qels99 ec landscape - school of mathematics and physics · co oled becs. p equations the fpe leads...

Quantum dynamics of evaporatively cooled

Bose�Einstein condensates

P� D� Drummond and J� F� Corney

The University of Queensland� Brisbane� Australia

May ��

http��xxx�lanl�gov�abs�cond�mat��

Drummond � Corney � Quantum dynamics of evaporative cooled BECs

Simulation of evaporative cooling

� 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

Drummond � Corney � Quantum dynamics of evaporative cooled BECs �

A quantum simulation�

� �Can a quantum system be probabilistically simulated by a classical �probabilistic� I�d

assume� universal computer� ��If you take the computer to be the classical kind I�ve

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��

� �Hence the implications of Feynman�s argument seems to be that we really cannot

simulate quantum dynamics on a local classical computer�� Actual calculations involve

only a few particles and very short propagations� �David M� Cerperle ��

� �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

than �ve particles� or to make serious approximations� �David M� Cerpele ��

�Simulating Physics with Computers� International Journal of Theoretical Physics ��

�The Simulation of Quantum Systems� Address at receiving the Feenberg Medal

�Lectures on Quantum Monte Carlo� May ��


�P representation

� expand quantum density matrix �� in o��diagonal coherent state projection

operators�

�� Z

P �� j��i h��j

h��j��i D��D��

� j�ii are coherent states of the atomic many�body operators

� D�� D�� are the functional integration measures

� P �� is a positive distribution function which exists for all density

matrices

� when the boundary terms in the integration vanish� P is governed by a

Fokker�Planck equation �FPE�


�P Equations

� the FPE leads to two stochastic phase�space equations�

�h��j

�t

� ��h�r��m� V �x�� i�h�x�� U�j��

��j �p

i�hU�j�t�x��j�t�x�

where�

� j � �� m � atomic mass

� V �x� � trapping potential

� �x� � loss rate

� U � atom�atom coupling

� �j�t�x� � stochastic term

� h��i �

D��

�

��E

�atom density


Correlations

� all quantum e�ects enter through the real stochastic noise terms �j�t�x�

� without these� the equations correspond to the approximate classical

mean� eld equations

� the noise terms are independent� Gaussian� and delta�correlated in space

and time�

h�i�t�x��j�t��x��i � �ij��x� x��t� t��


Potentials

�V�x�t��

��t�Vmax d j��sin�xj�L

j��

−0.

5

0

0.5

−0.

5

0

0.50

0.51

1.52

x

Tra

ppin

g P

oten

tial

y

V(x,y)

�thetrap

wallsarelowered

over

time

Drummond�

Corney

�Quantum

dynamicsofevaporativecooledBECs

�

Atom

loss

�hot

atom

sneartheedge

ofthetrap

escape

�ab

sorption

��x��

max d j��sin�xj�L

j��

−0.

5

0

0.5

−0.

5

0

0.50

0.51

1.52

x

Mod

ulat

ed A

bsor

ptio

n

y

Γ(x,y)

Drummond�

Corney

�Quantum

dynamicsofevaporativecooledBECs

Initial Conditions

� the precise initial conditions are not expected to be very signi cant

� subsequent quantum noise dominates initial thermal noise

� the simplest possible choice is made� a high�temperature Bose�Einstein

grand canonical ensemble

� in this initial state� no account is taken of the trapping or interparticle

potentials

Drummond � Corney � Quantum dynamics of evaporative cooled BECs

Parameters

� 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� � ��nm� trap size � ��m

� initial temperature T� � �� K

� initial number of atoms N� � �� in �d

� initial number of atoms N� � �� in �d


Evaporative cooling dynamics

� have done �d� �d and �d simulations

� nal distribution in k�space is tall �intense� and narrow� because of�

� ramped potential

� thermalising e�ect of collisions

� quantum e�ects dominate for for these parameters

� high atom loss rate initially

�� change trap shape

�� alter evaporative cooling procedure

Drummond � Corney � Quantum dynamics of evaporative cooled BECs ��

Three dimensional simulation � evolution of the momentum distribution

during a single trajectory�


Simulation of a three�dimensional Bose condensate� showing the ensemble

average �� paths� atom density hn�k�i along one dimension in Fourier

space versus time�


Con�nement Measure

� measurable �physical� quantities given by ensemble average

� want to measure the occupied volume in k�space

� use higher order correlations to do this

� de ne con nement parameter�

Q �R

dk h��k��k��k��

��k�i�Rdk h��k��k�i��

x��


0 20 40 60 80 100−5

0

5

10

15

20

25

Time

1/<∫ Ψ2*Ψ

1>

Q

Simulation of a three�dimensional Bose condensate� showing the ensemble

average evolution �� paths� of the con nement parameter Q�


Angular Momentum

� nonzero nal �angular� momentum �� vortices

� calculate evolution of angular momentum distribution

� occupation of angular modes is given by�

hn�j�i �X

n

h��jn��

jni

where

� n is the index for the radial modes

� j is the index for the angular modes


−10

−5

0

5

10

0 20 40 60 80 100

angu

lar

mom

entu

m (

j)

time (t)

−5

0

5

10

15

20

−10−5

05

10

0

50

100

−20

−10

0

10

20

30

angular momentum (j)time (t)

n(j))

Angular momentum distribution n�j�� during the condensation of a two�

dimensional BEC�


−10−5

05

10

0

50

100

0

2

4

6

8

10

12

angular momentum (j)time (t)

<n(

j)>

Ensemble average of the angular momentum distribution hn�j�i� during the

condensation of a two�dimensional BEC�


Conclusions

� rst principles quantum simulation of BEC

� �� atoms �initially� in �� trap modes� up to �� atoms in

condensate

� di�cult � not impossible with classical computers

� physics� evaporative cooling with variable potential

� condensate can form in excited mode

� can spontaneously form metastable vortices


qels99 ec landscape - school of mathematics and physics · co oled becs. p equations the fpe leads...

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