Stochastic projected Gross-Pitaevskii equation

Abstract

We have achieved the first full implementation of the stochastic projected Gross-Pitaevskii equation for a three-dimensional trapped Bose gas at finite temperature. Our work advances previous applications of this theory, which have only included growth processes, by implementing number-conserving scattering processes. We evaluate an analytic expression for the coefficient of the scattering term and compare it to that of the growth term in the experimental regime, showing the two coefficients are comparable in size. We give an overview of the numerical implementation of the deterministic and stochastic terms for the scattering process, and use simulations of a condensate excited into a large amplitude breathing mode oscillation to characterize the importance of scattering and growth processes in an experimentally accessible regime. We find that in such non-equilibrium regimes the scattering can dominate over the growth, leading to qualitatively different system dynamics. In particular, the scattering causes the system to rapidly reach thermal equilibrium without greatly depleting the condensate, suggesting that it provides a highly coherent energy transfer mechanism.

Stochastic projected Gross-Pitaevskii equation for spinor and multicomponent condensates

“…Full stochastic simulations of quenches and dissipative dynamics in spinor systems and mixtures are an important future aim. However we note that the purely damped equations, obtained by formally setting the noise in the SPGPE to zero, provide a qualitative description of the essential phenomenology of dissipative evolution [25,26,52,[75][76][77].…”

“…[26] shows that to a very good approximation the final term in Eq. (A16) is always inferior to the larger of the first two terms.…”

“…Equations (137), (138), (139), (140) give a generalization of the scalar SPGPE theory [22], as implemented in [23] and [26], in a form suitable for numerical simulations of dissipative dynamics and quench phenomena in spinor and multicomponent systems. We now apply this formalism to the simplest examples, namely the two-component mixture, and the spin-1 system.…”

“…This theory is valid across the phase transition to Bose-Einstein condensation, enabling studies of critical phenomena such as spontaneous vortex formation [50] (also see [53] and [54]). The theory has also been applied to the dissipative dynamics of topological excitations at high temperature [25,26,52,55,56]. A stochastic Gross-Pitaevskii equation formalism has also been developed by Stoof and coworkers [33,[57][58][59][60][61][62][63][64][65][66] that does not impose an explicit projector, but yields a similar description of the low energy region of the system near equilibrium (also see [67][68][69]).…”

“…Our approach is to extend the stochastic projected Gross-Pitaevskii equation (SPGPE) [22], which has emerged as a practical quantitative theory of high-temperature single component Bose-Einstein condensates (BECs) [23][24][25][26][27]. This extension of the SPGPE theory involves generalising the interactions to arbitrary binary contact interactions and accounting for the multi-component reservoirs that describe the high energy thermalised modes of the system.…”