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
In Chapter 6, we derived a generalized Gross–Pitaevskii condensate equation which is coupled to a kinetic equation for the distribution function for the thermal atoms. However, the kinetic equation in Chapter 6 is only valid in the semiclassical limit. It involves the assumption that the thermal energy k B T is much greater than the spacing between the harmonic trap energy levels ( k B T ≫ω 0 where ω 0 is the trap frequency) and also much greater than the average interaction energy ( k B T ≫ gn ). The ZNG model, based on HF excitations, is still expected to be adequate down to quite low temperatures in trapped Bose gases, as will be shown by the results in Chapter 12. However, the ZNG model will break down at very low temperatures, where the Hartree–Fock excitations must be replaced by the Bogoliubov spectrum. To deal with this, one has to derive a kinetic equation for the Bogoliubov quasiparticle excitations. This is the goal of the present chapter. In this chapter, we use the second-order Beliaev approximation to discuss the nonequilibrium dynamics of a trapped Bose-condensed gas at finite temperatures. In doing to, we combine the second-order Beliaev self-energies with the lower-order Bogoliubov excitation spectrum, including off-diagonal single-particle propagators but still omitting the anomalous correlation functions. This last condition defines the Bogoliubov–Popov approximation. In this chapter, we consider only the damping which arises from collisions. We will not explicitly calculate corrections that are second-order in g to the quasiparticle energy spectrum or to the condensate chemical potential, both of which are associated with the real parts of the second-order Beliaev self-energies. The present chapter is a natural generalization of work presented in Chapter 6 for the simpler HF excitation spectrum assumed in the ZNG analysis.
We present a generalization of the Hartree-Fock Bogoliubov (HFB) theory in which the coupling between one and two quasi-particles is taken into account.This is done by writing the excitation operators as linear combinations of one and two HFB quasi-particles. The excitation energies and the quasi-particle amplitudes are given by generalized Bogoliubov equations. The excitation spectrum has two branches. The first one is a discrete branch which is gapless and has a phonon character at large wave-length and, contrarily to HFB, is always stable. This branch is detached from a continuum branch whose threshold, at fixed total momentum, coincides with the two quasi-particle threshold of the HFB theory. The gap between these two branches at P=0 is equal to two times the HFB gap, which then provides for the relevant energy scale. We also give numerical results for a specific case.
This work is concerned with a microscopic model of a nonideal boson gas. The Hamiltonian is treated in a rigorous manner free of all approximations and, in particular, the Bogoliubov approximation of replacing the zero-momentum single-particle creation and destruction operators by a
c number is entirely avoided. Utilizing a method due to Wentzel, the Hamiltonian is replaced by one involving products of two second quantized operators which, in general, can be diagonalized. This replacement is shown to be rigorously valid in the infinite volume limit for all temperatures despite the avoidance of the Bogoliubov approximation. It is found that the assumption of a positive-definite quasiparticle excitation spectrum $\ensuremath{\epsilon}(\mathrm{k})$ becomes untenable for temperatures
T less than a critical value
Tc, if the interparticle potential is described by a certain kernel which possesses positive eigenvalues only, corresponding to a sufficiently repulsive interparticle force. Assuming that $\ensuremath{\epsilon}(\mathrm{k})=0$ only for k=0, it is found that for
T<Tc the system undergoes an Einstein condensation into the k=0 single-particle state, and the integral equations characterizing the system are precisely the same as would be obtained if the Bogoliubov approximation were made, proving within the framework of this model the strict validity of this approximation procedure. The well-known criterion for an Einstein condensation
v(0)>0, where
v(k) is the Fourier transform of the interparticle potential, is shown to be a weaker statement of our eigenvalue criterion. Further, in agreement with Girardeau and Arnowitt, it is found that an energy gap separates $\ensuremath{\epsilon}(0)$ and $\ensuremath{\epsilon}(\mathrm{k})$ for
k>0. An approximation method is developed for solving the integral equations which describe those systems which undergo an Einstein condensation, and it is shown that this method can be justified if $|T\ensuremath{-}{T}_{c}|\ensuremath{\lesssim}\frac{{T}_{c}}{10}$, and in cases of short-range repulsive interparticle forces and systems for which $\ensuremath{\rho}v(0)$ possesses a certain prescribed upper bound, $\ensuremath{\rho}$ being the number of particles per unit volume. The transition temperature is found to be lower than the corresponding quantity for the ideal boson gas of the same density. A detailed discussion of the thermal properties of the system in the vicinity of
Tc is also given. In an Appendix a general type of smeared Einstein condensation is assumed, and it is found that for systems in which this property occurs $\ensuremath{\epsilon}(\mathrm{k})$ is linear for small
k. This ansatz is shown to be tenable only if
v(0)<0. Finally, by assuming an appropriate pseudopotential representation of a hard repulsive core and by including a weak, longranged attractive force between particles, $\ensuremath{\epsilon}(\mathrm{k})$ shows a nonmonotonic behavior of the same qualitative type as observed in liquid
He4.
The paper deals with the properties of Bose-gases in inhomogeneous external fields at very low temperatures, when a Bose–Einstein condensation occurs. This particularly refers to trapped alkali gases, which are of great experimental and theoretical interest. The system is considered here in the frame of the developed approach close to the Bogoliubov one with necessary modifications. In particular the generalization of Bogoliubov transformations made it possible to find the spectrum of quasiparticles (low-energy thermal excitations) with account of purely quantum parts of ψ-operators in first order.