Quantized hydrodynamic model and the dynamic structure factor for a trapped Bose gas

TL;DR

This paper develops a quantized hydrodynamic model for a trapped Bose gas at zero temperature, enabling calculation of collective modes, condensate depletion, and inelastic scattering, bridging hydrodynamics and Bogoliubov theory.

Contribution

It introduces a quantized hydrodynamic framework based on the Gross-Pitaevskii equation, providing a new method to analyze collective excitations and observable properties of trapped Bose gases.

Findings

Calculated condensate depletion at T=0.

Derived inelastic light-scattering cross section.

Connected hydrodynamic and Bogoliubov descriptions.

Abstract

We quantize the recent hydrodynamic analysis of Stringari for the low-energy collective modes of a trapped Bose gas at $T=0$. This is based on the time-dependent Gross-Pitaevskii equation, but omits the kinetic energy of the density fluctuations. We diagonalize the hydrodynamic Hamiltonian in terms of the normal modes associated with the amplitude and phase of the inhomogeneous Bose order parameter. These normal modes provide a convenient basis for calculating observable quantities. As applications, we calculate the depletion of the condensate at $T=0$ as well as the inelastic light-scattering cross section $S({\bf q},\omega)$ from low-energy condensate fluctuations. The latter involves a sum over all normal modes, with a weight proportional to the square of the ${\bf q}$ Fourier component of the density fluctuation associated with a given mode. Finally, we show how the Thomas-Fermi hydrodynamic description can be derived starting from the coupled Bogoliubov equations.