initially a hot multimode system interacting particles in a nite trap quantum uctuations important near critical point what state does the evaporation

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1 Quantum dynamics of evaporatively cooled Bose-Einstein condensates P. D. Drummond and J. F. Corney The University of Queensland, Brisbane, Australia May 19, Drummond & Corney - Quantum dynamics of evaporative cooled BECs

2 initially a hot multimode system interacting particles in a nite trap quantum uctuations important near critical point what state does the evaporation process lead to? Is it really in thermal equilibrium? =) Must solve a quantum many-body problem in the time-domain! =) use quantum phase-space methods (quasi-probabilities) retains quantum features =) allows multimode simulation =) Simulation of evaporative cooling Drummond & Corney - Quantum dynamics of evaporative cooled BECs 1

3 \Can a quantum system be probabilistically simulated by a classical (probabilistic, I'd universal computer?...if you take the computer to be the classical kind I've assume) \Hence the implications of Feynman's argument seems to be that we really cannot quantum dynamics on a local classical computer... Actual calculations involve simulate 2 The Simulation of Quantum Systems, Address at receiving the Feenberg Medal 3 Lectures on Quantum Monte Carlo, May 1996 A quantum simulation? described so far, (...) and there're no changes in any laws, and there's no hocus-pocus, the answer is certainly, No!" (Richard P. Feynman 1 ) only a few particles and very short propagations." (David M. Cerperle 2 ) \The equivalent to Molecular Dynamics (Quantum Molecular Dynamics) does not exist in any practical sense... One is forced to either simulate very small systems (i.e. less ve particles) or to make serious approximations." (David M. Cerpele 3 ) 1 than Physics with Computers, International Journal of Theoretical Physics 21, 467 (1982) Simulating Drummond & Corney - Quantum dynamics of evaporative cooled BECs 2

4 representation +P expand quantum density matrix ^ in o-diagonal coherent state projection ( 1 2 ) j 1ih 2 j P 2 j 1 i D 1D 2 h P ( 1 2 ) is a positive distribution function which exists for all density matrices when the boundary terms in the integration vanish, P is governed by a equation (FPE) Fokker-Planck operators: Z ^ = j i i are coherent states of the atomic many-body operators D 1, D 2 are the functional integration measures Drummond & Corney - Quantum dynamics of evaporative cooled BECs 3

5 the FPE leads to two stochastic phase-space j j = 1 2 m = atomic mass V (x) = trapping potential ;(x) = loss rate U = atom-atom coupling + ^ E ^ 1 2 +P Equations = [;h 2 r 2 =2m + V (x) ; ih;(x)=2 + U j 3;j + pihu j (t x)] j (t where: j (t x) = stochastic term h 2 1i = =atom density D Drummond & Corney - Quantum dynamics of evaporative cooled BECs 4

6 all quantum eects enter through the real stochastic noise terms j (t x) without these, the equations correspond to the approximate classical equations mean-eld the noise terms are independent, Gaussian, and delta-correlated in space time: and Correlations h i (t x) j (t x )i = ij (x ; x )(t ; t ) Drummond & Corney - Quantum dynamics of evaporative cooled BECs 5

7 Potentials V (x t) = (1 ; t)v max d j=1 [sin(x j=l j )] 2 Trapping Potential V(x,y) y.5.5 x the trap walls are lowered over time Drummond & Corney - Quantum dynamics of evaporative cooled BECs 6

8 Atom loss hot atoms near the edge of the trap escape absorption: ;(x) = ; max d j=1 [sin(x j=l j )] 5 Modulated Absorption Γ(x,y) y.5.5 x Drummond & Corney - Quantum dynamics of evaporative cooled BECs 7

9 the precise initial conditions are not expected to be very signicant subsequent quantum noise dominates initial thermal noise the simplest possible choice is made: a high-temperature Bose-Einstein canonical ensemble grand in this initial state, no account is taken of the trapping or interparticle potentials Initial Conditions Drummond & Corney - Quantum dynamics of evaporative cooled BECs 8

10 try to choose parameters used in the experiments however computational constraints on the lattice size limited number of atoms =) limited to either narrow deep traps, or wide shallow traps =) go for a compromise: a = :6nm trap size ' 1m initial temperature T = 2:4 1 ;7 K initial number of atoms N ' 5 in 2d initial number of atoms N ' 15 in 3d Parameters Drummond & Corney - Quantum dynamics of evaporative cooled BECs 9

11 have done 1d, 2d and 3d simulations nal distribution in k-space is tall (intense) and narrow, because of: ramped potential thermalising eect of collisions quantum eects dominate for for these parameters high atom loss rate initially change trap shape =) alter evaporative cooling procedure =) Evaporative cooling dynamics Drummond & Corney - Quantum dynamics of evaporative cooled BECs 1

12 dimensional simulation - evolution of the momentum distribution Three a single trajectory. during Drummond & Corney - Quantum dynamics of evaporative cooled BECs 11

13 of a three-dimensional Bose condensate, showing the ensemble Simulation (15 paths) atom density hn(k)i along one dimension in Fourier average space versus time. Drummond & Corney - Quantum dynamics of evaporative cooled BECs 12

14 measurable (physical) quantities given by ensemble average want to measure the occupied volume in k-space use higher order correlations to do this dene connement parameter: R 1 1 (k) 2 (k) 2 (k)i dk h (k) ;R (k) dk x 1 h (k)i Connement Measure Q = Drummond & Corney - Quantum dynamics of evaporative cooled BECs 13

15 of a three-dimensional Bose condensate, showing the ensemble Simulation evolution (11 paths) of the connement parameter Q. average /< Ψ 2 *Ψ 1 > Q Time Drummond & Corney - Quantum dynamics of evaporative cooled BECs 14

16 calculate evolution of angular momentum distribution occupation of angular modes is given by: hn(j)i = n is the index for the radial modes j is the index for the angular modes ^ jn ^ jni h Angular Momentum nonzero nal (angular) momentum =) vortices X n where Drummond & Corney - Quantum dynamics of evaporative cooled BECs 15

17 momentum distribution n(j), during the condensation of a twodimensional Angular BEC. 1 2 angular momentum (j) n(j)) time (t) 5 5 time (t) angular momentum (j) Drummond & Corney - Quantum dynamics of evaporative cooled BECs 16

18 average of the angular momentum distribution hn(j)i, during the Ensemble of a two-dimensional BEC. condensation <n(j)> time (t) angular momentum (j) Drummond & Corney - Quantum dynamics of evaporative cooled BECs 17

19 rst principles quantum simulation of BEC 1 atoms (initially) in 32 trap modes dicult - not impossible with classical computers physics: evaporative cooling with variable potential condensate can form in excited mode can spontaneously form metastable vortices Conclusions up to 2 atoms in condensate Drummond & Corney - Quantum dynamics of evaporative cooled BECs 18

- thus, the total number of atoms per second that absorb a photon is

- thus, the total number of atoms per second that absorb a photon is Stimulated Emission of Radiation - stimulated emission is referring to the emission of radiation (a photon) from one quantum system at its transition frequency induced by the presence of other photons