We review phase space techniques based on the Wigner representation that provide an approximate description of dilute ultra-cold Bose gases. In this approach the quantum field evolution can be represented using equations of motion of a similar form to the Gross-Pitaevskii equation but with stochastic modifications that include quantum effects in a controlled degree of approximation. These techniques provide a practical quantitative description of both equilibrium and dynamical properties of Bose gas systems. We develop versions of the formalism appropriate at zero temperature, where quantum fluctuations can be important, and at finite temperature where thermal fluctuations dominate. The numerical techniques necessary for implementing the formalism are discussed in detail, together with methods for extracting observables of interest. Numerous applications to a wide range of phenomena are presented.

In a major extension of our previous model [Phys. Rev. Lett. 79, 1793] of condensate growth, we take account of the evolution of the occupations of lower trap levels, and of the full Bose-Einstein formula for the occupations of higher trap levels. We find good agreement with experiment, especially at higher temperatures. We also confirm the picture of the "kinetic" region of evolution, introduced by Kagan et al., for the time up to the initiation of the condensate. The behavior after initiation essentially follows our original growth equation, but with a substantially increased rate coefficient W 1 .[ S0031-9007(98) Although the first Bose condensed atomic vapor was produced in a magnetic trap only in 1995 [1][2][3], the kinetics of condensate formation has long been a subject of theoretical study [4,5]. There is now intense theoretical work on Bose-Einstein condensation, which is excellently summarized in [6]. Most theoretical studies of condensate growth either have not treated trapping or have considered only traps which are so broad that the behavior of the vapor is not essentially different from the untrapped situation. Furthermore, they have given only qualitative estimates of condensate growth. Our previous paper [7] introduced a simplified formula for the growth of a Bose-Einstein condensate, in which growth resulted from stimulated collisions of atoms in a thermal reservoir, where one of the atoms enters the lowest trap eigenstate, whose occupation thus grows to form a condensate. We thus included the trap eigenfunctions as an essential part of our description, and gave the first quantitative formula for the growth of a condensate. The growth rate was of the order of magnitude of that estimated from experiments current at that time.This direct stimulated effect must be very important once a significant amount of condensate has formed, but in the initial stages there will also be a significant number of transitions to other low lying trap levels whose populations will then also grow. As well as this, there will be interactions between the condensate, the atoms in these low lying levels, and the atomic vapor from which the condensate forms. This paper will extend the description of the condensate growth to include these factors, and will compare the results with experimental data on condensate growth obtained at MIT [8].As in our previous work, we divide the states in the potential into the condensate band, R C , which consists of the energy levels significantly affected by the presence of a condensate in the ground state, and the noncondensate band, R NC , which contains all the remaining energy levels above the condensate band. The division between the two bands is taken to be at the value, e max . The situation is illustrated in Fig. 1. The picture we shall use assumes that R NC consists of a large "bath" of atomic vapor, in thermal equilibrium, whose distribution function is given by a time-independent equilibrium Bose-Einstein distribution ͕exp͓͑E 2 m͒͞k B T ͔ 2 1͖ 21 with positive chemical potential...

The formation process of a Bose-Einstein condensate in a trap is described using a master equation based on quantum kinetic theory, which can be well approximated by a description using only the condensate mode in interaction with a thermalized bath of noncondensate atoms. A rate equation of the form ᠨ n 2W 1 ͑n͓͒°1 2 e ͕mn2m͖͞kT ¢ n 1 1͔ is derived, in which the difference between the condensate chemical potential m n and the bath chemical potential m gives the essential behavior. Solutions of this equation give a characteristic latency period for condensate formation and appear to be consistent with the observed behavior of both rubidium and sodium condensate formation. [S0031-9007(97) The experiments on Bose-Einstein condensation of dilute atomic gases [1-3] have stimulated theoretical effort, which has, however, not produced any definitive result for the growth of the condensate from the vapor, although there have been significant theoretical contributions [4][5][6][7][8]. This Letter will present a quantitative and experimentally testable description of the growth process, based on quantum kinetic theory [9,10], which can be simplified to a single first-order differential equation for the number n of atoms in the condensate.Our formulation contains the following principal features. We use the HamiltonianThe potential function u͑x 2 x 0 ͒ is as usual not the true interatomic potential, but rather a short range potentialapproximately of the form ud͑x 2 x 0 ͒-which reproduces the correct scattering length [11].We divide the condensate into two regions called the condensate band R C , and the noncondensate band R NC , as in Fig. 1. We treat R NC as being thermalized, representing the majority of the atoms as a heat bath which provides the source of atoms for condensate growth. The condensate band is the region of energy levels less than a value E R , which includes not only the ground state, in which the condensate forms, but also those levels which would be significantly affected by the presence of a condensate. [12] In the noncondensate band, with energy levels greater than E R , there is no significant such effect.The behavior in R C is treated fully quantummechanically, and a description in terms of trap levels modified by the presence of a condensate is used. At any time there is a given number N of atoms in R C , and the energy levels in such a situation can be described using the number-conserving Bogoliubov method devised by one of us [13], so that the state of R C is fully described by the total number of atoms N in R C , and the quantum state of the quasiparticles within R C . In this formulation we can write the condensate band field operator in the formThe quasiparticles, of energy e m N , are described by annihilation operators b m , while B y is the creation operator which takes the R C system, for any N, from the ground state with N atoms to the ground state with N 1 1 atoms. The condensate wave function is j N ͑x͒, and this satisfies the Gross-Pitaevskii equationThe amplitudes f m ͑x͒, g m ͑x͒ a...

We give a simple unified theory of vortex nucleation and vortex lattice formation which is valid from the initiation process up to the final stabilization of the lattice. We treat the growth of vortex lattices from a rotating thermal cloud, and their production using a rotating trap. We find results consistent with previous work on the critical velocity or critical angular velocity for vortex formation, and predict the initial number of vortices expected before their self assembly into a lattice. We show that the thermal cloud plays a crucial role in the process of vortex lattice nucleation. [3,4,5,6,7,8,9] have focussed attention on the mechanisms of vortex formation.One experimental method for creating vortices is to stir a condensate with an anisotropic potential, and a number of theoretical analyses of this scenario have been made in terms of the Gross-Pitaevskii equation, e.g. [10,11,12,13]. The view of these treatments is that the perturber causes a mixing of the condensate ground state and excited condensate states (with angular momentum values determined by the stirrer geometry). Typically, a perturbative calculation is used to obtain a critical rotational speed (or a critical linear speed at the Thomas-Fermi radius) of the stirrer at which the mixing becomes effective. The argument is then made that an instability will lead to growth of the vortex state. A more complete non-perturbative calculation for a localized rotating stirrer [14] shows that in this case the mixing is predominantly between the ground state and an l = 1 vortex state, and that as the relative amplitudes evolve by coherent "Rabi cycling", the full condensate exhibits a vortex cycling from infinity to the central regions of the condensate. No coherent mixing mechanisms, however, can explain the formation of a vortex lattice, since an energy barrier exists between the superposition state and the vortex lattice state, and some additional mechanism is required to remove the energy liberated when the vortex lattice is formed.An alternative method of producing vortices, in which a vapor of cold atoms is evaporatively cooled so as to preserve its angular momentum has been demonstrated by Haljan et al. [5]. In this experiment, which involves no stirring potential, the mechanism of formation of the vortices must involve a modification of the theory of condensate growth (which now exists in a reasonably good quantitative form; [15] and references therein) to take account of the non-zero angular momentum of the vapor from which the condensate is formed. In this paper we will show that the mechanism thus demonstrated is the fundamental process in those experiments in which the stirring of the condensate apparently generates a vortex lattice. Weak stirring, in the absence of any thermal cloud, only produces "Rabi cycling" of angular momentum in and out of the condensate. However, if the stirring also produces a rotating thermal cloud, then, by the same process responsible for the growth of a condensate [16], angular momentum is transferred irreve...