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International Journal of Modern Physics B

LOCALIZATION OF PLANE WAVES IN DILUTE BOSE–EINSTEIN GASES

A. Kwang-Hua Chu

Published Jan 10, 2007• 4 citations
Publisher

Abstract

Discrete Uehling–Uhlenbeck models together with the introduction of the disorder or an orientation-free parameter are adopted to study the possible localization of plane waves propagating in dilute hard-sphere Bose–Einstein gases under very-low temperature environment based on the acoustical analog. The curves of the localization length versus the disorder and Pauli-blocking measure we obtained here resemble qualitatively those proposed before.

DOI
10.1142/s021797920703590x

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Discrete Uehling–Uhlenbeck models together with the introduction of the disorder or an orientation-free parameter are adopted to study the possible localization of plane waves propagating in dilute hard-sphere Bose–Einstein gases under very-low temperature environment based on the acoustical analog. The curves of the localization length versus the disorder and Pauli-blocking measure we obtained here resemble qualitatively those proposed before.
The Ühling-Uhlenbeck equations and the discrete velocity model are used to study the propagation of plane waves in a dilute system of Bose particles with hard-sphere interactions. The Ühling-Uhlenbeck collision sum, as it is highly nonlinear, is first linearized and then decomposed by using the plane wave assumption. We address the dispersion relations thus obtained by the relevant parameter θ or B which describes the Bose particles for the quantum analogue of the discrete Boltzmann system when B is positive (and θ = 1). The attenuation of plane waves propagating in dilute Bose gases in the low density or temperature regime shows a rather sharp change which is completely different from that in dilute Boltzmann gases.
We demonstrate the effects of an induced disorder (or a free-orientation : θθθ which is related to the relative direction of scattering of particles w.r.t. to the normal of the propagating plane-wave front) upon the possible resonance of the plane (sound) wave propagating in Bose gases by using the quantum kinetic equations. We firstly present the diverse dispersion relations obtained by the relevant Pauli-blocking parameter BBB (which describes the Bose particles when BBB is positive) and the free-orientation θθθ and then, based on the acoustic analog, address the possible resonant states.
We consider a dilute hard-sphere Bose gas in random external potentials at low temperatures, in D=3, using the technique of pseudopotentials and the Bogoliubov transformation. At absolute zero, the random potentials can deplete the Bose condensate, though not completely. On the other hand, they generate an amount of normal fluid equal to 4/3 the condensate depletion. This is a localization effect that can destroy superfluidity at absolute zero. General features of the superfluid density in the neighborhood of this transition point agree qualitatively with experimental results on helium in porous media.
We investigate the thermodynamic properties of a dilute Bose gas in a correlated random potential using exact path integral Monte Carlo methods. The study is carried out in continuous space and disorder is produced in the simulations by a three-dimensional (3D) speckle pattern with tunable intensity and correlation length. We calculate the shift of the superfluid transition temperature due to disorder and we highlight the role of quantum localization by comparing the critical chemical potential with the classical percolation threshold. The equation of state of the gas is determined in the regime of strong disorder, where superfluidity is suppressed and the normal phase exists down to very low temperatures. We find a T 2 dependence of the energy in agreement with the expected behavior in the Bose glass phase. We also discuss the major role played by the disorder correlation length and we make contact with a Hartree–Fock mean-field approach that holds if the correlation length is very large. The density profiles are analyzed as a function of temperature and interaction strength. Effects of localization and the depletion of the order parameter are emphasized in the comparison between local condensate and total density. At very low temperature, we find that the energy and the particle distribution of the gas are very well described by the T =0 Gross–Pitaevskii theory, even in the regime of very strong disorder.
We revisit the problem of Anderson localization in a trapped Bose–Einstein condensate in 1D and 3D in a disordered potential, applying Quantum Monte Carlo technique because the disorder cannot be treated accurately in a perturbative way as even a small amount of disorder can produce dramatic changes in the physical properties of the system under investigation. Till date no unambiguous evidence of localization has been observed for matter waves in 3D. Matter waves made up of cold atoms are good candidates for such investigations. Simulations are performed for Rb gas in continuous space using canonical ensemble in the case of random and quasi-periodic potentials. To realize random and quasiperiodic potentials numerically we use speckle and bichromatic potentials, respectively. Owing to the high degree of control over the system parameters we specifically study the interplay of disorder and interaction in the system. A dilute Bose gas placed in a random environment falls into a fragmented localized state and the ergodicity (the repetitiveness of the wave function) is lost. An arbitrary Interaction can slowly overcome the effect of disorder and restore the ergodicity again. We observe that as the interaction strength increases, the wave functions become more and more delocalized. Since vanishing of Lyapunov exponent is only a necessary but not a sufficient condition for delocalization for probing the localization we calculate the mean square displacements as an alternative measure of localization. The path integral Monte Carlo technique in this paper numerically establishes the existing predictions of the scaling theory so far and paves a clear path for the further investigation of scaling theory to calculate more complicated properties like ‘critical exponents’ etc. in disordered quantum gases.