Next Article in Journal
Interfacial Crystallization within Liquid Marbles
Previous Article in Journal
Ferroelectricity, Superconductivity, and SrTiO3—Passions of K.A. Müller
Open AccessArticle

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

Moscow Institute of Physics and Technology (MIPT), Institutskiy Pereulok 9, 141701 Dolgoprudny, Moscow Region, Russia
*
Author to whom correspondence should be addressed.
Condens. Matter 2020, 5(4), 61; https://doi.org/10.3390/condmat5040061
Received: 7 September 2020 / Revised: 4 October 2020 / Accepted: 12 October 2020 / Published: 15 October 2020
(This article belongs to the Section Condensed Matter Theory)
Here, 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 the GP equation is studied in details. The proper classes 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. Plots of condensate density functions for the exponential-trigonometric form factor are given. For the sake of completeness, in this paper we consider the GP equation with an optical lattice potential in the limit of small condensate densities. This limit does not distinguish between dipolar, quadrupolar and octupolar gases. 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 Gross–Pitaevskii functional), a functional describing the perturbations of the condensate is derived in detail. This problem is an analog of the Bogolubov transformation used in the study of quantum Bose gases in operator formalism. 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, and plots of the corresponding series for the exponential-trigonometric form factor are given. 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. View Full-Text
Keywords: quantum field theory (QFT); scalar QFT; nonlocal QFT; Euclidean QFT; quantum gases; dipoles; quadrupoles; octupoles; Gross–Pitaevskii (GP) formalism; form factor quantum field theory (QFT); scalar QFT; nonlocal QFT; Euclidean QFT; quantum gases; dipoles; quadrupoles; octupoles; Gross–Pitaevskii (GP) formalism; form factor
Show Figures

Figure 1

MDPI and ACS Style

Alexandrov, A.A.; Badamshina, A.U.; Ogarkov, S.L. Quantum Gases of Dipoles, Quadrupoles and Octupoles in Gross–Pitaevskii Formalism with Form Factor. Condens. Matter 2020, 5, 61.

Show more citation formats Show less citations formats
Note that from the first issue of 2016, MDPI journals use article numbers instead of page numbers. See further details here.

Article Access Map by Country/Region

1
Back to TopTop