Bose gases, Bose–Einstein condensation, and the Bogoliubov approximation

Abstract

We review recent progress towards a rigorous understanding of the Bogoliubov approximation for bosonic quantum many-body systems. We focus, in particular, on the excitation spectrum of a Bose gas in the mean-field (Hartree) limit. A list of open problems will be discussed at the end. C ⃝ 2014 AIP Publishing LLC.

Effective Hamiltonian and excitation spectrum of harmonically trapped bosons

“…In the analysis of Bosecondensates in optical lattices the Bose-Hubbard model is often applied [16][17][18]. Mathematical aspects of the Bogoliubov approximation [19] for excitation spectra of weakly-interacting bosons are discussed in [20,21]. Theoretical description of Bogoliubov-type excitations in exciton-polariton Bose-condensates is given in [22].…”

Mean Field Derivation of DNLS from the Bose–Hubbard Model

“…A reference role in the literature is played by those recent works dealing with the rigorous version of the Bogoliubov theory of superfluids; see, e.g., the review [38]. The Gross-Pitaevskii equation is rigorously deduced, for example, in [14], whereas the fluctuations around it are studied in [13,17].…”

Excitation spectrum for the Lee-Huang-Yang model for the bose gas in a periodic box

“…In pursuing a rigorous analysis of the Lee-Huang-Yang model, we do not address the validity of quadratic models for BEC in this work. See [19] for a detailed discussion of these matters in the periodic setting. As such, we rely on the ubiquity of these approximations in many areas of physics for our motivation.…”

“…Both of these areas of study display mathematical methods which were originally inspired by techniques developed in physics. [3,17,19] A less popular method in the physical study of the Bose gas involves the non-unitary transformation of approximate Hamiltonians [5,20]. Such transformations describe a pairwise-scattering process (pair excitation) which results in the partial depletion of the condensate to excited states, and is hypothesized to contribute to corrections to the groundstate wavefunction beyond mean-field descriptions [7].…”