Thermalisation and vortex formation in a mechanically ... - Physics
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Thermalisation and vortex formation in a mechanically ... - Physics

Thermalisation and vortex formationin amechanically perturbed condensateJack DoddCentreACQAOR.J. Ballagh and Tod WrightJack Dodd centre for Photonicsand Ultra-cold AtomsDepartment of PhysicsUniversity of OtagoNew ZealandBlair BlakieCrispin GardinerTod WrightAshton BradleyMatthew DavisThis TalkDynamics of finite temperature condensates1] Vortex lattice via stirring a condensate‣ classical field formalism‣ formation and role of thermal cloud2] BKT transition in an optical latticeVortex Lattice Formation(I) by condensationExperiment of Cornell’s group (evaporatively cool a rotating thermal cloud)• Cannot simulate formation with unitary equation (e.g. GPE)• Need dissipation• Thermal cloud plays a key role‘Fudge’ Equation‣ recent experiment of Cornell’s group‣ preliminary dynamical description

trapVortex Lattice Formation(II) by stirring a condensateKey experiments: Ketterle MIT, Dalibard ENS,The ‘Classical Field’ method‘scattering’ termsFormal developmentMaster equation (eliminate R NC )Wigner functionStochastic PDE for stochastic field‘growth terms’50ms 150ms 300ms• Creation of thermal cloud by stirring• Various theoretical models …..• Wish to understand and model this mechanismInteractionswith the bathneglectedneglectedneglectedField is restricted to condensate band• stochastic initial condition(vacuum and thermal noise)Region of Validity in Practice• Sufficiently high particle density in position space.• Initially most modes unoccupied and some modes highly occupiedSolve in 2D in rotating frameCut-off energyProjector enforces thisrestricted evolutionInitial stateT = 0 condensate + vacuum noisea fewi.e. we are examining dynamics of thermalisationin the condensate bandBasis modes Y n (x) : Gauss-Laguerrecutoff at

Ensemble of 100 simulationsCondensateFrom one-body density matrixThermal cloud(everything else)largest eigenvalue (and eigenvector)

Evolution of• condensate fraction• mean angular momentumMomentum DistributionExtracting thermodynamic quantities of the thermal fieldEnergy distribution over non-interacting modesConceptualquestion arises:Is the Penrose-Onsager definition ofcondensed fractionappropriate for nonequilibrium?Fit Bose distributionHave neglected meanfield shift of mode energies

Evolution ofand TPart TwoBKT transition in an optical lattice• Substantial thermalisation of field in 2D• May require 3D treatment to describerelaxation to a rigid vortex latticeBerezinskii, Kosterlitz and Thouless (BKT) transitionBKT In a lattice of 2D Bose condensates• is a 2D phenomenone.g.Optical Lattice• vortices are central objectsThermalTemperaturePrevious treatment• based on Bose-Hubbard model• tight binding, localised mode expansionSimula & BlakiePRL 96,020404,2006free vorticesBKTT BKTN 0 atoms per well.‘bound’ vortex pairsT cBECInfluence ofharmonic trapAt T=0quantum fluctuations dominateMott insulator transition at

Condensates at each lattice site ( ); phaseFinite TemperatureThermal fluctuations dominateRecent experiment from Cornell’s groupSchweikhard, Tung, Cornell, cond-mat 07040289Experiment is in 2D hexagonal optical latticeDefineThen forandBose-HubbardThis is the ‘quantum phase model’ Trombettoni, et al , NJP 7 (2005) 57Quantum Monte Carlo shows(i) At low temperatures, , all phases same(ii) Atphases unlock. Identify as BKT transitionClosely related experimentfrom Anderson’s groupPRL 98,110402,2007Key resultOur Classical field simulationUse projected GP in 2DDynamical evolutionInitial state:Equilibrium state of harmonic trapat temperature TTechnical issue: What mode basis for evolution ?e.g.first excited statefor V OL = 320Gauss-Laguerre basisPlane wave basis

Initial state kT = 186 t =10; V OL = 20% t =35; V OL = 37%ParametersV OL = 280kT = 186recoil energy = 3.8initial = 6.5t =35; V OL = 100% t =79.5; V OL = 0%t =79.5; V OL = 10%Units: in terms of Harmonic trap: [energy]Remark: some technical issues for constructing projectorPhase of Field T=186 : Above the BKT transitionPhase of Field T=0 : Below the BKT transitionQuantum noise only

Vortex CountingAfter lattice ramp-downSummary‣ Development and application of classical field methodsfor finite temperature condensates‣ Revisit formation of vortex lattice by stirring• modelled development of thermal cloud- no vortex lattice formed‣ BKT transition of condensates in a latticeExperimentalresults• promising tool for quantitative description• projector needs to be developed

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