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 $L^2$-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. $\textit{Full-text embargoed until: 2026-01-01}$
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
This thesis is devoted to the rigorous study of dilute Bose gases with two-body and three-body interactions at zero temperature in two and three dimensions. Bose gases are many-body quantum systems composed of bosons, which notably do not obey the Pauli exclusion principle. As a consequence, at extremely low temperatures, a macroscopic fraction of the particles can occupy the same one-body quantum state. This phenomenon, known as Bose–Einstein condensation, was predicted in the 1920s by Bose and Einstein for noninteracting Bose gases, and first realised experimentally by the groups of Cornell and Wieman, and Ketterle, earning them the Nobel Prize for Physics in 2001. However, proving the occurrence of Bose–Einstein condensation in general interacting bosonic systems remains a major open problem and constitutes a central theme in this thesis. Another important property of bosonic matter is that it is generally more unstable than fermionic matter. In particular, for systems interacting via the Coulomb force, the Pauli exclusion principle is essential for explaining the stability of the second kind, which states that the energy and volume scale linearly with the number of particles. Nonetheless, bosonic systems with attractive but less singular interactions can also be stable, and the rigorous proof of this fact forms a second focus of this thesis. In three dimensions, we consider predominantly repulsive interactions. Firstly, we prove a leading order upper bound on the thermodynamic ground state energy of Bose gases with hard-core three-body interactions, which matches an existing lower bound. In addition, we obtain a simpler proof for integrable potentials. Secondly, we derive the leading order of the thermodynamic ground state energy of Bose gases with combined two-body and three-body interactions. More specifically, we show that, at the leading order, the ground state energy is entirely characterised by the scattering length of the two-body potential and the scattering hypervolume of the three-body potential. Thirdly, we study the time evolution of Bose–Einstein condensates with three-body interactions in the so-called Gross–Pitaevskii regime. We show that Bose–Einstein condensation is preserved under many-body evolution and that the condensate wavefunction evolves according to the quintic Gross–Pitaevskii equation. In particular, we show that the effective coupling constant is universal in the sense that it depends only on the scattering hypervolume of the interaction potential. As a by-product of our proof, we derive a stability of the second kind result for Bose gases with attractive two-body interactions and repulsive three-body interactions. In two dimensions, we consider trapped Bose gases with mean-field-type interactions, including the purely attractive case. We prove the stability of the second kind of the system and the convergence of the ground state energy per particle to that of a non-linear Schrödinger energy functional. As a consequence of the stability of the second kind, we also obtain Bose–Einstein condensation for the many-body ground states.
The study of out-of-equilibrium dynamics in ultra-cold gases is a new and exciting field, driven largely by the recent experimental advances in controlling and imaging cold clouds. The experimental and theoretical work thus far has been somewhat exploratory and largely numerical in nature, as the very paradigms for thinking about these systems are not well established. In this thesis I consider several different scenarios of ultra-cold bosonic and fermionic gases driven out of equilibrium and study their properties. In Chapter 1, I provide an overview of the phenomenology of ultra-cold gases, highlighting the timescales governing these systems and how the experimentalist can tune them. I discuss how cold gases can be cooled and trapped and discuss the basic physics behind optical lattices. I also discuss experimental probes of these gases, in particular the new high resolution imaging techniques developed recently at Chicago, Munich and Harvard. In Chapter 2, I discuss an early experiment (circa. 2008) which observed long lived spin dynamics in a thermal spin-1/2 Fermi gas. This experiment is an nice illustration of interesting physics resulting from the separation of timescales between spin and collisional dynamics. In my opinion, it is an excellent example of why cold gases are naturally suited to studying non-equilibrium dynamics. I simulate the experiment numerically using a collisionless Boltzmann equation and explain the observed spin dynamics both qualitatively and quantitively. In Chapter 3, I continue the discussion of spin waves in thermal gases by extending previous works on spin-1/2 gases to spin-1 Bose gases. In contrast to Chapter 2, the bulk of the work in this Chapter is analytic in nature. In particular, I find a spin wave instability in the thermal spin-1 Bose gas, which is the high temperature analog of the polar to ferromagnetic transition in a spin-1 Bose Einstein condensate. In Chapter 3, I turn my attention to bosonic systems and briefly review the the Bogoliubov mean-field theory. I calculate the momentum distribution and density-density correlation function of an interacting Bose gas within the Bogoliubov framework. Then I consider bosons in an optical lattice, and introduce the Bose Hubbard model. I calculate the mean-field phase diagram of the Bose Hubbard model and then consider fluctuations about the mean field, and derive the excitation spectrum of the lattice gas in the superfluid and insulating regimes. In Chapter 4, I ask what we learn by studying the dynamics of correlation functions following a sudden change in the interactions in a superfluid. Using the Bogoliubov theory developed in Chapter 3, I will show how the underlying excitation spectrum influences the long and short time behavior of the correlation functions. By considering a lattice dispersion, I study the analogous problem in a weak optical lattice and discuss how the lattice dispersion leads to additional features in the correlation functions. I will also discuss the timescale governing the revival of the condensate fraction in a quantum depleted gas. In Chapter 5, I derive equations of motion governing the dynamics of one and two body correlation functions in the single-band Bose Hubbard model, applicable to bosons in deep lattices. I then consider a simple quench from a Mott insulating initial state to a weakly interacting final state and produce analytic expressions describing the dynamics of correlations following such a quench. I discuss the timescale for the development of long range order following such a quench. I study the problem of chapter 4 using an equations of motion approach. This approach complements the Bogoliubov approach of Chapter 4. First, I derive exact expressions for a quench to a non-interacting state. I then consider how interactions redistribute quasi-momentum to first order in perturbation theory in different dimensions. In Chapter 6, I calculate the relevant timescales for local and global dynamics in trapped lattice Bose gases, a work done in collaboration with Dr. Kaden R. A Hazzard. Using a time-dependent Gutzwiller mean-field theory, I show that the timescale for local equilibration in these systems is fast in experimental terms. I then show that due to the spatial inhomogeneities inherent to cold gases, achieving global equilibrium can be quite complicated, sometimes taking longer than the lifetime of the experiment, an issue of practical importance to current day experiments. I continue this discussion in Chapter 7 which is a collaborative work with experimentalists David McKay and Prof. Brian DeMarco from the University of Toronto and the University of Illinois, Urbana Champaign. Using experimental and numerical methods, we show that the rapid timescales for local dynamics in interacting systems invalidates a frequently used cold atom technique for mapping out the momentum distribution of atoms in an optical lattice.
Abstract This volume introduces the basic concepts of Bose–Einstein condensation and superfluidity. It makes special reference to the physics of ultracold atomic gases; an area in which enormous experimental and theoretical progress has been achieved in the last twenty years. Various theoretical approaches to describing the physics of interacting bosons and of interacting Fermi gases, giving rise to bosonic pairs and hence to condensation, are discussed in detail, both in uniform and harmonically trapped configurations. Special focus is given to the comparison between theory and experiment, concerning various equilibrium, dynamic, thermodynamic, and superfluid properties of these novel systems. The volume also includes discussions of ultracold gases in dimensions, quantum mixtures, and long-range dipolar interactions.
1. Define Bose-Einstein condensation in an interacting boson system in term of macroscopic occupation. 2. Introduce the Gross-Pitaevskii equation and its applicable conditions. 3. Derive the hydrodynamic equations and various physical consequences of the hydrodynamics with or without harmonic trap. 4. Introduce sound wave and emphasize the difference between the wave behavior and the diffusive behavior. 5. Introduce the criterion for superfluidity and determine the superfluid critical velocity. 6. Introduce two-fluid hydrodynamics. 7. Introduce the Bogoliubov theory. 8. Introduce various concepts related to quasi-particle, including ground state as vacuum of quasi-particles, the zero-point energy of quasi-particle, quasi-particle lifetime and vacuum fluctuation. 9. Introduce healing length as the characteristic length scale for low-energy physics of an interacting Bose condensate. 10. Summarize the difference between an interacting and non-interacting Bose condensate. 11. Introduce the Bragg spectroscopy. 12. Show that the Bogoliubov theory fails in one-dimension due to the infrared divergence. 13. Discuss the basic idea of the Bethe-ansatz solution and the Tonks-Girardeau limit. 14. Introduce scale invariant quantum gases. 15. Introduce the Josephson effect and the self-trapping as two different transport phenomena driven by phase coherence. 16. Discuss the conjugate relation between the relative phase fluctuation and the relative particle number fluctuation, and show strong enough phase fluctuation renders the system into a Fock state. 17. Discuss an example of quantum measurement that can project two independent condensates to a coherent superposition state. 18. Discuss interference effects in density and density-density correlations. 19. Discuss the relation between the instability of the Sch\"odinger Cat state and the stability of symmetry breaking.
Bose-Einstein condensates have been experimentally realized in cold Bose gases for the first time in 1995.This discovery has renewed interest in the theory of interacting Bose gases; however, the mathematical understanding of Bose-Einstein condensation is still incomplete.In this thesis we derive effective theories for the dynamical and spectral properties of Bose gases.Our theoretical description is based on scaling regimes where the number of particles N becomes large and the two-body interaction potential scales as N 3β-1 V (N β •), for a fixed β ∈ (0, 1].The case β = 1 is known as Gross-Pitaevskii regime.Our first result concerns the time evolution of Bose-Einstein condensates after the trapping potential is removed.In previous works, it was shown that the many-body dynamics can be approximated by a one-body nonlinear Schrödinger equation at the level of reduced density matrices.We provide a more precise approximation of the time evolution: in a second-quantized setting, we construct a unitary effective dynamics, which, in the limit of large N , approximates the fully evolved state in Fock space norm.Our result comes with an explicit bound on the rate of convergence.It is valid for β < 1.Our second result addresses the properties of the ground state wave function.It has been shown before that the ground state wave function exhibits condensation in the limit N → ∞, i.e. the one-particle density matrix associated to the ground state approaches a projection.We prove the optimal bound for the convergence rate, for a translation invariant system.To do this, we estimate uniformly in N the number of excited particles.This result is valid for β = 1.Our last result concerns the ground state energy and the excitation spectrum of the translation invariant system.We prove the validity of Bogoliubov theory; this allows us to compute the ground state energy and the low-lying excitation spectrum up to an error that vanishes for large N .This result holds for β < 1.This thesis is based on the articles [11], [9] and [10].i
Bose-Einstein condensation is a unique phase transition in that it is not driven by interparticle interactions, but can theoretically occur in an ideal gas, purely as a consequence of quantum statistics. This chapter addresses the question, ‘ How is this ideal Bose gas condensation modified in the presence of interactions between the particles?’ This seemingly simple question turns out to be surprisingly difficult to answer. Here we outline the theoretical background to this question and discuss some recent measurements on ultracold atomic Bose gases that have sought to provide some answers. Introduction Unlike the vast majority of phase transitions, Bose-Einstein condensation (BEC) is not driven by interparticle interactions but can theoretically occur in an ideal (noninteracting) gas, purely as a consequence of quantum statistics. However, in reality, interactions are needed for a Bose gas to remain close to thermal equilibrium. It is thus interesting to discuss if something close to ideal gas BEC can be observed in a real system and what happens in the vicinity of the BEC transition in the presence of interparticle interactions. These simple questions have not been easy to answer, either theoretically or experimentally. The theoretical foundations for studying the effect of interactions on Bose condensed systems were laid over half a century ago by Bogoliubov [1], Penrose and Onsager [2], and Belieav [3], among others. These works initially focused on zerotemperature properties and were extended to nonzero temperature in the pioneering papers of Lee, Huang, and Yang [4, 5, 6, 7]. At that time, the main experimental system was liquid He-4 in which the inter-particle interactions are strong, making connections with theory difficult. The realisation in 1995 of BEC in weakly interacting ultracold atomic gases [8, 9] thus opened up the possibility to experimentally revisit some of these long-discussed questions. This was further aided by, among other advances, the use of Feshbach resonances to tune the interaction strength in atomic gases [10, 11]. Thus, in the last twenty years the study of ultracold Bose gases has been very successful (see Chapter 3 for an insightful account of selected topics and surprising developments over this exciting period). However, the fact that, until recently, ultracold atoms were confined using harmonic potentials has hindered the study of the BEC transition itself.