# Control and transport of matter waves in an optical lattice

Nathan Dupont

► To cite this version:

Nathan Dupont. Control and transport of matter waves in an optical lattice: Optimal control, ratchet effect and study of an emergent supercrystalline order. Quantum Physics [quant-ph]. Université Toulouse 3 - Paul Sabatier, 2022. English. NNT: tel-03997401v3

HAL Id: tel-03997401

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Université Fédérale

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Toulouse Midi-Pyrénées

THÈSE

En vue de l'obtention du
DOCTORAT DE L'UNIVERSITÉ DE TOULOUSE

Délivré par l'Université Toulouse 3 - Paul Sabatier

Présentée et soutenue par

Nathan DUPONT

Le 15 décembre 2022

Contrôle et transport d'ondes de matière dans un réseau
optique : Contrôle optimal, effet ratchet et étude de l'émergence
d'un ordre supercristallin

Ecole doctorale : SDM - SCIENCES DE LA MATIERE - Toulouse

Spécialité : **Physique**

Unité de recherche :

LCAR - Laboratoire Collisions Agrégats Réactivité

Thèse dirigée par
**David GUERY-ODELIN**

Jury

Mme Caroline CHAMPENOIS, Rapporteure

M. Jérôme BEUGNON, Rapporteur

M. Éric SURAUD, Examinateur

M. Nathan GOLDMAN, Examinateur

M. David GUERY-ODELIN, Directeur de thèse
<figure><img src="image_3.png" /></figure>
<table>
  <tbody>
    <tr>
      <td>Conclusion</td>
      <td>95</td>
    </tr>
    <tr>
      <td>4 Non-diffusive Hamiltonian ratchet</td>
      <td>97</td>
    </tr>
    <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>4.2 Quantum ratchet transport along regular classical trajectories</td>
      <td>108</td>
    </tr>
    <tr>
      <td>4.3 Experimental ratchet transport of matter waves</td>
      <td>112</td>
    </tr>
    <tr>
      <td>Conclusion</td>
      <td>118</td>
    </tr>
    <tr>
      <td>III Beyond the one-body wave function</td>
      <td>119</td>
    </tr>
    <tr>
      <td>5 Emergence of a tunable supercrystalline order in a Floquet-Bloch system</td>
      <td>121</td>
    </tr>
    <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>5.3 Experimental results</td>
      <td>140</td>
    </tr>
    <tr>
      <td>5.4 Discussion on the realized state</td>
      <td>146</td>
    </tr>
    <tr>
      <td>Conclusion</td>
      <td>149</td>
    </tr>
    <tr>
      <td>General conclusion</td>
      <td>151</td>
    </tr>
    <tr>
      <td>IV Appendices</td>
      <td>157</td>
    </tr>
    <tr>
      <td>A Rubidium 87 D<sub>2</sub> line</td>
      <td>159</td>
    </tr>
    <tr>
      <td>B Dipole beams preparation and control</td>
      <td>161</td>
    </tr>
    <tr>
      <td>C Lattice beams preparation and control</td>
      <td>163</td>
    </tr>
    <tr>
      <td>D Effect of the control duration in our QOC protocol</td>
      <td>167</td>
    </tr>
    <tr>
      <td>E Additional optimal control preparations</td>
      <td>169</td>
    </tr>
    <tr>
      <td>E.1 Momentum distribution control from the resting BEC</td>
      <td>169</td>
    </tr>
    <tr>
      <td>E.2 Control of the relative phases between three momentum components</td>
      <td>169</td>
    </tr>
    <tr>
      <td>E.3 Squeezed and rotated Gaussian states</td>
      <td>170</td>
    </tr>
    <tr>
      <td>F Résumé du manuscrit en français</td>
      <td>173</td>
    </tr>
    <tr>
      <td>Bibliography</td>
      <td>197</td>
    </tr>
  </tbody>
</table>
List of Figures

<table><tbody><tr><td>1.1</td><td>One-dimensional sine potential and corresponding phase space</td><td>14</td></tr><tr><td>1.2</td><td>Construction of a stroboscopic phase portrait</td><td>15</td></tr><tr><td>1.3</td><td>Examples of stroboscopic phase portraits</td><td>16</td></tr><tr><td>1.4</td><td>Subperiod evolution of a stroboscopic phase portrait</td><td>17</td></tr><tr><td>1.5</td><td>Band structures and Bloch eigenstates</td><td>22</td></tr><tr><td>1.6</td><td>Band structure and quasi-energy spectra</td><td>27</td></tr><tr><td>1.7</td><td>Wigner representation of Bloch eigenstates in the phase space of the static sine potential</td><td>30</td></tr><tr><td>1.8</td><td>Husimi representation of Bloch eigenstates in the phase space of the static sine potential</td><td>32</td></tr><tr><td>1.9</td><td>Husimi representations of Floquet-Bloch states</td><td>35</td></tr><tr><td>2.1</td><td>Sketch of the old and new dipole beam geometries</td><td>44</td></tr><tr><td>2.2</td><td>Schematic of absorption imaging</td><td>47</td></tr><tr><td>2.3</td><td>Imaging axes</td><td>48</td></tr><tr><td>2.4</td><td>Adiabatic lattice loading and matter wave diffraction</td><td>52</td></tr><tr><td>2.5</td><td>Calibration of the optical lattice depth from Kapitza-Dirac diffraction</td><td>54</td></tr><tr><td>2.6</td><td>Calibration of the optical lattice depth from a sudden lattice translation</td><td>56</td></tr><tr><td>2.7</td><td>Sketch of the band-mapping technique</td><td>57</td></tr><tr><td>"Hello World"</td><td>written with matter diffraction</td><td>61</td></tr><tr><td>3.1</td><td>Schematic of a gradient ascent iteration for the computation of a discretized control field</td><td>72</td></tr><tr><td>3.2</td><td>Non-unicity of the control fields for the preparation of a given target</td><td>74</td></tr><tr><td>3.3</td><td>Controlling momentum distributions: single momentum components</td><td>76</td></tr><tr><td>3.4</td><td>Controlling momentum distributions: two-component equiprobable and arbitrary momentum superpositions</td><td>78</td></tr><tr><td>3.5</td><td>Control and measurement of the phase between two momentum components</td><td>81</td></tr><tr><td>3.6</td><td>Preparations of Bloch eigenstates and superpositions thereof</td><td>83</td></tr><tr><td>3.7</td><td>Quantum state reconstruction in a Bloch system by likelihood maximization</td><td>87</td></tr><tr><td>3.8</td><td>Experimental preparation and measurement of non-squeezed Gaussian states</td><td>88</td></tr><tr><td>3.9</td><td>Evolution of the momentum distribution of superpositions of Gaussian states in the static lattice</td><td>89</td></tr><tr><td>3.10</td><td>Experimental preparation and measurement of squeezed Gaussian states</td><td>90</td></tr></tbody></table>
<table>
  <tbody>
    <tr>
      <td>3.11 Evolution of the momentum distribution of a highly squeezed state in the static lattice</td>
      <td>91</td>
    </tr>
    <tr>
      <td>3.12 Initial states for dynamical tunneling</td>
      <td>93</td>
    </tr>
    <tr>
      <td>3.13 Quantum-optimal control applied to dynamical tunneling experiments</td>
      <td>95</td>
    </tr>
    <tr>
      <td>4.1 Diagram of a mechanical rotating ratchet</td>
      <td>99</td>
    </tr>
    <tr>
      <td>4.2 Parameter maps for initial guess determination in the search for ratchet modulation parameters</td>
      <td>106</td>
    </tr>
    <tr>
      <td>4.3 Classical non-diffusive ratchet trajectory</td>
      <td>107</td>
    </tr>
    <tr>
      <td>4.4 Subperiod evolution of the stroboscopic phase portrait and ratcheting island</td>
      <td>108</td>
    </tr>
    <tr>
      <td>4.5 Ratcheting Floquet state and influence of the effective Planck constant</td>
      <td>111</td>
    </tr>
    <tr>
      <td>4.6 Floquet state mixing phenomenon illustrated on the ratcheting Floquet state</td>
      <td>112</td>
    </tr>
    <tr>
      <td>4.7 Semi-classical non-diffusive ratchet transport from the ground state</td>
      <td>114</td>
    </tr>
    <tr>
      <td>4.8 Experiments of ratchet transport from the ground state of the lattice</td>
      <td>116</td>
    </tr>
    <tr>
      <td>4.9 Ratchet transport experiments of ratcheting Floquet state prepared by QOC</td>
      <td>117</td>
    </tr>
    <tr>
      <td>5.1 Non-resonant modulation in the one-body wave function framework</td>
      <td>126</td>
    </tr>
    <tr>
      <td>5.2 Experimental protocol and nucleation of decimals peaks</td>
      <td>127</td>
    </tr>
    <tr>
      <td>5.3 Partial fits of Floquet spectra with the two-level tight-biding model</td>
      <td>130</td>
    </tr>
    <tr>
      <td>5.4 Typical results of the two-band tight-binding model with interactions for realistic parameters</td>
      <td>138</td>
    </tr>
    <tr>
      <td>5.5 Predictions from the two-band tight-binding model with interactions</td>
      <td>139</td>
    </tr>
    <tr>
      <td>5.6 Measurement of the quasi-momentum of the decimals peaks through band-mapping</td>
      <td>141</td>
    </tr>
    <tr>
      <td>5.7 Instability displacement with the modulation frequency</td>
      <td>142</td>
    </tr>
    <tr>
      <td>5.8 Measurement of the nucleation time of the instability</td>
      <td>144</td>
    </tr>
    <tr>
      <td>5.9 Survival time of the nucleated pattern</td>
      <td>145</td>
    </tr>
    <tr>
      <td>5.10 Instability nucleation through amplitude modulation of the lattice</td>
      <td>146</td>
    </tr>
    <tr>
      <td>5.11 Structure of the reduced one-body density matrix after nucleation of the instability</td>
      <td>148</td>
    </tr>
    <tr>
      <td>5.12 Correlations and coherence after nucleation of the instability</td>
      <td>149</td>
    </tr>
    <tr>
      <td>6.1 Summary figure of my PhD work</td>
      <td>152</td>
    </tr>
    <tr>
      <td>A.1 Rubidium 87 D<sub>2</sub> line</td>
      <td>159</td>
    </tr>
    <tr>
      <td>B.1 Legend for the optical diagram</td>
      <td>161</td>
    </tr>
    <tr>
      <td>B.2 Optical diagram for the dipole beams</td>
      <td>162</td>
    </tr>
    <tr>
      <td>C.1 Bode plot for the amplitude modulation of the lattice</td>
      <td>164</td>
    </tr>
    <tr>
      <td>C.2 Optical diagram for the lattice beams</td>
      <td>165</td>
    </tr>
    <tr>
      <td>D.1 Effect of the control duration</td>
      <td>167</td>
    </tr>
    <tr>
      <td>E.1 Experimental preparation of the plane wave with momentum p = 6 ħk<sub>L</sub> from the resting BEC</td>
      <td>170</td>
    </tr>
    <tr>
      <td>E.2 Preparation of three-momentum component states</td>
      <td>171</td>
    </tr>
    <tr>
      <td>E.3 Preparation and measurement of rotated squeezed Gaussian states</td>
      <td>171</td>
    </tr>
  </tbody>
</table>
<table><tr><td>F.1</td><td>Exemples de portraits de phase stroboscopiques</td><td>177</td></tr><tr><td>F.2</td><td>Structure de bande et spectres de quasi-énergie</td><td>178</td></tr><tr><td>F.3</td><td>Représentations de Husimi d'états de Floquet-Bloch</td><td>179</td></tr><tr><td>F.4</td><td>Chargement adiabatique du réseau optique et diffraction d'onde de matière</td><td>181</td></tr><tr><td>F.5</td><td>"Controle" écrit avec des diffractions d'ondes de matière</td><td>183</td></tr><tr><td>F.6</td><td>Reconstruction d'un état quantique dans un système de Bloch par maximisation de la vraisemblance</td><td>184</td></tr><tr><td>F.7</td><td>Préparation expérimentale et reconstruction d'états gaussiens comprimés</td><td>185</td></tr><tr><td>F.8</td><td>Évolution sous-stroboscopique d'un portrait de phase stroboscopique d'un système présentant un effet ratchet régulier</td><td>188</td></tr><tr><td>F.9</td><td>Influence de la constante de Planck effective sur le transport ratchet</td><td>189</td></tr><tr><td>F.10</td><td>Transport ratchet semi-classique et non-diffusif à partir de l'état fondamental du réseau optique</td><td>190</td></tr><tr><td>F.11</td><td>Protocole expérimental et observation typique de nucléation d'ordres décimaux</td><td>192</td></tr><tr><td>F.12</td><td>Résultats de l'étude de la stabilité du système pour des paramètres de modèle réalistes</td><td>193</td></tr><tr><td>F.13</td><td>Déplacement de l'instabilité avec la fréquence de modulation</td><td>194</td></tr></table>
<figure><img src="image_4.png" /></figure>
Remerciements

Ce manuscrit retrace le travail de recherche auquel j'ai eu la chance de prendre part au sein de l'équipe Atomes Froids du LCAR de février 2019 à décembre 2022. S'il m'arrive de dire "je" pour y narrer cette aventure, qu'on reste assuré du réel travail d'équipe que requièrent l'obtention, l'analyse et la compréhension des résultats j'y présentent. Hors du laboratoire, vous êtes pareillement nombreux à avoir participé à mon épanouissement lors de ces années passées à Toulouse. Aussi, je tiens à remercier ceux qui, directement ou indirectement, par leur travail ou<fnref n="1" /> leur amitié, ont contribué à la bonne réalisation de doctorat.

En premier lieu, merci aux membres de mon jury Jérôme Beugnon, Caroline Champ-
enois, Nathan Goldman et Éric Suraud pour votre temps, l'intérêt que vous avez porté à mon
manuscrit et la bienveillance qui transparaît dans vos rapports et dont vous avez fait montre
lors de la soutenance. Les échanges que nous y avons eu constituent pour moi une conclusion
enrichissante à ce travail de thèse, et ils demeureront un riche souvenir de ce rite de passage.

Merci ensuite aux trois chercheurs qui ont rendu possible ce doctorat : à David, pour ton solide encadrement et ta remarquable disponibilité malgré tes charges et activités de recherche transversales. Je loue ton intuition physique ainsi que ta passion communicative pour la recherche, et je te suis grandement reconnaissant pour ton soutien dans mes projets. À Juliette pour ton expertise expérimentale, ta bienveillance et ta pédagogie si précieuse à mon arrivée dans l'équipe. Enfin à Bruno, modèle de rigueur ! L'intérêt que je portais à ma thèse s'est vu décuplé lorsque tu as rejoins le groupe, et je te remercie pour ta patience et ton aide qui m'auront permis de m'approprier toujours davantage les sujets sur lesquels nous avons travaillé. Je conserverai un excellent souvenir de nos sessions de désossage d'antiques boîtiers électroniques, ainsi que de nos divers et longs échanges, fussent-ils scientifiques ou non.

Vous formez un solide et beau trio, et je me sais heureux d'avoir pu réaliser ce doctorat
avec vous. Je vous tiens et garderai chacun en très haute estime.

Je remercie grandement les autres membres, de passage comme moi, du groupe Atomes
Froids : merci à Maxime A. pour ta pédagogie et ton goût pour la physique atomique expérimentale
que tu as si bien su me transmettre. Merci à Gabriel C., fervent condisciple de Michel Dumas,
pour ces grands et nombreux moments de partage autour et hors de la science, à "travailler
sérieusement sans se prendre au sérieux". Tu m'as beaucoup appris, et j'ai grandement apprécié
travailler avec toi ! Merci à Lucas pour m'avoir épaulé sur la deuxième moitié de ma thèse,
et pour le cruciverbisme méridien (remerciant en passant Philippe Dupuis !). Merci enfin à
Floriane et Nicolas pour votre application, vos relectures zélées de codes et surtout pour votre
sympathie. La rapidité avec laquelle vous vous êtes approprié l'expérience et les formalismes

<footnote n="1">¹Un “ou” bien-sûr inclusif !</footnote>
théoriques propres à nos sujets de recherche m'impressionnent et m'assurent que la relève est amplement assurée ! On aura, par sous-ensembles, formé de fines équipes à la hauteur des exigences de transmission de savoir nécessaire à ce que continue de tourner sans trop faillir une expérience d'une telle technicité, et je suis fier d'avoir été, grâce à vous, un petit maillon de cette jolie chaîne.

Je remercie sincèrement Maxime Martinez, Bertrand Georgeot, Gabriel Lemarié, Olivier Giraud, Denis Ullmo, Peter Schlagheck et Dominique Sugny, avec qui nous avons collaboré au travers des différents sujets de recherche abordés durant mon doctorat.

Merci aux membres du LCAR, un laboratoire dans lequel il fait bon travailler, et notamment à Jean-Marc, Christine, Carole, Anna, Stéphane, Alexandre et Baptiste, ainsi qu'aux collègues doctorants et post-doctorants Qi, Ashley, Tangui, Romain, Léo, Julien A., Olivier T. et Julie R.. Ç'a été un réel plaisir de vous côtoyer au quotidien !

Plus largement à l'échelle de feue l'IRSAMC et de la FéRMI, je remercie chaleureusement Nicolas Destainville pour m'avoir, en 2018, accueilli en stage au LPT. C'est par cette introduction au milieu de la recherche que j'ai pris goût au travail de laboratoire et que s'est confirmé mon souhait de faire de la recherche fondamentale. Je remercie également Patricia, Nicolas E., Xavier et Sylvain. Un immense merci à Maxime M. pour tes qualités de chercheur et d'ami, et particulièrement pour les soirées pizzas (et celle à manger des lentilles, d'accord, un chouïa trop cuites), les longues discussions nocturnes, et ta promesse tenue de m'emmener voir une étape du Tour ! Merci à Nicolas M. et Théo pour les groupes de musique blues-jazz manouche-trad' irlandaise (oui) et ces nombreuses discussions captivantes sur la bande-dessinée, l'Histoire et autres sciences. Merci aussi à Julie C., Faedi, Ulysse, Agustin, Claire, Naimo, Denis R., Thomas B. et Thomas R., Hugo, Jordan, Benjamin E. et Benjamin R., Olivier G., Bhupen, Ravi, Noam, Julien H., Sasank, Juraj, Jeanne, Maen, Élise, Martin, Vincent et à tous les autres doctorants et post-doctorants de la FéRMI avec qui j'ai eu la chance d'interagir durant cette thèse.

Merci à l'École Doctorale Sciences de la Matière, et notamment à Éric Benoist et Jérôme Cuny pour votre implication et votre disponibilité.

Merci à Gabriel C., Maxime M., Ulysse, Maud T., Corentin, Tangui, Anaïs, Pierre, Sébastien, Sarah et Sophie pour les après-midi *binge watching* de séries, jeux de société et les goûters de crêpes !

Merci du fond du cœur à Christine, François, Damien et Mélitta pour vos accueils toujours chaleureux, les copieux repas et ces randonnées dans les Pyrénées, tous grandement appréciés durant mes cinq années passées à Toulouse.

Merci à ma famille pour leur simple et inconditionnel soutien depuis le début.

Enfin, un immense merci à Amélie, pour ta douceur au quotidien et ton infaillible accompagnement lors de la rédaction de ce manuscrit. Vivement le prochain chapitre en Belgique !

Bruxelles, 15 février 2023.
# General introduction

The use of light and magnetic fields to control the external degrees of freedom of atoms (their position and speed) has revolutionized the field of atomic physics. Benefiting from the development of lasers in the 1960s, techniques of laser trapping and cooling [1, 2, 3] have resulted in unprecedented control abilities of atomic particles, to such an extent that, in 1997, S. Chu, C. Cohen-Tannoudji and W. Phillips were awarded the Nobel Prize in Physics for their contributions to the development of these experimental techniques. Using them, one is able to cool gases of atoms near the absolute zero, at temperatures where the unusual laws of quantum physics begin to prevail. An emblematic example of this is Bose-Einstein condensation. Through this quantum phase transition first described by A. Einstein around 1925 [4, 5], a bosonic gas goes from an ensemble of particles classically described by Maxwell-Boltzmann statistics to a macroscopically populated matter wave whose evolution is governed by quantum mechanics: a Bose-Einstein condensate (BEC).

Bose-Einstein condensation was achieved for the first time in 1995, 70 years later, in the groups of E. Cornell and C. Wieman [6] and in that of W. Ketterle [7], from the combination of laser and evaporative coolings. This experimental achievement, rewarded by the Nobel prize in 2001, marked the birth of the field of quantum gases which has been constantly growing ever since. Ultracold quantum gases, which may be bosonic or fermionic, constitute very versatile platforms for the experimentalist to study and harness quantum mechanics:

* Using laser and magnetic fields, one is able to realize a wide variety of potentials [8, 9] in which to study the evolution of quantum systems. It is for instance possible to manipulate single atoms in optical tweezers [10, 11] and realize controllable periodic Bloch potential using e.g. optical lattices [12] or spatial light modulators [13, 14].

* Effective Hamiltonians [15, 16] can be designed in Floquet systems, where one periodically modulates the potentials in order to design specific energy level structures and dynamics. Such Floquet engineering is for example an approach to create artificial magnetism for neutral atoms [17, 18].

* A powerful tool provided by ultracold atoms is the ability to tune the strength and sign of short-range interatomic interactions using magnetic fields and Feshbach resonances [19]. Also, a strong experimental effort has recently been devoted to the realization and control of systems with longer-range interactions, such as dipolar gases [20, 21, 22] and Rydberg atom ensembles [23, 24, 25].

This high degree of control offered by quantum gas setups is of particular interest in the fields of quantum simulation and quantum metrology.
Quantum simulation [26, 27] consists in using a highly controllable quantum system to simulate the model Hamiltonian of another one, less controllable or more difficult to characterize. This idea was first suggested by R. Feynman in 1982 [28], and, since then, quantum simulation has been implemented on many platforms, such as superconducting circuits [29], photonic systems [30], trapped ions [31] and ultracold neutral atoms [32, 33, 34]. We are here interested in this last platform, on which in the last twenty years, major experiments of quantum simulation with ultracold atoms were realized. One can cite the three-dimensional implementation of the Bose-Hubbard model, on which the phase transition between a superfluid and a Mott insulator could be observed [35], the realization of the Anderson localization in one [36, 37], two [38, 39] and three-dimensional [40, 41, 42] systems with simulated disorder, and the Berezinskii-Kosterlitz-Thouless transition in two-dimensional gases [43, 44].

Another field of application that benefits from the great controllability offered by quantum gases is quantum metrology. There, a key example of the advantage of manipulating fully a quantum system is given by the use of squeezed states, which allow to reach sensitivities closer to the fundamental limits intimately related to Heisenberg uncertainty principle [45, 46]. This is famously the case of squeezed photonic states, used to enhanced spectroscopy [47] and interferometry experiments [48, 49], but there is also a long going theoretical and experimental effort to take advantage of squeezing with matter for enhanced matter wave interferometry [50, 51, 52, 53, 54, 55].

All these applications call for the development of new optimization methods to control and manipulate quantum states in cold atom systems. Several modern approaches have been proposed for this purpose, among which we can cite quantum feedback [56], shortcut to adiabaticity [57], machine learning [58] and optimal control theory [59, 60].

The Cold Atoms research group in which I did my PhD at *Laboratoire Collisions, Agrégats et Réactivité* (LCAR, Toulouse) takes part in the effort to develop techniques to control ultracold atom systems. My PhD work reported in this manuscript deals with both the development of such methods and their application to quantum simulation studies in a Floquet-Bloch system (periodic in space and time). The LCAR Cold Atoms group developed an ultracold atom experiment producing BECs of $^{87}$Rb that are placed in a far-detuned one-dimensional optical lattice. Along the axis of the lattice, the atoms experience a sine potential whose depth and position can be arbitrarily controlled in time by the means of acousto-optic modulators. This system constitutes a powerful and versatile platform to develop new methods for the control of quantum states, as well as to experimentally study the dynamics of matter waves in a sine potential with controllable time dependencies, in a regime of weakly interacting particles.

## Summary of my work and outline of this manuscript

This manuscript presents the research work in which I participated from 2019 to 2022 in the Cold Atoms group at LCAR. It is divided into three parts: a first part introducing theoretical, numerical and experimental methods required for the studies presented in the thesis, a second
part featuring results about the control and transport of quantum states apprehended in the one-body wave function formalism and a third part presenting a study beyond one-body physics in our system. New results are presented in the parts two and three of this thesis. There, a detailed introduction on the addressed subject is given in the beginning of the corresponding chapter.

**Part 1 - Methods for the study of wave function dynamics in a one-dimensional periodic potential.** The first part, made up of two chapters, is dedicated to the introduction of methods to study the dynamics of wave functions in a one-dimensional periodic potential whose parameters can be periodically modulated in time. **Chapter 1** consists in reminders of theoretical concepts and numerical tools for the study of such one-dimensional Floquet-Bloch systems. I then present in **Chapter 2** the BECs experiment and the controllable optical lattice used to implement this system in the laboratory, and with which the results presented in the following parts were obtained.

**Part 2 - Control and transport of the BEC one-body wave function in the optical lattice.**

The second part of this manuscript features two research works where the evolution of the atomic state is described by the Schrödinger equation and the formalism introduced in Chap. 1. In **Chapter 3**, I present the experimental implementation of a quantum-optimal control protocol in our experiment: by continuously shifting the optical lattice in an optimal way, we demonstrate our ability to prepare arbitrary momentum distributions and motional quantum states for the BECs in the lattice. This allowed us to prepare squeezed Gaussian states up to more than four times narrower in position than the ground state of the optical lattice, a state that would be unattainable using standard adiabatical methods. To characterize the states that we prepare, we implemented a quantum state tomography based on likelihood maximization which I then detail. I conclude this chapter with the presentation of a concrete application of quantum-optimal control to a simulation of dynamical tunneling in a Floquet system. **Chapter 4** features a study (both numerical and experimental) of the ratchet effect, which consists in the emergence of a directed current of particles in a potential periodic in space and time. I begin with a classical mechanics study in which I show how to design an integrable ratchet effect in a Hamiltonian system by correlating the modulations of amplitude and phase of a symmetric sine potential. I then consider quantum mechanics in this system, and detail how quantum transport along the ratcheting classical trajectories depends, in a non-monotonous way, on the effective Planck constant. Following these studies, we experimentally realized this system, and I report on the observation of non-diffusive ratchet transport of matter waves in the optical lattice.

**Part 3 - Beyond the one-body wave function.** The last part of this manuscript is composed of **Chapter 5**, featuring a study of the emergence of a supercrystalline order that results from interatomic interactions in the modulated optical lattice. In the experiments presented in this chapter, we load the atoms in the ground state of the lattice before modulating its phase at a frequency that couples the ground band to an excited band, that coupling being not resonant for
the initial state. In this configuration, we show how two-body collisions occurring between atoms
of the ground state lead to the appearance of atomic population at symmetric quasi-momenta
through four-wave mixing, at a position in the Brillouin zone that can be tuned by Floquet
engineering the quasi-energy spectrum of the system. This produces a new periodicity in the
distribution of atoms in the lattice, larger and non-commensurable with the lattice spacing.

The work presented in Chap. 3 results from a collaboration with Prof. D. Sugny from the
*Laboratoire Interdisciplinaire de Carnot Bourgogne* (Dijon), and the work presented in Chap. 5
results from a collaboration with Prof. P. Schlagheck from the CESAM research unit at the
University of Liège.
List of publications

I present a list of the publications of our group to which I contributed during my thesis:

[61] M. Arnal, G. Chatelain, M. Martinez, N. Dupont, O. Giraud, D. Ullmo, B. Georgeot, G. Lemarié, J. Billy, and D. Guéry-Odelin. *Chaos-assisted tunneling resonances in a synthetic *Floquet superlattice*, Science Advances (2020),

[62] G Chatelain, N Dupont, M Arnal, V Brunaud, J Billy, B Peaudecerf, P Schlagheck, and D Guéry-Odelin. *Observation and control of quantized scattering halos*, New Journal of Physics (2020),

[63] N. Dupont, G. Chatelain, L. Gabardos, M. Arnal, J. Billy, B. Peaudecerf, D. Sugny, and D. Guéry-Odelin. *Quantum State Control of a Bose-Einstein Condensate in an Optical Lattice*, PRX Quantum (2021),

[64] N. Dupont, F. Arrouas, L. Gabardos, N. Ombredane, J. Billy, B. Peaudecerf, D. Sugny, and D. Guéry-Odelin. *Phase-space distributions of Bose-Einstein condensates in an optical lattice: Optimal shaping and reconstruction*, currently under review (2022).

[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).

[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).

I detail in this manuscript the results presented in the last four works listed above: [63] and [64] are addressed in Chap. 3, [65] in Chap. 4 and [66] in Chap. 5. Experimental details about our results on chaos-assisted tunneling [61] and quantized scattering halos [62] can be found in the theses of M. Arnal [67] and G. Chatelain [68], previous PhD students of the group.
<figure><img src="image_5.png" /></figure>
# Part I

## Methods for the study of wave function dynamics in a one-dimensional periodic potential
<figure><img src="image_6.png" /></figure>
# Chapter 1 - Elements of theory for the study of
a wave function in a
one-dimensional potential

Si vous me savez peu de gré de ce que je vous dis, sachez-m'en beaucoup de ce que je ne vous dis pas.

Denis Diderot (Jacques le Fataliste et son maître, ca. 1778)

## Contents

<table><tr><td>Introduction</td><td>11</td></tr><tr><td>1.1 Classical particle in a one-dimensional periodic potential</td><td>11</td></tr><tr><td>1.1.1 Basics on analytical mechanics</td><td>11</td></tr><tr><td>1.1.2 Phase space of the one-dimensional sine potential</td><td>12</td></tr><tr><td>1.1.3 Symmetries of the phase space.</td><td>18</td></tr><tr><td>1.2 Wave function in a time-independent sine potential</td><td>18</td></tr><tr><td>1.2.1 Bloch's theorem in one dimension</td><td>19</td></tr><tr><td>1.2.2 The central equation for the sine potential and the band structure</td><td>20</td></tr><tr><td>1.2.3 Quantum state evolution in a Bloch system</td><td>23</td></tr><tr><td>1.3 Wave function in a time-dependent sine potential</td><td>24</td></tr><tr><td>1.3.1 Time-dependent sine potential: general case</td><td>24</td></tr><tr><td>1.3.2 Time-periodic Hamiltonians: Floquet formalism</td><td>25</td></tr><tr><td>1.4 Quantum states in the phase space</td><td>28</td></tr><tr><td>1.4.1 Wigner and Husimi representations and the effective Planck constant</td><td>29</td></tr><tr><td>1.4.2 Floquet states representation in the phase space</td><td>34</td></tr><tr><td>Conclusion</td><td>35</td></tr></table>
<figure><img src="image_7.png" /></figure>
# Introduction

The goal of this first chapter is to briefly recapitulate elements of theory, concepts and numerical tools used to study one-body wave functions in one-dimensional periodic potentials without interactions. The methods that we recall will then be used throughout this manuscript. Some of the developments presented in this chapter are inspired from [69].

## 1.1 Classical particle in a one-dimensional periodic potential

In this first section, we present some reminders on analytical mechanics. The main objective is to introduce the notion of phase space, and the construction of the stroboscopic phase portrait of a one-dimensional system periodically modulated in time. In the end of this first chapter, after discussing quantum dynamics in a one-dimensional sine potential, we revisit the notion of phase space to introduce the phase space representation of a quantum state (Sec. 1.4).

### 1.1.1 Basics on analytical mechanics

**Lagrangian mechanics.** We consider a classical system with $N$ degrees of freedom. In the Lagrangian formulation of analytical mechanics, the state of the system is completely described by $N$ independent generalized coordinates $q = (q_1, ..., q_N)$ and their time derivatives $\dot{q} = (\dot{q}_1, ..., \dot{q}_N)$. The dynamics of the system is ruled by the Lagrangian

$$L(q, \dot{q}, t) = T - V, \qquad (1.1)$$

with $T$ and $V$ being respectively the kinetic and potential energies. Given an initial condition $(q(t_0), \dot{q}(t_0))$, the classical trajectory $q(t)$ that the system will follow over the time interval $[t_0, t_1]$ is, among all the possible trajectories, the one that minimizes the action $S$, a functional of $q(t)$ that reads

$$S[q(t), t_0, t_1] = \int_{t_0}^{t_1} L(q, \dot{q}, t) \, dt. \qquad (1.2)$$

The minimization of $S$ leads to $N$ Euler-Lagrange equations:

$$\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_i} \right) - \frac{\partial L}{\partial q_i} = 0, \qquad (1.3)$$

which in turn gives $N$ equations of motion.
**Hamiltonian mechanics.** To introduce the phase space of the system, we switch to the Hamiltonian formulation of analytical mechanics by introducing the generalized momenta $p = (p_1, ..., p_N)$ and the Hamiltonian $H$ through the Legendre transformation:

$$p_i = \frac{\partial L}{\partial \dot{q}_i}, \qquad H(\mathbf{q}, \mathbf{p}, t) = \sum_{i=0}^{N} \dot{q}_i p_i - L. \qquad (1.4)$$

From the Hamiltonian, the equations of motion are given by the Hamilton equations:

$$\frac{dq_i}{dt} = \frac{\partial H}{\partial p_i} \quad \text{and} \quad \frac{dp_i}{dt} = -\frac{\partial H}{\partial q_i}. \qquad (1.5)$$

For a particle of mass $m$, a Cartesian coordinate system, a kinetic energy $T(\dot{\mathbf{q}}) = m\dot{\mathbf{q}}^2/2$ and a potential $V(\mathbf{q}, t)$ that does not depend on the velocities $\dot{\mathbf{q}}$, the Hamiltonian is

$$H(\mathbf{q}, \mathbf{p}, t) = T(\mathbf{p}) + V(\mathbf{q}, t), \qquad (1.6)$$

where the generalized momenta are the usual momenta $p = m\dot{q}$.

**Phase space.** In the Hamiltonian formalism, the state of the system at a given time $t_0$ is a point $(\mathbf{q}(t_0), \mathbf{p}(t_0))$ in the phase space of the system. In this space of dimension $2N$, the conservation of the mechanical energy constrains trajectories $(\mathbf{q}(t), \mathbf{p}(t))$ on a hypersurface of dimension $2N - 1$. For such a constant Hamiltonian system, the determinism of the dynamics implies that there cannot be intersection of trajectories in the phase space, since then the state at the hypothetical crossing would have multiple possible evolutions.

**Integrability of a dynamical system.** The dynamics of an Hamiltonian system is fully integrable if there are as many conserved and independent constants of motion as there are degrees of freedom [70, 71, 69]. As we will see for the time-independent sine-potential in one dimension, the phase space of an integrable system only displays regular trajectories. On the other hand, irregular chaotic trajectories can emerge in the phase space of systems with fewer constants of motion than degrees of freedom. As for most of the dynamical systems studied in this thesis, the majority of physical systems present a mixed dynamics, with the coexistence of integrable and chaotic trajectories.

### 1.1.2 Phase space of the one-dimensional sine potential

The Hamiltonian that will interest us in most of this manuscript is the Hamiltonian for the one-dimensional sine potential of spatial period $d$ whose depth and position can be varied along time:
$$H(x, p, t) = \frac{p^2}{2m} - A(t) \times \frac{V_0}{2} \cos \left( \frac{2\pi x}{d} + \varphi(t) \right). \qquad (1.7)$$

with $A(t)$ a dimensionless scaling function of order 1. In this section, we study the classical dynamics in this system by successively considering the case where $H$ is time-independent, the case where it is time-dependent and the special case where $H$ is periodic in time.

**Time-independent sine potential.** We first consider the time-independent Hamiltonian with modulation functions $A(t) = 1$ and $\varphi(t) = 0$ in Eq. (1.7):

$$H(x, p) = \frac{p^2}{2m} - \frac{V_0}{2} \cos \left( \frac{2\pi x}{d} \right), \qquad (1.8)$$

which corresponds to the Hamiltonian of the simple pendulum. To represent the phase space of the system, the general strategy is to compute<fnref n="1" /> several trajectories $(x(t), p(t))$ for several initial conditions $(x_0, p_0)$ using the Hamilton equations (1.5). However, as the time-independent Hamiltonian (1.8) conserves the mechanical energy, we can circumvent the explicit numerical integration of individual trajectories, evaluate the Hamiltonian over the discretized two-dimensional space $(x, p)$ and display lines of iso-mechanical energy. These lines (also called orbits, which we draw in Fig. 1.1) correspond to trajectories $(x(t), p(t))$ of a classical particle described by the Hamiltonian (1.8). As the ordinate $p$ of a point informs on the time evolution of its abscissa $x$, orbits are followed in a given direction (clockwise in a $(x, p)$ plot where $x$ and $p$ are defined as increasing to the right and upwards respectively, as in Fig. 1.1).

We display the separatrix, the limit trajectory of infinite period that separates periodic trajectories bounded within a site of the potential from unbounded trajectories that travel between sites. We define the momentum $p_{\text{sep}} = \sqrt{2mV_0}$ of the separatrix in $x = 0$. We note that, displaying the position in units of $d$ and the momentum in units of $p_{\text{sep}}$, the phase space of Fig. 1.1(b) describes the classical dynamics of the Hamiltonian (1.8) for all specific parameters $m, V_0$ and $d$. As we discuss in Sec. 1.4, this scaling invariance of the dynamics does not hold in the quantum case.

In this manuscript, we use different sets of characteristic quantities to scale the dynamics. In Fig. 1.1, the $(x, p)$ phase space is plotted with respect to $x/d$ and $p/p_{\text{sep}}$. In a following paragraph concerning time-modulated potentials, the time scale in the scaling of the momentum is proportional to the period of modulation. A summary on our two main dynamical scalings is presented at the end of this chapter (p. 33).

<footnote n="1">¹In the work presented in this manuscript, we use the Runge-Kutta “RK4” method to integrate classical dynamics.</footnote>
<figure><img src="image_8.png" /><figcaption>**Figure 1.1 | One-dimensional sine potential and corresponding phase space.** (a) One dimensional sine potential over three spatial cells (solid blue line). We draw a particle moving in the potential (black marker and arrow). (b) $(x, p)$ phase space over one spatial cell, with bounded trajectories (plain black lines), unbounded trajectories (dashed black lines) and the separatrix (dotted red line). We mark the phase space coordinate of the particle in panel (a) (black marker) with the direction of its evolution (black arrow).</figcaption></figure>

**Time-dependent sine potential.** The situation becomes more complicated for time-dependent systems. First of all, the mechanical energy is not conserved anymore. Even in one dimension, we have fewer constant(s) of motion than degree(s) of freedom, so we lose the certainty of having integrable dynamics. Furthermore, in the general case of time-dependent Hamiltonians $H(x, p, t)$, one cannot simply adjoin integrated trajectories in the $(x, p)$ plane in order to display the phase space of the system. Indeed, as the states in $(x_0, p_0, t_1)$ and $(x_0, p_0, t_2)$ generally experience different potentials $V(x_0, t_1)$ and $V(x_0, t_2)$, they evolve along different trajectories. A naive $(x, p)$ plot of the trajectories would result in crossing trajectories artifacts. Therefore, for a time-dependent system, one has to consider the $(x, p, t)$ phase space, with an additional effective dimension.

**Periodically time-modulated sine potential.** A specific type of time-dependent systems that interests us in this manuscript is the case of the periodically modulated sine potential in one dimension, with a Hamiltonian of the form $H(x, p, t + T) = H(x, p, t)$ (with $T$ the modulation period). Here, the periodicity of the Hamiltonian allows for a valid and informative representation of the $(x, p, t)$ phase space in the $(x, p)$ plane. To compare trajectories when the potential is identical, one only displays the states $(x(t), p(t))$ of the system at times $t = t_0 + nT$. As depicted in Fig. 1.2(a) for one trajectory, this amounts to only considering the trajectories when they
<figure><img src="image_9.png" /><figcaption>**Figure 1.2** | Construction of a stroboscopic phase portrait. (a) Intersections (blue dots) between a trajectory (blue solid line) of the $(x, p, t_0 + nT)$ phase space and the planes $(x, p, t_n)$ with $t_n = t_0 + nT$ (see text). (b) Resulting stroboscopic phase portrait. Figure inspired from [69].</figcaption></figure>

intersect the planes $(x, p, t_0 + nT)$, with $0 \le t_0 < T$ and $n \in \mathbb{N}$. The resulting plot (Fig. 1.2(b)) is called the *stroboscopic phase portrait* and is an example of Poincaré section, a method to represent the phase space of a system in a subspace of lower dimensionality [69].

We now give examples of stroboscopic phase portraits. We consider the time-dependent Hamiltonian (1.7) whose depth and position reference are modulated periodically: $A(t + T) = A(t)$ and $\varphi(t + T) = \varphi(t)$. We define the dimensionless units

$$ \tilde{x} = \frac{2\pi}{d} x, \quad \tilde{t} = \frac{2\pi}{T} t, \quad \tilde{p} = \frac{d\tilde{x}}{d\tilde{t}} = \frac{T}{md} p \quad \text{and} \quad \tilde{H} = \frac{T^2}{md^2} H. \qquad (1.9) $$

From Eqs. (1.7) and (1.9), the dimensionless Hamiltonian reads:

$$ \tilde{H}(\tilde{x}, \tilde{p}, \tilde{t}) = \frac{\tilde{p}^2}{2} - A(\tilde{t}) \times \gamma \cos(\tilde{x} + \varphi(\tilde{t})) \qquad (1.10) $$

with the scaled potential amplitude $\gamma = V_0 T^2 / 2md^2$. In Fig. 1.3, we consider two cases:

(a) The offset<fnref n="2" /> modulation of amplitude with one harmonic:

$$ \begin{aligned} A(\tilde{t}) &= 1 + \varepsilon_0 \cos(\tilde{t}) \\ \varphi(\tilde{t}) &= 0 \end{aligned} \qquad (1.11) $$

<footnote n="2">²In Chap. 2, we present the experimental realization of the systems discussed in this chapter. We then see how experimental constraints require $|A(\tilde{t})| > 0$, which explain the form of $A(\tilde{t})$ in Eq. (1.11), with $0 \le \varepsilon_0 \le 1$ in the experiment.</footnote>
<figure><img src="image_10.png" /><figcaption>**Figure 1.3** | Examples of stroboscopic phase portraits. Top: periodically modulated potentials over three spatial periods. Bottom: stroboscopic phase portraits. (a) Modulation of the depth of the potential, i.e. Eq. (1.10) for the modulation functions (1.11), with $\gamma = 0.375$, $\epsilon_0 = 0.24$ and a subperiod time of stroboscopic observation $\tilde{t}_0 = 0$ (see text). (b) Modulation of the position reference of the potential, i.e. Eq. (1.10) for the modulation functions (1.12), with $\gamma = 0.44$, $\phi_0 = 2\pi/24 = 15^\circ$ and a subperiod time of stroboscopic observation $\tilde{t}_0 = 0$.</figcaption></figure>

(b) The phase modulation with one harmonic:

$$ 
\begin{aligned}
A(\tilde{t}) &= 1 \\
\varphi(\tilde{t}) &= \varphi_0 \cos(\tilde{t})
\end{aligned} 
\qquad (1.12) 
$$

In the stroboscopic phase portraits of Fig. 1.3(a₂,b₂), we see three kinds of trajectories:

* We first have trajectories that live on closed lines inscribed inside one another. These regions of various shapes are called *regular islands* and consist of sections of inscribed tori<fnref n="3" /> made of surfaces of regular trajectories in the $(x, p, t)$ phase space.

* At large dimensionless momentum ($|\tilde{p}| \gtrsim 1$ in Fig. 1.3), we see the unbounded trajectories that we already encountered in the case of the static potential (Fig. 1.1(b)), with enough mechanical energy to only be slightly perturbed by the modulation.

* Finally, inside the unbounded trajectories and surrounding the regular islands is the *chaotic sea*, a region of irregular trajectories that ergodically span all the accessible region of the phase space.

<footnote n="3">³They are also called KAM tori [71], for A. Kolmogorov [72], V. Arnold [73] and J. Moser [74]. A classical trajectory cannot cross a KAM torus by virtue of determinism.</footnote>
<figure><img src="image_11.png" /><figcaption>Figure 1.4 | Subperiod evolution of a stroboscopic phase portrait. (a-e) Stroboscopic phase portraits of the same Hamiltonian (1.10) in the amplitude modulation case of Eq. (1.11). The difference between the panels is the reference time $\tilde{t}_0$ of the stroboscopic observations, with $\tilde{t}_0 = 0, \pi/2, \pi, 3\pi/2$ and $2\pi$ for (a) to (e) resp.</figcaption></figure>

The phase portraits of Fig. 1.3 give examples of mixed dynamics, with the coexistence of regular and chaotic trajectories. The degree of chaos (i.e. the ratio of phase space area between the chaotic sea and the regular islands) depends on the strength of the modulations [75, 69].

The stroboscopic phase portraits of Fig. 1.3 correspond to the specific choice of subperiod observation time $\tilde{t}_0 = 0$ (in accordance with our definitions of the modulation functions $A(\tilde{t})$ and $\phi(\tilde{t})$ in Eqs (1.11) and (1.12); see Fig.1.2 and the explanation of the construction of a stroboscopic phase portrait). This parameter can be varied to inform on the sub-stroboscopic evolution of the system. We illustrate this procedure in Fig. 1.4, where we draw, for different times $0 \le \tilde{t}_0 \le 2\pi$, the stroboscopic phase portraits in the case of the amplitude modulation of the sine potential (Fig. 1.3(a)). In Fig. 1.4, we can discriminate between two kinds of regular islands: the islands that are linked to a given site of the potential (e.g. the three islands on the $\tilde{p} = 0$ axis at $\tilde{t}_0 = 0$) and the islands that travel to neighboring sites between periods (e.g. the two islands in $(\tilde{x}, \tilde{p}) \approx (0, \pm 1.25)$ at $\tilde{t}_0 = 0$). In the dimensionless units (1.9), the average momentum of any trajectory from a given regular island is the same rational number$^4$:

$$ \langle \tilde{p}_{\text{reg}} \rangle = \lim_{n \to \infty} \frac{1}{n \times 2\pi} \int_{0}^{n \times 2\pi} \tilde{p}_{\text{reg}}(\tilde{t}) \, d\tilde{t} = w \in \mathbb{Q}, \quad (1.13) $$

for $\tilde{p}_{\text{reg}}(\tilde{t})$ the scaled momentum of a trajectory in the regular island. The number $w$ is the wind-ing number [76, 71] of the regular island, which corresponds to the number of sites traveled per modulation period. This observation is at the heart of Chap. 4, where, through the correlation of the amplitude and phase modulations (Eqs. (1.11) and (1.12)), we engineer a transporting regular island (i.e. with a winding number $w \neq 0$) that passes by $(x, p) = (0, 0)$.

<footnote>$^4$In the evaluation of this average, the necessity of the limit comes from the quasi-periodicity of trajectories in regular islands. With the exception of the trajectory at the center of a regular island, the result $\langle \tilde{p}_{\text{reg}} \rangle = w \in \mathbb{Q}$ is in general only obtained asymptotically.</footnote>
### 1.1.3 Symmetries of the phase space.

The symmetries of the phase space reflect the symmetries of the Hamiltonian. The two relevant symmetries for our dynamical study are the spatial symmetry $S_x$ and the time-reversal symmetry $S_t$:

$$S_x: \begin{array}{c} x \to -x \\ t \to t \end{array}, \quad \text{and} \quad S_t: \begin{array}{c} x \to x \\ t \to -t \end{array} \qquad (1.14)$$

with each symmetry transformation implying $p \to -p$. At time $t$, a given Hamiltonian $H(x, p, t)$ with trajectories $(x(t), p(t))$ has the symmetry $S$ if the symmetric trajectories $S(x(t), p(t))$ are solutions of the symmetric equations of motion deduced from $S(H(x, p, t))$.

For instance, the amplitude modulation of Eq. (1.11) never breaks the spatial symmetry $S_{\tilde{x}}$ of the Hamiltonian (1.10). Therefore, the stroboscopic phase portraits in Fig. 1.4 have the central symmetry $(\tilde{x}, \tilde{p}) \to (-\tilde{x}, -\tilde{p})$ for all observation times $\tilde{t}_0$. Moreover, the amplitude modulation being a function of $\cos(\tilde{t})$, the Hamiltonian has the symmetry $S_{\tilde{t}}$ only at times $\tilde{t}_0 = n\pi$ with $n \in \mathbb{N}$, which corresponds to the instants when the stroboscopic phase portraits have the axial symmetry $(\tilde{x}, \tilde{p}) \to (\tilde{x}, -\tilde{p})$ (Fig. 1.4(a,c,e)). The same analysis for the phase modulation (1.12) confirms the fact that the stroboscopic phase portrait of Fig. 1.3(b) displays the symmetry $S_{\tilde{t}}$ but not $S_{\tilde{x}}$ at $\tilde{t}_0 = 0$.

This concludes our reminders on classical mechanics. We now turn to the quantum regime for the study of a wave function in the sine-potential system.

## 1.2 Wave function in a time-independent sine potential

We first consider the case of a quantum particle in the static potential

$$V(x) = -\frac{V_0}{2} \cos \left( \frac{2\pi x}{d} \right). \qquad (1.15)$$

We define characteristic quantities that we use throughout this manuscript:

* the wave number of the sine potential $k_L = 2\pi/d$,

* the characteristic energy $E_L = \hbar^2 k_L^2 / 2m = h^2 / 2md^2$,

* the corresponding frequency $\nu_L = E_L / h = h / 2md^2$,

with $h$ the Planck constant. In the following, we express the depth $V_0$ of the potential in units of $E_L$ through the dimensionless parameter $s_0 = V_0 / E_L$. In Chap. 2, we discuss the experimental implementation of the potential (1.15) and we give the numerical values
of these quantities. In the experiment, realistic accessible values for $s_0$ are in the range $[0, 40]$.

When working with optical lattices, an energy scale commonly found in the literature is the recoil energy $E_r = h^2/8md^2 = E_L/4$, which is the kinetic energy acquired by an initially resting atom of mass $m$ when absorbing of a photon with wave number $k_L/2$. The realistic range for experimental lattice depths given in the previous paragraph corresponds to $V_0 \le 160E_r$.

The quantum treatment of periodic potentials falls under the study of wave functions in crystalline structures. A major result in this field is Bloch's theorem, that we briefly present in the one-dimensional case.

### 1.2.1 Bloch's theorem in one dimension

We want to determine the eigenvectors of a generic one-dimensional Hamiltonian:

$$ \hat{H} = \frac{\hat{p}^2}{2m} + \hat{V}(x) \qquad (1.16) $$

with $\hat{p} = -i\hbar\partial/\partial x$ and a spatially periodic potential $\hat{V}(x + d) = \hat{V}(x)$. We define the plane waves $|\chi_r\rangle$, eigenstate of the momentum operator $\hat{p}$ with eigenvalue $r\hbar k_L$:

$$ \langle x | \chi_r \rangle = \chi_r(x) = \frac{e^{i r k_L x}}{\sqrt{d}} \qquad (1.17) $$

with a normalization over one site of the potential. Since $\hat{H}$ is left invariant by the discrete spatial translation $x \to x - d$, it commutes with the spatial translation operator $\hat{T}_d = e^{-id\hat{p}/\hbar}$. Consequently, one can look for a basis of states in which $\hat{H}$ and $\hat{T}_d$ are diagonal operators, i.e. a common set of eigenstates for $\hat{H}$ and $\hat{T}_d$. Bloch's theorem [77, 78, 79] states that these eigenstates are the product of a plane wave $\chi_{q/k_L}(x)$ and a function with the periodicity of the potential. They are functions of the form:

$$ \psi_q(x) = u_q(x)e^{iqx}, \qquad (1.18) $$

where the wave number $q$ will be referred to as the quasi-momentum$^5$, and $u_q(x + d) = u_q(x)$. Considering smooth enough potentials $V(x)$, the functions $u_q(x)$ can be expressed as their Fourier series:

$$ u_q(x) = \sum_{\ell \in \mathbb{Z}} c_{q, \ell} \frac{e^{i \ell k_L x}}{\sqrt{d}} = \sum_{\ell \in \mathbb{Z}} c_{q, \ell} \chi_{\ell}(x), \qquad (1.19) $$

<footnote>$^5$Even though the quantity associated to $q$ that is homogeneous to a momentum is rather $\hbar q$.</footnote>
with $c_{q,l} \in \mathbb{C}$. From Eqs. (1.18) and (1.19), the Bloch eigenstates can be written as

$$ \psi_q(x) = \sum_{\ell \in \mathbb{Z}} c_{q,\ell} \frac{e^{i(\ell k_L + q)x}}{\sqrt{d}} = \sum_{\ell \in \mathbb{Z}} c_{q,\ell} \chi_{\ell+q/k_L}(x), \qquad (1.20) $$

as well as any state evolving in the potential at given quasi-momentum $q$, since the basis of the Bloch eigenstates spans the Hilbert space for all $q$. We refer to the coefficients $c_{q,\ell} \in \mathbb{C}$ as the plane wave coefficients of a state, with the normalization condition $\sum_{\ell} |c_{q,\ell}|^2 = 1$. Within a subspace of fixed $q$, a state in a Bloch system is fully defined by the infinite set of its plane wave coefficients, and we see from Eq. (1.20) that it consists of a comb of plane waves with momenta $p = \hbar(\ell k_L + q)$ (with $\ell \in \mathbb{Z}$). We see that a phenomenological interpretation for the quasi-momentum is that it is the quantity that sets the momentum reference of the plane wave comb in the rest frame of the potential. Furthermore, Eq. (1.20) implies that quantum states at a given quasi-momentum in Bloch systems are periodic under the discrete quasi-momentum translation $q \to q + k_L$ (with $c_{q+mk_L,\ell} = c_{q,\ell}$ for $m \in \mathbb{Z}$ in the plane wave expansion of a given state $\psi_q(x)$). This defines the first Brillouin zone, the relevant quasi-momentum cell in which to study the dynamics of a system, that we define as $-0.5 < q/k_L \le 0.5$.

In the following, we write $|\psi_q\rangle$ an arbitrary state at quasi-momentum $q$, and $|\phi_{q,n}\rangle$ the $n^{th}$ Bloch eigenstate at quasi-momentum $q$ (where the index $n$ sorts the states by increasing eigenvalues; see below).

### 1.2.2 The central equation for the sine potential and the band structure

**The central equation.** We now go back to the specific case of a wave function in a sine potential:

$$ V(x) = -s_0 \frac{E_L}{2} \cos(k_L x). \qquad (1.21) $$

From Bloch's theorem, we look for the eigenstates of this system in the form of Bloch waves $\phi_{q,n}(x) = \sum_{\ell} c_{q,\ell}^{(n)} \chi_{\ell+q/k_L}(x)$. Combining Eqs (1.16), (1.20) and (1.21), the time-independent Schrödinger equation leads to the central equation:

$$ \hat{H}\phi_{q,n}(x) = E_{q,n}\phi_{q,n}(x) \qquad (1.22) $$

$$ \left(\ell + \frac{q}{k_L}\right)^2 c_{q,\ell}^{(n)} - \frac{s_0}{4} \left(c_{q,\ell-1}^{(n)} + c_{q,\ell+1}^{(n)}\right) = \frac{E_{q,n}}{E_L} c_{q,\ell}^{(n)}, $$

with $E_{q,n}$ the eigenenergy associated with the Bloch eigenstate $|\phi_{q,n}\rangle$. We see that the Hamiltonian for the sine potential is tridiagonal in the basis of the plane waves, as it couples the plane wave $|\chi_{\ell}\rangle$ to the plane waves $|\chi_{\ell \pm 1}\rangle$. For given parameters ($s_0, q$), its diagonalization gives the
Bloch eigenstates and the associated eigenenergies. This is done numerically. As the Hilbert space dimension of this continuous system is infinite, $\ell$ needs to be truncated to $|\ell| \le \ell_{max}$, an upper bound for the plane waves of highest kinetic energy considered. For a given problem, $\ell_{max}$ is determined by making sure that the plane waves with the highest kinetic energy contributing to the dynamics are taken into account (i.e. by ensuring that the choice of a greater $\ell_{max}$ results in the same dynamics). The resulting Hilbert space dimension is $d_H = 2\ell_{max} + 1$. To model realistic situations in the experimental optical lattice (see Chap. 2), our simulations are typically done with $d_H \sim 30$.

**The band structure.** For a given $s_0$, the spectrum of the Hamiltonian as a function of the quasi-momentum gives the band structure. In this thesis, we adopt the nomenclature (s, p, d, f ...) for the bands of eigenenergy $n = (0, 1, 2, 3...)$). In Fig. 1.5, we draw the band structures for two non-zero potential depths, and we show examples of Bloch eigenstates by displaying their plane wave coefficients $c_{q,\ell}$ and their spatial density of probability $|\phi_{q,n}(x)|^2$. We make a series of remarks:

* We define the parity operator $\hat{P}$, whose action on an arbitrary state in the position representation is

$$ \hat{P}\psi_q(x) = \psi_q(-x). \qquad (1.23) $$

For a symmetric potential $V(x) = V(-x)$, the Hamiltonian commutes with $\hat{P}$. Therefore,

$$ \phi_{-q,n}(x) = \phi_{q,n}(-x). \qquad (1.24) $$

Consequently, we have $c_{q,\ell}^{(n)} = \pm c_{-q,-\ell}^{(n)}$, and moreover $c_{q,\ell}^{(n)} = \pm c_{q,-\ell}^{(n)}$ for $q \in \{0, 0.5\}$. This is illustrated in Fig. 1.5(b,c,d,g,h).

* For a Bloch system left invariant by the time-reversal symmetry $S_t$ (1.14) (such as the time-independent sine potential that we consider), the band structure is symmetric around $q = 0$, so we have $E_{q,n} = E_{-q,n}$ [79] (see Fig. 1.5(a,e)).

* Comparing the two ground states in Fig. 1.5(d) and (h), we see that, as the depth of the potential increases, the envelope of the $c_\ell$ gets wider and the probability dispersion of the spatial density gets narrower. This illustrates the uncertainty principle $\Delta x \Delta p \ge \hbar/2$ between the two incompatible observables $x$ and $p$. At the end of this chapter, we briefly discuss the uncertainty principle when we consider the representation of a wave function in the phase space (Sec. 1.4). This phenomenon is also highlighted in Chap. 3, where we perform state squeezing in the optical lattice.

* Figure 1.5(f,g) shows two states at $q \neq 0$. We see how the momentum reference of the plane wave comb is translated accordingly. In Chap. 5, we study the emergence of states
<figure><img src="image_12.png" /><figcaption>**Figure 1.5** | Band structures and Bloch eigenstates. (a,e) Band structures in the first Brillouin zone obtained from Eq. (1.22) with $s_0 = 1$ (a) and $s_0 = 5$ (e). The bands s, p, d and f are in blue, orange, green and red. The grey shaded areas indicate the energies below the maximum potential energy $V(\pm d/2) = s_0E_L/2$. $(b,c,d,f,g,h)_1$ Plane wave coefficients $c_\ell$ as a function of the plane waves momenta $p$ in the rest frame of the potential for the Bloch eigenstates identified on panels (a,e). $(b,c,d,f,g,h)_2$ Spatial densities of probability for the Bloch eigenstates identified on panels (a,e).</figcaption></figure>
coexisting at different quasi-momenta in a modulated optical lattice. The signature of the phenomenon is the population of plane waves on translated momentum comb.

**The limit $s_0 \to 0$.** In the absence of potential ($s_0 = 0$), the Hamiltonian in Eq. (1.22) is diagonal in the basis of the plane waves. This is expected since the plane waves are the eigenstates of the momentum operator $\hat{p}$. The band structure then simply becomes $E_{q,n} \to E_{q,\ell}/E_L = (\ell+q/k_L)^2$, i.e. the parabolic relation of dispersion for the free particle. In this situation, we get pairs of degenerate eigenstates at $q/k_L = 0$ and $q/k_L = 0.5$ as there then exist two relative integer values of $\ell$ such that $\ell + q/k_L$ takes the same value (with the exception of $\ell = 0$ at $q = 0$). This has experimental consequences when performing a band-mapping (see Sec. 2.4.3).

### 1.2.3 Quantum state evolution in a Bloch system

In the absence of interactions, the dynamics of a wave function is ruled by the time-dependent Schrödinger equation:

$$i\hbar \frac{\partial}{\partial t} \psi_q(x, t) = \hat{H}(t) \psi_q(x, t). \qquad (1.25)$$

From time $t_0$ to $t$, the evolution of a state $\psi_q(x, t_0)$ is given by

$$\psi_q(x, t) = \hat{U}(t, t_0) \psi_q(x, t_0). \qquad (1.26)$$

The evolution operator $\hat{U}(t, t_0)$ is a unitary operator that satisfies the Schrödinger equation

$$i\hbar \frac{\partial}{\partial t} \hat{U}(t, t_0) = \hat{H}(t) \hat{U}(t, t_0). \qquad (1.27)$$

The solution of this equation reads

$$\hat{U}(t, t_0) = \hat{T} \exp \left\{ -\frac{i}{\hbar} \int_{t_0}^{t} \hat{H}(t') dt' \right\}. \qquad (1.28)$$

where $\hat{T}$ is the time-ordering operator. For a time-independent periodic potential, it simplifies into

$$\begin{aligned} \hat{U}(t, t_0) &= e^{-i\hat{H} \times (t-t_0)/\hbar} \\ &= \sum_{n \in \mathbb{N}} e^{-iE_{q,n}(t-t_0)/\hbar} |\phi_{n,q}\rangle \langle\phi_{n,q}|. \end{aligned} \qquad (1.29)$$
We see that it is straightforward to obtain the evolution of an arbitrary state in the time-independent sine potential (1.21) from its expansion on the Bloch eigenstates:

$$ \begin{gathered} |\psi_q(t_0)\rangle = \sum_{n \in \mathbb{N}} d_{q,n}(t_0) |\phi_{q,n}\rangle \qquad (1.30) \\ \Rightarrow \hat{U}(t, t_0) |\psi_q(t_0)\rangle = \sum_{n \in \mathbb{N}} d_{q,n}(t) |\phi_{q,n}\rangle \end{gathered} $$

with $d_{q,n}(t) = e^{-iE_{q,n}(t-t_0)/\hbar} d_{q,n}(t_0)$.

In Chap. 2, we present a method to calibrate the depth of our optical lattice that consists in probing the eigenenergies $E_{q,n}$ by triggering an out-of-equilibrium evolution in the sine potential.

## 1.3 Wave function in a time-dependent sine potential

### 1.3.1 Time-dependent sine potential: general case

Let us now consider the case of a sine potential whose depth and phase reference can be arbitrarily varied in time:

$$ V(x, t) = -A(t) \times s_0 \frac{E_L}{2} \cos(k_L x + \varphi(t)). \qquad (1.31) $$

The diagonal and off-diagonal matrix elements of the Hamiltonian in the central equation (1.22) become

$$ \langle \chi_{\ell+q/k_L \pm 1} | \frac{\hat{H}(t)}{E_L} | \chi_{\ell+q/k_L} \rangle = - \frac{A(t) s_0 e^{\mp i \varphi(t)}}{4}. \qquad (1.32) $$

We cannot develop further in the case of arbitrary functions $A(t)$ and $\varphi(t)$. We can however make numerical simulations to study a given dynamics, for instance by iteratively constructing the evolution operator from a series of Hamiltonians considered constant over a small enough time intervals $\delta t$:

$$ \hat{U}(t, t_0) = \lim_{\delta t \to 0} \prod_{n=0}^{\lfloor (t-t_0)/\delta t \rfloor} \exp \left\{ \frac{-i \hat{H} (t - n \delta t) \times \delta t}{\hbar} \right\}. \qquad (1.33) $$

with $\lfloor \cdot \rfloor$ denoting the integer part. In Chap. 3, we show how the optical lattice can be continuously moved in space in order to bring an initial state to a given target state. The quantum
optimal control algorithm used for the determination of the control field $x_0(t)$ notably relies on Eq. (1.33) to integrate the dynamics in the lattice.

In the next section, we address the specific case of time-periodic modulations of the sine potential (1.31).

### 1.3.2 Time-periodic Hamiltonians : Floquet formalism

We consider a potential that is periodically modulated in time:

$$V(x, t) = V(x, t + T). \qquad (1.34)$$

**Floquet operator, Floquet states and quasi-energies.** In this system, a natural basis [80, 16] over which to decompose a state to study its evolution is the set of Floquet states $\{|\varphi_m^{(t_0)}\rangle\}$, the eigenstates of the evolution operator from $t_0$ to $t_0 + T$ (i.e. a one-period evolution). This operator, called the Floquet operator, reads

$$U_F^{(t_0)} = U(t_0 + T, t_0) = \hat{T} \exp\left\{\frac{-i}{\hbar} \int_{t_0}^{t_0+T} \hat{H}(t)dt\right\}, \qquad (1.35)$$

with the subperiod reference time $0 \le t_0 < T$. As the evolution operator is unitary, its eigenvalues are complex numbers with norm one:

$$U_F^{(t_0)} |\varphi_m^{(t_0)}\rangle = \lambda_m |\varphi_m^{(t_0)}\rangle = e^{-i\epsilon_m T/\hbar} |\varphi_m^{(t_0)}\rangle. \qquad (1.36)$$

where we have defined the *quasi-energies* $\epsilon_m$ given by $\epsilon_m = -\arg\{\lambda_m\}\hbar/T$. We see that the quasi-energies are periodic under the translation

$$\epsilon_m \rightarrow \epsilon_m + \frac{2\pi\hbar}{T} = \epsilon_m + h\nu, \qquad (1.37)$$

with $h\nu$ the energy of one phonon at the frequency $\nu = 1/T$. Therefore, we only consider the quasi-energies in the interval $-h\nu/2 < \epsilon_m \le h\nu/2$. This “quasi-energetic Floquet cell” is the translation of the Brillouin zone in the time domain [16]. For a given Floquet potential (Eq. (1.34)), two useful properties (see [16]) are that:

* the Floquet states for a different time reference $t_1$ can be determined from the Floquet states $|\varphi_m^{(t_0)}\rangle$ by propagating them from $t_0$ to $t_1$, i.e.

$$|\varphi_m^{(t_1)}\rangle = \hat{U}(t_1, t_0) |\varphi_m^{(t_0)}\rangle, \qquad (1.38)$$

* the quasi-energies $\epsilon_m$ of $U_F^{(t_0)}$ do not depend on the reference time $t_0$.
From the Floquet states and their associated quasi-energies, we rewrite the Floquet opera- tor (1.35)

$$ (\hat{U}_{\text{F}}^{(t_0)})^n = \sum_{m \in \mathbb{N}} e^{-in \times \epsilon_m T/\hbar} |\varphi_m^{(t_0)}\rangle \langle \varphi_m^{(t_0)}| \qquad (1.39) $$

for $n \in \mathbb{N}$ modulation periods. Comparing Eqs. (1.29) and (1.39), we see that, in the Floquet formalism, the discrete, stroboscopic dynamics of a quantum system with a time periodic Hamiltonian is analogous to the continuous dynamics with a time independent Hamiltonian ; the quasi-energies and the Floquet states playing the role of the eigenenergies and the Bloch eigenstates. A potential that is both periodic in space and time is called a Floquet-Bloch potential.

**Quasi-energy spectrum.** The analog of the band structure for a Floquet-Bloch Hamiltonian is the quasi-energy spectrum (or Floquet spectrum) as a function of the quasi-momentum. As examples, we draw in Fig. 1.6 two quasi-energy spectra (computed from Eqs. (1.32) and (1.33)) for the amplitude modulated Floquet-Bloch potential

$$ V(x,t) = -[1 + \epsilon_0 \cos(2\pi\nu t)] s_0 \frac{E_L}{2} \cos(k_L x), \qquad (1.40) $$

with parameters $s_0 = 4$, $\nu = 3.2 \nu_L$ and the two amplitudes of modulation $\epsilon_0 = 0$ and $\epsilon_0 = 0.15$. For $\epsilon_0 = 0$ (Fig. 1.6(b)), we are actually in the situation of the static potential, so the Floquet states are simply the Bloch eigenstates, with the eigenenergies correspondence

$$ \epsilon_n \equiv E_n \mod (h\nu) + \Delta\epsilon \qquad (1.41) $$

(the dependence on $q$ is implicit, and $\Delta\epsilon$ is an arbitrary offset of all the quasi-energies corresponding on the arbitrary global phase in Eq. (1.36)). This equivalence modulo the phononic energy $h\nu$ (Eq. (1.37)) is well illustrated by the comparison between the band structure of the static potential (Fig. 1.6(a)) and the quasi-energy spectrum in absence of modulation (Fig. 1.6(b)). We see that the quasi-energy spectrum consists of superimposed slices of the band structure with energy height $h\nu$. In the absence of modulation (Fig. 1.6(b)), the quasi-energy levels cross at the quasi-momenta where the Bloch bands in the band structure are separated by $k \times h\nu$ with $k \in \mathbb{N}^*$. The integer $k$ is directly the order of the phononic transition implied. For $\epsilon_0 > 0$ (Fig. 1.6(c)), the inter-band couplings induced by the modulation of the potential leads to Bloch eigenstates hybridizations (see below), with quasi-energy bands displaying avoided crossings whose width increases with the modulation amplitude.
<figure><img src="image_13.png" /><figcaption>**Figure 1.6 | Band structure and quasi-energy spectra.** (a) Band structure (plain colored lines) of the static sine potential for $s_0 = 4$. The s, p, d, f and g bands are in blue, orange, green, red and purple respectively. The arrows are placed at the quasi-momenta where the Bloch bands are separated by an energy $|E_{i,q} - E_{j,q}| = k \times h\nu$ (see text), with $k = 1$ (plain back arrow) and $k = 2$ (dotted grey arrow). Arrows are only shown for $q < 0$ for the figure to remain legible. The horizontal dashed lines correspond to the energy period in the quasi-energy spectra (b,c) (see text). (b,c) Quasi-energy spectra for the time-periodic potential (1.40) for two different amplitudes of modulation $\epsilon_0$ and other parameters $s_0 = 4$ and $\nu = 3.2 \nu_L$. The amplitudes of modulation are $\epsilon_0 = 0$ (b) and $\epsilon_0 = 0.15$ (c).</figcaption></figure>

**Identification of the relevant Floquet states.** The static band structure of Fig. 1.6(a) has been obtained numerically in a Hilbert space of truncated dimension $d_H$ (in this case $d_H = 31$, see Sec. 1.2.2). After the diagonalization of the Hamiltonian, one can directly identify the $n^{th}$ Bloch eigenstate as it is the state corresponding to the $n^{th}$ greatest eigenenergy. The amplitude modulated sine potential (1.40) is also a continuous system with an infinite Hilbert space dimension. As for the static case, the Floquet spectra of Fig. 1.6(b,c) have been obtained working with a basis of the $d_H$ plane waves (from Eqs. (1.32) and (1.33) with $d_H = 31$). Therefore, for each quasi-momentum considered in Fig. 1.6(b,c), we actually have $d_H$ states, with no obvious ordering between the quasi-energies $-h\nu/2 < \epsilon_m \le h\nu/2$ (which are moreover defined up to an arbitrary offset). To identify relevant Floquet states, an efficient method is to sort them by their overlap with known states. We use this method, in Fig. 1.6(b,c) where, for all the quasi-momenta considered, we only display the quasi-energies of the $N = 5$ Floquet states maximally alike the first five Bloch eigenstates. Furthermore, the color$^6 C_m^{(\varphi)}$ with which we draw the quasi-energy $\epsilon_m$ of a given Floquet state $|\varphi_{q,m}\rangle$ is the weighted average of the colors $C_n^{(\phi)}$ associated to the first five Bloch bands $|\phi_{q,n}\rangle$, with weights corresponding to the overlaps between the Bloch eigenstates and the Floquet state considered:

<footnote>$^6$For instance encoded in RGB: $C_i = [r_i, g_i, b_i]$ with $0 \le r_i, g_i, b_i \le 1$.</footnote>
$$C_m^{(\varphi)} = \sum_{n=1}^{N} C_n^{(\phi)} |\langle\phi_{q,n}|\varphi_{q,m}\rangle|^2 \qquad (1.42)$$

Doing so, the above-mentioned hybridization of Bloch eigenstates induced by the modulation of the potential can directly be observed at the avoided crossings of Fig. 1.6(c), through which the color of a Floquet band smoothly transitions between the colors associated with the two static Bloch bands.

## 1.4 Quantum states in the phase space

In Fig. 1.5, we have drawn band structures and represented typical Bloch eigenstates both in the *p* and *x* representations. In this section, we discuss the endeavor of giving both informations at once, i.e. representing the quantum state of a one-dimensional system in the (*x*,*p*) phase space.

In the beginning of this chapter (Sec. 1.1), we recalled that the state of a classical system at a given time is a point in the position-momentum phase space of the system. In the quantum regime, the position and momentum of a particle are not defined point-wise. Instead, if the state of a particle is defined by the ket $|\psi\rangle$, the Born rule states that the probability of finding the particle between *x* and *x* + *dx* is $|\langle x|\psi\rangle|^2dx = |\psi(x)|^2dx$, and the probability of finding the particle with a momentum between *p* and *p* + *dp* is $|\langle p|\psi\rangle|^2dp = |\psi(p)|^2dp$. Moreover, the wave function $\psi(x)$ and $\psi(p)$ for a particle in a given state are related by the Fourier transform:

$$\psi(p) = \int_{-\infty}^{\infty} \frac{\psi(x) e^{-ixp/\hbar}}{\sqrt{2\pi\hbar}} dx \qquad (1.43)$$

with the Heisenberg uncertainty relation between the standard deviation<fnref n="7" /> $\Delta x$ of $\psi(x)$ and the standard deviation $\Delta p$ of $\psi(p)$:

$$\Delta x \Delta p \geq \frac{\hbar}{2}. \qquad (1.44)$$

This inequality implies that the minimal “area” $\Delta x \Delta p$ of a state in the phase space is typically $\hbar$, and corroborates that a wave function cannot be described by a vector in the phase space.

<footnote n="7">⁷For an observable $\hat{A}$, we have $\Delta A = \sqrt{\langle \hat{A}^2 \rangle_{\psi} - \langle \hat{A} \rangle_{\psi}^2}$,</footnote>
### 1.4.1 Wigner and Husimi representations and the effective Planck constant

**Wigner representation.** To represent a quantum state $|\psi\rangle$ in the phase space, E. Wigner [81] introduced the function $W_\psi(x, p) : \mathbb{R}^2 \to \mathbb{R}$:

$$W_\psi(x, p) = \frac{1}{\pi \hbar} \int_{-\infty}^{\infty} \psi^*(x+u)\psi(x-u) \exp\left\{-i\frac{2pu}{\hbar}\right\} du. \quad (1.45)$$

such that its integration along the $x$ axis or the $p$ axis (called the marginal distributions) yields the density of probability:

$$\int_{\Omega_p} W_\psi(x, p) dp = |\psi(x)|^2 \quad (1.46)$$

$$\int_{\Omega_x} W_\psi(x, p) dx = |\psi(p)|^2$$

where $\Omega_{x,p}$ are the intervals over which $\psi(x)$ and $\psi(p)$ are normalized (in the case of our sine potential, we work with $\Omega_x = [-d/2, d/2)$ and $\Omega_p = (-\infty, \infty)$). However, it is known that the Wigner function can take negative values [82], so one speaks of a *quasiprobability distribution*. For a state $\psi_q(x)$ in the subspace of quasi-momentum $q$ written over the basis of the plane waves in an infinite sine potential (Eq. (1.20)), the Wigner function reads [83]:

$$W_{\psi_q}(x, p) = \begin{cases} \frac{1}{2\pi} \sum_{\ell \in \mathbb{Z}} c_{\ell}^{*} c_{m-\ell} e^{-i(2\ell-m)k_{\mathrm{L}}x}, & p/\hbar = q + \frac{m}{2}k_{\mathrm{L}} \\ 0, & \text{elsewhere} \end{cases} \quad (1.47)$$

with $m \in \mathbb{Z}$. We note that $W_{\psi_q}(x, p)$ takes non-zero values for integer and half integer values of $p - \hbar q$, whereas $|\psi_q(p)|^2$ is non-zero only for integer values of $p - \hbar q$ (Sec. 1.2.1). In Fig. 1.7, we plot the Wigner representations and marginal distributions of the first two Bloch eigenstates for a lattice of depth $s_0 = 3$ at quasi-momentum $q = 0$. We see that the contribution of $W_{\psi_q}(x, p)$ for half integer values of $p$ cancels out when integrating along $x$ for the evaluation of the marginal distribution $|\psi_q(p)|^2$, as could also be deduced from (1.47).

**Husimi representation.** The alternative to the Wigner representation that we choose to use in this manuscript is the Husimi representation [84] commonly used for dynamical studies in the semi-classical regime (see below). The Husimi function $Q_\psi(x, p) : \mathbb{R}^2 \to [0, 1]$ can be expressed as

$$Q_\psi(x, p) = \frac{1}{2\pi\hbar} |\langle g(x, p) | \psi \rangle|^2. \quad (1.48)$$
<figure><img src="image_14.png" /><figcaption>**Figure 1.7** | Wigner representation of Bloch eigenstates in the phase space of the static sine potential (1.21) (a) Wigner representation $W_ψ(x, p)$ (Eq. (1.45)) of the ground state $|ψ⟩ = |φ_{q,n=0}⟩$ at depth $s_0 = 3$ and quasi-momentum $q = 0$. For visibility, the resulting horizontal stripes are plotted with width $ħk_L/2$ but are in reality infinitesimally thin. (b) Densities of probability in the momentum representation obtained either from the marginal distribution of $W_ψ(x, p)$ (dark blue bars) or directly from the known plane wave coefficients (light blue bars). (c) Densities of probability in the position representation obtained either from the marginal distribution of $W_ψ(x, p)$ (dark blue line) or directly from the Fourier series of $ψ(x)$ (Eq. (1.20)) with the known plane wave coefficients (curve delimiting the light blue area). (d,e,f) Same as (a,b,c) for the first excited state $|ψ⟩ = |φ_{q=0,n=1}⟩$. The classical trajectories are plotted in black lines.</figcaption></figure>

i.e. the evaluation of the overlap between $|ψ⟩$ and a Gaussian state $|g(x, p)⟩$ centered in $(⟨\hat{x}⟩_{g(x,p)}, ⟨\hat{p}⟩_{g(x,p)}) = (x, p)$. In the position representation, this state can be written

$$⟨x|g(x_0, p_0)⟩ = \frac{1}{\sqrt[4]{2\pi(\Delta x)^2}} \exp\left\{-\frac{(x-x_0)^2}{4(\Delta x)^2} + i\frac{xp_0}{\hbar}\right\}. \qquad (1.49)$$

The Gaussian state $|g(x_0, p_0)⟩$ is a state that minimizes the uncertainty inequality (1.44), with $Δp = ħ/2Δx$. The evaluation of $Q_ψ(x, p)$ (1.48) informs on the probability of measuring the particle described by the state $|ψ⟩$ in a typical phase space area of ~ ħ around the point $(x, p)$, with normalization $\iint Q_ψ(x, p) dx dp = 1$. There is however a somewhat arbitrary choice that needs to be made on the standard deviation $Δx$ of the Gaussian state in the Husimi function (1.48). In the case of a harmonic oscillator of angular frequency $ω$, one usually takes the ground state of the system, i.e. a Gaussian state with standard deviation $Δx = \sqrt{ħ/2mw}$. In the case of the sine potential, we make a similar choice by taking the ground state of the harmonic oscillator that approximates the bottom of each site of the potential. At depth $s_0$, the corresponding angular frequency is $ω_0 = \sqrt{s_0} \times 2\pi v_L$ (see Sec. 1.2). The plane wave coefficients of this periodic Gaussian state are [85]:
$$c_{q,\ell}(x_0, p_0) = \left(\frac{2}{\pi\sqrt{s_0}}\right)^{1/4} \exp\left\{-i(\ell k_L + q)x_0 - \frac{(\ell\hbar k_L + \hbar q - p_0)^2}{\sqrt{s_0(\hbar k_L)^2}}\right\} \quad (1.50)$$

with standard deviations $\Delta x = s^{-1/4}k_L^{-1}$ and $\Delta p = s^{1/4}\hbar k_L/2$, verifying the uncertainty relation (1.44). Such Husimi representations are shown in Fig. 1.8.

Equation (1.50) only yields normalized states $|g_q(x_0, p_0)\rangle = \sum_\ell c_{q,\ell}(x_0, p_0)|\chi_{\ell+q/k_L}\rangle$ when the standard deviation $\Delta p$ of the envelope of the plane wave coefficients is far greater than the spacing $\hbar k_L$ of the momentum comb:

$$\Delta p \gg \hbar k_L \Leftrightarrow s \gg 1. \qquad (1.51)$$

<figure><img src="image_15.png" /></figure>

At depth $s_0 \ge 5$, one can compute that the error of normalization is at most

$$\left| \sum_{\ell \in \mathbb{Z}} |c_{q,\ell}(x_0, p_0)|^2 - 1 \right| \le 3.25 \cdot 10^{-5}. \qquad (1.52)$$

As a precaution, we systematically renormalize the states obtained from the definition (1.50).

**The effective Planck constant.** In Fig. 1.8, we plot, alongside trajectories from classical dynamics, the Husimi functions of the first two Bloch eigenstates in the sine potential for the two depths $s_0 = 3$ and $s_0 = 20$. We see that, while the classical trajectories scaled by $k_L^{-1}$ and $p_{sep}$ do not depend on the parameters of the potential, the effect of the variation of the depth of the potential is well visible on the quantum states' extent in the phase space. The difference in behavior between the classical and quantum descriptions can be shown formally: we once again consider the one-dimensional Hamiltonian

$$H(x, p) = \frac{p^2}{2m} - \frac{s_0 E_L}{2} \cos(k_L x) \qquad (1.53)$$

and the dimensionless units

$$\tilde{x} = \frac{x}{x_0}, \quad \tilde{p} = \frac{p}{p_0} \quad \text{and} \quad \tilde{t} = \frac{p_0 t}{m x_0} = \frac{t}{t_0}, \qquad (1.54)$$

with the scalings

$$x_0 = k_L^{-1} \quad \text{and} \quad p_0 = p_{\text{sep}} = \sqrt{2m s_0 E_L} = \hbar k_L \sqrt{s_0}. \qquad (1.55)$$

The scaled Hamiltonian is
<figure><img src="image_16.png" /><figcaption>Figure 1.8 | Husimi representation of Bloch eigenstates in the phase space of the static sine potential (1.21). (a-b) Husimi representation $Q_ψ(x, p)$ of the first two Bloch eigenstates $|ψ⟩ = |φ_{q,n=0,1}⟩$ in a sine potential of depth $s_0 = 3$ at $q = 0$. (c-d) Same as (a,b) in a sine potential of depth $s_0 = 20$. The classical trajectories are plotted in black lines. Please note that two momentum axes are presented on each side of the panels: the first one on the left is in absolute units of $ħk_L$, and the second one on the right is relative to a given classical trajectory (in this case the separatrix, see Fig. 1.1). Having fixed here the relative momentum axes, classical trajectories are invariant under variation of the depth of the potential, whereas Husimi representations of the shown states display important differences, especially in extent. These scaling behaviors are embedded in the notion of the effective Planck constant (see text).</figcaption></figure>

$$ \tilde{H}(\tilde{x}, \tilde{p}) = \frac{\tilde{p}^2}{2} - \frac{1}{4} \cos(\tilde{x}) \qquad (1.56) $$

with $\tilde{H} = Hm/p_0^2 = H/(2s_0E_L)$. One can verify that the Hamilton equations (1.5) hold:

$$ \frac{d\tilde{x}}{d\tilde{t}} = \frac{\partial \tilde{H}}{\partial \tilde{p}} \quad \text{and} \quad \frac{d\tilde{p}}{d\tilde{t}} = -\frac{\partial \tilde{H}}{\partial \tilde{x}}. \qquad (1.57) $$

The resulting equations of motion describe the scaled classical dynamics for the Hamilto-
nian (1.56) which now depends on zero parameter. This demonstrates the scaling invariance of the classical dynamics (see Fig. 1.8). However, if we now consider the quantum dynamics, the scaled Schrödinger equation reads:

$$i \frac{\hbar}{x_0 p_0} \frac{\partial |\psi\rangle}{\partial \tilde{t}} = \left( \frac{\hat{p}^2}{2} - \frac{1}{4} \cos(\hat{x}) \right) |\psi\rangle \quad (1.58)$$

We see that the reduced Planck constant is scaled by an action parameter $S_0 = x_0 p_0$. This leads to define an effective Planck constant:

$$\hbar_{\text{eff}} = \frac{\hbar}{S_0}, \quad (1.59)$$

a tunable parameter that sets the typical extent of quantum states in the scaled phase space (Fig. 1.8), as well as the prevalence and timescales of quantum phenomena in the system (such as tunneling between the sites of the potential).

For the time-independent sine potential, we find in the end, with our choice of scaling,

$$\hbar_{\text{eff}} = \frac{1}{\sqrt{s_0}}. \quad (1.60)$$

## Scaling of the time-independent sine potential

$$\begin{gathered} H(x, p) = \frac{p^2}{2m} - s_0 \frac{E_L}{2} \cos(k_L x) \quad (1.61) \\ \rightarrow \tilde{H}(\tilde{x}, \tilde{p}) = \frac{\tilde{p}^2}{2} - \frac{1}{4} \cos(\tilde{x}) \end{gathered}$$

with

<figure><img src="image_17.png" /></figure>

$$\tilde{x} = k_L x, \quad \tilde{p} = \frac{p}{p_{\text{sep}}} = \frac{p}{\hbar k_L \sqrt{s_0}}, \quad \tilde{t} = 4\pi\nu_L \sqrt{s_0} t, \quad \text{and} \quad \tilde{H} = \frac{H}{2s_0 E_L}. \quad (1.62)$$

* The scaled classical dynamics does not depend on any parameter.

* The scaled quantum dynamics depends on

$$\hbar_{\text{eff}} = \frac{1}{\sqrt{s_0}}. \quad (1.63)$$

The same procedure can be followed to scale the dynamics of the periodically modulated sine potential (see p. 15). In this case, it is the frequency of modulation that tunes $\hbar_{\text{eff}}$, and thus the “quantumness” of the dynamics.
Scaling of the sine potential periodically modulated in depth and position

$$
\begin{align}
H(x, p, t) &= \frac{p^2}{2m} - A(t) \times s_0 \frac{E_L}{2} \cos(k_L x + \varphi(t)) \nonumber \\
&\qquad \qquad \qquad \qquad \qquad \qquad A(t+T) = A(t) \nonumber \\
&\qquad \qquad \qquad \qquad \qquad \qquad \varphi(t+T) = \varphi(t) \tag{1.64} \\
&\qquad \qquad \qquad \qquad \qquad \qquad T = \nu^{-1} \nonumber
\end{align}
$$

$$
\rightarrow \tilde{H}(\tilde{x}, \tilde{p}, \tilde{t}) = \frac{\tilde{p}^2}{2} - A(\tilde{t}) \times \gamma \cos(\tilde{x} + \varphi(\tilde{t}))
$$

with

$$
\tilde{x} = k_L x, \quad \tilde{p} = \frac{T}{md} p, \quad \tilde{t} = \frac{2\pi}{T} t, \quad \text{and} \quad \tilde{H} = \frac{T^2}{md^2} H. \qquad (1.65)
$$

The classical dynamics depends on

$$
A(\tilde{t}), \quad \varphi(\tilde{t}) \quad \text{and} \quad \gamma = s_0 \left(\frac{\nu_L}{\nu}\right)^2. \qquad (1.66)
$$

The quantum dynamics depends, in addition, on

$$
\hbar_{\text{eff}} = 2 \frac{\nu_L}{\nu}. \qquad (1.67)
$$

One can vary $\hbar_{\text{eff}}$ in a fixed scaled stroboscopic phase space (with a fixed $\gamma$ (1.66)) by tuning $\nu$ and adjusting $s_0 = 4\gamma/\hbar_{\text{eff}}^2$ accordingly.

1.4.2 Floquet states representation in the phase space

In this final section, we combine the notions of stroboscopic phase portraits (Sec. 1.1.2), Floquet states (Sec. 1.3.2) and Husimi function (Sec. 1.4.1) to give an example of phase space representation of Floquet states. We define the semi-classical regime ($\hbar_{\text{eff}} \to 0$), in which the extent of the quantum states gets much smaller than the area of the regular islands in the stroboscopic phase portrait. In this regime, the Floquet states can be divided between states localized on regular islands and states spread over the chaotic sea [86, 87]. We illustrate this phenomenon in Fig. 1.9, in the case of the amplitude-modulated potential (Eq. (1.11)) and for a small value of $\hbar_{\text{eff}}$.

For finite values of $\hbar_{\text{eff}}$, coupling can occur between Floquet states, resulting in new states
that can span both regular and chaotic regions of the phase space. This mechanism is discussed
in Chap. 4, in which we determine optimal values of $\hbar_{\text{eff}}$ for the transport of matter waves via
a transporting regular island in the phase space. The quantum coupling between regular and
chaotic regions of the phase space is also at the origin of chaos-assisted tunneling (CAT) [88, 89].
During my thesis, we made the first observations of CAT resonances [61] with matter waves. I do
not present this work in this manuscript, as it is already detailed in previous theses, focusing on
both the experimental [67, 68] and theoretical aspects [69] of the study. We however re-address
this subject at the end of Chap. 3, where we apply a state-control protocol to the study of
dynamical tunneling (of which CAT is a particular case).
<figure><img src="image_18.png" /><figcaption>**Figure 1.9** | Husimi representations of Floquet-Bloch states in the amplitude-modulated poten-
tial (1.11) with parameters $\gamma = 0.375$, $\varepsilon_0 = 0.24$ and $\hbar_{\text{eff}} = 0.2$. (a) A Floquet state localized on regular islands. (b) A Floquet state spread over the chaotic sea.</figcaption></figure>

**Conclusion**

In this first chapter, I briefly introduced concepts and numerical tools for the study of wave
functions in a time-periodic potential. During my thesis, I have worked with Bose-Einstein
Condensates placed in a controllable one-dimensional optical lattice for the simulation of this
system. The next chapter is dedicated to the presentation of the experiment.
<figure><img src="image_19.png" /></figure>
# Chapter 2 - Experimental setup

C'était l'époque des diodes, on en foutait sous tous les boutons, ça s'vendait comme des p'tits pains.

Karim Debbache (CROSSED - 11 - WarGames, 2013)

## Contents

<table><tr><td>Introduction</td><td>39</td></tr><tr><td>2.1 Bose-Einstein condensation</td><td>39</td></tr><tr><td>2.2 Experimental sequence for Bose-Einstein condensation</td><td>40</td></tr><tr><td>2.2.1 Magneto-optical traps</td><td>41</td></tr><tr><td>2.2.2 Magnetic trapping and micro-wave evaporation</td><td>42</td></tr><tr><td>2.2.3 Dipole trapping and evaporation</td><td>43</td></tr><tr><td>2.2.4 Characterization of the hybrid trap at the end of the evaporative cooling</td><td>45</td></tr><tr><td>2.3 Observing the system</td><td>46</td></tr><tr><td>2.3.1 Absorption imaging</td><td>46</td></tr><tr><td>2.3.2 Time-of-flight</td><td>48</td></tr><tr><td>2.4 The one-dimensional optical lattice</td><td>49</td></tr><tr><td>2.4.1 Adiabatic loading of the lattice ground state</td><td>51</td></tr><tr><td>2.4.2 Optical lattice calibration</td><td>53</td></tr><tr><td>2.4.3 Band-mapping</td><td>55</td></tr><tr><td>Conclusion</td><td>57</td></tr></table>
<figure><img src="image_20.png" /></figure>
Introduction

In the first chapter, we recalled elements of theory to study the dynamics of a wave function in a sine potential. In this chapter, we describe the ultracold atom setup used at LCAR to realize this system with Bose-Einstein condensates (BECs) of $^{87}$Rb in a one-dimensional optical lattice. We start with a brief introduction on Bose-Einstein condensation. We then present our experimental sequence for the obtention of BECs and how we access the atomic momentum distribution. We conclude with a description of the optical lattice, its control, and common experimental techniques that are used in the experiments presented in the following chapters of this manuscript.

2.1 Bose-Einstein condensation

In this section, we briefly introduce concepts and quantities relative to Bose-Einstein con-
densation. For details and derivations of these results, see e.g. [90, 91].

Bose-Einstein condensation is a quantum phase transition first predicted for massive parti-
cles in seminal works of A. Einstein in 1924 [4] and 1925 [5]. In an ideal gas of N bosons, there
exists a critical temperature T_C below which a macroscopic number N_0 of particles condenses
into the ground state |ψ_0⟩ of the trap in which they are confined. In the zero temperature limit
(see below), all the particles indexed by i are in the ground state |ψ_0⟩ of the system, and the
BEC state is

$$
|\Psi\rangle = \bigotimes_{i=1}^{N} |\psi_{0}^{(i)}\rangle . \qquad (2.1)
$$

In a gas of density *n*, the average distance between particles is *l* = *n*⁻¹/³. Bose-Einstein con-
densation occurs when the typical extent of the particle wave functions becomes of the order of
*l*. For particles of mass *m* in a gas at temperature *T*, this typical wave function size is given by
the thermal de Broglie wavelength

$$
\Lambda_T = \sqrt{\frac{2\pi\hbar^2}{mk_B T}}, \qquad (2.2)
$$

with $k_B$ the Boltzmann constant. The condition for Bose-Einstein condensation is thus $\Lambda_T \ge l$,
and $T_C$ is defined as the temperature under which this condition is met. In order to reach this
regime, one needs to cool the gas while keeping the atomic density high enough for $\Lambda_T$ to become
greater than $l$.

For a dilute (see below) gas of *N* bosons in a three-dimensional harmonic trap with average
frequency $\bar{\Omega} = (\Omega_x\Omega_y\Omega_z)^{1/3}$, it can be shown [90] that, in the semi-classical limit where $k_B T \gg$
$\hbar\Omega_j$ (with $j = x,y,z$; i.e. when the level spacing in the harmonic potential is much smaller than the thermal energy), $T_C$ is approximately given by:

$$k_B T_C \approx 0.94 \hbar \bar{\Omega} N^{1/3}, \qquad (2.3)$$

with a number of condensed atoms

$$N_0(T) \approx N \left[ 1 - \left( \frac{T}{T_C} \right)^3 \right], \quad \text{for } T < T_C. \qquad (2.4)$$

The experimental setup presented in this chapter produces BECs of $N \approx 5 \cdot 10^5$ atoms in a three-dimensional harmonic trap with average angular frequency $\bar{\Omega} \approx 2\pi \times 36$ Hz (see Sec. 2.2.4). From Eq. (2.3), the critical temperature is approximately

<figure><img src="image_21.png" /></figure>

$$T_C \approx 130 \text{ nK.} \qquad (2.5)$$

Working at temperatures $T \lesssim T_C$, the above-mentioned semi-classical limit is valid, since $\hbar\bar{\Omega}/k_B \sim 1$ nK.

Experimentally, Bose-Einstein condensation was first achieved in 1995, more than 70 years after its prediction, in the group of E. Cornell and C. Wieman with $^{87}$Rb [6] and in the group of W. Ketterle with $^{23}$Na [7]. In 2001, they were awarded the Nobel Prize in Physics for this experimental achievement. Since then, the number of research groups working with quantum gases has exploded [92].

The version of the experimental setup used for the studies presented in this manuscript is in operation at LCAR since 2016. Previous PhD students of the group have extensively detailed in their thesis [93, 94, 67, 68] the construction of the experiment as well as the physics behind the different cooling steps towards Bose-Einstein condensation. In this manuscript, I choose to simply list the steps that compose the cooling sequence, with brief descriptions and updated key figures. During my thesis, we modified the geometry of the dipole trap in which the BECs are finally obtained. In the corresponding section (Sec. 2.2.3), I describe and characterize this new configuration.

## 2.2 Experimental sequence for Bose-Einstein condensation

We start with a vapor of Rb atoms in a vacuum chamber [93]. Our atomic cooling sequence for the obtention of BECs is composed of three main phases:

(2.2.1) - an initial cooling stage in magneto-optical traps,
(2.2.2) - a first step of forced evaporation, by selectively inducing micro-wave transitions in a strong magnetic gradient,

(2.2.3) - a second step of forced evaporation in a dipole trap (with a remaining low magnetic gradient).

In the section dedicated to each phase, we begin with general considerations about the cooling process involved, followed by a paragraph with specific experimental details.

**Note on evaporative cooling.** The last two cooling steps are based on evaporative cooling [95, 96]: through elastic collisions in a trap of given depth, particles sometimes acquire enough energy to leave the trap, so the total energy of the gas decreases. Further elastic collisions lead to the thermalization of the ensemble at a lower temperature. In the experiment, we perform forced evaporation, which consists in actively facilitating the loss of the most energetic atoms (see Secs. 2.2.2 and 2.2.3). In our case, given the relatively low energy of the colliding bosons, collisions occur in *s*-waves [90, 97, 68]. In this case, the cross section of collisions is $\sigma = 8\pi a_s^2$, where the *s*-wave scattering length $a_s$ describes the effective range of inter-particle interaction<fnref n="1" />. Furthermore, we work in the regime of dilute Bose gases, meaning that the average distance between particles is much larger than the *s*-wave scattering length ($na_s^3 \ll 1$).

## 2.2.1 Magneto-optical traps

A magneto-optical trap (MOT) combines Doppler cooling, with laser beams slightly red-detuned from an atomic resonance, and Zeeman shifts induced by a magnetic field gradient to create a non-conservative trap in which the atoms are cooled and confined [90, 100]. We work with the $D_2$ line<fnref n="2" /> of $^{87}$Rb, between the two atomic levels $5^2S_{1/2}$ and $5^2P_{3/2}$. The cooling transition is between the states $|F=2\rangle \rightarrow |F'=3\rangle$ and this two-state cycle is closed by a repumping beam tuned on the $|F=1\rangle \rightarrow |F'=2\rangle$ transition.

**Experimental details.** Our MOT setup is composed of a two-dimensional MOT (made by the SYRTE lab (Paris)) that feeds a home-made three-dimensional MOT at the center of a rectangular glass cell. For the 3D MOT, the intensity of the six cooling beams is ~2 mW/cm<fnref n="2" />, and a magnetic field gradient of $b' \approx 13$ G/cm is created by two coils (located on the cell sides [93], along the *x* axis of the cell shown in Fig. 2.1) in which current flows in opposite directions. The 3D MOT is loaded during 5 s, and we get<fnref n="3" /> a cloud of $N \sim 4 \cdot 10^9$ atoms at $T \sim 400$ µK. After a dark MOT [101, 67] and an optical molasses [102, 90, 67], we shut down

<footnote n="1">¹For the atomic state in which the atoms are when we perform evaporative cooling (see below), we have $a_s \approx 5.23$ nm [98, 99].</footnote>

<footnote n="2">²We present details of this line in App. A. For more information, see [99].</footnote>

<footnote n="3">³The amount of atoms in the MOT saturates absorption images (see Sec. 2.3) and these numbers are estimates. Reliable references are taken for a shortened 3D MOT loading time. After 1 s (instead of 5), we have $N \approx 1.9 \cdot 10^9$ atoms at $T \approx 300$ µK</footnote>
the magnetic gradient and the repumping beams for 5 ms. The atoms accumulate in the three $m_F$ levels of the state $|F = 1\rangle$. At this stage, we have $N \sim 4 \cdot 10^9$ atoms at $T \lesssim 100$ µK.

### 2.2.2 Magnetic trapping and micro-wave evaporation

The second cooling step consists in capturing the atoms in a quadrupole magnetic trap and selectively transferring the most energetic atoms toward a non-trapped state [95, 103]. In a magnetic field $\mathbf{B}(\mathbf{r})$, an atom in the state $|F, m_F\rangle$ experiences the potential

$$U_{\text{mag}}(\mathbf{r}) = g_F m_F \mu_B |\mathbf{B}(\mathbf{r})|, \qquad (2.6)$$

where $g_F$ is the Landé g-factor (dependent on the state considered, see App. A) and $\mu_B = 9.27 \cdot 10^{-24}$ J/T is the Bohr magneton. We generate a magnetic quadrupole field (see below), which near the field zero can be written as

$$\mathbf{B}(x, y, z) = b' \begin{pmatrix} x/2 \\ y/2 \\ -z \end{pmatrix}. \qquad (2.7)$$

with gravity along $-e_z$. The atoms that we capture are in the low-field-seeking state $|F = 1, m_F = -1\rangle$ (with $g_F = -1/2$ for $F = 1$), and we work with the transition toward the non-trapped high-field-seeking state $|F = 2, m_F = -1\rangle$ (with $g_F = 1/2$ for $F = 2$). In the absence of magnetic field, the micro-wave frequency associated with this transition is $\nu_0 \approx 6.835$ GHz [99]. As in the MOT, the gradient of the quadrupole makes the energy levels of the different $m_F$ states spatially dependent. This allows us to tune the frequency of the micro-waves shone on the atoms to only induce the transition at a controlled distance from the center of the trap, in regions only explored by the most energetic atoms. This forced evaporation, followed by a subsequent thermalization, causes the temperature of the atom cloud to decrease, as described above (p. 41).

**Experimental details.** This second phase happens in a conservative quadrupole trap produced by three pairs of water-cooled coils in the anti-Helmholtz configuration where current$^4$ flows in opposite direction. They are located above and below the glass cell [93], along the $z$ axis of the cell (Fig. 2.1). The coils withstand 160 A of current, which corresponds to a total maximum gradient of $\approx 300$ G/cm. Among the atoms falling from the 3D MOT, only the ones in the low-field-seeking state $|F = 1, m_F = -1\rangle$ are captured$^5$. To match the size of the expanding cloud falling during 5 ms from the 3D MOT, the initial gradient of magnetic field is set to $\approx 60$ G/cm. A compensation coil allows to adjust the position of the magnetic field zero along the $y$

<footnote>$^4$The six coils are powered by six SM 15-200D supplies by Delta Elektronika.</footnote>

<footnote>$^5$As Bose-Einstein condensation is achieved reliably on the experiment, increasing the number of atoms captured in the quadrupole trap through spin polarization is not required.</footnote>
axis of the cell (Fig. 2.1). In 500 ms, we load $1.2 \cdot 10^9$ atoms at a temperature that has increased to $T \approx 170$ µK. To increase the interatomic collision rate for the evaporation, we compress adiabatically the cloud by ramping the gradient of magnetic field to $\approx 250$ G/cm in 200 ms. The temperature rises to $T \approx 220$ µK. During 8.5 s, we then shine micro-waves on the atoms, at a frequency swept from 6.720 to 6.818 GHz<fnref n="6" />. At this stage, we have $N \approx 7 \cdot 10^7$ atoms at $T \approx 40$ µK.

Majorana losses [90, 93] near the magnetic field zero at the center of the quadrupole (2.7) prevent us from pushing the cooling further with this method. The final cooling step is then performed in an optical dipole trap.

### 2.2.3 Dipole trapping and evaporation

For the third and final cooling stage, the atoms are loaded in an optical dipole trap. The dipole force results from the interaction between an atom and a light beam of inhomogeneous intensity, which is far-detuned from an atomic transition. Considering a light beam of intensity $I(r)$ and optical frequency $\nu$, the dipole potential [8, 79] experienced by the atom is proportional to $I(r)$ and inversely proportional to the frequency detuning $\Delta = \nu - \nu_0$ (where $\nu_0$ is the frequency associated to the atomic transition considered):

$$U_{\text{dip}}(\mathbf{r}) \propto \frac{I(\mathbf{r})}{\Delta}. \qquad (2.8)$$

Depending on the sign of $\Delta$, the resulting dipole force $\mathbf{F} = -\nabla V(\mathbf{r})$ is either attractive or repulsive. In the experiment, we work with linearly polarized Gaussian beams (see below) of wavelength $\lambda = 1064$ nm, i.e. red-detuned from the $D_2$ line of $^{87}$Rb ($\lambda_0 \approx 780.2$ nm, see App. A). This creates an attractive force towards the intensity maxima. In this situation, Eq. (2.8) becomes $U_{\text{dip}}(\mathbf{r}) = \zeta I(\mathbf{r})$, with $\zeta \approx -2.1 \cdot 10^{-36}$ J/(W/m²) [104]. Finally, the intensity of a Gaussian beam propagating along the $x$ axis is:

$$I(x, y, z) = \frac{2P}{\pi w^2(x)} \exp\left\{-\frac{2(y^2 + z^2)}{w^2(x)}\right\}, \qquad (2.9)$$

where $P$ is the power of the beam and $w(x)$ is the distance to the $x$ axis at which the intensity is $1/e^2$ times its value on the axis<fnref n="7" />. The end of the evaporation takes place in a dipole trap made of crossing red-detuned laser beams, whose powers are decreased in time to force the evaporation [93].

<footnote n="6">⁶For the generation of the micro-wave, see [93]. The final frequency $\nu_f = 6.818$ GHz cuts the depth of the quadrupole trap to $h|\nu_f - \nu_0| \approx k_B \times 580$ µK, with $h$ the Planck constant and $\nu_0 = 6.83$ GHz the frequency associated to the transition between the levels in the absence of magnetic field (i.e. in the center of the trap).</footnote>

<footnote n="7">⁷This distance is $w(x) = w_0\sqrt{1 + \left(\frac{x}{x_R}\right)^2}$, with $w_0$ the beam waist and $x_R = \pi w_0^2/\lambda$ the Rayleigh length.</footnote>
<figure><img src="image_22.png" /><figcaption>**Figure 2.1** | Sketch of the old and new dipole beam geometries. The dark red rectangles represent the dipole beams and the pale blue rectangle represents the glass cell (neither the beam sizes nor their divergence is to scale with the size of the cell). To prevent reflections, the incidence angle of the horizontal beam $H_1$ on the uncoated cell is ~ 1° [93]. (a) Old geometry illustrated with an axial section of the cell. The angle between the recycled vertical beam $V_1$ and the z axis was $\theta_a \approx 30°$. (b) New geometry illustrated in the horizontal plane. The angle between the two independent horizontal beams $H_1$ and $H_2$ is $\theta_b \approx 16°$, so the principal axes $x'$ and $y'$ of the resulting trap make an angle $\theta' \approx 8°$ with x and y.</figcaption></figure>

**Experimental details.** After the micro-wave evaporation, we decompress the cloud by lowering the magnetic gradient from $b' \approx 250$ G/cm to $b' \approx 22$ G/cm in 1.1 s. This final gradient compensates approximately 72% of gravity, and the atoms slowly fall into a dipole trap positioned approximately 200 µm below the center of the quadrupole [17]. The dipole trap is made of two crossed Gaussian beams<fnref n="8" /> of wavelength $\lambda = 1064$ nm. Figure 2.1 shows the previous and new geometries of the dipole trap.

**Previous configuration** - An horizontal beam $H_1$ (with waist $w_0 = 75$ µm and initial power $P_0 = 10.3$ W) went through the cell a first time. It was recycled into a vertical beam $V_1$ ($w_0 = 35$ µm, $P_0 = 6$ W) that went back through the cell in its $(x, z)$ plane from top to bottom. This recycled beam made an angle $\theta \approx 30°$ with the vertical axis. In this recycled configuration, only the intensity of $H_1$ was locked with a PID<fnref n="9" /> controller acting on the power of the radio-frequency (RF) driving the acousto-optic modulator (AOM) of this beam.

**New configuration** - Two independent beams $H_{1,2}$ ($w_0 = 45$ µm, $P_0 = 4$ W) intersect in the horizontal plane with an angle $\theta \approx 16°$. The reasons for the change are an increased volume for the dipole trap as well as the ability to independently diagnose the individual effect of the two beams on the atoms (which was impossible in the recycled configuration). In this configuration, both the intensities of $H_1$ and $H_2$ are controlled with PIDs. To minimize the reflection on the uncoated cell, the polarization of both beams is horizontal (i.e. in the $(x, y)$ plane of the cell). To prevent interferences between the beams, we work with opposite AOM diffraction orders $\pm 1$ to detune the beams frequencies by about 80 MHz (see App. B for details).

<footnote n="8">⁸See App. B for technical details and the optical diagram for the preparation of the dipole beams.</footnote>

<footnote n="9">⁹PID: proportional–integral–derivative</footnote>
We load $N \approx 1.2 \cdot 10^7$ atoms at $T \approx 40$ µK into the resulting hybrid trap (composed of the dipole trap, the magnetic trap and gravity). We force the evaporation by lowering the power of the dipole beams using the smooth power law [105]

$$P(t) = P_0 \left(1 + \frac{t}{\tau}\right)^{-\alpha}, \qquad (2.10)$$

with $\tau = 1.75$ s and $\alpha = 4$. After 5.4 s of evaporation, the depth of the dipole trap is reduced by a factor $P_0/P_f \approx 275$ and Bose-Einstein condensation is achieved. The experiment produces BECs of $N \approx 5 \cdot 10^5$ $^{87}$Rb atoms$^{10}$ in the lowest hyperfine level $|F=1, m_F = -1\rangle$ every 22 s. Such a BEC is shown in Fig. 2.2(c).

### 2.2.4 Characterization of the hybrid trap at the end of the evaporative cooling

At the end of the evaporative cooling, the BEC is in a hybrid trap made of

* the two dipole beams with powers $P_f \approx 20$ mW,

* the quadrupole (2.7) with a magnetic gradient $b' \approx 22$ G/cm and whose center is located approximately 200 µm above the crossed dipole trap,

* the gravitational potential.

We locally approximate this hybrid trap with the three-dimensional harmonic potential

$$U_{\text{hyb}}(x', y', z) \approx \frac{1}{2} m [\Omega_{x'}^2(x')^2 + \Omega_{y'}^2(y')^2 + \Omega_{z'}^2 z^2], \qquad (2.11)$$

where $(x', y', z) = (0, 0, 0)$ is the center of the crossed dipole trap and the primed $x'$ and $y'$ lie along the principal axes of the trap, i.e. at an angle $\theta' \approx 8^\circ$ with $x$ and $y$ (see Fig. 2.1(b)). To measure the angular frequencies $\Omega_j$, we trigger out-of-equilibrium evolutions$^{11}$ and measure the periods of oscillation in the corresponding directions (we present the method for the observation of the system in the next section 2.3). We find

$$\Omega_{x'} = 2\pi \times (10.4 \pm 0.2) \text{ Hz}, \qquad (2.12)$$

$$\Omega_{y'} = 2\pi \times (68 \pm 4) \text{ Hz},$$

$$\Omega_z = 2\pi \times (66 \pm 2) \text{ Hz},$$

with an average angular frequency $\bar{\Omega} = (\Omega_x \Omega_{y'} \Omega_{z'})^{1/3} \approx 2\pi \times 36$ Hz.

<footnote>$^{10}$For information, we made BECs of up to $N = 2 \cdot 10^5$ atoms every 30 s with the previous dipole trap geometry.</footnote>

<footnote>$^{11}$We make three independent experiments where we trigger oscillations by pulsing current into coils approximately placed along the right axis, i.e. one of the 3D MOT coils to measure $\Omega_{x'}$, the compensation coil for $\Omega_{y'}$ and one of the magnetic trap coils for $\Omega_z$ (see Sec. 2.2).</footnote>
**Life time in the hybrid trap and reduction of the number of atoms in the BECs.** We measure the life time in the hybrid trap by monitoring the number of atoms (see following section) in the hybrid trap during a holding time at the end of the evaporation. During this holding time, we observe a spontaneous evaporation leading to an exponential decrease of the number of atoms in the BECs. We measure a characteristic time of $\tau \approx 20$ s for this exponential atomic loss. This method can be used to artificially reduce the number of atoms in the BECs for experiments that require a control on that parameter. This is the case in Sec. 3.4 and 5.

## 2.3 Observing the system

### 2.3.1 Absorption imaging

In this section, we explain how we access the atomic density distribution using absorption imaging. Its principle is the following: in due time, we illuminate the atoms with a laser beam that is resonant with an atomic transition and light is absorbed. From the shadow of the cloud, we determine its position and size (Fig. 2.2). The number of atoms in the cloud is deduced from the fraction of light absorbed.

As depicted in Fig. 2.2, we consider a resonant beam traveling along a fictitious $z$ axis. In the regime where the beam intensity is low compared to the atomic saturation intensity, the intensity that remains after passing through the cloud is given by the Beer-Lambert law:

$$I_{\text{out}}(x, y) = I_{\text{in}}(x, y) e^{-\sigma_0 n_c(x, y)} \qquad (2.13)$$

where $I_{\text{in}}(x, y)$ is the initial intensity, $\sigma_0$ is the cross section of absorption and $n_c(x, y) = \int n(x, y, z) dz$ is the atomic column density traversed by the light. By taking pictures with and without atoms, we respectively access $I_{\text{out}}(x, y)$ ad $I_{\text{in}}(x, y)$. From Eq. (2.13), the column density simply reads

$$n_c(x, y) = \frac{1}{\sigma_0} \text{OD}(x, y), \qquad (2.14)$$

with the optical density $\text{OD}(x, y) = \ln\{I_{\text{in}}(x, y)/I_{\text{out}}(x, y)\}$. The number of atoms in a cloud is $N = \iint n_c(x, y) dx dy$. On the example of Fig. 2.2(c), we measure a BEC of $N = 5.11 \cdot 10^5 \pm 3\%$ atoms over 10 realizations (where the uncertainty corresponds to one standard deviation of the statistics).
<figure><img src="image_23.png" /><figcaption>**Figure 2.2** | **Schematic of absorption imaging.** (a) We illuminate the atom cloud (black disk) with a resonant beam (red area), light is absorbed (white area) and $I_{\text{out}}(x, y)$ is measured by taking an image on the screen (thin white rectangle). (b) $I_{\text{in}}(x, y)$ is measured by taking an image shortly after without the cloud. (c) Experimental example of $I_{\text{in}}(x, y)$, $I_{\text{out}}(x, y)$ and the resulting optical density when performing the absorption image of a BEC after a 35 ms time-of-flight (see Sec. 2.3.2). The coordinate system (x, y, z) is that of the imagery axis and does not correspond to the coordinate system of the cell (see Fig. 2.3). Figure inspired from [69].</figcaption></figure>

**Experimental details.** The frequency of the imaging beam is tuned on the transition $|F = 2\rangle \rightarrow |F' = 3\rangle$ (i.e. the cooling transition in the MOT, see Sec. 2.2.1 and App. A). We recall that, after the MOT stage, the atoms are in the state $|F = 1\rangle$. Before imaging the cloud, we repump the atoms in the state $|F = 2\rangle$ by shining the repumping beam for a few ms. For $^{87}$Rb atoms equally distributed in the five $m_F$ levels of this state and an imaging beam with linear polarization and wavelength $\lambda \approx 780$ nm, the cross section of absorption is $\sigma_0 \approx 0.136 \ \mu\text{m}^2$ [61]. To correct imperfections (extraneous light, defaults on the optical path, etc.), we take three images$^{12}$: (i) a first image with the atoms and the imaging beam ($I_{\text{atoms}}$), (ii) an image of the imaging beam without the atoms ($I_{\text{beam}}$) and (iii) a background image ($I_{\text{background}}$) without the beam. We compute the OD with $I_{\text{in}} = I_{\text{beam}} - I_{\text{background}}$ and $I_{\text{out}} = I_{\text{atoms}} - I_{\text{background}}$. Figure 2.2(c) shows examples of $I_{\text{in}}$, $I_{\text{out}}$ and the resulting OD when performing the absorption image of a BEC after time-of-flight (see Sec. 2.3.2).

Three imaging setups (see Fig. 2.3) are installed on the experiment to observe the atoms along different axes during the experimental sequence:

<footnote>$^{12}$Images are taken using a CCD camera A102f from Basler. The screen is made of 1392 × 1040 square pixels with side 6.45 µm. The signal is a photoelectron count per pixel proportional to the imaging beam intensity at that point.</footnote>
<figure><img src="image_24.png" /><figcaption>**Figure 2.3 | Imaging axes** in the horizontal plane (with a 90° rotation from Fig. 2.1(b)). The identified angles are $\theta_1 \approx 16^\circ$ and $\theta_2 \approx 30^\circ$. The “vert. BEC” imaging beam goes through the dipole cross from the bottom to the top of the cell. The red lines and arrows indicate the crossed dipole trap with the direction of propagation of the beams and the pale blue rectangle represents the cell (not to scale).</figcaption></figure>

* the “MOT” axis, in the horizontal plane, with magnification $M \approx 0.33$,

* the “horizontal BEC” axis, in the horizontal plane, with magnification $M \approx 2.5$,

* the “vertical BEC” axis, with magnification $M \approx 4.2$. This axis was added during my thesis to visualize the new crossed dipole trap.

Every experimental image presented in this manuscript is taken with the “horizontal BEC” imaging axis.

### 2.3.2 Time-of-flight

The observation of the atoms in the experiment is generally performed after a time-of-flight (TOF): after a sudden switch-off of all trapping potentials, the atom cloud starts to fall due to gravity, with a constant expansion in the center of mass frame that carries information on the *in situ* velocity distribution. We take an absorption image after a time $t_{\text{TOF}}$.

For a thermal atomic ensemble ($T \gg T_c$, see Sec. 2.1) described by the Maxwell-Boltzmann statistics, the standard deviation of the cloud size along a given axis $x$ following a TOF is

$$ \Delta x(t_{\text{TOF}}) = \sqrt{\Delta x_0^2 + (\Delta v_x t_{\text{TOF}})^2}. \qquad (2.15) $$

with $\Delta x_0$ and $\Delta v_x$ the *in situ* standard deviations of the position and velocity respectively. By measuring the expansion of a thermal cloud as $t_{\text{TOF}}$ increases, Eq. (2.15) allows to access the cloud temperature $T$, as the slope of $\Delta x^2(t_{\text{TOF}}^2)$ is proportional to $T$ according to the equipartition theorem:
$$ \frac{1}{2} m \Delta v_x^2 = \frac{1}{2} k_B T. \qquad (2.16) $$

Moreover, after sufficiently long TOF ($t_{TOF} \gg \Delta x_0 / \Delta v_x$), one can neglect the initial size of the cloud, and the final density distribution reflects directly the velocity distribution of the cloud before the TOF.

For quantum gases, TOF imagery allows to dilute the clouds whose observation is made difficult by their small size and high optical density. In a similar way as for the thermal case, one measures a momentum density of probability when observing the system after a sufficiently long ballistic TOF. For a BEC composed of atoms condensed in the state $|\psi\rangle$, one even statistically samples $|\langle p|\psi\rangle|^2$ in a single TOF measurement of the system (see for instance next section 2.4).

## 2.4 The one-dimensional optical lattice

In this last section, we describe our one-dimensional optical lattice and introduce experimental methods that we frequently use in the rest of this manuscript to study matter waves dynamics in the optical lattice.

An optical lattice is a periodic dipole potential (Eq. (2.8)) resulting from a spatially periodic variation of light intensity. Our one-dimensional optical lattice results from the interference of two Gaussian beams (Eq. (2.9)) with the same wavelength and linear polarization, that are counterpropagating along the $x$ axis (see below). Neglecting the divergence of the beams$^{13}$, their electric fields along the $x$ axis are the real part of the phasors

$$ \underline{E}_1(x, t) = \underline{E}_0 e^{i(-kx - \omega t - \varphi_1)} e_\perp \quad \text{and} \quad \underline{E}_2(x, t) = \underline{E}_0 e^{i(kx - \omega t - \varphi_2)} e_\perp, \qquad (2.17) $$

where $k = 2\pi/\lambda$ is the wavenumber of the laser beams and $e_\perp$ is a constant unitary vector in the $(y, z)$ plane. The light intensity resulting from the interference of these fields is proportional to the square modulus of the total field $\underline{E} = \underline{E}_1 + \underline{E}_2$:

$$ \begin{aligned} I(x) &\propto |\underline{E}(x, t)|^2 = 4 |\underline{E}_0|^2 \cos^2 \left( kx + \frac{\varphi}{2} \right) \qquad (2.18) \\ &= 2 |\underline{E}_0|^2 \cos \left( \frac{2\pi x}{\lambda/2} + \varphi \right) + 2 |\underline{E}_0|^2 \end{aligned} $$

with $\varphi = \varphi_1 - \varphi_2$ the phase difference between the beams. We see that the step of the periodic variation of intensity is $d = \lambda/2$. Dropping the constant term that stems from the linearization of the quadratic cosine, we finally get the attractive dipole potential for the lattice

<footnote n="13">13 e.g. near the focal point, for values of $x$ small compared to the Rayleigh length $x_R$ of the beams (see below).</footnote>
$$V(x) = -\frac{V_0}{2} \cos(k_L x + \varphi), \qquad (2.19)$$

where we used the wave number of the periodic potential $k_L = 2\pi/d$ defined in Chap. 1. $V_0$ is proportional to the maximum intensity of the lattice standing wave of Eq. (2.18). Controlling the intensity and the relative phase $\varphi$ of the interfering beams, we get the sine potential studied in Chap. 1:

$$V(x,t) = -A(t) \times s_0 \frac{E_L}{2} \cos(k_L x + \varphi(t)). \qquad (2.20)$$

**Experimental details.** We here briefly present the experimental implementation of this controlled one-dimensional optical lattice. Technical details and the optical diagram for the preparation of the beams can be found in App. C. The two beams for the lattice are obtained from the same laser source with wavelength $\lambda = 1064$ nm. They are counterpropagating along the $x$ axis of the cell and focused on the crossed dipole trap, with waist $w_0 = 150$ µm and a maximum power $P = 2.3$ W. Using four synchronized arbitrary waveform generators and three AOMs (see App. C), we are able to continuously control $A(t)$ and $\varphi(t)$ in Eq. (2.20).

We remark that, by only controlling the intensity of the lattice beams, we have $0 \le A(t) \le s_{max}/s_0$ in Eq. (2.20), where $s_{max}$ is the maximum lattice depth achievable ($s_{max} \approx 40$ for our setup).

Given the atomic mass $m$ of $^{87}$Rb and the lattice spacing $d$:

$$m \approx 1.443 \cdot 10^{-25} \text{ kg} \quad \text{and} \quad d = \lambda/2 = 532 \text{ nm}, \qquad (2.21)$$

we evaluate the characteristic lattice quantities defined in the previous chapter (sec. 1.2):

* $k_L = 2\pi/d \approx 11.81 \cdot 10^6 \text{ m}^{-1}$,

* $E_L = \hbar^2 k_L^2 / 2m \approx 5.375 \cdot 10^{-30} \text{ J}$,

* $\nu_L = E_L/h \approx 8111.3 \text{ Hz}$.

Our one-dimensional optical lattice constrains the dynamics along the $x$ axis into lattice sites associated with the harmonic oscillator frequency $\nu = \sqrt{s_0} \nu_L$ (see Sec. 1.4.1). This frequency is to be compared with the frequencies (2.12) of the three-dimensional hybrid trap in which the atoms are held. We see that, regardless the lattice depth, the typical timescales for the dynamics in the lattice ($T \approx 123 \text{ µs}/\sqrt{s_0}$) are much smaller than the timescales in the hybrid trap ($T_j = 2\pi/\Omega_j > 15 \text{ ms}$). As the typical duration of the experiments presented in this manuscript is $\sim 1$ ms, the dynamics along the $x$ axis is uncoupled from the dynamics along the perpendicular axes, which allows us to assume a one-dimensional system in our studies. In
particular in the following, we consider along the $x$ axis only the shared wave function (2.1) of a BECs evolving in the time-dependent Bloch system (2.20), i.e. Bloch waves (see Chap. 1)

$$ \psi_q(x,t) = \sum_{\ell \in \mathbb{Z}} c_{q,\ell}(t) |\chi_{\ell+q/k_L}\rangle, \qquad (2.22) $$

with the plane waves defined in Sec. 1.2.1 as $\langle x | \chi_{\ell q/k_L} \rangle = e^{i(\ell k_L + q)x}/\sqrt{d}$.

We now present a series of established experimental techniques when working with ultracold atoms in optical lattices.

## 2.4.1 Adiabatic loading of the lattice ground state

The first technique we present is a protocol to load the ground state of the static lattice. For a given lattice depth $s_0$, this state is the Bloch eigenstate $|\phi_{q=0,n=0}^{(s_0)}\rangle$ associated to the lowest eigenenergy $E_{q=0,n=0}^{(s_0)}$ (see Sec.1.2.1). Initially, the BEC state is the resting plane wave that corresponds to the ground state of a lattice of null depth $|\chi_0\rangle = |\phi_{0,0}^{(s_0=0)}\rangle$, and we want to drive this state to $|\phi_{0,0}^{(s_0)}\rangle$. One can do so adiabatically<fnref n="14" /> by ramping up the lattice depth $s_0$ with a timescale $\tau$ much larger than the largest timescales of the system given its energy levels [106]. This smallest timescale is the period associated to the transition towards the plane waves of second lowest kinetic energy $|\chi_{\pm 1}\rangle$, which have an energy difference $E_L$ with the BEC. The timescale in comparison to which the lattice loading needs to be slow is $h/E_L = \nu_L^{-1} \approx 123$ µs. To this end, we ramp up the lattice depth $s(t) = A(t) \times s_0$ with the smooth amplitude function

$$ A(t) = (\kappa + 1) \left(\frac{t}{\tau}\right)^{\kappa} - \kappa \left(\frac{t}{\tau}\right)^{\kappa+1}. \qquad (2.23) $$

Experimentally, we use the characteristic loading time $\tau = 1$ ms (sufficiently large in comparison to $h/E_L$) and slope parameter $\kappa = 11$. The lattice depth follows the loading curve (2.23) during the interval $0 \le t \le t_{load} = \tau$, such that $0 \le A(t) \le 1$.

We show in Fig. 2.4 an experiment of adiabatic lattice loading. The matter wave diffraction observed in Fig. 2.4(e) illustrates how the TOF measurement of BEC in an optical lattice (i.e. a Bloch wave (2.22)) amounts to sampling its momentum distribution: during the TOF expansion, the planes waves separate into spatially resolved momentum components<fnref n="15" />, with a momentum difference $\Delta p = \hbar k_L = h/d$. When we shine the imaging beam, an atom is measured in the plane wave $\ell$ with probability $|c_{q,\ell}|^2$, which we access by measuring the fraction of atoms in the $\ell^{th}$ momentum component. In the specific case of Fig.2.4, we adiabatically load the ground state of the lattice, and we see that the relative atomic populations in the momentum orders are in

<footnote n="14">¹⁴The term “adiabatic” must be understood in the quantum sense of an eigenstate follow-up as a parameter of the system is varied. On the other hand, a diabatic process projects the initial state onto several non-degenerate eigenstates of the final system.</footnote>

<footnote n="15">¹⁵In the form of diffraction orders that we often refer to as “momentum orders”.</footnote>
<figure><img src="image_25.png" /><figcaption>**Figure 2.4** | Adiabatic lattice loading and matter wave diffraction. (a) Lattice depth curve (2.23) to adiabatically load the ground state of the lattice of depth $s_0$ (see text). (b) Numerical simulation of the lattice loading showing the momentum distribution as a function of time. (c) Experimental TOF absorption image of the BEC before loading the lattice. The grey rectangles delimit the regions in which the plane wave populations are extracted (see (e), Sec. 2.3 and text for details). (d) Bar diagram comparing the experimental (blue) and numerical (red) plane wave populations (i.e. momentum distribution). (e) Experimental TOF absorption image at the end of the loading. (f) Experimental (blue) and numerical (red) momentum distributions. The lattice depth, determined after the loading (see Sec. 2.4.2), is $s_0 = 8.06 \pm 0.10$. The duration of the TOF is $t_{\text{TOF}} = 35$ ms and the colormaps for the OD on panels (c,e) are truncated to 0.8 of their respective maximum value in order to see details.</figcaption></figure>

agreement with a numerical computation (see Sec. 1.2.2) of momentum distribution $|\langle p | \phi_{0,0}^{(s_0)} \rangle|^2$ of the ground state (where the experimental $s_0$ is determined in the next section).

The method of adiabatic lattice loading is limited to the preparation of the ground state of the lattice. In the next chapter (Chap. 3), we apply quantum optimal control to extend the range of quantum states reachable in an optical lattice.

**Note on measurement of the momentum component population.** Here, the plane wave population is extracted from a count of the atoms in the rectangular regions shown in Fig. 2.4(c,e). In Fig. 2.4(e), we see atomic signal between the diffraction orders 0 and $\pm 1$. This signal comes from scattering halos populated by pairs of atoms undergoing a two-body collision at the beginning of the TOF [107, 62] (see also Fig. 2.5 where this phenomenon is more obvious). As the momentum distribution of the atoms in the periodic lattice is quantized (see Chap. 1), these halos (that also expand during the TOF) are seen in between the diffraction orders (mostly in
between the most populated ones). During my thesis, we have studied these halos and developed a protocol to observe them between given momentum orders. This method, which relies on a sudden lattice shift followed by a precisely timed evolution in the lattice in order to populate given momentum orders, is detailed in our publication [62] as well as in a previous thesis [68]. This study is at the root of our initiative to implement quantum-optimal control on the experiment, which is the subject of the next chapter of this manuscript. Importantly, scattering halos can make the extraction of momentum distributions more complicated as (i) momentum orders are depopulated during the process [68] and (ii) the counting region associated to a given order can be polluted by scattering atoms coming from adjacent orders. This latter effect can be mitigated by subtracting to each order population the halo signal measured in the two lateral regions between that order and its neighbors.

### 2.4.2 Optical lattice calibration

In Sec. 2.4 (with details in App. C), we presented the optical lattice with its amplitude and phase control by the mean of three AOMs. The experiments of quantum state preparation and transport in the optical lattice that we present in the next two chapters require to know the depth $s_0$ of the optical lattice with great precision (typically a few percents). In this section, we introduce two methods to measure the lattice depth.

**Kapitza-Dirac diffraction.** The first method that we use relies on pulsing the lattice of unknown depth on the BEC [108]. For a sufficiently short pulse (see below), one can consider that the atoms get their momentum affected by the potential without enough time to start accumulating a displacement. This is the Raman-Nath approximation [109, 110], notably used for efficient computation of the dynamics in the kicked-rotor [111], a simple yet powerful model to study quantum chaos. Formally, one neglects the kinetic terms in the Hamiltonian when integrating the time-dependent Schrödinger equation during the pulse. In the position representation, we have

$$i\hbar \frac{\partial}{\partial t} \psi(x,t) \approx \hat{V}(x) \psi(x,t) \quad \Rightarrow \quad \psi(x,t) \approx \exp \left\{ \frac{it}{\hbar} \times s_0 \frac{E_L}{2} \cos(k_L x) \right\} \psi(x,0) \quad (2.24)$$

with $\psi(x,0) = \chi_0(x) = 1/\sqrt{d}$ the initial wave function just before pulsing the lattice. To compute this phase impression, we use the Jacobi-Anger identity:

$$e^{i z \cos(\theta)} = \sum_{n=-\infty}^{\infty} i^n J_n(z) e^{in\theta}, \quad (2.25)$$

where $J_n(z)$ are the Bessel functions of the first kind. The evolution (2.24) becomes
<figure><img src="image_26.png" /><figcaption>**Figure 2.5** | Calibration of the optical lattice depth from Kapitza-Dirac diffraction. Horizontal stack of experimental TOF absorption images as the duration of the Raman-Nath pulse increases.</figcaption></figure>

$$ \psi(x, t) \approx \sum_{\ell \in \mathbb{Z}} i^{\ell} J_{\ell}(\pi \nu_{\mathrm{L}} s_{0} t) \frac{e^{i \ell k_{\mathrm{L}} x}}{\sqrt{d}} = \sum_{\ell \in \mathbb{Z}} c_{\ell}(t) \chi_{\ell}(x) \qquad (2.26) $$

with plane wave coefficients $c_{\ell}(t) = i^{\ell}J_{\ell}(\pi\nu_{\mathrm{L}}s_{0}t)$.

A quick method to estimate the lattice depth $s_0$ is to experimentally determine the first pulse time $\tau_0$ for which the zero$^{th}$ plane wave is totally depleted (i.e. a complete depletion of the zero$^{th}$ momentum order). Once this time is found, and knowing the first zero $z_0 \approx 2.405$ of $J_0(z)$, we get

$$ \begin{aligned} |c_0(\tau_0)|^2 = 0 &\Leftrightarrow |J_0(\pi \nu_L s_0 \tau_0)|^2 = 0 \qquad (2.27) \\ &\Leftrightarrow s_0 = \frac{z_0}{\pi \nu_L \tau_0} \approx \frac{94.38 \ \mu\mathrm{s}}{\tau_0} \end{aligned} $$

Figure 2.5 shows experimental Kapitza-Dirac diffraction of matter wave in an optical lattice of fixed depth for an increasing pulse duration. We see that the zero$^{th}$ momentum order is depleted for $\tau_0 \approx 2.6 \ \mu\mathrm{s}$, which gives a lattice depth $s_0 \approx 36.3$ according to Eq. (2.27).

In Eqs. (2.24), (2.26) and (2.27), we made the approximation that the lattice is pulsed for a duration negligible compared to the time needed for the atoms to start moving in the potential. In a semi-classical picture, a lower bound for this timescale is approximately given by the period $T_0$ of the harmonic oscillator that approximates the bottom of the lattice sites. In the previous chapter (see Sec. 1.4.1), we found the frequency of this harmonic oscillator to be $\nu_0 = \sqrt{s_0} \nu_L$.

We have $T_0 \propto s_0^{-1/2}$ while $\tau_0 \propto s_0^{-1}$ (Eq. (2.27)). These scalings are favorable for this first lattice calibration, which works if $s_0$ is large enough for the Raman-Nath approximation
to hold up to $\tau_0$ [94]. We state the following condition for the validity of this calibration method:

$$ \frac{\tau_0}{T_0} \approx \frac{z_0}{\pi\sqrt{s_0}} \ll 1 \Leftrightarrow \sqrt{s_0} \gg 1 \qquad (2.28) $$

This calibration technique is a fast way to estimate the depth of a sufficiently deep lattice. For more precision (notably for $s_0 < 10$ where most of the experiments presented in this manuscript are performed), we use a second method that relies on fitting an out-of-equilibrium evolution of the atoms in the lattice.

**Oscillation of the translated ground state.** A second, longer but more precise method has been proposed by our group and published before my thesis [112, 113]. It consists in loading the ground state (Sec. 2.4.1) of the lattice of unknown depth $s_0$ before applying a sudden phase translation $\phi_0$ such that the atoms are placed on the flank of the lattice wells and start oscillating. In the quantum picture, the translated ground state<fnref n="16" /> is projected onto the Bloch eigenstates of the lattice and its evolution is totally determined by this projection according to Eq. (1.29). Using $s_0$ as an adjustable parameter<fnref n="17" />, we perform a least-square fit of the experimental evolution of the atomic momentum distribution. In Fig. 2.6, we show an example of lattice calibration with the experimental data and numerical fit. The lattice depth determined with the experiment of Fig. 2.6 was used in Fig. 2.4 to compare the experimental loading of the ground state with the numerical simulation. This lattice calibration method is used before and after each experiment presented in the rest of this manuscript (to verify that the lattice depth, which notably depends on the lattice beams alignment, has not shifted).

A strength of this method is that, given the adiabatic loading of the lattice, we can infer with confidence that the initial state is the shifted ground state of the lattice of unknown depth. In Chap. 3 about quantum-optimal control, in order to certify our quantum state preparation protocol, we revisit this method of parameter determination through an evolution in the static lattice. However, the point of view is there reversed, as the unknown parameter is the initial quantum state that begins to evolve in a static lattice of previously calibrated depth.

### 2.4.3 Band-mapping

Band-mapping is a well established experimental technique for cold atoms in optical lattices [107, 114, 79], by which the weights of a state on the different Bloch bands are mapped on

<footnote n="16">¹⁶We recall (see Sec. 1.2.1) the expression of the translation operator $\hat{T}_{\Delta x} = e^{-i\Delta x\hat{p}/\hbar}$ with $\hat{p}|\chi_r\rangle = r\hbar k_L|\chi_r\rangle$. The translation of a state $|\psi\rangle$ in a Bloch system reads $\hat{T}_{\Delta x}|\psi\rangle = \sum_{\ell} c_{\ell}e^{-i\Delta x\ell k_L}|\chi_{\ell}\rangle$ with $c_{\ell} = \langle\chi_{\ell}|\psi\rangle$. According to our expression of the lattice potential (2.19), a phase shift $\phi_0$ translates the lattice by a distance $\Delta x = -\phi_0/k_L$ in the laboratory frame of reference which translates an arbitrary state $|\psi\rangle$ by a distance $-\Delta x = \phi_0/k_L$ in the lattice.</footnote>

<footnote n="17">¹⁷For each lattice depth considered, the Bloch eigenstates and eigenenergies are computed numerically as described in Sec. 1.2.2.</footnote>
<figure><img src="image_27.png" /><figcaption>Figure 2.6 | Calibration of the optical lattice depth from a sudden lattice translation. (a) Stack of experimental integrated TOF absorption images as a function of the holding time showing the momentum evolution of the initial state (which was the ground state up until the sudden lattice shift) in the translated lattice. (b) Numerical evolution at the lattice depth that fits (a). (c) Relative plane wave populations (with panel ($c_{\ell}$) featuring $|c_{\ell}|^2$) as a function of time, with experimental data of (a) (blue points) and fitting numerical simulation of (b) (solid red line). The known parameter was the initial lattice phase translation $\phi_0 = 30^{\circ}$ and we measured $s_0 = 8.06 \pm 0.10$.</figcaption></figure>

the plane wave populations. It consists in an adiabatic unloading of the lattice, during which a state $|\psi_q\rangle$ has the norm of its Bloch coefficients $|\langle\phi_{q,n}|\psi_q\rangle|$ conserved (with $|\phi_{q,n}\rangle$ the $n^{th}$ Bloch eigenstate at quasi-momentum $q$). For degenerate Bose gases resolved in quasi-momentum, this methods allows to access the probability distribution of a state over the basis of the Bloch waves. In $q/k_L = 0$ and $q/k_L = 0.5$ however$^{18}$, the closings of the gaps lead to eigenstate degeneracies that mix the information for two consecutive bands (with the exception of the ground state at $q/k_L = 0$). In the end, an atom initially in the $n^{th}$ Bloch band at quasi-momentum $q$ ends up in the plane wave(s) of reduced momentum

$$p/\hbar k_L = \begin{cases} \frac{q}{k_L} + \operatorname{sgn}(q) (-1)^n \lfloor \frac{n+1}{2} \rfloor, & q \notin \{0, 0.5\} \\ \pm \left( \frac{q}{k_L} + (-1)^n \lfloor \frac{n+1}{2} \rfloor \right), & q \in \{0, 0.5\} \end{cases} \qquad (2.29)$$

where the integer part $\lfloor \cdot \rfloor$ and the factors $(-1)^n$ and $\operatorname{sgn}(q)$ stem from the unfolding of the band structure. We draw in Fig. 2.7 an example for three different states in $q/k_L = 0$ (blue), $q/k_L = 0.5$ (green) and $q/k_L \notin \{0, 0.5\}$ (yellow) before and after a fictitious band-mapping protocol.

This method is used in Chap. 5 where we study the emergence of unstable modes in the

<footnote>$^{18}$We recall that we express the quasi-momentum in the first Brillouin zone $q/k_L \in (-0.5, 0.5]$, see Sec. 1.2.1.</footnote>
<figure><img src="image_28.png" /><figcaption>**Figure 2.7** | Sketch of the band-mapping technique. (a) Band structure of a static one-dimensional optical lattice at depth $s_0 = 1$ (solid lines). (b) Equivalent, unfolded representation of the same band structure (solid and dotted lines). (c) Unfolded representation of the band structure of a lattice of depth $s_0 = 0$ (solid and dotted lines) i.e. relation of dispersion of the free particle (solid line). We draw three hypothetical states resolved in quasi-momentum: the blue and yellow states are superpositions of the first three Bloch states in $q/k_L = 0$ and $q/k_L = 0.25$ respectively, and the green state is a superposition of the first two Bloch states in $q = 0.5$. Between the three superpositions considered, only the yellow state in $q/k_L \notin \{0, 0.5\}$ has the norm of all its Bloch coefficients unambiguously accessible through band-mapping.</figcaption></figure>

Brillouin zone of the phase-modulated optical lattice. These instabilities lead to the population of states coexisting at different quasi-momenta, which we highlight through band-mapping.

## Conclusion

This brief presentation of the experimental setup concludes this first part of methods. Once again, I redirect an unfulfilled reader to the theses of my predecessors [93, 94, 115, 67, 68] where the experimental setup has been more detailed. The next parts of this manuscript present results obtained on the ultracold atom experiment described in this chapter and that will be apprehended in the theoretical and numerical framework discussed in Chap. 1.
<figure><img src="image_29.png" /></figure>
# Part II

## Control and transport of the BEC one-body wave function in the optical lattice
<figure><img src="image_30.png" /></figure>
# Chapter 3 - Quantum optimal control of matter waves in a one-dimensional optical lattice

<figure><img src="image_31.png" /><figcaption>Horizontal stack of 54 independent momentum distributions (see Sec. 3.2)</figcaption></figure>

## Contents

<table><tr><td>Introduction</td><td>63</td></tr><tr><td>3.1 Optimal-control formalism</td><td>65</td></tr><tr><td>3.1.1 Introduction to optimal control in the classical case</td><td>65</td></tr><tr><td>3.1.2 Application to one-dimensional Bloch systems</td><td>67</td></tr><tr><td>3.1.3 Numerical method for control field computation</td><td>70</td></tr><tr><td>3.1.4 Numerical results, local optima and control duration</td><td>73</td></tr><tr><td>3.2 Controlling momentum distributions</td><td>74</td></tr><tr><td>3.2.1 Fidelities between momentum distributions</td><td>74</td></tr><tr><td>3.2.2 Preparing single momentum components</td><td>75</td></tr><tr><td>3.2.3 Preparing arbitrary superpositions of momentum components</td><td>78</td></tr><tr><td>3.3 Wave function control</td><td>79</td></tr><tr><td>3.3.1 Control of the relative phases between two momentum components</td><td>79</td></tr><tr><td>3.3.2 Preparation of Bloch eigenstates and superpositions thereof</td><td>80</td></tr><tr><td>3.3.3 Quantum state reconstruction by likelihood maximization</td><td>82</td></tr><tr><td>3.3.4 Non-squeezed Gaussian states</td><td>86</td></tr><tr><td>3.3.5 Squeezed Gaussian states</td><td>89</td></tr><tr><td>3.4 An application to quantum simulation</td><td>91</td></tr><tr><td>3.4.1 Introduction to dynamical tunneling</td><td>92</td></tr><tr><td>3.4.2 Experimental results</td><td>94</td></tr><tr><td>Conclusion</td><td>95</td></tr></table>
<figure><img src="image_32.png" /></figure>
Introduction

In the first part of this manuscript, we recalled basics of quantum mechanics to study wave
function dynamics in a one-dimensional Floquet-Bloch system (Chap. 1), and we presented our
experimental setup, in which we place BECs in a controllable one-dimensional optical lattice
(Chap. 2) to implement this system in the laboratory. This setup is simple and therefore
a powerful system on which to develop methods of quantum state manipulation, notably for
quantum simulation, as we will show in this chapter. To perform the simulation of a quantum
system [26, 27], one has to engineer the Hamiltonian to be studied, prepare the system in the
desired initial state and in due time measure the state of system. In this chapter, we present
research work that we undertook in 2021, which is part of the development and implementation
of techniques to face the last two of the above-mentioned requirements: initial state preparation
and final state measurement.

**Quantum state control.** The preparation of a desired quantum state generally calls for manipulating a quantum dynamical system by means of external time-dependent parameters. In practice, it amounts to steering that system from a state easily provided by an experimental setup to the desired one with high fidelity, and in a reasonable, or even minimum time, while satisfying experimental constraints and limitations. Several approaches have been developed for quantum state control, allowing to reach states unachievable by adiabatical means. We can cite the use of composite pulses [116, 117], shortcuts to adiabaticity [57] or machine-learning schemes [58]. Another approach, which we use in this chapter, is the application of optimal-control theory to control quantum systems.

Optimal-control theory, of which we introduce the formalism in Sec. 3.1, arguably goes
back to the Brachistochrone problem, submitted by J. Bernoulli in 1696 to the mathematicians
of his time [118]. Considering a system that can, more generally, be controlled with external
parameters, optimal-control theory is a mathematical framework with which one can compute
the optimal variation of control parameters in order for the system to follow optimal trajectories
that minimize some cost functionnal. The evolution of this field is intertwined with the genesis
of analytical mechanics in the 18th century, before the crowning contributions of L. Pontryagin [119] in the 1950s and later R. Bellman [120]. Applications of optimal control could hardly be
more diverse, with examples in aerospace with satellite maneuvering [121, 122], communication
protocols in computer science [123], finance [124] and biology [125].

Quantum optimal control (QOC) is the application of optimal-control theory to quantum systems [126], with proposals and implementations e.g. in the fields of quantum chemistry, quantum information, magnetic resonance and NV-centers, (see [59, 60] and references therein). QOC is well suited for quantum state control of cold and ultracold atoms for interferometry [127, 128, 129] and quantum simulation [130, 131]. We present in this chapter the implementation of QOC in our experiment, where we use the time variation of the phase of the optical lattice (Sec. 2.4) as our single control parameter in order to control the motional state $|\psi_{q,\ell}\rangle = \sum_{\ell} c_{q,\ell} |\chi_{\ell+q/k_L}\rangle$ of BECs in the lattice. We prepare firstly momentum probability distributions (targeting sets of real elements $\{|c_{q,\ell}|^2\}$, i.e. regardless the relative phases between
coefficients $c_{q,l}$), before fully addressing specific quantum states in the Hilbert space of our Bloch system (targeting sets of complex amplitudes $\{c_{q,l}\}$).

**Observation of the system and certification of the preparation.** When performing measurement on a quantum system, one never accesses the wave function of the system, but rather a possible outcome of the measurement of an observable with a probability distribution given by the squared modulus of the wave function. This is well illustrated by our TOF measurement, through which we access the momentum distribution of the BEC in the lattice, i.e. the set $\{|c_{q,l}|^2\}$ of the squared moduli of the plane wave amplitude (see Sec. 2.4.1). Such a measurement, based on a single observation of the system, is sufficient to evaluate a fidelity to a target defined purely as a momentum probability distribution, but cannot ensure proper quantum state preparation. Indeed, to certify our ability to prepare a given quantum state $|\psi_{q,l}\rangle$ associated to a unique set $\{c_{q,l}\}$ of complex coefficients, we need to access all the information of these measured coefficients $c_{q,l}$, that is both their norm $|c_{q,l}|$ and phase $\arg\{c_{q,l}\}$. Full quantum state tomography [132] is a matter of great relevance to quantum simulation and computation. Several methods to solve that problem have been put forward, using e.g. mappings from the motional state to internal degrees of freedom [133, 134, 135], while recent proposals in the context of many-body systems exploit randomized measurements [136] or neural networks [137]. In our case, we reconstruct the motional state of the BECs in the lattice with a maximum likelihood iterative method inspired by quantum optics [138, 139, 140, 141], using measurements of the subsequent evolution of the prepared state in the static lattice potential.

This chapter is organized as follow: in the first section 3.1, we give an introduction to the basics of optimal-control theory, we detail the application of QOC to our one-dimensional Bloch system, and we present our numerical approach to compute the lattice phase curve for the preparation of a given target. The second section features experimental results on the preparation of readily-observable momentum distributions (Sec. 3.2). In the third section, we target specific quantum states and detail the state reconstruction algorithm (Sec. 3.3). To conclude, we present in the Sec. 3.4 an application of our QOC implementation to the quantum simulation of dynamical tunneling [142, 89, 61] in the depth-modulated optical lattice. Some of the developments in Sec. 3.1 are inspired from [68, 143].

The research work presented in this chapter results from a collaboration with Prof. Dominique Sugny from the Laboratoire Interdisciplinaire de Carnot Bourgogne (Dijon), who specializes in the application of optimal control to quantum systems. It has lead to the publications:

[63] N. Dupont, G. Chatelain, L. Gabardos, M. Arnal, J. Billy, B. Peaudecerf, D. Sugny and D. Guéry-Odelin, *Quantum State Control of a Bose-Einstein Condensate in an Optical Lattice*, PRX Quantum, 2:040303, (2021)

[64] N. Dupont, F. Arrouas, L. Gabardos, N. Ombredane, J. Billy, B. Peaudecerf, D. Sugny and D. Guéry-Odelin. *Phase-space distributions of Bose-Einstein condensates in an optical lattice: Optimal shaping and reconstruction*, currently under review (2022)
# 3.1 Optimal-control formalism

## 3.1.1 Introduction to optimal control in the classical case

Optimal-control theory deals with systems described by parameterized equations of evolution:

$$ \dot{x}(t) = f(x(t), u(t), t), \qquad (3.1) $$

with the initial condition $x(0) = x_0$ and a given scalar cost functional $J[x(t), u(t), t]$ (see below). In Eq. (3.1),

* $x(t) \in \mathbb{R}^n$ is the state vector of the system (for a classical system, it can for example be phase space coordinates, with $n = 2N$ if there are $N$ degrees of freedom; Sec. 1.1),

* $f$ is a function encapsulating the evolution (e.g. the Hamilton equations for a dynamical system),

* $u(t) \in \mathbb{R}^m$ are $m$ time-dependent control parameters used to drive the initial state $x_0$ to a target state $x_T$ (with $m \neq n$ in general).

The optimal-control problem is to determine the control $u(t)$ over the time interval $0 \le t \le t_c$ (with $t_c$ the control duration, a parameter let free for now) that drives $x_0$ into $x(t_c) \simeq x_T$ while minimizing the cost $J$. This cost can generally be written

$$ J[x(t), u(t), t] = \phi(x(t_c), t_c) + \int_0^{t_c} f_0(x(t), u(t), t) \, dt \qquad (3.2) $$

where the first term accounts for the cost in reaching the target, and in the time required, and the second term is a cost functional dependent on the path $x(t)$ and resource $u(t)$ to reach the final state (it can for example be a resource cost of the form $\int_0^{t_c} |u(t)|^2 dt$).

**Pontryagin maximum principle.** To solve such optimal-control problems, L. Pontryagin introduced in 1956 what is now refered to as the Pontryagin maximum principle (PMP). We here state this principle without demonstration (for more information, see [119, 144, 143]). We start by introducing a scalar, Hamiltonian-like function $H_P$, named Pontryagin Hamiltonian, that can be written as

$$ H_P(x(t), p(t), u(t), t) = -f_0(x(t), u(t), t) + p(t) \cdot f(x(t), u(t), t), \qquad (3.3) $$
where the adjoint momentum-like<fnref n="1" /> vector $\mathbf{p}(t) \in \mathbb{R}^n$ is nothing but a time-dependent Lagrange multiplier. This adjoint state has its evolution governed by Hamilton equations deduced from the Pontryagin Hamiltonian:

$$ \dot{\mathbf{x}} = \frac{\partial H_P}{\partial \mathbf{p}} \quad \text{and} \quad \dot{\mathbf{p}} = -\frac{\partial H_P}{\partial \mathbf{x}}. \qquad (3.4) $$

The PMP states that optimal controls<fnref n="2" /> $\mathbf{u}^*(t)$, that minimize the cost, maximize $H_P$ along the trajectory:

$$ H_P(\mathbf{x}(t), \mathbf{p}(t), \mathbf{u}^*(t), t) = \max_{\mathbf{u}(t)} \{H_P(\mathbf{x}(t), \mathbf{p}(t), \mathbf{u}(t), t)\} \quad \text{for } 0 \le t \le t_c, \qquad (3.5) $$

or, equivalently

$$ \left. \frac{\partial H_P}{\partial \mathbf{u}} \right|_{\mathbf{u}=\mathbf{u}^*} = 0 \quad \text{for} \quad 0 \le t \le t_c. \qquad (3.6) $$

Solving an optimal-control problem amounts to solving Eq. (3.6) for $\mathbf{u}^*(t)$. However, we so far do not have enough information to evaluate $H_P$ and $\mathbf{p}(t)$. One tackles a given problem depending on the constraints that enter the definition of the cost function (3.2) [143, 68]. This leads to the definition of a series of cases:

(i) If neither $\mathbf{x}(t_c)$ nor $t_c$ are fixed, we set

$$ \mathbf{p}(t_c) = - \left. \frac{\partial \phi(\mathbf{x}(t), t)}{\partial \mathbf{x}} \right|_{t=t_c} \quad \text{and} \quad \left. \frac{\partial \phi(\mathbf{x}(t), t)}{\partial t} \right|_{t=t_c} = H_P(\mathbf{x}(t_c), \mathbf{p}(t_c), \mathbf{u}(t_c), t_c). \qquad (3.7) $$

(ii) If $\mathbf{x}(t_c)$ is fixed and $t_c$ is free, we set

$$ \left. \frac{\partial \phi(\mathbf{x}(t), t)}{\partial t} \right|_{t=t_c} = H_P(\mathbf{x}(t_c), \mathbf{p}(t_c), \mathbf{u}(t_c), t_c). \qquad (3.8) $$

(iii) If, on the contrary, $t_c$ is fixed and $\mathbf{x}(t_c)$ is free, we set

$$ \mathbf{p}(t_c) = - \left. \frac{\partial \phi(\mathbf{x}(t), t)}{\partial \mathbf{x}} \right|_{t=t_c} \qquad (3.9) $$

(iv) Finally if both $\mathbf{x}(t_c)$ and $t_c$ are fixed, Eqs. (3.3) and (3.4) contain enough information to solve for $\mathbf{u}^*(t)$ in Eq. (3.6).

<footnote n="1">¹We stress that $\mathbf{p}$ is a mathematical construction, adjoint vector to the state vector $\mathbf{x}$. In the classical dynamics example taken above, we have $\mathbf{x} = (q_1, ..., q_N, p_1, ..., p_N)^T$, comprising itself the true generalized momenta $p_i$ (Sec. 1.1).</footnote>

<footnote n="2">²We consider the case where there are no constraints on $\mathbf{u}(t)$.</footnote>
**Our constraints.** In our case, we will only require for the final state $\boldsymbol{x}(t_c)$ to reach $\boldsymbol{x}_T$ the most accurately possible in a fixed amount of time $t_c$ and regardless of the path taken. This puts us in the third case detailed above, with a cost function that becomes

$$J[\boldsymbol{x}(t), \boldsymbol{u}(t), t] \rightarrow J(\boldsymbol{x}(t_c)) = \phi(\boldsymbol{x}(t_c)) = -\mathcal{F}(\boldsymbol{x}_T, \boldsymbol{x}(t_c)), \quad (3.10)$$

where $\mathcal{F}(\boldsymbol{x}_T, \boldsymbol{x}(t_c))$ is a fidelity to the target to be maximized. Thus, we need to determine $\boldsymbol{u}^*(t)$ from the system composed of Eqs. (3.4) and (3.6) with the boundary conditions $\boldsymbol{x}(0) = \boldsymbol{x}_0$ (defined as our initial state) and $\boldsymbol{p}(t_c)$ (Eq. (3.9)). However, this task is generally not easy. We therefore proceed using an iterative algorithm. One can show that the transformation $\boldsymbol{u}(t) \rightarrow \boldsymbol{u}(t) + \delta\boldsymbol{u}(t)$ with $\delta\boldsymbol{u}(t) = \varepsilon \partial H_P / \partial \boldsymbol{u}$ (with $\varepsilon$ a small positive parameter) increases the fidelity [143], with $\boldsymbol{u}^*(t)$ a fixed point of the transformation (according to Eq. (3.6)). In our implementation, we use a numerical iterative gradient-ascent method based on this property. The procedure is detailed in Sec. (3.1.3).

### 3.1.2 Application to one-dimensional Bloch systems

In order to apply the previous discussion to the control of BECs in the optical lattice, we need to get expressions for the state vector, adjoint vector (with its final condition) and $\partial H_P / \partial \boldsymbol{u}$ in our system. In the experiment, the atoms experience the controllable lattice potential

$$V(x,t) = -A(t)s_0 \frac{E_L}{2} \cos(k_L x + \varphi(t)). \quad (3.11)$$

In Sec. 1.2, we recalled Bloch's theorem in one dimension, from which it follows that a quantum state at a given quasi-momentum $q$ in a one-dimensional potential of period $d$ is uniquely expanded over the basis of the plane waves as

$$|\psi_q(t)\rangle = \sum_{\ell \in \mathbb{Z}} c_{q,\ell}(t) |\chi_{\ell+q/k_L}\rangle, \quad (3.12)$$

with $\langle x | \chi_{\ell+q/k_L} \rangle = e^{i(\ell k_L + q)x} / \sqrt{d}$ and $k_L = 2\pi/d$. We state the uniqueness of this expansion with the isomorphism

$$|\psi_q(t)\rangle \leftrightarrow C_q(t) = \begin{pmatrix} \vdots \\ c_{q,\ell}(t) \\ \vdots \end{pmatrix}. \quad (3.13)$$

In the following, we represent an arbitrary state $|\psi_q(t)\rangle$ using its associated coefficient vector $C_q(t)$, and we drop the explicit dependence on $q$. In vector form, the time-dependent Schrödinger equation reads
$$i\dot{\mathcal{C}}(t) = M(u(t))\mathcal{C}(t), \qquad (3.14)$$

with $M(u(t)) = H(u(t))/E_L$ (see Eqs. (1.22) and (1.32)) and $u(t)$ is a parameter of the controlled lattice potential (3.11). In the experiments presented in Secs. 3.2, 3.3 and 3.4, the control parameter $u(t)$ is the lattice phase $\varphi(t)$, and the control is performed in a lattice of constant depth ($A(t) = 1$). In what follows, we keep a general narrative with the parameter $u(t)$ as long as our development allows it.

The formalism of optimal-control theory is introduced for real-valued state vectors. We connect our quantum state $\mathcal{C}(t)$ to the real vector $\mathbf{X}(t)$ defined as

$$\mathbf{X}(t) = \begin{pmatrix} \mathrm{Re}\{\mathcal{C}(t)\} \\ \mathrm{Im}\{\mathcal{C}(t)\} \end{pmatrix} = \begin{pmatrix} \vdots \\ \mathrm{Re}\left\{ \begin{array}{c} \vdots \\ c_{\ell}(t) \end{array} \right\} \\ \vdots \\ \mathrm{Im}\left\{ \begin{array}{c} \vdots \\ c_{\ell}(t) \end{array} \right\} \end{pmatrix}. \qquad (3.15)$$

We get a real-valued vector $\mathbf{X}(t) \in \mathbb{R}^{2d_H}$, with $d_H$ the Hilbert space dimension (see Sec. 1.2.2). The evolution equation for $\mathbf{X}(t)$ in the form of (3.1) is obtained from the Schrödinger equation (3.14):

$$\dot{\mathbf{X}}(t) = \begin{pmatrix} \mathrm{Im}\{M(u(t))\} & \mathrm{Re}\{M(u(t))\} \\ -\mathrm{Re}\{M(u(t))\} & \mathrm{Im}\{M(u(t))\} \end{pmatrix} \mathbf{X}(t) = \mathbf{f}(\mathbf{X}(t), u(t)). \qquad (3.16)$$

Following optimal-control formalism, we define the Pontryagin Hamiltonian for this problem:

$$H_P = \mathbf{P}(t) \cdot \mathbf{f}(\mathbf{X}(t), u(t)) = \mathbf{P}^T(t) \times \mathbf{f}(\mathbf{X}(t), u(t)), \qquad (3.17)$$

where we have denoted the matrix product with $\times$. In a similar way as the relation (3.15) between $\mathbf{X}(t)$ and $\mathcal{C}(t)$, we associate a complex-valued vector $\mathbf{D}(t)$ to the adjoint state $\mathbf{P}(t)$:

$$\mathbf{P}(t) = \begin{pmatrix} \mathrm{Re}\{\mathbf{D}(t)\} \\ \mathrm{Im}\{\mathbf{D}(t)\} \end{pmatrix}. \qquad (3.18)$$

From Eqs (3.17) and (3.18), the Pontryagin Hamiltonian for the one-dimensional Bloch system becomes
$$
\begin{align}
H_P &= \operatorname{Re}\{\mathbf{D}\} \operatorname{Re}\{\dot{\mathbf{C}}\} + \operatorname{Im}\{\mathbf{D}\} \operatorname{Im}\{\dot{\mathbf{C}}\} \tag{3.19} \\
&= \operatorname{Re}\{(\operatorname{Re}\{\mathbf{D}^\mathrm{T}\} - i \operatorname{Im}\{\mathbf{D}^\mathrm{T}\}) \times (\operatorname{Re}\{\dot{\mathbf{C}}\} + i \operatorname{Im}\{\dot{\mathbf{C}}\})\} \nonumber \\
&= \operatorname{Re}\{\mathbf{D}^\dagger \times \dot{\mathbf{C}}\}. \nonumber
\end{align}
$$

where the time dependence is implicit. Using Eq. (3.14), we finally get for $H_P$:

$$
\begin{align}
H_P &= \operatorname{Im}\{\mathbf{D}^{\dagger}(t) M(u(t)) \mathbf{C}(t)\} \nonumber \\
    &\Rightarrow \frac{\partial H_P}{\partial u} = \operatorname{Im}\{\mathbf{D}^{\dagger}(t) \frac{\partial M(u(t))}{\partial u} \mathbf{C}(t)\} \tag{3.20}
\end{align}
$$

Finally, the boundary condition $\mathbf{D}(t_c)$ is obtained from the evolution of $\mathbf{P}(t)$ given by the Hamilton equations applied to $H_P$. Using Eqs. (3.16) and (3.17):

$$
\begin{align}
\dot{\mathbf{P}} = - \left( \frac{\partial H_P}{\partial \mathbf{X}} \right)^\mathrm{T} &= \left[ \mathbf{P}^\mathrm{T} \begin{pmatrix} -\operatorname{Im}\{M(u(t))\} & -\operatorname{Re}\{M(u(t))\} \\ \operatorname{Re}\{M(u(t))\} & -\operatorname{Im}\{M(u(t))\} \end{pmatrix} \right]^\mathrm{T} \tag{3.21} \\
&= \begin{pmatrix} \operatorname{Im}\{M(u(t))\} & \operatorname{Re}\{M(u(t))\} \\ -\operatorname{Re}\{M(u(t))\} & \operatorname{Im}\{M(u(t))\} \end{pmatrix} \mathbf{P} \nonumber
\end{align}
$$

where we have used the fact that $M(u(t)) = H(u(t))/E_L$ is Hermitian $^3$. We get that $\mathbf{D}(t)$ evolves as $\mathbf{C}(t)$:

$$i \dot{\mathbf{D}}(t) = M(u(t)) \mathbf{D}(t), \qquad (3.22)$$

with a final condition obtained from Eqs. (3.9) and (3.10):

$$
\begin{align}
\mathbf{P}^{\mathrm{T}}(t_c) &= + \left. \frac{\partial \mathcal{F}}{\partial \mathbf{X}} \right|_{t=t_c} \nonumber \\
\Rightarrow \mathbf{D}(t_c) &= \left. \frac{\partial \mathcal{F}}{\partial \mathbf{C}^{\mathrm{T}}} \right|_{t=t_c}, \tag{3.23}
\end{align}
$$

The adjoint state $\mathbf{D}(t)$ is evaluated for all time $0 \le t \le t_c$ by propagating its final state (3.23) backward in time using Eq. (3.22). We also need to define the fidelity function $\mathcal{F}$ to the target

<footnote>$^3$Which implies for our development $\operatorname{Re}\{M^\mathrm{T}\} = \operatorname{Re}\{M\}$ and $\operatorname{Im}\{M^\mathrm{T}\} = -\operatorname{Im}\{M\}$.</footnote>
$|\psi_T\rangle \leftrightarrow C_T$. This choice depends on the states targeted (see below). For most of our targets, we will use the quantum fidelity between pure states

$$ \mathcal{F}(|\psi_T\rangle, |\psi(t)\rangle) = |\langle\psi_T|\psi(t)\rangle|^2 = |C_T^\dagger \times C(t)|^2. \qquad (3.24) $$

With the expressions for the state $C(t)$ and its adjoint $D(t)$, their boundary condition (a free choice for $C_0$ and Eq. 3.23 for $D(t_c)$), the knowledge that they both evolve according to Schrödinger equation (Eqs. (3.14) and (3.22)) and the expression for $\partial H_P/\partial u$ along the trajectory (Eq. (3.20)), we have all the ingredients to iteratively compute control fields.

### 3.1.3 Numerical method for control field computation

As discussed in the previous section, our control parameter in the studies presented thereafter is the lattice phase $\varphi(t)$. We now explicit this choice with $u(t) \to \varphi(t)$. For numerical computation, we discretize the time vector $t \to t_n = t_0, t_1, \dots, t_N$ during the state preparation, with $t_0 = 0$ and $t_N = N\Delta t = t_c$ (with $\Delta t$ a small time interval; see below). The lattice phase $\varphi(t)$ is defined constant by part over $\Delta t$ (see Fig. 3.1 below), and we write $\varphi(t_n) = \varphi_n$. The evolution operator from $t_{n-1}$ to $t_n$ is

$$ U(t_n, t_{n-1}) = U_n = \exp\left\{\frac{-i\Delta t H_n}{\hbar}\right\} = \exp\{-i2\pi\nu_L\Delta t M_n\}, \qquad (3.25) $$

where we have used $H(t_n) = H_n = E_L M_n = h\nu_L M_n$. We want to drive an initial state $C(0)$ to a target $C_T$. We start with the initialization of the control field $\varphi_n^{(0)}$ (an initial guess), which prepares a certain final state

$$ C^{(0)}(t_c) = U_N^{(0)} U_{N-1}^{(0)} \dots U_2^{(0)} U_1^{(0)} C(0), \qquad (3.26) $$

that has a fidelity $F^{(0)}$ to the target. If we vary of the control field $\varphi_n \to \varphi_n + \delta\varphi_n$, the variation of fidelity is

$$ \delta F = \sum_{n=1}^{N} \frac{\partial F}{\partial \varphi_n} \delta \varphi_n \qquad (3.27) $$

The heart of the gradient ascent method [145, 146, 147] that we use is to choose $\delta\varphi_n$ so that $\delta F \ge 0$ in order for $F$ to increase. We see that if we modify the control field with the corrections

$$ \delta\varphi_n = \varepsilon \frac{\partial F}{\partial \varphi_n}, \qquad (3.28) $$
with $\epsilon > 0$, we make sure to have $\delta\mathcal{F} \ge 0$ since the right-hand-side of Eq. (3.27) becomes $\sum_n \epsilon(\partial\mathcal{F}/\partial\varphi_n)^2$ where each term is greater than or equal to zero. As we will see below, the corrections (3.28) correspond to the general control field corrections $\delta\mathbf{u}(t) = \epsilon\partial H_P/\partial\mathbf{u}$ discussed at the end of Sec. 3.1.1.

We now show how one can compute $\delta\mathcal{F}/\delta\varphi_n$. As the fidelity only depends on the final state reached $\mathcal{C}(t_c)$ (for a fixed target), the dependence of $\mathcal{F}$ on the control field $\varphi_n$ is that of $\mathcal{C}(t_c)$. The resulting complex derivative reads<fnref n="4" />

$$ \begin{aligned} \frac{\partial \mathcal{F}}{\partial \varphi_n} &= \frac{\partial \mathcal{C}^{\dagger}(t_c)}{\partial \varphi_n} \frac{\partial \mathcal{F}}{\partial \mathcal{C}^{\dagger}(t_c)} + \frac{\partial \mathcal{F}}{\partial \mathcal{C}(t_c)} \frac{\partial \mathcal{C}(t_c)}{\partial \varphi_n} \\ &= 2 \operatorname{Re} \left\{ \frac{\partial \mathcal{C}^{\dagger}(t_c)}{\partial \varphi_n} \frac{\partial \mathcal{F}}{\partial \mathcal{C}^{\dagger}(t_c)} \right\}. \end{aligned} \quad (3.29) $$

The first factor in the argument of the real-part function is

$$ \begin{aligned} \frac{\partial \mathcal{C}^{\dagger}(t_c)}{\partial \varphi_n} &= \mathcal{C}_0^{\dagger} U_1^{\dagger} \dots U_{n-1}^{\dagger} \frac{\partial U_n^{\dagger}}{\partial \varphi_n} U_{n+1}^{\dagger} \dots U_N^{\dagger} \\ &\approx i 2 \pi \nu_L \Delta t \mathcal{C}_0^{\dagger} U_1^{\dagger} \dots U_{n-1}^{\dagger} \frac{\partial M_n}{\partial \varphi_n} U_{n+1}^{\dagger} \dots U_N^{\dagger} \end{aligned} \quad (3.30) $$

where we expanded $U_n \approx 1_{d_H} - i2\pi\nu_L\Delta t M_n$ (arguing that $\Delta t$ is a small time interval) and used $M_n^\dagger = M_n$. As the second factor in the argument of the real-part function in Eq. (3.29), we recognize the boundary condition $\mathbf{D}(t_c) = \mathbf{D}_N$ (Eq. (3.23)). Equation (3.29) thus becomes<fnref n="5" />

$$ \begin{aligned} \frac{\partial \mathcal{F}}{\partial \varphi_n} &\approx 2 \operatorname{Re} \left\{ i 2 \pi \nu_L \Delta t \mathcal{C}_0^\dagger U_1^\dagger \dots U_{n-1}^\dagger \frac{\partial M_n}{\partial \varphi_n} U_{n+1}^\dagger \dots U_N^\dagger \mathbf{D}_N \right\} \\ &\approx -4 \pi \nu_L \Delta t \operatorname{Im} \left\{ \mathcal{C}_{n-1}^\dagger \frac{\partial M_n}{\partial \varphi_n} \mathbf{D}_n \right\} = 4 \pi \nu_L \Delta t \operatorname{Im} \left\{ \mathbf{D}_n^\dagger \frac{\partial M_n}{\partial \varphi_n} \mathcal{C}_{n-1} \right\}, \end{aligned} \quad (3.31) $$

where we have propagated $\mathcal{C}_0^\dagger$ forward up to $\mathcal{C}_{n-1}^\dagger$, and $\mathbf{D}_N$ backward up to $\mathbf{D}_n$. From Eqs. (3.27) and (3.28), we see that doing this operation for all $n$ such that $0 \le t_n \le t_c$ gives the corrections to be applied to each $\varphi_n$ to increase the fidelity.

In Eq. (3.31), we recognize a discrete version of the derivative of the Pontryagin Hamiltonian with respect to the control parameter (3.20). Our gradient ascent (3.28) is effectively performed by iteratively modifying the control field with the corrections

<footnote n="4">⁴with $z + z^* = 2 \operatorname{Re}\{z\}$; $z \in \mathbb{C}$.</footnote>

<footnote n="5">⁵with $\operatorname{Re}\{iz\} = -\operatorname{Im}\{z\}$ and $\operatorname{Im}\{z^*\} = -\operatorname{Im}\{z\}$.</footnote>
<figure><img src="image_33.png" /><figcaption>Figure 3.1 | Schematic of a gradient ascent iteration for the computation of a discretized control field showing the field $\varphi_n$ at time $t_n$ at the current step (gray line) and at the next step (black line), after applying the corrections $\delta\varphi_n$ (red arrows). Our discretization convention is such that the evolution operator $U(\varphi_n)$ transforms $C_{n-1}$ into $C_n$.</figcaption></figure>

$$ \delta\varphi_n = \varepsilon 4\pi\nu_L\Delta t \operatorname{Im}\left\{D_n^\dagger \frac{\partial M_n}{\partial \varphi_n} C_{n-1}\right\} \approx \varepsilon' \frac{\partial H_P}{\partial \varphi_n} \qquad (3.32) $$

as prescribed by optimal-control theory through the PMP (Sec. 3.1.1). The small positive parameter $\varepsilon$ is initialized to 1 and then slightly adjusted following a line-search method [147]. Our numerical approach is summarized in a list of concrete steps in the following box, and an iteration of this algorithm is schematically depicted in Fig. 3.1.

## Iterative gradient-ascent algorithm

0. Set a initial state $C_0$, a target $C_T$, a fidelity function and a control duration$^a$ $t_c$ (we state typical values for this parameter in the next section 3.1.4). Set also a break condition on the fidelity or the number of iterations.

1. Initialize a guess field $\varphi_n^{(0)}$.

2. Propagate $C_0$ forward to obtain $C_n$ for all $n$ using the time dependent Schrödinger equation (Eqs. (3.26) and (3.25)).

3. Evaluate the adjoint state at final time $D_N$ using Eq. (3.23) and propagate it backward up to $t = 0$ using Schrödinger equation to obtain $D_n$ for all $n$.

4. Compute the corrections $\delta\varphi_n$ for all $n$ using Eq. (3.32) and the integration of $C_n$ and $D_n$. Transform $\varphi_n \rightarrow \varphi_n + \delta\varphi_n$.

5. Repeat from step 2 (with a possible adaptation of the correction amplitude $\varepsilon$) until reaching the desired fidelity or the maximum number of iterations.

<footnote>$^a$Which needs to be discretized into a number $N$ of sufficiently small time steps $\Delta t$ in order for the approximation done in Eq. (3.30) to be valid, as well as for the time-discretized fidelity gradient (3.31) to approximate the gradient of the Pontryagin Hamiltonian (3.20) in the continuous-time limit.</footnote>
### 3.1.4 Numerical results, local optima and control duration

We now present numerical results of control field computation using the algorithm presented in the previous section. As an example, we target, in the subspace of quasi-momentum $q = 0$, the state

$$|\psi_T\rangle = \frac{1}{\sqrt{2}}(|\chi_{-3}\rangle + |\chi_3\rangle) \qquad (3.33)$$

i.e. the superposition of plane waves with momenta $(-3, 3) \hbar k_L$ and a relative phase $\arg\{c_{-3}/c_3\} = 0$. Our initial state for the control is the ground state of the lattice at depth $s_0 = 5$ (a parameter that is fixed during the QOC preparation). We use the quantum fidelity (3.24), with a break condition of $\mathcal{F} > 0.995$ for the algorithm. Here, and in the following of this manuscript, we define $|\psi_{QOC}\rangle$ as the theoretical state obtained from the numerical propagation of the initial state with the optimized control field. Results are shown in Fig. 3.2 for two different resulting control fields for the preparation of the state (3.33). The control fields have been obtained from two different initial guesses, from which, in both cases, the gradient ascent algorithm has converged to an optimal control (which means that we effectively get $\mathcal{F} > 0.995$). This illustrates the general fact in optical-control theory that if there exist multiple local optima for the control parameter with respect to the cost function, a gradient-based approach may only converge towards one of these optima depending on the initialization of the control field [126, 143]. In this case, both optimal control fields $\varphi(t)$ give a satisfactory result, which mean that the local fidelity maxima near our two initializations were such that $\mathcal{F} \approx 1$.

**Remark on the control duration.** In our numerical approach of this optical-control problem, we fix the control duration $t_c$. This quantity has to be large enough for the target to become reachable, and preferably from a vast region of control field initialization (see previous paragraph). In our system, the evolution of the atoms in the lattice is governed by the band structure (Sec. 1.2.2). As our initial state is the ground state of the optical lattice, we define as the relevant timescale the period $T_0$ associated to the transition between the two lowest bands $s$ and $p$ at the center of the Brillouin zone ($q = 0$):

$$T_0(s_0) = \frac{h}{E_{0,1}^{(s_0)} - E_{0,0}^{(s_0)}}, \qquad (3.34)$$

with $E_{q,n}^{(s_0)}$ the $n^{th}$ Bloch eigenenergy at quasi-momentum $q$ and lattice depth $s_0$. For $t_c \ll T_0$, not enough time is given for the atomic state to eventually evolve toward the target. For $t_c \gg T_0$, it is easier for the numerical algorithm to converge. The problem then becomes an experimental one, as experimental fluctuations have more time to accumulate and limit the fidelity of the preparation. We present a brief study of the control duration effect on experimental preparations in App. D. In our experiments, we typically use control durations of a few $T_0$ (mainly 1.5 to
<figure><img src="image_34.png" /><figcaption>**Figure 3.2** | Non-unicity of the control fields for the preparation of a given target. ($\mathbf{a}_1, \mathbf{b}_1$) Examples of lattice phase evolutions as a function of time optimized by optimal-control theory (solid black line) for the preparation of $|\psi_T\rangle$ (3.33), starting from two different initial guesses (dashed black line). ($\mathbf{a}_2, \mathbf{b}_2$) Corresponding numerical evolutions of momentum distribution. ($\mathbf{a}_3, \mathbf{b}_3$) Comparisons between the momentum distribution of $|\phi_{QOC}\rangle$ (red bars) and the target (grey bars). Resulting numerical quantum fidelities (Eq. (3.24)) are $F \gtrsim 0.996$ for (a,b). The lattice depth is $s_0 = 5$ and time is given in units of $T_0(s_0 = 5) \approx 62$ µs (see text).</figcaption></figure>

$2T_0$). Most of the results presented in this chapter are obtained with lattice depths around $s_0 = 5$, for which $T_0 \approx 62$ µs.

## 3.2 Controlling momentum distributions

We now turn to experimental results. In first sets of experiments, we control the momentum distribution $|\langle p|\psi\rangle|^2$ of BECs in the optical lattice. We start by introducing two new fidelity definitions, which are less restrictive than (3.24) given our targets.

### 3.2.1 Fidelities between momentum distributions

Our goal in this first section of results is to target given momentum distributions $\{|c_\ell|^2\}$ at quasi-momentum $q = 0$. As our targets only depend on the norms $\{|c_\ell|\}$, we note that it is unnecessarily restrictive to require the preparation of a specific state whose $c_\ell$ coefficients have imposed relative phases. Indeed, a given momentum distributions $\{|c_\ell|^2\}$ is achieved if we
drive our initial state to the Hilbert subspace composed of the infinitely large number of states whose plane wave coefficient norms are {|c<sub>ℓ</sub>|}. We define two fidelities, less restrictive than the quantum fidelity (3.24), that allow to compute control fields towards this subspace:

$$ \mathcal{F}^{(\alpha)} (\{|c_{\ell}|\}, \{|d_{\ell}|\}) = 1 - \frac{1}{2} \sum_{\ell \in \mathbb{Z}} (|c_{\ell}| - |d_{\ell}|)^2 \quad (3.35) $$

$$ \mathcal{F}^{(\beta)} (\{|c_{\ell}|\}, \{|d_{\ell}|\}) = \left( \sum_{\ell \in \mathbb{Z}} |c_{\ell}| |d_{\ell}| \right)^2, \quad (3.36) $$

with c<sub>ℓ</sub>, d<sub>ℓ</sub> the plane wave coefficients of the two states between which the fidelities are computed, and 0 ≤ F<sup>(α,β)</sup> ≤ 1 for normalized states ∑<sub>ℓ</sub> |c<sub>ℓ</sub>|<sup>2</sup> = ∑<sub>ℓ</sub> |d<sub>ℓ</sub>|<sup>2</sup> = 1. This duplicity of definition for the fidelity between momentum distributions is fortuitous: for the results presented in the following sections 3.2.2 and 3.2.3, control fields have been computed with the first definition F<sup>(α)</sup> (3.35). For the treatment of the experimental data, we however use the second definition F<sup>(β)</sup> (3.36), as it corresponds to the quantum fidelity (3.24) in the specific case of complex phase matching between the two states coefficients of the two states<sup>6</sup>. We have verified that the results of momentum distribution presented in the following could have been obtained with either definition, whether for the control field calculation or data treatment.

<figure><img src="image_35.png" /></figure>

The choice for the fidelity function impacts steps 3 and 5 of our iterative gradient-ascent algorithm (p. 72), namely the final condition for the adjoint state and the break condition.

Experimental results of momentum distribution control are shown in the next two sections. In Sec. 3.2.2, we present preparations of single momentum components. In Sec. 3.2.3, we present equiprobable superpositions of two momentum components and an arbitrarily populated momentum state.

### 3.2.2 Preparing single momentum components

We first target momentum distributions given by {|c<sub>ℓ</sub>|<sup>2</sup>} = {δ<sub>m,ℓ</sub>}, with m the reduced momentum p/ħk<sub>L</sub> of the targeted plane waves. Results are shown in Fig. 3.3 for m = 0, 2, 4, 10 and quantitative details can be found in Table 3.1. The measured experimental fidelities are very close to the numerical ones, and both show a similar trend, with slowly decreasing values as we go from low momentum targets to high momentum targets (see Table 3.1). At the same time, the control phase gets more complex, as seen in Fig.3.3(d). To reach high single momentum states, such as p = 10 ħk<sub>L</sub> (Fig.3.3(d)) we increase the time of the control ramp to ensure the convergence of the algorithm. We note that we reach momenta much higher than the separatrix momentum p<sub>sep</sub> = √s<sub>0</sub> ħk<sub>L</sub> (Sec. 1.1.2), which corresponds to the classical limit with a sudden lattice shift [62].

<footnote n="6"><sup>6</sup>Given the coefficient norms of two states, F<sup>(β)</sup> (3.36) is therefore an upper bound of F (3.24).</footnote>
<figure><img src="image_36.png" /><figcaption>**Figure 3.3** | Controlling momentum distributions: single momentum components. Left: Control field as a function of time. Right: TOF absorption images, where the OD colormaps have been truncated to 0.5 of their maximum respective value (which saturates the center of the mainly populated orders) to reveal preparation defects. (a,b,c₁,d) QOC preparations. (c₂) Lattice acceleration (see text). See Table 3.1 for details.</figcaption></figure>

We stress that, as any image of atoms featured in this thesis, the ODs shown in panels (a,b,c₁,d) are TOF absorption images of BECs directly dropped from the optical lattice (in this case, once the propagation procedure has been applied). Strikingly, the usual momentum quantization induced by the lattice periodicity imprinted in the wave function is suppressed. While this is expected as we target single plane waves, these experiments interestingly amount to the realization of a blazed grating for matter waves (in direct analogy with optics), as the measured momentum distributions correspond to an *in situ* constant phase gradient $\Delta\phi(x) = mk_L x$ across the lattice sites (see for instance [90]).

Preparing the single plane wave with momentum $m \times \hbar k_L$ corresponds to accelerating all the atoms of the BEC up to that momentum. We can compare our QOC method to the more standard protocol of accelerating the lattice to impart momentum to the atoms. In Fig. 3.3(c₂), we illustrate this comparison in the case of $m = 4$, where we uniformly accelerate the lattice up to a velocity of $4h/md$ for the same amount of time $t_c$ and lattice depth $s_0$ used for the corresponding QOC experiment of Fig. 3.3(c₁). After the acceleration, the fidelity to the target state, i.e. the single momentum state $p = 4\hbar k_L$, is only $F_{c_2}^{(\beta)} = 0.49 \pm 0.03$, which is much lower than the one obtained using the QOC method ($F_{c_1}^{(\beta)} = 0.89 \pm 0.01$, see Table 3.1). This is clearly visible on the experimental absorption image of Fig. 3.3(c₂), with atomic population on the neighboring momentum orders. One could argue that the fidelity to the target state using the acceleration method can be increased by working adiabatically and performing Bloch oscillations [148, 149, 79]. Numerical simulations show however that reaching a fidelity $F^{(\beta)} > 90\%$ to a given single momentum state requires low lattice depths $s_0 < 1$ and that the adiabaticity condition [79] then
<table><caption>**Table 3.1 | Detail table related to the preparation of single momentum components.** This table refers to Fig. 3.3. $s_0$ is the lattice depth, $t_c/T_0$ is the control duration rescaled by the period $T_0$ (Eq. (3.34), see text) and $m$ is the momentum (rescaled by $\hbar k_L$) of the targeted plane wave. The fidelity $F^{(\beta)}$ is defined in Eq. (3.36). $F_{\text{num}}^{(\beta)}$ is evaluated between the target and $|\psi_{\text{QOC}}\rangle$. $F_{\text{exp}}^{(\beta)}$ is evaluated between the target and the state experimentally prepared by the application of optimal control fields (with an average over 10 realizations, and an uncertainty corresponding to one standard-deviation).</caption><thead><tr><th>Fig. 3.3</th><th>a</th><th>b</th><th>c</th><th>d</th></tr></thead><tbody><tr><td>s<sub>0</sub></td><td>5.1±0.2</td><td>5.2±0.2</td><td>5.2±0.2</td><td>5.1±0.2</td></tr><tr><td>t<sub>c</sub>/T<sub>0</sub></td><td>1.5</td><td>1.5</td><td>1.5</td><td>4.25</td></tr><tr><td>m</td><td>0</td><td>2</td><td>4</td><td>10</td></tr><tr><td>F<sub>num</sub><sup>(β)</sup></td><td>0.99</td><td>0.99</td><td>0.89</td><td>0.87</td></tr><tr><td>F<sub>exp</sub><sup>(β)</sup></td><td>0.94±0.01</td><td>0.94±0.01</td><td>0.89±0.01</td><td>0.74±0.05</td></tr></tbody></table>

leads to a generally much longer control time $t_c$ (see below). The control time will also grow linearly with the momentum of the targeted state, reaching $t_c = 1.7$ ms for the $p = 4 \hbar k_L$ state considered in our comparison (to be compared with $t_c = 1.5 T_0 \approx 92$ µs using QOC). Such long control times also mean that the atoms move much further away from the center of the trap (Sec. 2.2.4) during the lattice acceleration, with expected deleterious effects as this additional potential is not taken into account by our model. In Table 3.2, we summarize the results of the comparison between QOC and adiabatic acceleration for several momentum orders. We conclude that the QOC method is both a fast and accurate procedure to populate single momentum states with a high fidelity.

<table><caption>**Table 3.2 | QOC versus adiabatic lattice acceleration.** Comparison of the control time $t_c$ required to reach a given single momentum target ($p = m \times \hbar k_L$) with the same fidelity using either our QOC protocol or an adiabatical Bloch oscillation scheme with a uniformly accelerated lattice ("Acc"). $F_{\text{exp}}^{(\beta)}$ refers to the experiments Fig. 3.3(b,c,d) as well as preparation of the 8th plane wave not featured in Fig. 3.3. We indicate the lattice depth $s_0$ at which the experiments were performed (for the QOC case), or which would be required (for the accelerated case).</caption><thead><tr><th></th><th>m<br>F<sub>exp</sub><sup>(β)</sup></th><th>2</th><th>4</th><th>8</th><th>10</th></tr></thead><tbody><tr><td rowspan="2">QOC</td><td>s<sub>0</sub>(±0.2)</td><td>5.2</td><td>5.2</td><td>5.1</td><td>5.1</td></tr><tr><td>t<sub>c</sub> (µs)</td><td>91.7</td><td>91.7</td><td>260</td><td>260</td></tr><tr><td rowspan="2">Acc</td><td>s<sub>0</sub></td><td>0.75</td><td>1.1</td><td>2.1</td><td>2.3</td></tr><tr><td>t<sub>acc</sub> (µs)</td><td>1.7 · 10<sup>3</sup></td><td>1.7 · 10<sup>3</sup></td><td>1.2 · 10<sup>3</sup></td><td>1.4 · 10<sup>3</sup></td></tr></tbody></table>

The comparison between the control fields $\phi(t)$ in both methods (QOC and lattice acceleration) also sheds light on the way the optimal phase is designed: the artificially-made folded quadratic growth pattern in the control phase for the acceleration method (Fig. 3.3(c<sub>2</sub>)) can also be found in the QOC phases, for instance in Fig. 3.3(c<sub>1</sub>,d). In the case of panel (c<sub>1</sub>), the optimal control field $\phi(t)$ can be interpreted as a first acceleration stage towards the targeted momentum state (for $t/T_0 < 0.75$), and a second "correction" stage to reduce the population in unwanted momentum states.
<figure><img src="image_37.png" /><figcaption>**Figure 3.4** | Controlling momentum distributions: two-components equiprobable and arbitrary momentum superpositions. Left: Control field as a function of time. Right: TOF absorption images, where the OD colormaps have been truncated to 0.5 of their maximum respective value (which saturates the center of the mainly populated orders) to reveal preparation defects. See Table 3.3 for details.</figcaption></figure>

### 3.2.3 Preparing arbitrary superpositions of momentum components

In this last section of results concerning momentum distribution control, we shape the lattice phase $\varphi(t)$ to realize arbitrary momentum superpositions. Results are shown in Fig. 3.4, with details in Table 3.3. We begin with equiprobable superpositions of two momentum states, varying their relative momentum. We show the case of neighboring momentum states (Fig. 3.4(a)), opposite momentum states (Fig. 3.4(b)) and an arbitrary pair of momentum states (Fig. 3.4(c)). We also prepare an arbitrary momentum distribution corresponding to an “ascending staircase” distribution ($|c_{-2}|^2, |c_{-1}|^2, |c_0|^2, |c_1|^2, |c_2|^2) = (1, 2, 3, 4, 5)/15$ (Fig. 3.4(d)). In each case shown, we achieve good experimental fidelities to the ideal target (larger than 88%, see Table 3.3), slightly below the corresponding numerical fidelities.

To further demonstrate the robustness and versatility of our QOC method for the preparation of momentum distributions, we prepared all the $2^7 = 128$ equal-weight superpositions of momenta between $p = -3\hbar k_L$ and $p = 3\hbar k_L$ in the lattice of depth $s_0 \approx 5$. As an amusing way of presenting the results, we constructed an alphabet by concatenating the resulting TOF absorption images, allowing us to write words and sentences. An example of such a printout is shown in the beginning of this chapter (p. 61).

All experimental results of QOC featured in this chapter are obtained with the ground state of the lattice as the initial state for the control. The arbitrariness of this choice is illustrated in App. E, where we show an experiment of momentum distribution control with the resting BEC ($c_\ell = \delta_{0,\ell}$) as the initial state.

We recall that the targets in this section were defined regardless the relative phases between their $c_\ell$ coefficients. In the next section, we experimentally target specific quantum states, and
<table><caption>**Table 3.3 |** Detail table related to the preparation of two-component equiprobable and arbitrary momentum superpositions. This table refers to Fig. 3.4. $s_0$ is the lattice depth, $t_c/T_0$ is the control duration reduced by the period $T_0$ (Eq. (3.34), see text), $\{m\}$ is the set of momenta of the targeted components (reduced by $\hbar k_L$) and $\{|c_m|^2\}_T$ are the square moduli of the corresponding coefficients. The fidelity $F^{(\beta)}$ is defined in Eq. (3.36). $F_{\text{num}}^{(\beta)}$ is evaluated between the target $|\psi_{\text{QOC}}\rangle$. $F_{\text{exp}}^{(\beta)}$ is evaluated between the target and the state experimentally prepared by the application of optimal control fields (with an average over 10 realizations, and an uncertainty corresponding to one standard-deviation).</caption><thead><tr><th>Fig. 3.4</th><th>a</th><th>b</th><th>c</th><th>d</th></tr></thead><tbody><tr><td>s<sub>0</sub></td><td>4.6±0.2</td><td>5.0±0.1</td><td>4.6±0.2</td><td>5.1±0.1</td></tr><tr><td>t<sub>c</sub>/T<sub>0</sub></td><td>1.5</td><td>1.5</td><td>≈ 2</td><td>1.5</td></tr><tr><td>{m}</td><td>{-3, -2}</td><td>{-4, 4}</td><td>{-3, 2}</td><td>{-2, -1, 0, 1, 2}</td></tr><tr><td>|c<sub>m</sub>|<sup>2</sup>}</td><td></td><td>{0.5, 0.5}</td><td></td><td>{1, 2, 3, 4, 5}/15</td></tr><tr><td>F<sub>num</sub><sup>(β)</sup></td><td>0.95</td><td>0.92</td><td>0.98</td><td>0.99</td></tr><tr><td>F<sub>exp</sub><sup>(β)</sup></td><td>0.88±0.03</td><td>0.93±0.01</td><td>0.93±0.02</td><td>0.94±0.01</td></tr></tbody></table>

present how we manage to assess the prepared states.

## 3.3 Wave function control

In this second section of results, we tackle the preparation of specific quantum states $|\psi\rangle = \sum_l c_{q,l}|\chi_{l+q/k_L}\rangle$. To compute control fields, we therefore use the usual fidelity between pure quantum states $F = |\langle\psi|\phi\rangle|^2$. The difficulty now lies in the measurement of the prepared complex wave function in order to certify our preparation protocol. Our general procedure to deduce the prepared quantum states is to hold them in the static lattice after the preparation. We then use the data of their evolution in the optical lattice as information to identify the state. In the following, we first show that we can prepare and identify superpositions of two momentum components with an arbitrary relative phase (Sec.3.3.1) and specific eigenstates of the lattice potential as well as eigenstate superpositions (Sec.3.3.2). We then turn to the more general preparation of arbitrary states in the system, through the target of Gaussian states (for which we develop a systematic reconstruction method) that we translate and superpose (Sec. 3.3.4) or squeeze (Sec. 3.3.5) in the $(x, p)$ phase space of each lattice site.

### 3.3.1 Control of the relative phases between two momentum components

To prepare a given quantum state in our Bloch system, we need to control the complex phase of each$^7$ coefficient $c_{q,l}$. We first demonstrate this control by targeting the simple momentum superposition

$$|\psi_T^{(\Delta\phi)}\rangle = \frac{1}{\sqrt{2}} (|\chi_1\rangle + e^{i\Delta\phi} |\chi_{-1}\rangle). \qquad (3.37)$$

<footnote>$^7$Actually each coefficient minus one, since states are defined up to a global phase.</footnote>
For several target values of the relative phase $\Delta\phi = j \times \pi/8$ (with $j \in \{0, 1, ..., 15\}$) we compute an optimal-control ramp that prepares the corresponding superposition $|\psi_{QOC}^{(\Delta\phi)}\rangle$ with numerical quantum fidelity $F_{\text{num}} > 0.995$. With each ramp, we also evaluate the prepared relative phase $\Delta\phi_{\text{prep}} = \arg\{c_{-1}/c_1\}$ from a simulation of the optimal control preparation. To measure the relative phase $\Delta\phi_{\text{meas}}$ of the experimentally prepared superpositions, we use the subsequent evolutions of the momentum distribution in the static lattice. These evolutions are measured over an extra 110 µs ($\approx 1.75 T_0$). Results are shown in Fig. 3.5.

For each of the five evolutions presented, we see that the initial momentum distributions (at $t=0$) are the same, which is the expected superposition $(-1, 1) \hbar k_L$ given the addressed state (3.37). The subsequent evolutions then strongly differ depending on the prepared relative phase $\Delta\phi_{\text{prep}}$. This allows us to measure relative phases $\Delta\phi_{\text{meas}}$ with precision by fitting, on the experimental data, numerical evolutions of the ideal state (3.37) with the relative phase $\Delta\phi$ as the adjustable parameter. This procedure is illustrated in Fig. 3.5(c$_{-2}$-c$_2$). In Fig. 3.5(f), we compare the prepared and measured relative phases. The good agreement between the two (see the caption of Fig. 3.5 for details) demonstrates our ability to engineer the phase of momentum superpositions reliably. The residual mismatch may be attributed to at least two different effects: on the first hand, lattice depth fluctuations has a double deleterious effect on this experimental scheme, as they impact both the QOC preparations and the subsequent evolutions. On the other hand, the adjustments of wave function dynamics performed here suffer from only having the relative phase $\Delta\phi$ as adjustable parameter. Indeed, we here effectively assume that the prepared states at the beginning of the evolution were always the ideal state (3.37), with an equal-weight superposition of the sole momentum components $(-1, 1) \hbar k_L$. Even if the preparations were ideal, we know for a fact that this is inaccurate, as the numerical fidelities to the targets only are $F \gtrsim 0.995$. This necessarily limits the measurement potential of our fit. Later in this chapter (Sec. 3.3.3), we overcome this limitation with a reconstruction method that only require a constant lattice depth $s_0$ as an assumption.

We have also performed similar experiments with three-momentum superpositions and different relative phases. These experiments are presented in App. E.

### 3.3.2 Preparation of Bloch eigenstates and superpositions thereof

To further demonstrate our ability to prepare quantum states with specific relative phases between their $c_\ell$ coefficients, we target two Bloch eigenstates of the static lattice and two given superpositions of Bloch eigenstates. Our target are computed from the diagonalization of the static lattice Hamiltonian in the basis of the plane waves (see Sec. 1.2.2). Writing $|\phi_{q,n}^{(s_0)}\rangle$ the $n^{\text{th}}$ Bloch eigenstate (ranked by increasing eigenenergies) at quasi-momentum $q$ and lattice depth $s_0$, we target the states$^8$ (with indices corresponding to the panels of Fig. 3.6)

<footnote>$^8$The specific depth values indicated for these targets correspond to the depths calibrated before each experiment (Sec. 2.4.2).</footnote>
<figure><img src="image_38.png" /><figcaption>**Figure 3.5** | Control and measurement of the phase between two momentum components. (a-e) Top: stacks of integrated experimental images (blue) showing the evolution of the momentum distribution during a 110 µs holding time in a static lattice after applying a control field $\phi(t)$ preparing the momentum superposition $|\psi_{\Delta\phi}\rangle$ of momentum components $(-1, 1) \hbar k_L$ (Eq. (3.37)) with an expected relative phases $\Delta\phi_{\text{prep}} = 3^\circ, 46^\circ, 96^\circ, 184^\circ$ and $276^\circ$. Bottom: numerical propagation (red) in a static lattice of the same momentum superposition with a relative phase adjusted by least-squares fitting of the experimental data, yielding respectively $\Delta\phi_{\text{meas}} = (11 \pm 6)^\circ, (48 \pm 7)^\circ, (84 \pm 6)^\circ, (192 \pm 7)^\circ$ and $(285 \pm 6)^\circ$. (c-2-c2) Detail of the evolution of momentum populations in (c), with panel (cℓ) featuring the ℓth momentum component, and showing the experimental data (blue dots) and numerical propagation of the superposition $|\psi_{\Delta\phi_{\text{meas}}}\rangle$ with relative phase $\Delta\phi_{\text{meas}}$ determined by least-square fitting (continuous red line). (f) Measured relative phase $\Delta\phi_{\text{meas}}$ as a function of QOC prepared relative phase $\Delta\phi_{\text{prep}}$ for data (a-e) and more. All data shown were obtained for a calibrated lattice depth $s_0 \approx 5$. The error bars represent the 95 % confidence interval for the fitted value of $\Delta\phi_{\text{meas}}$. The grey dotted line is of slope one.</figcaption></figure>

$$ 
\begin{aligned}
|\psi_T^{(b)}\rangle &= |\phi_{q=0,n=1}^{(s_0=8.15)}\rangle \\
|\psi_T^{(c)}\rangle &= |\phi_{q=k_L/4,n=2}^{(s_0=8.26)}\rangle \\
|\psi_T^{(e)}\rangle &= \frac{1}{\sqrt{2}} \left( |\phi_{q=0,n=1}^{(s_0=5.60)}\rangle + |\phi_{q=0,n=3}^{(s_0=5.60)}\rangle \right) \\
|\psi_T^{(f)}\rangle &= \frac{1}{\sqrt{2}} \left( |\phi_{q=k_L/4,n=0}^{(s_0=5.66)}\rangle + |\phi_{q=k_L/4,n=1}^{(s_0=5.66)}\rangle \right).
\end{aligned}
\qquad (3.38) $$

**Expectations.** In the case of the one-eigenstate targets $|\psi_T^{(b,c)}\rangle$, we expect to observe no evolution of the momentum distributions as we target steady states of the system. For the two eigenstate superpositions $|\psi_T^{(e,f)}\rangle$, we are preparing the two-level superposition:

$$ |\psi_q(t=0)\rangle = \frac{1}{\sqrt{2}} \left( |\phi_{q,n}^{(s_0)}\rangle + |\phi_{q,m}^{(s_0)}\rangle \right), \qquad (3.39) $$
whose evolution can be written as (Sec. 1.2.3)

$$|\psi_q(t)\rangle = \frac{e^{-iE_{q,n}^{(s_0)}t/\hbar}}{\sqrt{2}} \left( |\phi_{q,n}^{(s_0)}\rangle + e^{-i(E_{q,m}^{(s_0)} - E_{q,n}^{(s_0)})t/\hbar} |\phi_{q,m}^{(s_0)}\rangle \right). \quad (3.40)$$

We therefore expect to detect only one frequency of evolution $\nu_{q,n,m}^{(s_0)} = |E_{q,m}^{(s_0)} - E_{q,n}^{(s_0)}|/\hbar$. We compute $\nu_{(e)} \approx 33$ µs and $\nu_{(f)} \approx 59$ µs.

For these expectations to hold, the eigenstates that go into our target definitions (3.38) need indeed to be eigenstates (see below). While this is directly the case when targeting $|\psi_T^{(b,e)}\rangle$ (as we apply our QOC protocol in the subspace of quasi-momentum $q=0$), we need, in the case of $|\psi_T^{(c,f)}\rangle$, to put our preparations in the subspace $q = k_L/4$ after the QOC stage. A state associated with a particle of mass $m$ initially in the subspace of quasi-momentum $q$ is projected onto the subspace of quasi-momentum $q'$ by instantaneously giving it a speed $v = \hbar(q' - q)/m$ with respect to the Bloch system in which it evolves. In the experiment, we induce this instantaneous speed shift by putting the lattice at speed $v = -\hbar(q' - q)/m$ with a linear lattice phase curve once the QOC preparation is done<fnref n="9" />.

We present the results of these experiments in Fig. 3.6. The control duration is $t_c = 1.5 T_0$ and we obtain control fields with numerical fidelities $F \gtrsim 0.99$. In Fig. 3.6(b,c,e,f), we compare the evolutions of the states prepared experimentally and the theoretical states $|\psi_{QOC}\rangle$ prepared numerically. As expected, the plane wave populations for the one-eigenstate preparations do not evolve in time (Fig. 3.6(b,c)), demonstrating that we have indeed generated eigenstates. For the eigenstate superpositions (Fig. 3.6(e,f)), we strikingly observe the evolution of all momentum orders with mainly one frequency in each case (one can contrast these with e.g. Fig. 3.5(c)). The observed frequency are in agreement with our expectations (see above).

We point out that similar experiments of eigenstate preparations where performed in one, two and three dimensions with control sequences of pulsed lattices [130].

We have so far demonstrated quantum state control with simple two-momentum component superpositions (Sec. 3.3.1) and here with the preparation of lattice eigenstates. To prove our ability to prepare arbitrary quantum states, we need a systematic state reconstruction method to measure our preparations.

In the rest of this chapter, we work in the subspace of quasi-momentum $q=0$.

### 3.3.3 Quantum state reconstruction by likelihood maximization

We present in this section the implementation of an iterative likelihood maximization method that allows us to systematically reconstruct prepared states from their evolution in the static lattice after the preparation. As we will see, a strength of this method is that no

<footnote n="9">⁹For the targets $|\psi_T^{(c,f)}\rangle$, the lattice is given a speed of $v = -\hbar k_L/4m \approx -2.158$ mm/s after the QOC stage.</footnote>
<figure><img src="image_39.png" /><figcaption>**Figure 3.6** | Preparations of Bloch eigenstates and superpositions thereof. (a,d) Lattice band structures (color lines) for the two depths $s_0 = 8.2$ (a) and $s_0 = 5.6$ (d) over one Brillouin zone, and identification of the eigenstates targeted in panels (b,c) and eigenstate superpositions targeted in panels (e,f) (black annotations). The grey areas indicate the energies below the maximum potential energy of the lattice. (b) Preparation of the eigenstate of the *p* band at quasi-momentum $q = 0$ for $s_0 = 8.15 \pm 0.30$. ($b_1$) Experimental data showing the evolution of the prepared state in the lattice. ($b_2$) Numerical evolution of $|\psi_{QOC}\rangle$. ($b_3$) Time-averaged experimental (blue) and theoretical (red) momentum distributions. The error bars represent the standard deviation along the evolution. (c) Same as (b) but for the eigenstate on the *d* band at quasi-momentum $q/k_L = 0.25$ for $s_0 = 8.26 \pm 0.10$. (e) Same as (b) but for the equal-weight superposition of eigenstates of the *p* and *f* bands at quasi-momentum $q = 0$ for $s_0 = 5.60 \pm 0.15$. (f) Same as (b) but for the equal-weight superposition of eigenstates of the *s* and *p* bands at quasi-momentum $q/k_L = 0.25$ for $s_0 = 5.66 \pm 0.15$. This experimental figure can be compared with the theoretical figure 1.5 p. 22.</figcaption></figure>