A number-conserving approach to a minimal self-consistent tr

We describe a number conserving approach to the dynamics of Bose-Einstein condensed dliute atomic gases. This builds upon the works of Gardiner [C. W. Gardiner, Phys. Rev. A 56, 1414 (1997)], and Castin and Dum [Y. Castin and R. Dum, Phys. Rev. A 57, 3008

a r X i v :c o n d -m a t /0610623v 3 [c o n d -m a t .o t h e r ] 6 J u l 2007A number-conserving approach to a minimal self-consistent treatment of condensate and

non-condensate dynamics in a degenerate Bose gas

S.A.Gardiner

Department of Physics,Durham University,Rochester Building,South Road,Durham DH13LE,United Kingdom

S.A.Morgan Department of Physics and Astronomy,University College London,Gower Street,London WC1E 6BT,United Kingdom ?(Dated:February 6,2008)We describe a number conserving approach to the dynamics of Bose-Einstein condensed dliute atomic gases.This builds upon the works of Gardiner [C.W.Gardiner,Phys.Rev.A 56,1414(1997)],and Castin and Dum [Y .Castin and R.Dum,Phys.Rev.A 57,3008(1998)].We consider what is e ?ectively an expansion in powers of the ratio of non-condensate to condensate particle numbers,rather than inverse powers of the total number of particles.This requires the number of condensate particles to be a majority,but not necessarily almost equal to the total number of particles in the system.We argue that a second-order treatment of the relevant dynamical equations of motion is the minimum order necessary to provide consistent coupled condensate and non-condensate number dynamics for a ?nite total number of particles,and show that such a second-order treatment is provided by a suitably generalized Gross-Pitaevskii equation,coupled to the Castin-Dum number-conserving formulation of the Bogoliubov-de Gennes equations.The necessary equations of motion can be generated from an approximate third-order Hamiltonian,which e ?ectively reduces to second order in the steady state.Such a treatment as described here is suitable for dynamics occurring at ?nite temperature,where there is a signi?cant non-condensate fraction from the outset,or dynamics leading to dynamical instabilities,where depletion of the condensate can also lead to a signi?cant non-condensate fraction,even if the non-condensate fraction is initially negligible.PACS numbers:03.75.Nt,67.40.Db,05.30.Jp I.INTRODUCTION By de?nition,a dilute atomic gas that has undergone Bose-Einstein condensation [1,2,3]has a large number of compo-nent particles occupying the same mode [2,3,4,5,6,7,8].E ?ects associated with such a macroscopic occupation were ?rst observed in super?uid helium and in superconducting metals [9].The importance of interactions in such com-paratively dense condensed-matter systems means that the condensate fraction,although important,is substantially less than the non-condensate fraction.In systems composed of laser and magnetically cooled and trapped dilute atomic gases [10,11,12]the situation is often very di ?erent;the atomic gas can be su ?ciently cold and dilute for the condensate fraction to be a large proportion of the total number of atoms.It is for this reason that the Gross-Pitaevskii equation [13,14,15,16],originally conceived to develop a qualitative understanding of processes in super?uid helium,has achieved the status of a quantitatively useful description of degenerate dilute gases of bosonic atoms.

The Gross-Pitaevskii equation is essentially a classical ?eld

approximation to an underlying quantum ?eld.Notwithstand-

ing its broad utility,there are many situations where a more

accurate description is required.Super?uid to Mott-insulator

phase-transitions in optical lattices [17],and dimer formation

via controlled manipulation of magnetic ?elds (in order to

ex-

We describe a number conserving approach to the dynamics of Bose-Einstein condensed dliute atomic gases. This builds upon the works of Gardiner [C. W. Gardiner, Phys. Rev. A 56, 1414 (1997)], and Castin and Dum [Y. Castin and R. Dum, Phys. Rev. A 57, 3008

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extensions have been proposed.These include generaliza-tions[58,59,60,61,62,63,64]of linear response the-ory[65,66],stochastic interpretations of the Gross-Pitaevskii equation[67,68,69,70,71],Hartree-Fock-Bogoliubov ap-proaches[72,73,74,75,76,77,78,79,80],a variety of kinetic theories[81,82,83,84,85,86,87,88,89,90,91,92], and a cumulant-based formalism[21,93,94,95].

The description presented here is within a number-conserving formalism,and builds on the works of Gardiner [44],and Castin and Dum[45],which are essentially equiv-alent to each other.Symmetry-breaking formulations,which automatically violate particle number conservation,have met with considerable success in describing the observed proper-ties of Bose-Einstein condensed dilute atomic gases.Here symmetry-breaking is de?ned as the breaking of the U(1) global phase symmetry whose Noether charge is the total par-ticle number.Technically,however,symmetry-breaking for-mulations require a coherent superposition of di?erent num-bers of particles.One could argue that the actual particle number is only known statistically in any real experiments, and should be considered an ensemble average from multi-ple realizations of the same experiment.Even given this,it is di?cult to see how shot-to-shot number-coherences could be built up.It is therefore important to understand any dif-ferences which might appear between number-conserving and symmetry-breaking formulations.

A signi?cant di?erence that does occur,in the number-conserving formulation presented in this paper,is the presence of nonlocal terms in the equations of motion for the conden-sate and non-condensate fractions.These arise from de?ning the two fractions in such a way that they must be mutually or-thogonal.This orthogonality is not in general ful?lled if one de?nes the condensate fraction as being the expectation value of a?eld operator,as occurs in symmetry-breaking formula-tions.The no …… 此处隐藏：46045字，全部文档内容请下载后查看。喜欢就下载吧 ……