusoïdal finement contrôlable dans le temps, et cela dans un régime de faibles interactions interatomiques.

Ce manuscrit de thèse est organisé de la manière suivante:

**Partie 1.** Nous commençons par une première partie de méthodes, composée des chapitres 1 et 2. Dans le **Chapitre 1**, nous présentons des rappels de théorie pour l'étude de la dynamique d'une fonction d'onde dans un système de Floquet-Bloch. Dans le **Chapitre 2**, nous présentons notre montage expérimental pour l'obtention de CBE, ainsi que le réseau optique et la manière dont nous le contrôlons, ce qui nous permet de réaliser dans le laboratoire le système décrit au premier chapitre.

**Partie 2.** La seconde partie présentent deux études expérimentales pour lesquelles les échelles de temps des expériences nous permettent de négliger les interactions interatomiques dans les CBE. Le **Chapitre 3** porte sur la mise en place d'un protocole de contrôle optimal quantique afin de contrôler arbitrairement l'état quantique externe des CBE dans le réseau optique (c'est-à-dire leur état de mouvement). Dans le **Chapitre 4**, nous détaillons l'étude, à la fois théorique et expérimentale, d'un effet *ratchet* (ou rochet) non-diffusif (c'est-à-dire le long de trajectoires classiques non-chaotiques) dans un système hamiltonien.

**Partie 3.** La troisième et dernière partie de ce manuscrit est composée du **Chapitre 5** où nous nous plaçons dans un régime expérimental dans lequel il n'est plus possible de négliger les interactions entre les atomes qui constituent le CBE. Nous y présentons l'étude de l'émergence d'une nouvelle structuration des atomes dans le réseau optique modulé en phase, qu'il nous est possible de contrôler par l'ingénierie, dans l'espace des impulsions, d'instabilités dynamiques dues à un phénomène de mélange à quatre ondes.

Nous dressons ci-après un bref résumé de ces cinq chapitres.

# Chapitre 1 : Éléments de théorie pour l'étude d'une fonction d'onde dans un potentiel unidimensionnel

Dans ce premier chapitre, nous rappelons des éléments de théorie pour étudier la dynamique d'une fonction d'onde dans un potentiel unidimensionnel $V(x, t)$ dont les paramètres peuvent dépendre du temps. La forme générale de potentiel qui nous intéresse dans ce manuscrit est la suivante :