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