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
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.
The first- and second-order correlation functions of trapped interacting Bose-Einstein condensates are investigated numerically on a many-body level from first principles. Correlations in real space and momentum space are treated. The coherence properties are analyzed. The results are obtained by solving the many-body Schr\"odinger equation. It is shown in an example how many-body effects can be induced by the trap geometry. A generic fragmentation scenario of a condensate is considered. The correlation functions are discussed along a pathway from a single condensate to a fragmented condensate. It is shown that strong correlations can arise from the geometry of the trap, even at weak interaction strengths. The natural orbitals and natural geminals of the system are obtained and discussed. It is shown how the fragmentation of the condensate can be understood in terms of its natural geminals. The many-body results are compared to those of mean-field theory. The best solution within mean-field theory is obtained. The limits in which mean-field theories are valid are determined. In these limits the behavior of the correlation functions is explained within an analytical model.
The evolution of Bose-Einstein condensates is amply described by the time-dependent Gross-Pitaevskii mean-field theory which assumes all bosons to reside in a single time-dependent one-particle state throughout the propagation process. In this work, we go beyond mean field and develop an essentially exact many-body theory for the propagation of the time-dependent Schr\"odinger equation of
N interacting identical bosons. In our theory, the time-dependent many-boson wave function is written as a sum of permanents assembled from orthogonal one-particle functions, or orbitals, where both the expansion coefficients and the permanents (orbitals) themselves are time-dependent and fully determined according to a standard time-dependent variational principle. By employing either the usual Lagrangian formulation or the Dirac-Frenkel variational principle we arrive at two sets of coupled equations of motion, one for the orbitals and one for the expansion coefficients. The first set comprises of first-order differential equations in time and nonlinear integrodifferential equations in position space, whereas the second set consists of first-order differential equations with time-dependent coefficients. We call our theory multiconfigurational time-dependent Hartree for bosons, or
MCTDHB(M), where
M specifies the number of time-dependent orbitals used to construct the permanents. Numerical implementation of the theory is reported and illustrative numerical examples of many-body dynamics of trapped Bose-Einstein condensates are provided and discussed. The convergence of the method with a growing number
M of orbitals is demonstrated in a specific example of four interacting bosons in a double well.
In this thesis we study and characterize the behavior of Bose-Einstein condensates in a BJJ, using two different theoretical formalisms: the mean-field approximation (with the Gross-Pitaevskii equation) and many-body calculations (based on Bose-Hubbard models). \n\t\t\t\t With single-component Bose-Einstein condensates, we have focused on the study of the structure of the ground state as a function of the system parameters. We have looked for strongly correlated states, that cannot be described with mean-field theories, and we have proposed a variational wave function that captures the structure of the ground state for a broad interval of the system parameters. We have also studied the nonlinear effects of the Gross-Pitaevskii equation, visible when atom-atom interactions are strong. \n\t\t\t\t In the case of binary mixtures of Bose-Einstein condensates we have performed an intensive study of the different regimes that can arise and in which conditions. The standard two-mode approximation is one of the most used in the study of the Josephson effect, as it gives simple analytic equations that capture, to a great extend, the behavior of the system. When the link between condensates is not weak enough, one has to consider a correction to this approximation, namely, the improved two-mode approximation. In this thesis, we have derived this last approximation for the binary mixture and we have checked its validity comparing it with numerical simulations of the three-dimensional Gross-Pitaevskii equation. Moreover, as the Josephson dynamics is almost one-dimensional, we have considered the two most common reductions of the dimensionality of the Gross-Pitaevskii equation. In this case, we have also compared, using simulations, these one-dimensional reductions with the three-dimensional equation. \n\t\t\t\t We have also studied spinor condensates in an external BJJ. We have focused on condensates formed by atoms with spin
F=1, which can be in any of the three internal states
mF=0,pm1. Furthermore, in contrast to the binary mixture, spin interchange is allowed, so that the number of particles of each component becomes a dynamic variable. First, we have studied this system within the mean-field framework, using the Gross-Pitaevskii equation. We have derived the two-mode approximation equations and we have focused in studying the decoupling of the Josephson effect and the population transfer dynamics. In this case, we also compare the results with numerical simulations of the three dimensional Gross-Pitaevskii equation. Second, we have studied the spinor BJJ using the Bose-Hubbard formalism, because some features of quantum fluctuations are better captured than with the mean-field. We have characterized the ground state, paying special attention to the regions where it is strongly correlated. We have seen how the spin singlet formation (strongly correlated state between two particles) affects the structure of the ground state. \n\n\t\t\t\t Finally, we have studied finite temperature effects on spinor Bose-Einstein condensates in the presence of a magnetic field, for two different cases. First, we have analyzed a condensate formed by
F=1 atoms with contact interactions. We have considered that the condensate was formed by atoms in the internal state
mF=0 and we have studied the dependence of the fluctuations of the other two components
mF=pm1 as a function of temperature. We have used the Bogoliubov formalism applied to an homogeneous system, and then, we have generalized the result to an harmonic trap by using the local density approximation. Second, we have studied a condensate formed by particles with
F=3 with contact interactions and moreover, dipolar interactions. In a similar way as the previous case, we consider that the condensate is formed by particles with
mF=−3 and we study the fluctuations in
mF=−2 and
mF=−3.