Number-conserving approaches ton-component Bose-Einstein condensates

Abstract

We develop the number-conserving approach that has previously been used in a single component Bose-Einstein condensed dilute atomic gas, to describe consistent coupled condensate and noncondensate number dynamics, to an n-component condensate. The resulting system of equations comprises, for each component, of a generalised Gross-Pitaevskii equation coupled to modified Bogoliubov-de Gennes equations. Lower-order approximations yield general formulations for multi-component Gross-Pitaevskii equations, and systems of multicomponent Gross-Pitaevskii equations coupled to multi-component modified number-conserving Bogoliubovde Gennes equations. The analysis is left general, such that, in the n-component condensate, there may or may not be mutually coherent components. An expansion in powers of the ratio of noncondensate to condensate particle numbers for each coherent set is used to derive the self-consistent, second-order, dynamical equations of motion. The advantage of the analysis developed in this article is in its applications to dynamical instabilities that appear when two (or more) components are in conflict and where a significant noncondensed fraction of atoms is expected to appear.

Nonequilibrium kinetic theory for trapped binary condensates

“…It is interesting to note that both the multi-component number conserving work of [111] as well as the stochastic treatment of [107] have independently identified a condensate to non-condensate "exchange" collisional event, physically analogous to that presented in this work by C kj 12 . In the former case such terms appear as in the present work as off-diagonal normal pair averages of fluctuation operators in the corresponding generalized Gross-Piteavskii equation, with such averages however defined in terms of number-conserving operators, for example see Eq.…”

“…This leaves us with the off-diagonal normal pair averages of the form δ † jδ k . In general, these could be thought of as describing coherences between the two physical systems and could be treated on equal footing to condensate and excited state populations [111,115]. However, in the absence of any external coupling, one would expect such terms to evolve on the more rapid collisional timescale, and thus be suitable candidates for adiabatic elimination.…”

Stochastic projected Gross-Pitaevskii equation for spinor and multicomponent condensates

“…The one-body Wigner function (49) can now be used to evaluate the one-body reservoir correlation functions approximately. In the local density approximation, the thermal equilibrium solution at temperature T = (βk B ) −1 , and spin-dependent chemical potential µ σ is the Bose-Einstein distribution…”

“…At low to moderate temperatures generalised mean field theories have been developed, and successfully modelled a number of experimental scenarios. The Zaremba-Nikuni-Griffin (ZNG) [28][29][30][31][32][33], projected Gross-Pitaevskii equation (PGPE) [34][35][36][37][38][39][40][41][42][43][44] (including applications to spinor condensates [45,46]), and number conserving [47][48][49] theories each have advantages for describing BEC evolution, namely, relative ease of handling thermal cloud dynamics, inclusion of many appreciably populated coherent modes, and inclusion of off-diagonal long range order, respectively. At temperatures well below the BEC transition (T c ), these effects are essential aspects of finite-temperature BEC physics.…”

Low-dimensional stochastic projected Gross-Pitaevskii equation

“…These fall broadly into two categories: generalized mean-field theories that treat the condensate within a symmetry breaking approach [53], and theories stemming from a phase-space representation of the field theory [1,54]; for reviews see [30,55]. A notable exception is given by the number conserving approach to the mean field theory, that avoids breaking U(1) symmetry, while describing the dynamics of the condensate and its Bogoliubov fluctuations [56][57][58]. A stochastic Gross-Pitaevskii treatment of the high temperature BEC derived via the Keldysh approach to non-equlibrium dynamics [54] has also been used in several studies [34,37,[59][60][61][62][63][64][65].…”