We consider a gas of indistinguishable bosons, confined to a ring of radius R, and interacting via a pair-wise cosine potential. This may be thought of as the quantized Hamiltonian Mean Field (HMF) model for bosons originally introduced by Chavanis as a generalization of Antoni and Ruffo’s classical model. This thesis contains three parts: In part one, the dynamics of a Bose-condensate are considered by studying a generalized Gross-Pitaevskii equation (GGPE). Quantum effects due to the quantum pressure are found to substantially alter the system’s dynamics, and can serve to inhibit a pathological instability for repulsive interactions. The non-commutativity of the large-N , long-time, and classical limits is discussed. In part two, we consider the GGPE studied above and seek static solutions. Exact solutions are identified by solving a non-linear eigenvalue problem which is closely related to the Mathieu equation. Stationary solutions are identified as solitary waves (or solitons) due to their small spatial extent and the system’s underlying Galilean invariance. Asymptotic series are developed to give an analytic solution to the non- linear eigenvalue problem, and these are then used to study the stability of the solitary wave mentioned above. In part three, the exact solutions outlined above are used to study quantum fluctuations of gapless excitations in the HMF model’s symmetry broken phase. It is found that this phase is destroyed at zero temperature by large quantum fluctuations. This demonstrates that mean-field theory is not exact, and can in fact be qualitatively wrong, for long-range interacting quantum systems, in contrast to conventional wisdom.
We consider a gas of indistinguishable bosons, confined to a ring of radius R, and interacting via a pair-wise cosine potential. This may be thought of as the quantized Hamiltonian Mean Field (HMF) model for bosons originally introduced by Chavanis as a generalization of Antoni and Ruffo’s classical model. This thesis contains three parts: In part one, the dynamics of a Bose-condensate are considered by studying a generalized Gross-Pitaevskii equation (GGPE). Quantum effects due to the quantum pressure are found to substantially alter the system’s dynamics, and can serve to inhibit a pathological instability for repulsive interactions. The non-commutativity of the large-N , long-time, and classical limits is discussed. In part two, we consider the GGPE studied above and seek static solutions. Exact solutions are identified by solving a non-linear eigenvalue problem which is closely related to the Mathieu equation. Stationary solutions are identified as solitary waves (or solitons) due to their small spatial extent and the system’s underlying Galilean invariance. Asymptotic series are developed to give an analytic solution to the non- linear eigenvalue problem, and these are then used to study the stability of the solitary wave mentioned above. In part three, the exact solutions outlined above are used to study quantum fluctuations of gapless excitations in the HMF model’s symmetry broken phase. It is found that this phase is destroyed at zero temperature by large quantum fluctuations. This demonstrates that mean-field theory is not exact, and can in fact be qualitatively wrong, for long-range interacting quantum systems, in contrast to conventional wisdom.
The Hamiltonian mean-field (HMF) model describes particles on a ring interacting via a cosine interaction, or equivalently, rotors coupled by infinite-range XY interactions. Conceived as a generic statistical mechanical model for long-range interactions such as gravity (of which the cosine is the first Fourier component), it has recently been used to account for self-organization in experiments on cold atoms with long-range optically mediated interactions. The significance of the HMF model lies in its ability to capture the universal effects of long-range interactions and yet be exactly solvable in the canonical ensemble. In this work we consider the quantum version of the HMF model in one dimension and provide a classification of all possible stationary solutions of its generalized Gross-Pitaevskii equation (GGPE), which is both nonlinear and nonlocal. The exact solutions are Mathieu functions that obey a nonlinear relation between the wave function and the depth of the mean-field potential, and we identify them as bright solitons. Using a Galilean transformation these solutions can be boosted to finite velocity and are increasingly localized as the mean-field potential becomes deeper. In contrast to the usual local GPE, the HMF case features a tower of solitons, each with a different number of nodes. Our results suggest that long-range interactions support solitary waves in a novel manner relative to the short-range case.
We study the thermodynamics of quantum particles with long-range interactions at T = 0. Specifically, we generalize the Hamiltonian mean-field (HMF) model to the case of bosons. We consider the Hartree approximation that becomes exact in a proper thermodynamic limit with a coupling constant k ∼ 1/ N . The equilibrium configurations are solutions of the mean-field Schrödinger equation with a cosine interaction. We show that the homogeneous phase, which is unstable in the classical regime, becomes stable in the quantum regime. The homogeneous phase is stabilized by the Heisenberg uncertainty principle. This takes place through a second-order phase transition where the control parameter is the normalized Planck constant. The homogeneous phase is unstable for and stable for . The inhomogeneous phase is stable for and disappears for . We point out analogies between the bosonic HMF model and the concept of boson stars in astrophysics. We also discuss the differences between bosons and fermions for what concerns the thermodynamic limit, the order of the phase transition and the form of the density profiles.
It is widely believed that mean-field theory is exact for a wide range of classical long-range interacting systems. Is this also true once quantum fluctuations have been accounted for? As a test case we study the Hamiltonian mean-field (HMF) model for a system of bosons which is predicted (according to mean-field theory) to undergo a second-order quantum phase transition at zero temperature. The ordered phase is characterized by a spontaneously broken O(2) symmetry, which, despite occurring in a one-dimensional model, is not ruled out by the Mermin-Wagner theorem due to the presence of long-range interactions. Nevertheless, a spontaneously broken symmetry implies gapless Goldstone modes whose large fluctuations can restore broken symmetries. In this work we study the influence of quantum fluctuations by projecting the Hamiltonian onto the continuous subspace of symmetry-breaking mean-field states. We find that the energetic cost of gradients in the center-of-mass wave function inhibits the breaking of the O(2) symmetry, but that the energetic cost is very small, scaling as O(1/N^{2}). Nevertheless, for any finite N, no matter how large, this implies that the ground state has a restored O(2) symmetry. Implications for the finite-temperature phases, as well as the classical limit, of the HMF model are discussed.
The central theme of this thesis is that of coherence in bosonic matter-wave fields. Analogous to optical coherence; the measure of the capacity of a photonic beam to produce fringes upon interference; matter-wave coherence is in many ways the defining characteristic of the BEC phase of identical massive bosons. Its study therefore plays a critical role in the quest to better understand the physics of ultra-cold quantum gases, which has been the goal of this research. This work approaches coherence from a number of different angles and is for this reason presented in two parts. The first of these delves into the physics of perfectly coherent matterwave fields which are adequately described by ‘mean field’ equations, while the second is devoted to origins and repercussions of the diminished coherence evident in actual atom laser experiments. The presentation of the original work of this project begins with a detailed study of the interaction between an atomic and a molecular condensate, coherently coupled at zero temperature via one of several possible mechanisms (a Feshbach resonance, for example) in three spatial dimensions. Despite the presence of repulsive scattering interaction between the atoms, the mean-field equations of motion are known to support bright solitary waves — time-independent solutions which remain localised even in the absence of a trapping potential. (These solutions are the matter-wave analogue of spatio-temporal solitons in quadratic nonlinear media, but can also represent stationary states of the Schr¨odinger-Newton equations describing bosons interacting via a gravitational field.) Ignoring all interactions save the repulsive inter-atomic scattering mentioned, a variational ansatz approach is employed to analytically approximate these stationary solutions using Gaussian density distributions of the atomic and molecular clouds. By evolving these approximate solutions numerically, a comprehensive map of the accessible parameter space is constructed, showing primarily stable localised propagation. Parameter space regions where the approximate Gaussian solutions display strong ‘breathing’ oscillations are identified, and a modified ansatz having exponentially decaying tails is found to dramatically reduce the amplitude of these oscillations. Finally, by seeding a numerical relaxation algorithm with our Gaussian solutions, we find that the only parameter space region in which the approximate solutions diffuse under propagation is also devoid of exact localised stationary solutions. The discussion then proceeds beyond the scope of mean field theory, and turns to a detailed treatment of the quantum mechanical observable corresponding to the centre of mass of systems of identical particles. Firstly, careful consideration is given to its operational definition in the presence of fluctuating particle numbers. Possible pitfalls due to naive approximations when the number fluctuations are large are also addressed. Attention is then given to the fundamental quantum noise associated with this variable, which provides a physical mechanism for diminished first order coherence in degenerate Bose gases, even at zero temperature. Characteristic lower bounds (standard quantum limits) to the resulting mean position uncertainty are identified both for systems of identical bosons and identical fermions with comparable density profiles. Interestingly, fermions are found to have an intrinsically lower mean-position uncertainty than bosons, a fact which is attributed to the action of Pauli repulsion between atoms confined to a fixed density distribution. The results of numerical simulations of the exact quantum evolution of mean position variance are then presented. These demonstrate firstly the quantum diffusion of the mean position of one dimensional matter-wave solitons and secondly the evolution of variance in the centre of mass of a Bose gas undergoing forced evaporative cooling. Finally, attention is paid to the fact that the Penrose-Onsager criterion, which essentially equates Bose condensation with perfect first-order matter-wave coherence, fails to correctly detect the presence of BEC when a priori knowledge of the exact condensate mode is imperfect or unavailable. Insight is gained by drawing analogies between this criterion and entropic measures of entanglement between particular system subspaces. This motivates employment of the concept of entanglement of formation as a means to providing a more accurate measure of condensation in the general case of mixed system states. Unfortunately, this new measure remains adversely sensitive to superpositions of condensate modes. For this reason, alternative criteria based on higher order particle number correlation functions are also proposed.
We study the thermodynamics of quantum particles with long-range interactions at T=0. Specifically, we generalize the Hamiltonian Mean Field (HMF) model to the case of fermions and bosons. In the case of fermions, we consider the Thomas-Fermi approximation that becomes exact in a proper thermodynamic limit. The equilibrium configurations, described by the Fermi (or waterbag) distribution, are equivalent to polytropes with index n=1/2. In the case of bosons, we consider the Hartree approximation that becomes exact in a proper thermodynamic limit. The equilibrium configurations are solutions of the mean field Schrödinger equation with a cosine interaction. We show that the homogeneous phase, that is unstable in the classical regime, becomes stable in the quantum regime. This takes place through a first order phase transition for fermions and through a second order phase transition for bosons where the control parameter is the normalized Planck constant. In the case of fermions, the homogeneous phase is stabilized by the Pauli exclusion principle while for bosons the stabilization is due to the Heisenberg uncertainty principle. As a result, the thermodynamic limit is different for fermions and bosons. We point out analogies between the quantum HMF model and the concepts of fermion and boson stars in astrophysics. Finally, as a by-product of our analysis, we obtain new results concerning the Vlasov dynamical stability of the waterbag distribution.
This thesis is about the derivation of effective mean field equations and their next-order corrections starting from nonrelativistic many-body quantum theory. Mean field equations provide an approximate ansatz for the description of interacting many-particle systems. In this ansatz the interaction between the particles is replaced by a self-consistent external potential leading to an effective one-body description of the many-particle system. Next-order corrections provide an approximation which goes one step further and tries to capture also subleading effects that are not resolved by the mean field ansatz. We present mathematical proofs for the validity of such effective theories for different models that are motivated, e.g., from the theory of ultracold atoms (the bosonic Hartree equation and the corresponding Bogoliubov theory) and from plasma physics (the motion of a tracer particle through a degenerate and dense electron gas). Starting from a many-body Schrödinger equation, our goal is to show that the solutions converge in a particular limit to the solutions to an effective mean field equation and its next-order corrections. After a short introduction and a summary in Chapter one, we present the main part of this work in three self-contained chapters. In Chapter two we analyze the dynamics of a large number N of nonrelativistic bosons in the weak coupling limit, i.e., for a coupling constant g_N=1/N. It is well known that in the limit of infinite particle number, the Hartree equation emerges as an effective one-particle theory of the Bose gas. This is closely related to the remarkable physical phenomenon of Bose-Einstein condensation at low temperature, namely that the majority of particles in a Bose gas occupies the same copy of a single one-particle quantum state. Our emphasis lies in the description of the few particles that fluctuate around the Bose-Einstein condensate. We show convergence of the fully interacting dynamics to an auxiliary time evolution in the norm of the N-particle space. This result allows us to prove several other assertions. Among other things, it is used to derive the Hartree equation with optimal speed of convergence 1/N for initial states that are close to ground states of interacting systems and also to prove convergence of the N-particle solution towards a time evolution obtained from the Bogoliubov Hamiltonian on Fock space. Chapter three is about the low energy properties of the weakly interacting homogeneous Bose gas. Here, we derive a novel estimate for low energy eigenfunctions stating that the probability for finding l particles out of their total number N not in the condensate is exponentially small in the number l. Using this bound, we then prove that the ground state wave function of the microscopic model satisfies certain quasifree type properties. The exponential decay is moreover used to provide an alternative proof for the validity of Bogoliubov's approximation for the low-lying energy eigenvalues. Bogoliubov theory states that the excitation energies of the Bose gas are given by excitations of free quasiparticles obeying an effective energy-momentum dispersion relation which is linear for small momentum. The linearity of the effective dispersion relation is an essential ingredient for the explanation of the superfluid character of the Bose gas. In Chapter four we study the time evolution of a single particle coupled through a pair potential to a dense and homogeneous ideal Fermi gas in two spatial dimensions. This type of model is well known in plasma physics where it is used to describe the energy loss of ions moving through a dense and degenerate electron gas. We analyze the model for a coupling parameter g=1 and prove closeness of the time evolution to an effective dynamics for large densities of the gas and for long time scales of the order of some power of the density. The effective dynamics is generated by the free Hamiltonian with a large but constant energy shift. To leading order, this energy shift is given by the spatially homogeneous mean field potential produced by the gas particles, whereas at next-to-leading order, one has to consider an additional correction to the mean field energy which is due to so-called recollision processes.