This thesis describes a collection of experiments that explore interacting ultracold Bose gases, both in and out of equilibrium. Our experiments are performed using a gas of $^{39}$K or $^{87}$Rb confined in the uniform potential of an optical box trap, a novel testbed for quantum many-body phenomena. Our work focuses on weakly interacting non-equilibrium systems, moderately interacting systems that are still in equilibrium, and the unitary Bose gas which is both strongly interacting and eludes equilibrium. We begin with studies of weakly-interacting gases far from equilibrium, which feature ties to nonlinear wave phenomena. Highlights of our experiments include the direct measurement of turbulent-cascade fluxes, which (alongside realizing a tuneable dissipation scale) allow us to demonstrate the zeroth law of turbulence, and the first observation of weak collapse, a general type of nonlinear wave collapse predicted over 40 years ago. We then turn to moderately strong interactions, confronting existing theories of interacting quantum fluids. One of our most important results is the first quantitative measurement of the quantum depletion of a Bose–Einstein condensate, confirming a 70-year-old theory first developed to describe liquid helium. The culmination of our work explores the unitary Bose gas, where interparticle interactions are as strong as allowed by the laws of quantum mechanics. This strongly-correlated state promises tantalizing possibilities, including emergent universal behavior set solely by the gas density and novel forms of superfluidity. However, the strong interactions also lead to a complex interplay between coherent and dissipative dynamics. By disentangling these two processes, we have caught a glimpse of the promises that the unitary Bose gas holds. In particular, we observe the emergence of universal behavior, and find that the gas features a well defined quasi-equilibrium state, with a non-zero condensed fraction.
This thesis describes a collection of experiments that explore interacting ultracold Bose gases, both in and out of equilibrium. Our experiments are performed using a gas of 39K or 87Rb confined in the uniform potential of an optical box trap, a novel testbed for quantum many-body phenomena. Our work focuses on weakly interacting non-equilibrium systems, moderately interacting systems that are still in equilibrium, and the unitary Bose gas which is both strongly interacting and eludes equilibrium. We begin with studies of weakly-interacting gases far from equilibrium, which feature ties to nonlinear wave phenomena. Highlights of our experiments include the direct measurement of turbulent-cascade fluxes, which (alongside realizing a tuneable dissipation scale) allow us to demonstrate the zeroth law of turbulence, and the first observation of weak collapse, a general type of nonlinear wave collapse predicted over 40 years ago. We then turn to moderately strong interactions, confronting existing theories of interacting quantum fluids. One of our most important results is the first quantitative measurement of the quantum depletion of a Bose–Einstein condensate, confirming a 70-year-old theory first developed to describe liquid helium. The culmination of our work explores the unitary Bose gas, where interparticle interactions are as strong as allowed by the laws of quantum mechanics. This strongly-correlated state promises tantalizing possibilities, including emergent universal behavior set solely by the gas density and novel forms of superfluidity. However, the strong interactions also lead to a complex interplay between coherent and dissipative dynamics. By disentangling these two processes, we have caught a glimpse of the promises that the unitary Bose gas holds. In particular, we observe the emergence of universal behavior, and find that the gas features a well defined quasi-equilibrium state, with a non-zero condensed fraction.
This thesis describes studies of out-of-equilibrium dynamics in an interacting homogeneous Bose gas. We use the versatile platform of an ultracold gas of 39K, prepared in an optical box trap. The experiment features both tuneable interaction strengths and a homogeneous atomic density. We examine first a weakly interacting near-equilibrium system, followed by far-from-equilibrium dynamics under moderate interactions, and finally combine strong interactions with highly non-equilibrium dynamics in studies of the unitary Bose gas. We start with a theoretical study of the effects of three-body recombination on the temperature of a weakly interacting, partially condensed gas. Contrary to the expectation of heating, we find that for appropriate values of the condensed fraction and gas parameter, the system temperature is expected to decrease. Not only do we predict lower temperatures, however, but also for the condensate fraction to grow. We then present experiments in which we engineer a far-from-equilibrium momentum distribution, and study the dynamics as the gas relaxes. As the distribution evolves, we observe bidirectional dynamic scaling, with a flow of particles to lower momenta while energy is transported to higher ones. These flows are characterised by spatiotemporal scaling exponents, which we find do not vary appreciably over a range of moderate interaction strengths. Finally, we look to the unitary Bose gas, quenching clouds into the strongly interacting regime. This regime offers insight into universal behaviour that depends only on the interparticle spacing, as well as being an avenue for the study of Efimov physics. However, few things come for free; strongly interacting Bose systems are plagued by high atom-loss and heating rates. We study atom-loss rates and the dynamics of correlations in our gases, isolate the coherent effects from the dissipative ones, and identify a quasi-equilibrium state that features a non-zero condensate fraction.
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
Advances in trapping and cooling of ultracold gases over the last several decades have made it possible to test many formerly outstanding predictions from disparate branches of physics. This thesis touches on three historical problems that have found new life recently in the context of ultracold Bose gases of alkali atoms. The first problem revolves around an outstanding prediction from Boltzmann over a century and half old that the breathing mode of a isotropically trapped classical gas should oscillate indefinitely. I analyze recent experimental results [Nat. Phys. 11, 1009 (2015)], and attribute observed damping sources to trap imperfections. The second question is about the analogue of first and second sound modes from liquid helium in trapped dilute gases. I present the results of a joint theoretical/experimental investigation of the breathing mode of a finite temperature Bose-Einstein condensate (BEC), attributing a striking collapse revival behavior of the resultant oscillation to in-phase and out-of-phase normal modes of the thermal cloud and condensate. The third problem is that of the formation of Borromean ring-like three-body bound states, referred to as Efimov trimers, in strongly-interacting few-body systems. I extend the predicted spectrum of Efimov states into the realm of many degenerate internal levels, and investigate the difficult three-body elastic scattering problem.\nThese questions are part of the broader theme of this thesis: How can our understanding of few-body physics in the ultracold limit be translated into statements about the bulk behavior of an ultracold gas? For weakly-interacting Bose gases, this translation is well-known: the many-body properties of the gas are well-described by the tracking just the one and two particle correlations. I analyze a generalization of this procedure to higher order correlations, the general connection between few-body physics and correlations in a dilute gas, and results for the emergence of Efimov physics in the magnetic phase of the strongly-interacting Bose gas.
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.
We review turbulence in ultracold quantum gases, using the scalar contact-interaction Bose-Einstein condensate as the reference system for quantized circulation, compressibility, vortices, sound, and cascades. We focus on the quantitative diagnostics that connect helium and classical phenomenology to microscopic wave-function dynamics: incompressible and compressible kinetic-energy spectra, wave-occupation spectra, spectral fluxes, vortex-resolved correlations, and velocity statistics. These diagnostics distinguish equilibrium vortex organization, decaying turbulent relaxation, forced cascade dynamics, and weak-wave turbulence, and show why power laws alone are insufficient evidence for a cascade. We survey experiments on two-dimensional Onsager clustering, three-dimensional vortex-line turbulence, box-trap wave cascades, engineered dissipation, and turbulent equations of state. We close by briefly placing the contact-interaction scalar superfluid system in a broader landscape of nonlocal, multicomponent, fermionic, and driven-dissipative quantum fluids, where turbulence concepts can be tested for universality.
We study non-equilibrium and strongly-correlated dynamics in two contexts. We begin by analysing quantum many-body systems out of equilibrium through the lens of cold atomic impurities in Bose gases. Such highly-imbalanced mixtures provide a controlled arena for the study of interactions, dissipation, decoherence and transport in a many-body quantum environment. Specifically we investigate the oscillatory dynamics of a trapped and initially highly-localised impurity interacting with a weakly-interacting trapped quasi low-dimensional Bose gas. This relates to and goes beyond a recent experiment by the Inguscio group in Florence. We witness a delicate interplay between the self-trapping of the impurity and the inhomogeneity of the Bose gas, and describe the dissipation of the energy of the impurity through phononic excitations of the Bose gas. We then study the transport of a driven, periodically-trapped impurity through a quasi one-dimensional Bose gas. We show that placing the weakly-interacting Bose gas in a separate periodic potential leads to a phononic excitation spectrum that closely mimics those in solid state systems. As a result we show that the impurity-Bose gas system exhibits phonon-induced resonances in the impurity current that were predicted to occur in solids decades ago but never clearly observed. Following this, allowing the bosons to interact strongly, we predict the effect of different strongly-correlated phases of the Bose gas on the motion of the impurity. We show that, by observing the impurity, properties of the excitation spectrum of the Bose gas, e.g., gap and bandwidth, may be inferred along with the filling of the bosonic lattice. In other words the impurity acts as a probe of its environment. To describe the dynamics of such a strongly-correlated system we use the powerful and near-exact time-evolving block decimation (TEBD) method, which we describe in detail. The second part of this thesis then analyses, for the first time, the performance of this method when applied to simulate non-equilibrium classical stochastic processes. We study its efficacy for a well-understood model of transport, the totally-asymmetric exclusion process, and find it to be accurate. Next, motivated by the inefficiency of sampling-based numerical methods for high variance observables we adapt and apply TEBD to simulate a path-dependent observable whose variance increases exponentially with system size. Specifically we calculate the expected value of the exponential of the work done by a varying magnetic field on a one-dimensional Ising model undergoing Glauber dynamics. We confirm using Jarzynski's equality that the TEBD method remains accurate and efficient. Therefore TEBD and related methods complement and challenge the usual Monte Carlo-based simulators of non-equilibrium stochastic processes.