Abstract Fragmented Bose–Einstein condensates are large systems of identical bosons displaying multiple macroscopic occupations of one-body states, in a suitable sense. The quest for an effective dynamics of the fragmented condensate at the leading order in the number of particles, in analogy to the much more controlled scenario for complete condensation in one single state, is deceptive both because characterising fragmentation solely in terms of reduced density matrices is unsatisfactory and ambiguous, and because as soon as the time evolution starts the rank of the reduced marginals generically passes from finite to infinite, which is a signature of a transfer of occupations on infinitely many more one-body states. In this work we review these difficulties, we refine previous characterisations of fragmented condensates in terms of marginals, and we provide a quantitative rate of convergence to the leading effective dynamics in the double limit of infinitely many particles and infinite energy gap.
Abstract Fragmented Bose–Einstein condensates are large systems of identical bosons displaying multiple macroscopic occupations of one-body states, in a suitable sense. The quest for an effective dynamics of the fragmented condensate at the leading order in the number of particles, in analogy to the much more controlled scenario for complete condensation in one single state, is deceptive both because characterising fragmentation solely in terms of reduced density matrices is unsatisfactory and ambiguous, and because as soon as the time evolution starts the rank of the reduced marginals generically passes from finite to infinite, which is a signature of a transfer of occupations on infinitely many more one-body states. In this work we review these difficulties, we refine previous characterisations of fragmented condensates in terms of marginals, and we provide a quantitative rate of convergence to the leading effective dynamics in the double limit of infinitely many particles and infinite energy gap.
preprint: SISSA 39/2007/mp Emergence and applications of the ubiquitous tool of reduced density matrices in the rigorous analysis of Bose Einstein condensation is reviewed, and new related results are added. The need and the nature of scaling limits of infinitely many particles is discussed, which imposes that a physically meaningful and mathematically well-posed definition of asymptotic condensation is placed at the level of marginals. The topic of correlations in the condensed state is addressed in order to show their influence at this level of marginals, both in the true condensed state and in the suitable trial functions one introduces to approximate the many-body structure and energy. Complete condensation is shown to be equivalently defined at any fixed kbody level, both for pure and mixed states. Further, it is proven to be equivalent to some other characterizations in terms of asymptotic factorization of the many-body state, which are currently present in the literature.
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 can be understood as a macroscopic occupation of a single-particle state. But an alternative definition, which requires only a macroscopic occupation of an arbitrarily small energy band of single-particle states, is also known, referred to as generalized Bose–Einstein condensation, and considered to be thermodynamically more stable. Depending on the number of macroscopically occupied single-particle states in the condensate, three types are then distin-guished. If the number of macroscopically occupied single-particle states is finite but at least one, then the condensation is said to be of type-I. A type-II condensate occurs if infinitely many single-particle states are macroscopically occupied. If a generalized Bose–Einstein condensation without any macroscopically occupied single-particle states is present, then one speaks of a type-III con-densation. In noninteracting Bose gases, it is generally easier to prove the occurrence of generalized Bose–Einstein condensation than to determine its type. While a main step for the former task is to show that the critical density is finite, quite detailed knowledge of the energy gaps of the cor-responding single-particle Hamiltonian at the lower edge of the spectrum is needed for the latter one. In this dissertation, we study the occurrence of a Bose–Einstein condensate in Bose gases that are placed in metrically transitive random potentials. We mainly consider the Poisson random potential. The thermodynamic limit and Dirichlet boundary conditions are used throughout this work. We explore noninteracting Bose gases in arbitrary dimension at positive temperatures and a one-dimensional interacting Bose gas at zero temperature. We provide fairly general sufficient conditions for the random potential that imply the almost sure occurrence of generalized Bose–Einstein condensation in noninteracting Bose gases. These conditions are fulfilled by a Poisson random potential, depending on its single-impurity potential. They are also met by the one-dimensional Luttinger–Sy model, which is a central system in the research area of Bose gases in random potentials, and a generalization thereof. Moreover, we state sufficient energy gap conditions such that one can conclude a type-I condensation in Poisson random potentials. We consider different probabilistic notions here, such as convergence in the rth mean, r P N, in probability, and with probability almost one. In particular, we prove that in the Luttinger–Sy model with infinitely strong random potential a type-I condensation occurs in the rth mean, r P N, if the particle density is greater than the critical one. For the Luttinger–Sy model with a finitely strong random potential, we show the same statement, but with probability almost one. Finally, we discuss the Luttinger–Sy model with a repulsive contact interaction between the particles at zero temperature. We provide sufficient requirements such that a generalized Bose–Einstein condensate almost surely occurs. We also show that the type of the condensate is altered by the interparticle interaction. In particular, we demonstrate a transition from a type-I to a type-III condensate when the strength of the contact interaction is increased. Furthermore, we prove that the interparticle interaction reduces the local particle density.
Linear response of simple (i.e., condensed) Bose-Einstein condensates is known to lead to the Bogoliubov--de Gennes equations. Here, we derive linear response for fragmented Bose-Einstein condensates, i.e., for the case where the many-body wave function is not a product of one, but of several single-particle states (orbitals). This gives one access to excitation spectra and response amplitudes of systems beyond the Gross-Pitaevskii description. Our approach is based on the number-conserving variational time-dependent mean-field theory [Alon, Streltsov, and Cederbaum, Phys. Lett. A 362, 453 (2007)], which describes the time evolution of best-mean-field states. Correspondingly, we call our linear-response theory for fragmented states LR-BMF. In the derivation it follows naturally that excitations are orthogonal to the ground-state orbitals. As applications excitation spectra of Bose-Einstein condensates in double-well potentials are calculated. Both symmetric and asymmetric double wells are studied for several interaction strengths and barrier heights. The cases of condensed and twofold fragmented ground states are compared. Interestingly, even in such situations where the response frequencies of the two cases are computed to be close to each other, which is the situation for the excitations well below the barrier, striking differences in the density response in momentum space are found. For excitations with an energy of the order of the barrier height, both the energies and the density response of condensed and fragmented systems are very different. In fragmented systems there is a class of ``swapped'' excitations where an atom is transferred to the neighboring well. The mechanism of its origin is discussed. In asymmetric wells, the response of a fragmented system is purely local (i.e., finite in either one or the other well) with different frequencies for the left and right fragments. This finding is in stark contrast to that for condensed systems.
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 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.