Re-examining the quadratic approximation in theory of a weakly interacting Bose gas with condensate: the role of nonlocal interaction potentials

Abstract

We derive and analyze the coupled equations of quadratic approximation of the Bogoliubov model for a weakly interacting Bose gas. The first equation determines the condensate density as a variational parameter and ensures the minimum of the grand thermodynamic potential. The second one provides a relation between the total number of particles and chemical potential. Their consistent theoretical analysis is performed for a number of model interaction potentials including contact (local) and nonlocal interactions, where the latter provide nontrivial dependencies in momentum space. We demonstrate that the derived equations have no solutions for the local potential, although they formally reproduce the well-known results of the Bogoliubov approach. At the same time, it is shown that these equations have the solutions for physically relevant nonlocal potentials. We show that in the regimes close to experimental realizations with ultracold atoms, the contribution of the terms originating from the quadratic part of the truncated Hamiltonian to the chemical potential can be of the same order of magnitude as from its c-number part. Due to this fact, in particular, the spectrum of single-particle excitations in the quadratic approximation acquires a gap. The issue of the gap is also discussed.Re-examining the quadratic approximation ...

Calculation of nonequilibrium thermodynamic potential of Bose system near the condensation point

“…It is established that the results obtained for Bose gas do not fit into the Landau theory of phase transitions of the second kind. In this regard it is noted that the literature suggests the inapplicability of the chemical potential as an independent TD variable and of the Gibbs grand canonical distribution for the consideration of the Bose condensation in general (see [9,12] and review [11]).…”

“…In this case, after the TD boundary transition, the integral in (35) has no singularity and converges. Further, in the literature there is an opinion on the inapplicability of chemical potential as an independent variable in the Bose condensation theory (see, for example, [9,12]). Moreover, in the literature, the applicability of the Gibbs grand canonical distribution to the study of Bose condensation is considered controversial (see a review in [11]).…”

“…The realization of this definition in a microscopic theory in the vicinity of equilibrium, as the construction of a nonequilibrium TD potential in general, is an actual task (see, for example, [10,11]). In this paper, we discuss the implementation of the Bogolyubov definition for the Bose system in the presence of condensate (in the literature there is an increase of interest to the study of Bose condensation (see, for example, [11,12])). In such a system, the order parameter is the occupation number of the one-particle state with the momentum 0  p…”

Broken-axisymmetry state and magnetic state diagram of spin-1 condensate through the prism of quadrupole degrees of freedom

“…We must also ensure that the radicands in Eqs. (17) are positive. Thus, we have the following three cases:…”

“…Moreover, for spinor condensates, this approximation does not reproduce the complete structure of the single-particle excitation spectrum [16]. The role of non-local interaction has been recently discussed in physics of ultracold Bose [17,18] and Fermi [19][20][21] gases.…”