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Physical Review A

Nucleation of Bose-Einstein condensation

Henk T. C. Stoof

Published Jun 1, 1992• 109 citations
Publisher

Abstract

Using a functional formulation of the Keldysh theory we investigate the time evolution of a dilute Bose gas through the critical point. We determine the critical region of the gas and show that if the system is quenched inside this region, Bose-Einstein condensation is nucleated on a time scale of O(\ensuremath{\Elzxh}/${\mathit{k}}_{\mathit{B}}$${\mathit{T}}_{\mathit{c}}$) by means of a coherent population of the one-particle ground state. We also examine the subsequent buildup of the condensate density, taking place on a much longer time scale. However, in the experimentally interesting case of spin-polarized atomic hydrogen or cesium it is still short compared to the lifetime of the gas.

DOI
10.1103/physreva.45.8398

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Using a functional formulation of the Keldysh theory we investigate the time evolution of a dilute Bose gas through the critical point. We determine the critical region of the gas and show that if the system is quenched inside this region, Bose-Einstein condensation is nucleated on a time scale of O(\ensuremath{\Elzxh}/{\mathit{k}}_{\mathit{B}}$${\mathit{T}}_{\mathit{c}}) by means of a coherent population of the one-particle ground state. We also examine the subsequent buildup of the condensate density, taking place on a much longer time scale. However, in the experimentally interesting case of spin-polarized atomic hydrogen or cesium it is still short compared to the lifetime of the gas.
Using magnetically trapped atomic hydrogen as an example, we investigate the prospects of achieving Bose-Einstein condensation in a dilute Bose gas. We show that, if the gas is quenched sufficiently far into the critical region of the phase transition, the typical time scale for the nucleation of the condensate density is short and of O(ℏ/kBTc)O(\hbar/k_{B}T_{c})O(ℏ/kB​Tc​). As a result we find that thermalizing elastic collisions act as a bottleneck for the formation of the condensate. In the case of doubly-polarized atomic hydrogen these occur much more frequently than the inelastic collisions leading to decay and we are lead to the conclusion that Bose-Einstein condensation can indeed be achieved within the lifetime of the gas.
Using magnetically trapped atomic hydrogen as an example, we investigate the prospects of achieving Bose–Einstein condensation in a dilute Bose gas. We show that, if the gas is quenched sufficiently far into the critical region of the phase transition, the typical time scale for the nucleation of the condensate density is short and of O(ħ/k B T c ). As a result we find that thermalizing elastic collisions act as a bottleneck for the formation of the condensate. In the case of doubly polarized atomic hydrogen these occur much more frequently than the inelastic collisions leading to decay and we are led to the conclusion that Bose–Einstein condensation can indeed be achieved within the lifetime of the gas. Introduction In the last few years it has been clearly demonstrated that not only charged ions but also neutral atoms can be conveniently trapped and cooled by means of electro-magnetic fields. Although the physics of the various ingenious scenarios developed to accomplish this is already interesting in itself [1], the opportunities offered by an atomic gas sample at very low temperatures are exciting in their own right. Examples in this respect are the performance of high-precision spectroscopy, the search for a violation of CP invariance by measuring the electric dipole moment of atomic cesium [2], the construction of an improved time standard based on an atomic fountain [3] and the achievement of Bose–Einstein condensation in a weakly interacting gas.
In this paper, we present a method for simulating the kinetic evolution of a dilute gas of atoms that are cooled below the critical temperature for Bose-Einstein condensation. Our method gives insight into the formulation of physical kinetics by illustrating directly the decomposition of the distribution function into an infinite sum of single-particle trajectories. This approach is valid for the entire range of phase-space densities, although we limit the discussion here to exclude the region where the condensate fraction is close to unity and the effect of the mean field is significant. We present explicit calculations of finite number effects on equilibrium, the dynamic build-up of the ground state, and simulations of evaporative cooling.
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].
The problem of understanding how a coherent, macroscopic Bose- Einstein condensate (BEC) emerges from the cooling of a thermal Bose gas has attracted significant theoretical and experimental interest over several decades. The pioneering achievement of BEC in weakly interacting dilute atomic gases in 1995 was followed by a number of experimental studies examining the growth of the BEC number, as well as the development of its coherence. More recently, there has been interest in connecting such experiments to universal aspects of nonequilibrium phase transitions, in terms of both static and dynamical critical exponents. Here, the spontaneous formation of topological structures such as vortices and solitons in quenched cold-atom experiments has enabled the verification of the Kibble-Zurek mechanism predicting the density of topological defects in continuous phase transitions, first proposed in the context of the evolution of the early universe. This chapter reviews progress in the understanding of BEC formation and discusses open questions and future research directions in the dynamics of phase transitions in quantum gases. Introduction The equilibrium phase diagram of the dilute Bose gas exhibits a continuous phase transition between condensed and noncondensed phases. The order parameter characteristic of the condensed phase vanishes above some critical temperature T c and grows continuously with decreasing temperature below this critical point. However, the dynamical process of condensate formation has proved to be a challenging phenomenon to address both theoretically and experimentally. This formation process is a crucial aspect of Bose systems and of direct relevance to all condensates discussed in this book, despite their evident system-specific properties. Important questions leading to intense discussions in the early literature include the time scale for condensate formation and the role of inhomogeneities and finite-size effects in “closed” systems. These issues are related to the concept of spontaneous symmetry breaking, its causes, and implications for physical systems (see, for example, Chapter 5 by Snoke and Daley). In this chapter, we give an overview of the dynamics of condensate formation and describe the present understanding provided by increasingly well-controlled cold-atom experiments and corresponding theoretical advances over the past twenty years. We focus on the growth of BECs in cooled Bose gases, which, from a theoretical standpoint, requires a suitable nonequilibrium formalism.
We examine the time evolution of a weakly interacting Bose gas in the course of the Bose-Einstein phase transition and show that, in contrast with previous claims in the literature, the relevant time scale for the appearance of the condensate is finite and, under the conditions we consider, of O(\ensuremath{\Elzxh}/kB{\mathit{k}}_{\mathit{B}}kB​Tc{\mathit{T}}_{\mathit{c}}Tc​), which is very small compared to the characteristic lifetime of the system due to inelastic collisions in the gas.