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CRC Press eBooks

Kinetics of Bose–Einstein Condensation

Boris V. Svistunov, Egor S. Babaev, Nikolay V. Prokof'ev

Published Apr 15, 2015• 1 citation
Publisher

Abstract

In this chapter,∗ we will analyze strongly nonequilibrium kinetics of Bose-Einstein condensate (BEC) formation in a weakly interacting Bose gas (WIBG). The process takes place as a result of self-evolution when occupation numbers in the initial state are large enough to ensure finite condensate density upon thermalization. In WIBG, the scenario of BEC formation involves several stages taking place at different wavenumber intervals. The key stages of the ordering process are of classicalfield character and are described by the Gross-Pitaevskii equation (GPE), with turbulent initial condition.

DOI
10.1201/b18346-21

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Cambridge University Press eBooks•Feb 28, 2000•0 citations
In this chapter,∗ we will analyze strongly nonequilibrium kinetics of Bose-Einstein condensate (BEC) formation in a weakly interacting Bose gas (WIBG). The process takes place as a result of self-evolution when occupation numbers in the initial state are large enough to ensure finite condensate density upon thermalization. In WIBG, the scenario of BEC formation involves several stages taking place at different wavenumber intervals. The key stages of the ordering process are of classicalfield character and are described by the Gross-Pitaevskii equation (GPE), with turbulent initial condition.
We consider the ordering kinetics in a strongly non-equilibrium state of a (weakly) interacting Bose gas, characterized, on one hand, by large occupation numbers, and, on the other hand, by the absence of long-range order. Up to higher-order corrections in inverse occupation numbers, the evolution is described by non-linear Schrodinger equation with a turbulent initial state. The ordering process is rather rich and involves a number of qualitatively different regimes that take place in different regions of energy space. Specially addressed is the case of evolution in an external potential.
Large scale numerical simulations of the Gross-Pitaevskii equation are used to elucidate the self-evolution of a Bose gas from a strongly nonequilibrium initial state. The stages of the process confirm and refine the theoretical scenario of Bose-Einstein condensation developed by Svistunov and co-workers [J. Mosc. Phys. Soc. 1, 373 (1991); Sov. Phys. JETP 75, 387 (1992); 78, 187 (1994)]: the system evolves from the regime of weak turbulence to superfluid turbulence via states of strong turbulence in the long-wavelength region of energy space.
We investigate formation of Bose-Einstein condensates under nonequilibrium conditions using numerical simulations of the three-dimensional Gross-Pitaevskii equation. For this, we set initial random weakly nonlinear excitations and the forcing at high wave numbers and study propagation of the turbulent spectrum toward the low wave numbers. Our primary goal is to compare the results for the evolving spectrum with the previous results obtained for the kinetic equation of weak wave turbulence. We demonstrate existence of a regime for which good agreement with the wave turbulence results is found in terms of the main features of the previously discussed self-similar solution. In particular, we find a reasonable agreement with the low-frequency and the high-frequency power-law asymptotics of the evolving solution, including the anomalous power-law exponent ${x}^{*}\ensuremath{\approx}1.24$ for the three-dimensional wave action spectrum. We also study the regimes of very weak turbulence when the evolution is affected by the discreteness of the Fourier space, and the strong turbulence regime when the emerging condensate modifies the wave dynamics and leads to formation of strongly nonlinear filamentary vortices.
The kinetics of the formation of coherent correlation properties associated with Bose condensation is studied in detail. The evolution of a nonequilibrium state with no “condensate-seed” is related to a hierarchy of relaxation times. At the first stage, a particle flux in energy space toward low energies sets in. The evolution in this case is described by a nonlinear Boltzmann equation, with a characteristic time given by interparticle collisions. When the particles which will later form the condensate have a kinetic energy which is less than the potential energy, a quasicondensate starts to form. In this stage, fluctuations of the density (but not of the phase) are suppressed and short-range coherent correlation properties are governed by the equation of motion for a quasiclassical complex field. The next stage is connected with the formation of the long-range order. The time for forming topological order and therefore genuine superfluidity proves to be dependent on the system size. The off-diagonal long-range order, arising after the attenuation of long-wave phase fluctuations, has a size-dependent relaxation time as well. Introduction The problem of Bose–Einstein Condensation (BEC) kinetics, being interesting in itself, has acquired a special significance in connection with experimental efforts to observe this condensation in a number of systems with particles with a finite lifetime. Such systems include spin-polarized atomic hydrogen [1], excitons [2] and biexcitons [3] in semiconductors and, more recently, laser-cooled atomic systems [4].
Condensate Formation in the Bose Gas An important question in connection with the experiments aimed at achieving BEC of a weakly interacting Bose gas with a finite lifetime is the time needed for the formation of the condensate. It is not at all evident that the condensation can always take place within the lifetime of the system. For the case of liquid He, the assumption of equilibrium seems satisfied because 4 He atoms, which do not decay, undergo roughly 10 11 interparticle-scattering events per second, assuming classical hard-sphere scattering. In other boson systems, however, nonequilibrium effects may have a significant effect. For excitons in a semiconductor, the lifetime of the particle may be only a few hundred scattering times. Spin-polarized hydrogen also has a finite lifetime to nonpolarized states [1], and alkali atoms can recombine into molecules [2]. Naively judging from classical behavior, one might assume that hundreds of scattering times may seem more than adequate time to establish equilibrium. The case of Bose condensation is unique, however, since a macroscopic number of particles must enter a single quantum state that by random-scattering processes alone is highly improbable. Therefore we need not assume that the time for Bose condensation is short. Early experiments with orthoexcitons in CU 2 O seemed to indicate that the excitons might not have enough time to condense within their lifetime [3]. As discussed in Subsection 1.4.3, the orthoexciton density increased to approach the BEC boundary given in Eq. (1.18) and highly degenerate Bose-Einstein momentum distributions were observed, but a significant condensate fraction did not appear. Instead the gas increased its temperature to accommodate extra particles, reaching temperatures well above the lattice temperature.