Summary form only given. The dilute gas experiments on the Bose-Einstein condensation of magnetically confined alkali vapors have already generated widespread interest across a broad spectrum of traditional fields of physics. They have provided an exciting stage on which to build bridges linking the theory of complex and collective phenomena in superconducting and superfluid systems with the single particle microscopic physics described in quantum optics and laser physics. The approach I discuss is based on a derivation of the quantum kinetic theory from first principles. Based on a microscopic picture of the weakly interacting bosonic gas, a generalized kinetic theory for a Markovian many-body density operator is derived. The density operator is functionally dependent on a few key variables, which must be carefully chosen. These quantities serve as master variables and determine the system's evolution on a coarse-grained time scale. The weak interactions allow a perturbative expansion of the evolution, from which kinetic equations that describe the dynamics of the expectation value of any single-time observable are obtained. The final structure of the theory has many analogies with the quantum theory of the laser, such as amplitude and phase noise for the macroscopic field, squeezing dynamics, and a nonlinear master equation describing the underlying evolution.
Summary form only given. The dilute gas experiments on the Bose-Einstein condensation of magnetically confined alkali vapors have already generated widespread interest across a broad spectrum of traditional fields of physics. They have provided an exciting stage on which to build bridges linking the theory of complex and collective phenomena in superconducting and superfluid systems with the single particle microscopic physics described in quantum optics and laser physics. The approach I discuss is based on a derivation of the quantum kinetic theory from first principles. Based on a microscopic picture of the weakly interacting bosonic gas, a generalized kinetic theory for a Markovian many-body density operator is derived. The density operator is functionally dependent on a few key variables, which must be carefully chosen. These quantities serve as master variables and determine the system's evolution on a coarse-grained time scale. The weak interactions allow a perturbative expansion of the evolution, from which kinetic equations that describe the dynamics of the expectation value of any single-time observable are obtained. The final structure of the theory has many analogies with the quantum theory of the laser, such as amplitude and phase noise for the macroscopic field, squeezing dynamics, and a nonlinear master equation describing the underlying evolution.
The most salient features of the Bose-Einstein condensation of a magnetically confined alkali vapor is the diluteness of the gas and the extremely weak effective interactions. From a theoretical point of view, the interesting aspect is the potential formulation of the many-body quantum theory for a non-uniform and potentially non-equilibrium system founded entirely on microscopic physics. The crucial postulate is the rapid attenuation of many particle quantum correlations in the dilute system which can be motivated from universal considerations. In principle, it will be possible to provide direct comparison between theory and experiment over all temperature scales with no phenomenological parameters — a challenge facing the theoretical community in the near future. The dilute gas experiments provide an exciting stage on which to build bridges linking the theory of complex and collective phenomena in superconducting and superfluid systems, with the single particle microscopic physics described in quantum optics and laser physics.
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
"Motivated by experiments on cold alkali atoms, I present a theoretical study of weakly interacting quantum degenerate particles. These experiments observe a wide range of phenomena and consequently this thesis has a broad scope, involving issues of coherence, stability, superfluidity, and kinetics. I explore six topics which exemplify the rich and exciting physics of cold alkali atoms: first, the effect of interactions on the transition temperature of a dilute Bose gas; second, the connection between broken non-gauge symmetries and the appearance of multiple condensates in a Bose gas; third, the role of thermally activated ""phase slip"" events in destroying superfluidity near the critical temperature of a Bose gas; fourth, mechanical instabilities in clouds of attractive bosons; fifth, the kinetics of a gas of partially condensed atoms; and sixth, the nonlinear optical properties of an atomic gases."
In the first part of this thesis, we present a unified kinetic theory that describes the finite-temperature, non-equilibrium dynamics of a Bose-Einstein condensed gas interacting with a thermal cloud in a trap. This theory includes binary interactions to second order in the interaction potential and reduces to a diagonal quantum Boltzmann equation for Bogoliubov quasiparticles. The Hartree-Fock-Bogoliubov interactions include the pairing field and are expressed as many-body T matrices to second order. The interactions thus include the correct renormalized scattering physics. This renormalized theory is automatically gapless. Thus, the excited Bogoliubov modes are naturally orthogonal to the condensate ground state. This kinetic theory is a complete second-order theory that reduces to the Gross-Pitaevskii equation and the quantum Boltzmann equation in the respective limits and thus is capable of describing the system over a wide temperature range. In the second part, we consider a many-body theory of a dilute Fermi gas near a Feshbach resonance. Experiments explore the crossover physics between the Bardeen-Cooper-Schrieffer (BCS) superfluidity of a two-spin Fermi gas, and the Bose-Einstein condensation (BEC) of composite bosons. We consider correlations between a composite boson and a fermion pair and show that such correlations are the minimal ingredients needed in a many-body theory to generate the correct boson-boson scattering length in the Bose-Einstein limit of the crossover. We also use imaginary-time propagation to find zero-temperature ground states in the BCS/BEC crossover. A cumulant expansion allows us to systematically include higher-order interactions between bosons and fermions. In particular, we calculate the Hartree term across the resonance. We further apply the cumulant-expansion method to thermal fermions and composite bosons interacting above the transition temperature in the normal phase. We numerically calculate the full time dependence in ramps across the resonance in this regime and find different two-body and many-body time scales in the system. We calculate molecular conversion efficiencies as a function of temperature and phase-space density, and find good agreement with results from JILA potassium experiments.
The condensation of weakly interacting alkali atom gases observed in 1995 is a manifestation of the macroscopic quantum phenomenon called Bose-Einstein condensation. In contrast to superconductivity and Helium superfluids, the dilute alkali atom condensates are directly observable using optical imaging techniques and they constitute highly controllable quantum systems. Hence, they offer an ideal test bench to investigate the applicability of quantum field theories and to predict the properties of the condensates starting from the first principles.\n\nSince the first experimental realizations of dilute Bose-Einstein condensates, a great interest has been focused on this field, resulting in numerous experimental studies concerning, for example, nonlinear dynamics and the elementary excitations of these many-particle quantum systems. The existence of topological phase singularities, quantized vortices, first predicted theoretically and finally observed experimentally in 1999, led to another series of pioneering theoretical and experimental studies.\n\nSeveral methods to create single-quantum vortices were realized already in 2001 but, however, the puzzle of creating multiply quantized vortices still remained. In this thesis, the so-called topological method to create two and four times quantized vortices is studied in detail. Since the instrumentation required for the experimental realization of the topological method had already been installed in many laboratories, the theoretical predictions of this thesis were experimentally verified shortly after their publication. Multiply quantized vortices are energetically unstable and tend to split into singly quantized vortices. However, the dissipation of energy in the condensates is minimal at low temperatures and, consequently, can lead to long lifetimes for the multiquantum vortices. Nevertheless, one prediction of this thesis is that the lifetimes of doubly quantized vortices are relatively short, owing to the known dynamical instability of the state. The splitting of the doubly quantized vortex has recently been observed experimentally.\n\nThe stability of vortices in dilute Bose-Einstein condensates is intimately related to the excitation spectrum of the stationary vortex state. However, developing an accurate, computationally feasible finite-temperature quantum field theory remains an open problem. In this thesis, we study the elementary excitations of an irrotational condensate within a recently developed systematic second order perturbation theory. The collapse and revival of certain elementary excitations is discovered.\n\nThe most recent studies of this thesis are devoted to stationary, but rotationally asymmetric vortex states. In weakly interacting condensates, a novel stationary state, a vortex tripole holding finite angular momentum is presented. Also the lowest elementary excitations of the so-called vortex dipole, tripole, and quadrupole states are studied.