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Physical Review A

Bose-Einstein condensates and the spectrum of excitations in a two-dimensional channel

I. V. Shunyaev, Andrei A. Elistratov, Yu. E. Lozovik

Published Nov 28, 2016• 1 citation
Publisher

Abstract

The transverse spatial distribution of weakly interacting Bose gas in a narrow channel is found. The corresponding analytical solution of the Gross-Pitaevskii equation has the form of a cnoidal wave. The possible physical realization of the model is a system of cold atoms or quasiparticles, e.g., excitons, cavity photons, exciton polaritons. The obtained formula is used for the numerical calculation of the spectrum of excitations and the critical velocity of the condensate. These numerical results do not coincide with the Bogoliubov spectrum, and we show that the strongly inhomogeneous condensate density leads to larger values of the critical velocity. We also make numerical estimation of the critical velocity for various types of considered particles or quasiparticles.

DOI
10.1103/physreva.94.053625

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The transverse spatial distribution of weakly interacting Bose gas in a narrow channel is found. The corresponding analytical solution of the Gross-Pitaevskii equation has the form of a cnoidal wave. The possible physical realization of the model is a system of cold atoms or quasiparticles, e.g., excitons, cavity photons, exciton polaritons. The obtained formula is used for the numerical calculation of the spectrum of excitations and the critical velocity of the condensate. These numerical results do not coincide with the Bogoliubov spectrum, and we show that the strongly inhomogeneous condensate density leads to larger values of the critical velocity. We also make numerical estimation of the critical velocity for various types of considered particles or quasiparticles.
The properties of traps for exciton polaritons in a semiconductor microcavity with an embedded quantum well have been considered. The behavior of the two-component Bose-Einstein condensate of photons and excitons described by the coupled system of Gross-Pitaevskii equations has been investigated. The analytical solutions for weak-confinement traps have been found in the Thomas-Fermi approximation. In the case of strong confinement, the behavior of the condensate has been investigated and constraints on the possible values of the chemical potential of the system have been obtained. The wavefunctions and generally different spatial profiles of the coupled photon and exciton condensates have been found.
We develop an approach for a description of collective excitations in a two-dimensional gas of interacting Bose particles in an external potential. We present a method of finding an approximate analytical solution for the spectra of collective excitations of a Bose gas in a linear potential and in a potential of the form u(r)=m-u₀cosh²x/l, where m is the chemical potential. Numerical study shows that the analytical solution corresponds to collective modes localized at the edge or at the low-density region. We investigate the influence of the external potential on a critical velocity of a superfluid flow. It is shown that the effect of strong suppression of the critical velocity takes place in a nonuniform Bose system. We discuss a possibility of Bose-Einstein condensation (BEC) in the systems under investigations at nonzero temperatures and find that in case of a finite number of the particles BEC can emerge.
We theoretically and numerically study Bogoliubov-wave turbulence in three-dimensional atomic Bose-Einstein condensates with the Gross-Pitaevskii equation, investigating three spectra for the macroscopic wave function, the density distribution, and the Bogoliubov-wave distribution. In this turbulence, Bogoliubov waves play an important role in the behavior of these spectra, so that we call it Bogoliubov-wave turbulence. In a previous study [Proment et al., Phys. Rev. A 80, 051603(R) (2009)], a $\ensuremath{-}3/2$ power law in the spectrum for the macroscopic wave function was suggested by using weak wave turbulence theory, but we find that another $\ensuremath{-}7/2$ power law appears in both theoretical and numerical calculations. Furthermore, we focus on the spectrum for the density distribution, which can be observed in experiments, discussing the possibility of experimental observation. Through these analytical and numerical calculations, we also demonstrate that the previously neglected condensate dynamics induced by the Bogoliubov waves is remarkably important.
We discuss the Bloch-state solutions of the stationary Gross-Pitaevskii equation and of the Bogoliubov equations for a Bose-Einstein condensate in the presence of a one-dimensional optical lattice. The results for the compressibility, effective mass and velocity of sound are analysed as a function of the lattice depth and of the strength of the two-body interaction. The band structure of the spectrum of elementary excitations is compared with the one exhibited by the stationary solutions (Bloch bands). Moreover, the numerical calculations are compared with the analytic predictions of the tight binding approximation. We also discuss the role of quantum fluctuations and show that the condensate exhibits 3D, 2D or 1D features depending on the lattice depth and on the number of particles occupying each potential well. We finally show how, using a local density approximation, our results can be applied to study the behaviour of the gas in the presence of harmonic trapping.
We study the behavior of exciton polaritons in an optical microcavity with an embedded semiconductor quantum well. We use a two-component exciton-photon approach formulated in terms of path integral formalism. In order to describe spatial distributions of the exciton and photon condensate densities, the two coupled equations of the Gross-Pitaevskii type are derived. For a homogeneous system, we find the noncondensate photon and exciton spectra, calculate the coefficients of transformation from the exciton-photon basis to the lower-upper polariton basis, and obtain the exciton and photon occupation numbers of the lower and upper polariton branches for nonzero temperatures. For an inhomogeneous system, the set of coupled equations of the Bogoliubov--de Gennes type is derived. The equations govern the spectra and spatial distributions of noncondensate photons and excitons.
With the help of a set of exact closed-form solutions to the stationary Gross—Pitaevskii (GP) equation, we calculate the collective excitation and quantum depletion of a weakly interacting Bose gas in the presence of a periodic array of quantum wells. The excitation spectrum (Bogoliubov spectrum) is obtained from the solution of the linearized time-dependent GP equation, which develops energy bands ħω j ( p ) periodic in quasi-momentum space. Moreover, we calculate the excitation strengths Z j ( p ) relative to the density operator and then the dynamic structure factor S ( p , ω). Accordingly, the analytical expressions of quantum depletion of the system are obtained. We find that the quantum depletion is enhanced when the interatomic interactions become larger and the potential is sufficiently deep. The conditions for the possible experimental realization of our scenario are also proposed.