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APPLICATION OF DISPERSIVE PDE TECHNIQUES TO THE STUDIES OF THE TIME-DEPENDENT HARTREE-FOCK-BOGOLIUBOV SYSTEM FOR BOSONS

Jacky J. Chong

Published Jan 1, 2019• 1 citation
Publisher

Abstract

The thesis provides rigorous quantitative analyses for studying quantum fluctuations of a non-relativistic Bose gas about a Bose-Einstein condensate. In recent years, the dynamics of the condensate and the excitations of a Bose gas was shown to be well approximated by the quasifree dynamics with governing equations given by a system of coupled nonlinear dispersive PDEs called the time-dependent Hartree-Fock-Bogoliubov (HFB) system (c.f. Grillakis and Machedon, J. Fix. Point Theory A., (2013), Grillakis and Machedon, Comm. PDEs, (2017), Bach et al. in arXiv:1602.05171v2, (2018), Benedikter et al., Ann. Henri Poincare, (2018)). However, both the quantitative and qualitative analysis of the time-dependent HFB system are still in their early stages of development. Thus, the primary purposes of the thesis are to further the development of some analytic tools necessary for studying the time-dependent HFB system and use these effective equations to provide quantitative estimates for the true dynamics of the Bose gas at absolute zero temperature in Fock space norm. The thesis comprises the entirety of the author's current and past projects on the time-dependent HFB system. Each project falls into one or more of two categories: studying the local or global well-posedness of the time-dependent HFB system, or obtaining global-in-time Fock space estimates for the error terms of the quasifree approximation to the dynamics of a system of interacting bosons. In the former category, the author employs techniques of dispersive PDE theory developed in the past three decades along with classical methods of harmonic analysis to study lower regularity solutions to the time-dependent HFB system. The lower regularity well-posedness of solutions for the time-dependent HFB system is necessary for studying norm approximation of the dynamics of a dilute Bose gas with strong interactions. In the latter category, we prove global a-priori estimates for the solutions to the HFB system and use them to obtain estimates for the error terms of the Fock space approximation.

DOI
10.13016/igti-hjun

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The thesis provides rigorous quantitative analyses for studying quantum fluctuations of a non-relativistic Bose gas about a Bose-Einstein condensate. In recent years, the dynamics of the condensate and the excitations of a Bose gas was shown to be well approximated by the quasifree dynamics with governing equations given by a system of coupled nonlinear dispersive PDEs called the time-dependent Hartree-Fock-Bogoliubov (HFB) system (c.f. Grillakis and Machedon, J. Fix. Point Theory A., (2013), Grillakis and Machedon, Comm. PDEs, (2017), Bach et al. in arXiv:1602.05171v2, (2018), Benedikter et al., Ann. Henri Poincare, (2018)). However, both the quantitative and qualitative analysis of the time-dependent HFB system are still in their early stages of development. Thus, the primary purposes of the thesis are to further the development of some analytic tools necessary for studying the time-dependent HFB system and use these effective equations to provide quantitative estimates for the true dynamics of the Bose gas at absolute zero temperature in Fock space norm. The thesis comprises the entirety of the author's current and past projects on the time-dependent HFB system. Each project falls into one or more of two categories: studying the local or global well-posedness of the time-dependent HFB system, or obtaining global-in-time Fock space estimates for the error terms of the quasifree approximation to the dynamics of a system of interacting bosons. In the former category, the author employs techniques of dispersive PDE theory developed in the past three decades along with classical methods of harmonic analysis to study lower regularity solutions to the time-dependent HFB system. The lower regularity well-posedness of solutions for the time-dependent HFB system is necessary for studying norm approximation of the dynamics of a dilute Bose gas with strong interactions. In the latter category, we prove global a-priori estimates for the solutions to the HFB system and use them to obtain estimates for the error terms of the Fock space approximation.
Abstract We introduce the map of dynamics of quantum Bose gases into dynamics of quasifree states, which we call the “nonlinear quasifree approximation”. We use this map to derive the time-dependent Hartree–Fock–Bogoliubov (HFB) equations describing the dynamics of quantum fluctuations around a Bose–Einstein condensate. We prove global well-posedness of the HFB equations for pair potentials satisfying suitable regularity conditions, and we establish important conservation laws. We show that the space of solutions of the HFB equations has a symplectic structure reminiscent of a Hamiltonian system. This is then used to relate the HFB equations to the HFB eigenvalue equations discussed in the physics literature. We also construct Gibbs equilibrium states at positive temperature associated with the HFB equations, and we establish criteria for the appearance of Bose–Einstein condensation.
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 lll particles out of their total number N not in the condensate is exponentially small in the number lll. 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.
This thesis proves certain results concerning an important question in non-equilibrium quantum statistical mechanics which is the derivation of effective evolution equations approximating the dynamics of a system of large number of bosons initially at equilibrium (ground state at very low temperatures). The dynamics of such systems are governed by the time-dependent linear many-body Schroedinger equation from which it is typically difficult to extract useful information due to the number of particles being large. We will study quantitatively (i.e. with explicit bounds on the error) how a suitable one particle non-linear Schroedinger equation arises in the mean field limit as number of particles N → ∞ and how the appropriate corrections to the mean field will provide better approximations of the exact dynamics. In the first part of this thesis we consider the evolution of N bosons, where N is large, with two-body interactions of the form N³ᵝv(Nᵝ⋅), 0≤β≤1. The parameter β measures the strength and the range of interactions. We compare the exact evolution with an approximation which considers the evolution of a mean field coupled with an appropriate description of pair excitations, see [18,19] by Grillakis-Machedon-Margetis. We extend the results for 0 ≤ β < 1/3 in [19, 20] to the case of β < 1/2 and obtain an error bound of the form p(t)/Nᵅ, where α>0 and p(t) is a polynomial, which implies a specific rate of convergence as N → ∞. In the second part, utilizing estimates of the type discussed in the first part, we compare the exact evolution with the mean field approximation in the sense of marginals. We prove that the exact evolution is close to the approximate in trace norm for times of the order o(1)√N compared to log(o(1)N) as obtained in Chen-Lee-Schlein [6] for the Hartree evolution. Estimates of similar type are obtained for stronger interactions as well.
This PhD thesis contains new results on the mathematical study of Bose-Einstein condensation and the main part of it is devoted to mixtures of condensates, i.e., systems composed of multiple bosonic species in interaction. We prove the validity of effective ground state theories for mixtures in the Gross-Pitaevskii and mean-field regime. We show that the ground state energy asymptotics, in the large-N limit, is captured by the minimum of a suitable one-body functional. Moreover, we prove that in the ground state all species exhibit Bose-Einstein condensation onto the minimizer of that functional. For mixtures in the mean-field regime, we provide a rigorous justification of Bogoliubov’s theory. This is done by computing the contribution to the ground state energy which is due to excited particles. We also prove a norm approximation for the ground state vector, in the Fock space norm. From the time-dependent viewpoint, we prove for the first time the validity of the effective equations that were previously known due to heuristic physical arguments, and that are confirmed by robust experimental evidence. Our results show that, for mixtures in the Gross-Pitaevskii and mean-field regime, the effective dynamics is governed by a system of non-linear Schrödinger equations, one for each species of the mixture. In the final part of the thesis we present additional results on problems and models related to the study on mixtures. We were able to derive the effective dynamics for spinor- and pseudo-spinor condensates. The equations that we obtain are precisely those of modern experiments with ultra-cold spin bosons. We also show that the mean-field model provide a time-dependent control of condensation that is very accurate for the typical duration times of experiments. A further result is the global well-posedness in the energy space of the singular Hartree equation. Last, we present new remarks on the adaptation of known techniques that one needs in order to prove the derivation of the magnetic Gross-Pitaevskii equation.
This thesis investigates the rigorous derivation of non-linear effective equations describing the time evolution of Bose-Einstein condensation from the N-particle linear Schrödinger equation, focusing on the one-particle reduced density matrix. The first chapter outlines the mathematical framework of many-body quantum mechanics, including the Fock space formalism. The second chapter studies the mean-field regime, proving that the condensed state converges to the non-linear Hartree equation by analyzing fluctuation dynamics to quantify the convergence rate. The third chapter addresses the physically realistic but mathematically challenging Gross-Pitaevskii regime, characterized by rare but strong interactions. Here, Bogoliubov transformations are used to capture crucial microscopic short-scale correlations, allowing for a quantitative derivation of the cubic non-linear Schrödinger equation. While the preceding chapters focus exclusively on repulsive interactions, the fourth chapter explores the focusing case with attractive potentials, which can lead to finite-time blow-up. Operating in an intermediate scaling regime, the effective focusing cubic non-linear Schrödinger equation is rigorously derived. Since the negative energy cannot bound many-body fluctuations, convergence is proven by approximating the dynamics via a quadratic Hamiltonian, in the spirit of Bogoliubov theory, and avoiding Grönwall estimates through the analysis of a coupled system of one-body linear equations for the one-particle density matrices.