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Introduction

The present chapter is devoted to the theory of homogeneous dilute Bose gases at zero temperature. The mean-field theory for the dilute Bose gas is discussed in the first part of the chapter. We derive the Gross-Pitaevskii equation for the order parameter and we use it to calculate the ground-state energy of the system. The small oscillations of the order parameter around the equilibrium solution provide us with the elementary excitation energies. The presentation of the material in this section follows closely the review [6]. In the second part of the chapter we discuss the Bogoliubov model, which is a theory beyond mean-field and takes into account the fluctuations of the order parameter. We introduce Bogoliubov effective Hamiltonian, discuss its diagonalization by means of the Bogoliubov transformation and we calculate the corrections to the of the ground state energy arising from the quantum fluctuations. The excitation spectrum predicted by the Bogoliubov model agrees with the one obtained from the time-dependent Gross-Pitaevskii equation. Results for the number of noncondensed particles (quantum depletion) and the one-body density matrix are also discussed. Much of the treatment of this part of the chapter parallels closely the one given in the book [11].
next up previous contents
Next: Mean-field description: Gross-Pitaevskii equation Up: Dilute Bose gas Previous: Dilute Bose gas   Contents