Stochastic projected Gross-Pitaevskii equation for spinor and multicomponent condensates

A stochastic Gross-Pitaevskii equation is derived for partially condensed Bose gas systems subject to binary contact interactions. The theory we present provides a classical-field theory suitable for describing dissipative dynamics and phase transitions of spinor and multicomponent Bose gas systems composed of an arbitrary number of distinct interacting Bose fields. A class of dissipative processes involving distinguishable particle interchange between coherent and incoherent regions of phase space is identified. The formalism and its implications are illustrated for two-component mixtures and spin-1 Bose-Einstein condensates. For systems composed of atoms of equal mass, with thermal reservoirs that are close to equilibrium, the dissipation rates of the theory are reduced to analytical expressions that may be readily evaluated. The unified treatment of binary contact interactions presented here provides a theory with broad relevance for quasiequilibrium and far-from-equilibrium Bose-Einstein condensates.

Figure 1

Schematic of reservoir-interaction processes for the binary contact interaction Hamiltonian. Collision processes described by (a) the one-field terms and (b) the two-field terms given by (132b) and (132c), respectively. In general the latter processes include grand-canonical interactions with the reservoir involving both number and energy exchange between $C$ and $I$.

Figure 2

Illustration of reservoir-interaction processes in two-component mixtures [(a), (b), (c)] and spin-1 condensates (d). (a) Processes contributing to one-field damping for the two-component mixture (the same processes occur in spinor systems for collisions between distinguishable particles). (b) Energy transfer processes arising from two-field damping. (c) Particle-transfer interactions arising from two-field damping. (d) Example dissipative processes in a spin-1 system stemming from spin-changing collisions in one- and two-field damping (respectively). Both processes generate particle exchange with the reservoir for the $-1$ component.