Many interesting physical systems, e.g. condensed matter systems, are hard to study. From the experimental point of view, in such systems most parameters are ''frozen" for a given sample. From the theoretical point of view, they often cannot be simply modelled from first principles. In contrast, laser-matter interactions are very well known theoretically, and a very high degree of control of laser beams is available for the experimentalist. In the last years, one learned to ''engineer" quantum hamiltonians using ultracold atoms interacting with electromagnetic fields. For example, laser beams may form an interference pattern that is ''seen" by the atoms as a periodic potential, thus creating an analog of a crystal. However, the analog has great advantages over the original system, as one has a larger control of its properties: One can vary the parameters of the potential by changing the properties of the laser beams, one can make a ''mesoscopic" quantum system by using a Bose-Einstein condensate, one can change it to a degenerate Fermi gas by using a fermionic isotope, one can control atom-atom interactions by using the so-called . Building on seminal ideas of Feynman, the notion of ''quantum simulator" -- a ''simpler" system able to mimic the physics of a complicate one -- recently became a reality. In a variety of situations of experimental interest, weakly interacting degenerate boson gases (Bose-Einstein condensates) are very well described by the Nonlinear Schödinger Equation with a cubic nonlinearity (called, in the present context, the {\it Gross-Pitaevskii equation}), which is much simpler than the underlying many-body problem. This opened a new field for both theoretical and experimental studies: The {\it nonlinear} quantum mechanics, where the familiar techniques like the spectral analysis in terms of eigenvalues and eigenvectors, which derive from the linearity of the usual Schödinger equation, cannot be used. The mathematical framework of this new quantum mechanics is still largely to be invented. Meanwhile, one can resort to numerical simulations -- and to experiments. In this talk, I will discuss two simple yet important examples illustrating these ideas. First, I will show how one can ''quantum simulate" with ultracold atoms the celebrated Anderson model, which describes quantum disordered systems. One can even enrich this system by adding particle-particle interactions in a controlled way. Second, I will show a simple model displaying ''quasiclassical" chaos, that is, chaotic behavior related to sensitivity to the initial conditions, which cannot exist in {\it linear} quantum mechanics. I will show that the observed behavior, although intrinsically quantum, can be described by the KAM theorem.
Many interesting physical systems, e.g. condensed matter systems, are hard to study. From the experimental point of view, in such systems most parameters are ''frozen" for a given sample. From the theoretical point of view, they often cannot be simply modelled from first principles. In contrast, laser-matter interactions are very well known theoretically, and a very high degree of control of laser beams is available for the experimentalist. In the last years, one learned to ''engineer" quantum hamiltonians using ultracold atoms interacting with electromagnetic fields. For example, laser beams may form an interference pattern that is ''seen" by the atoms as a periodic potential, thus creating an analog of a crystal. However, the analog has great advantages over the original system, as one has a larger control of its properties: One can vary the parameters of the potential by changing the properties of the laser beams, one can make a ''mesoscopic" quantum system by using a Bose-Einstein condensate, one can change it to a degenerate Fermi gas by using a fermionic isotope, one can control atom-atom interactions by using the so-called . Building on seminal ideas of Feynman, the notion of ''quantum simulator" -- a ''simpler" system able to mimic the physics of a complicate one -- recently became a reality. In a variety of situations of experimental interest, weakly interacting degenerate boson gases (Bose-Einstein condensates) are very well described by the Nonlinear Schödinger Equation with a cubic nonlinearity (called, in the present context, the {\it Gross-Pitaevskii equation}), which is much simpler than the underlying many-body problem. This opened a new field for both theoretical and experimental studies: The {\it nonlinear} quantum mechanics, where the familiar techniques like the spectral analysis in terms of eigenvalues and eigenvectors, which derive from the linearity of the usual Schödinger equation, cannot be used. The mathematical framework of this new quantum mechanics is still largely to be invented. Meanwhile, one can resort to numerical simulations -- and to experiments. In this talk, I will discuss two simple yet important examples illustrating these ideas. First, I will show how one can ''quantum simulate" with ultracold atoms the celebrated Anderson model, which describes quantum disordered systems. One can even enrich this system by adding particle-particle interactions in a controlled way. Second, I will show a simple model displaying ''quasiclassical" chaos, that is, chaotic behavior related to sensitivity to the initial conditions, which cannot exist in {\it linear} quantum mechanics. I will show that the observed behavior, although intrinsically quantum, can be described by the KAM theorem.
Nonlinear effects are ubiquitous in nature. Many interesting phenomena are described by differential equations that are nonlinear. Even richer dynamics can be observed with additional long-range spatial coupling. For example, an interesting type of pattern discovered recently can form in systems with nonlocal coupling. The pattern, called chimera states, is composed of both phase coherent and incoherent regions coexisting in the same system. Through numerical studies of oscillatory media with nonlocal diffusive coupling, I show here for the first time that stable chimera knot structures can exist in 3D. Knots were not previously known to be stable in oscillatory media, nor were such non-trivial chimera patterns known to exist in 3D. To realize different nonlinear phenomena in a controlled way in experiments, a flexible physical system is required. Ultracold atomic systems, specifically, Bose-Einstein condensates (BECs), are good candidates because of the high controllability of almost all parameters, including the nonlinearity, in real time. Hence, experimental studies can be carried out for a variety physical systems, including many classical and quantum field equations. In particular, in this thesis I study a setup of BECs with a third-order Kerr nonlinearity to generate Schr\"odinger cat states, which have applications in quantum metrology. I showed that cat states involving hundreds of atoms should be realizable in BECs. This requires careful optimization of the experimental parameters and analysis of the atom loss. Inspired by the previous two projects, it is an interesting question if chimera states can exist in BECs. By analyzing the underlying mechanism of the effective nonlocal diffusive coupling, I establish here a new analogous mechanism to achieve mediated nonlocal spatial hopping for particles in BECs with two interconvertible states. By adiabatically eliminating the fast mediating channel, I obtain the mean-field of Bose-Hubbard model with fully tunable hopping strength, hopping range, and nonlinearity. This is the first known conservative system exhibiting chimera patterns. More importantly, I show that the model should be implementable in BECs with current technology.
Quantum simulation is one of the most active trends in ultracold-atom physics. A quantum simulator is a system that mimics the dynamics of another system; the advantage being that the quantum simulator is simpler than the original, either from the experimental or the theoretical point of view (or both).The Anderson model is a paradigm for quantum disorder, originally introduced in the helm of solid-state physics; however, quantum solid-state disordered systems are very difficult to study experimentally, and even numerically (except in the one-dimensional case). The kicked rotor is a paradigm for classical and quantum Hamiltonian chaos, essentially because it is very easy to simulate it in a computer. Its quantum realization with ultracold atoms is also relatively simple, and is a “clean” (e.g. from decoherence) and controllable system. Amazingly, the quantum kicked rotor can be mathematically mapped onto the Anderson model. It is thus a quite subtle quantum simulator of the physics of disordered systems. In this talk I will describe a series of experiments we have performed in Lille taking advantage of the correspondence between disordered condensed matter systems and the atomic kicked rotor to investigate experimentally in great detail the Anderson metal-insulator transition. I will discuss the new possibilities opened by evolutions of this system, in particular by introducing nonlinearity via atom-atom interactions.
Quantum systems consisting of many particles are both interesting from the physics and the mathematics perspective. One aspect that makes them fascinating from the physics point of view is that they can exhibit quantum effects on macroscopic scales such as Bose-Einstein condensation. This phenomenon, first predicted 100 years ago by Bose [19] and Einstein [44], occurs when a dilute gas of bosons is cooled down to very low temperatures. Below a certain critical temperature a condensate forms, meaning that a macroscopic fraction of the particles is in the same one-particle state, and thus, quantum phenomena become relevant on length scales much larger than the individual atom. The occurrence of Bose-Einstein condensation is also related to superfluidity as suggested in [88]. Bose-Einstein condensation has been achieved experimentally independently by the groups of Cornell and Wieman [3] and Ketterle [40] earning them a Nobel prize in 2001. While Bose and Einstein analyzed theoretically the non-interacting gas, an effective theory describing weakly interacting Bose gases was proposed by Bogoliubov in 1947 with the goal of explaining the occurrence of superfluidity in liquid helium [18]. Using assumptions such as that condensation indeed occurs, his theory yields predictions for the low-energy states of the system. Seen from the mathematics perspective, describing the Bose gas is an interesting challenge due to the strong correlations between particles introduced by the interaction making this case much harder to study than the non-interacting one. In particular, a proof that Bose-Einstein condensation occurs in the thermodynamic limit for continuous systems is still out of reach. However, there has been a lot of work regarding the rigorous analysis of bosonic systems in various relevant regimes. This thesis contains three new contributions establishing rigorous results on bosonic systems in different, physically relevant settings. It is based on the articles [37, 27, 31]. The first result concerns the time-evolution of initially trapped Bose gases in the ultra-dilute Gross-Pitaevskii regime. Under some assumptions on the initial data satisfied in particular by a class of states that are expected to approximate the ground state in a trap, we derive a quasi-free approximation that is close to the true solution in L2-norm. In particular, we show that this approximate wave function is described in terms of a Bose-Einstein condensate in a one-particle state following the time-dependent Gross-Pitaevskii equation together with pair excitations with respect to this condensate. As a corollary, we establish a central limit theorem for bounded one-particle observables. The second result is a simple proof of Bose-Einstein condensation for interacting, ultra-dilute gases. More specifically, we consider scalings interpolating between the Gross-Pitaevskii regime and the thermodynamic limit. While condensation had been shown to hold in these regimes before, the advantage of the method used here is its simplicity. Lastly, the third result establishes an upper bound on the ground state energy of a dilute Bose gas in the thermodynamic limit. It captures rigorously the third order correction to the energy per unit volume for sufficiently low densities as predicted in the physics literature first by Wu and subsequently others. Full-text embargoed until: 2026-01-01
Remarkable experimental advances in the past few decades have paved the way for a new era of ultra-cold physics. Experimentalists can now cool large samples of atoms down to temperatures where quantum mechanical effects dominate, and control their bulk geometrical structure and microscopic interactions with ease. The result is a wide open testing ground for quantum many-body physics. In this talk, I will provide an overview of the important concepts and methods that form the foundations of this field, before steering the discussion to Bose gases, or gases with constituents that obey Bose-Einstein statistics. At ultra-cold temperatures, gases of bosons condense into a single macroscopicquantum mechanical state, forming a Bose-Einstein condensate. One of the more interesting properties of Bose-Einstein condensates is that they are superfluid, and can exhibit dissipationless flow. If the bosons possess large dipole moments, their interaction character changes signifi cantly, as does their collective superfluid behavior. I will discuss some of the theoret ical and computational tools used to model Bose-Einstein condensates, demonstrate how dipolar interactions play a profound role in the basic properties of these novel superfluids, and provide an outlook for the future of this exciting field.
In recent years ultracold atomic gases have emerged as a new and promising platform for exploring the physics of quantum many-body systems, allowing researchers to revisit fundamental problems of condensed-matter physics such as superconductivity or magnetism. By trapping ultracold atoms in the periodic potential created by interfering laser beams (a so-called optical lattice), it has indeed become possible to engineer very clean ?materials? whose parameters can be tuned in a wide range. This bottom-up approach comes very close to Feynman?s idea of a ?quantum simulator?, an
Many outstanding problems in quantum physics, such as high-Tc superconductivity or quark confinement, are still - after decades of research - awaiting commonly accepted explanations. One reason is that such systems are often difficult to control, show an intermingling of several effects, or are not easily accessible to measurement. To arrive at a deeper understanding of the physics at work, researchers typically derive simplified models designed to capture the most striking phenomena of the system under consideration. However, due to the exponential complexity of Hilbert space, even some of the simplest of such models pose formidable challenges to analytical and numerical calculations. In 1982, Feynman proposed to solve such quantum models with experimental simulation on a physically distinct, specifically engineered quantum system [Int. J. Theor.Phys. 21, 467]. Designed to be governed by the same underlying equations as the original model, it is hoped that direct measurements on these so called quantum simulators (QSs) will allow to gather deep insights into outstanding problems of physics and beyond. In this thesis, we identify four requirements that a useful QS has to fulfill, relevance, control, reliability, and efficiency. Focusing on these, we review the state of the art of two popular approaches, digital QSs (i.e., special purpose quantum computers) and analog QSs (devices with always-on interactions). Further, focusing on possibilities to increase control over QSs, we discuss a scheme to engineer quantum correlations between mesoscopic numbers of spinful particles in optical lattices. This technique, based on quantum polarization spectroscopy, may be useful for state preparation and quantum information protocols. Additionally, employing several analytical and numerical methods for the calculation of many-body ground states, we demonstrate the variety of condensed-matter problems that can be attacked with QSs consisting of ultracold ions or neutral atoms in optical lattices. The chosen examples, some of which have already been realized in experiment, include such diverse settings as frustrated antiferromagnetism, quantum phase transitions in exotic lattice geometries, topological insulators, non-Abelian gauge-fields, orbital order of ultracold Fermions, and systems with long-range interactions. The experimental realization of all of these models requires techniques which go beyond standard optical lattices, e.g., time-periodic driving of lattices with exotic geometry, loading ultracold atoms into higher bands, or immersing trapped ions into an optical lattice. The chosen models, motivated by important open questions of quantum physics, pose difficult problems for classical computers, but they may be amenable in the near future to quantum simulation with ultracold atoms or ions. While the experimental control over relevant models has increased dramatically in the last years, the reliability and efficiency of QSs has received considerably less attention. As a second important part of this thesis, we emphasize the need to consider these aspects under realistic experimental conditions. We discuss specific situations where terms that have typically been neglected in the description of the QS introduce systematic errors and even lead to novel physics. Further, we characterize in a generic example the influence of quenched disorder on an analog QS. Its performance for simulating universal behavior near a quantum phase transition seems satisfactory for low disorder. Moreover, our results suggest a connection between the reliability and efficiency of a QS: it works less reliable exactly in those interesting regimes where classical calculations are less efficient. If QSs fulfill all of our four requirements, they may revolutionize our approach to quantum-mechanical problems, allowing to solve the behavior of complex Hamiltonians, and to design nano-scale materials and chemical compounds from the ground up.