Quantum Gases of Dipoles, Quadrupoles and Octupoles in Gross-Pitaevskii Formalism with Form Factor

TL;DR

This paper develops a comprehensive theoretical framework for dipolar, quadrupolar, and octupolar Bose gases using the Gross-Pitaevskii formalism, analyzing interactions, stability, and excitations with potential experimental implications.

Contribution

It introduces a detailed derivation of the GP equation with form factors for complex interactions and provides solutions and stability analysis for non-uniform condensates.

Findings

Explicit interaction potentials in momentum space demonstrate attraction and repulsion zones.

Derived a general solution for the GP equation with a double series form factor.

Identified the threshold momentum for condensate instability and its dependence on system parameters.

Abstract

Classical and quantum field theory of dipolar, axisymmetric quadrupolar and octupolar Bose gases is considered within a general approach. Dipole, axisymmetric quadrupole and octupole interaction potentials in the momentum representation are calculated. These results clearly demonstrate attraction and repulsion areas in corresponding gases. Then the Gross-Pitaevskii (GP) equation, which plays a key role in the present paper, is derived from the corresponding functional. The zoology of the form factors appearing in GP equation is studied in details. The classes of proper for the description of spatially non-uniform condensates form factors are chosen. In the Thomas-Fermi approximation a general solution of the GP equation with a quasilocal form factor is obtained. This solution has an interesting form in terms of a double rapidly converging series that universally includes all the interactions considered. An important analysis of the condensate stability, in other words the study of condensate excitations, is also performed in this paper. In the Gaussian approximation (from the GP functional), a functional describing the perturbations of the condensate is derived in details. For a probe wave function in the form of a plane wave, a spectrum of Bogoliubov excitations was obtained, from which an equation describing the threshold momentum for the emergence of instability was derived. An important result of this paper is the dependence of the threshold on the momentum of a stationary condensate. For completeness of the presentation, the approximating expression in the form of a rapidly converging series is obtained for the corresponding dependence. Finally, in the conclusion a quantum hydrodynamic theory for dipolar, axisymmetric quadrupolar and octupolar gases is briefly presented, giving a clue to the experimental determination of the form factors.