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Integrable model in Integrable model in Bose-Einstein condensatesBose-Einstein condensates
Wu-Ming Liu(Institute of Physics, Chinese Academy of Science
s )http:// www.iphy.ac.cn
Email: [email protected]
Phone: 86-10-82649249
CollaboratorsCollaboratorsProf. S.T. Chui (Delaware Univ.)Prof. I. Kats (ILL, France)Prof. J.Q. Liang (Shanxi Univ.)Prof. B. A. Malomed (Tel Aviv Univ.)Prof. Q. Niu (Texas Univ.)Prof. Y.Z. Wang (SIOM, CAS)Prof. B. Wu (IOP, CAS)Prof. W.P. Zhang (East China Normal Univ.)Prof. W.M. Zheng (ITP, CAS)
OutlineOutline1. Introduction
2. BEC tunneling - instanton
3. BEC interference - long time solution
4. BEC near Feshbach resonance– solitonBEC near Feshbach resonance– soliton
5. BEC in optical lattice – discrete soliton5. BEC in optical lattice – discrete soliton
6. Two component BEC - soliton inelastic collision
7. Spinor BEC - soliton
8. Conclusion
1. Introduction 7Li 6Li
C. E. Wieman and E. A. Cornell, Science 269, 198 (1995).
40 Lab. Elements: Li, Na, K, H, Rb, He, Fermi gases Y.Z. Wang, BEC in China, March 2002, Shanghai, China
2. BEC tunneling- instanton2. BEC tunneling- instanton
W.M. Liu, W.B. Fan, W.M. Zheng, J.Q. Liang, S.T. Chui,
Quantum tunneling of Bose-Einstein condensates
in optical lattices under gravity,
Phys. Rev. Lett. 88, 170408 (2002).
Fig. 1. The effective optical-plus-gravitational potential U/ER for parameters used in our experiment (ER = 2k2/2m is the photon recoil energy with k = 2 / ). The horizontal oscillating curves illustrate de Broglie waves from the tunnel output of each well. In region A, the relative phases of the waves interfere constructively to form a pulse. Heavy lines illustrate the energies of the lowest bound states of harmonic oscillator potentials that match the shapes of the actual potentials near each local energy minimum.
B.P. Anderson et al., Science 282, 1686 (1998).
Figure 1. (A) Combined potential of the optical lattice and the magnetic trap in the axial direction. The curvature of the magnetic potential is exaggerated by a factor of 100 for clarity. (B) Absorption image of the BEC released from the combined trap. The expansion time was 26.5 ms and the optical potential height was 5ER.
..
F.S. Cataliotti, Science 293, 843 (2001).
HamiltonianHamiltonian
22 2
( , )sin ( )2 l
pH U x y z mgz
m
Landau-Zener tunnelingLandau-Zener tunneling
Wannier-Stark tunnelingWannier-Stark tunneling
Parameters:Wells: 30 or 200
Atoms number: 10³ /well
Density: n₀=10¹³ cm ³⁻
4 /MF BU K nK atom
157 /k BE K nK atom
Landau-Zener tunnelingLandau-Zener tunneling Barrier between lattices is low Localized level between lattices is coupling Miniband Adiabatic approximation Tunneling between delocalized states in different Bloch bands
Potential energy and Bloch bandsPotential energy and Bloch bands
Tilted bands and WS laddersTilted bands and WS ladders
Wannier-Stark tunnelingWannier-Stark tunneling An external field Wavefunction of miniband is localization Miniband is divided into discrete level Wannier-Stark ladder Tunneling between localized states in different individual wells
—Wannier-Stark localized states
Bloch bands and WS LadderBloch bands and WS Ladder
2
, ,
, ,
( )2
( ) ( )
( ) exp( )
( ) ( )
B
k k
k k
PH V x
mV x d V x
x ikx
x d x
2
0
( )2
( ) ( )
2
W
l
B
PH V x Fx
mV x d V x
E E ldF
TdF
WS Ladder
Wannier-Stark energy spectrumWannier-Stark energy spectrum
Resonances condition for discrete spectrum
mean energy of \alpha band
Actual energy spectrum for discrete spectrum
,
0, 1...lE dFl
l
, 2
/
l E dFl i
I.W. Herbst et al., Commun. Math. Phys. 80, 23(1981)J. Agler et al., ibid 100, 161 (1985)
J.-M. Combes et al., ibid 140, 291(1991)
Potential energy and energy bandsPotential energy and energy bands
2ImE
Decay rateDecay rate
2 2H EA e e iT
E: complex energyE: complex energy Transition amplitudeTransition amplitude
No crossing--conditionNo crossing--condition
top
top
0
( ( , ) )
/ 2
lU x y E
mg
Transition amplitudeTransition amplitude
*
2
( ) ( ) ( , ; , )
1( , ; , ) { }exp( [ ( ) ( )] )
2
f
i
E f E i f f i i f i
f f i i
A d d
dzD x m V z d
d
Periodic instanton represents pseudo-condensPeriodic instanton represents pseudo-condensed atom configuration responsible for tunnelined atom configuration responsible for tunneling under barrier at energy Eg under barrier at energy E
20 0 0
30 0
1 4( ) ( , ) cos ( )
21 4
( , )sin ( )3
l
l
V z V U x y z
U x y z
Potential V(z)Potential V(z)
0 0 0 0
1( , ) cos ( , )( )sin
3l lV U x y U x y 0 arcsin( )4 ( , )l
mg
U x y
Euler-Lagrange equation
21( ) ( ( ))
2
dzm V z E
d
Periodic instanton solution – solutPeriodic instanton solution – solutions of classical Euler-Lagrange ions of classical Euler-Lagrange equations in Euclidean space-timequations in Euclidean space-time with finite energye with finite energy
denote three roots of equation V(z)=E
1 2 3( ) ( ) ( )z E z E z E
23 2 3( ) ( ) ( )z z z z sn u k
All instanton contributionsAll instanton contributions
20 1 3
' 21
8 ( , )cos ( )2
( ) 3WlU x y z z
E i ek k mA e
342 22
0 1
4 22
64 2( , ) cos ( ) [(1 )( 2) ( )
15 3
2( 1) ( )
l
k mW U x Y z z k k k
k k k
Decay rate of metastable stateDecay rate of metastable state
'1
1 30
( )
2( )
3
EE
E
we
k
z zw w
is energy dependent frequencyis energy dependent frequency
20
0 2
16 ( , ) coslU x yw
m
is frequency of small oscillationsis frequency of small oscillations
Decay rate of nth low excited stateDecay rate of nth low excited state
0 0nE n E
max0
0
2max 0 0 0
4321( )
! 2
1( , ) cos cot
6
nn
l
V
n w
V V U x y
Harmonic approximationHarmonic approximation
Metastable ground stateMetastable ground state
max
0
36
5max0 0
0
312
2
V
wVw e
w
Tunneling rate of Landau-Zener regimeTunneling rate of Landau-Zener regime
2
2
4
8
cg
gLZ
c
mge
g
Atoms:Atoms:
Yale experimental Yale experimental
parametersparameters
87Rb
850
( , ) 2.1
50l R
LZ
nm
U x y E
ms
3 10
1
12.26 10
12.37
88LZ
LZ
s
s
ms
TheoryTheory
Atoms:Atoms: INFM (Istituto NazioINFM (Istituto Nazionale di Fisica della nale di Fisica della Materia, Italy)Materia, Italy)
87Rb
795
( , ) 5
0.3l R
LZ
nm
U x y E
s
TheoryTheory
3 10
1
2.63 10
2.60
0.39LZ
LZ
s
s
s
At high temperature:At high temperature:Arrhenius lawArrhenius law
max /0
2BV k T
AR e
Temperature dependenceTemperature dependence
0max0
0
432
0( ) (1 )
w
k TB
B
Vwe
wk TT e e
Crossover temperatureCrossover temperature
0
2
257
( , ) 2.1
crB
cr
l R
hwT
k
T nK
U x y E
At low temperature:At low temperature:Pure quantum tunnelingPure quantum tunneling
At intermediate temperature:At intermediate temperature:Thermally assisted tunnelingThermally assisted tunneling
Measure tunneling from lowest metastable stateMeasure tunneling from lowest metastable state
1. Turn on a potential which has only one state in each well.2. Accelerate potential in such a way that only band of state
s from these levels are swept along with potential, leaving all higher states behind (so they can be neglected).
3. Increase amplitude of potential, so that different wells become isolated from each other.
4. Tilt potential (by acceleration) to achieve Wannier-Stark regime described by present theory.
5. Observe how many atoms survive in time t.
0
0
( )n
nB
Et
k T
n
NN t e
Z
PopulationPopulationExperimental predictionExperimental prediction
Measure decays from excited Measure decays from excited states and at higher temperature states and at higher temperature 1. Starting with a thermal distribution of free
atom states, turn on potential to some amplitude, so that eventually there are n bands lying in wells.
2. Accelerate potential so that n bands are taken along with wells, leaving atoms in higher bands behind. The acceleration must be such that occupation number of each of n bands is not changed during this process.
3. Same as (3) above.4. Same as (4) above.5. Same as (5) above.
3. BEC interference–long time solution3. BEC interference–long time solution
W.M. Liu, B. Wu, Q. Niu,
Nonlinear effects in interference of Bose-Einstein condensates,
Phys. Rev. Lett. 84, 2294 (2000).
W. Ketterle, Science 275, 637 (1997).
Experimental parameters:
Separation of two BEC ~ 40 μm
Fringe spacing ~ 15 μm
Expanding time ~ 40 ms
Demonstration: 1. laser-like 2. coherent 3. long-range correlation
Implication: 1. atomic laser 2. Josephson effect
Many-body Hamiltonian
The mean field theory
Gross-Pitaevskii equation
rrrrVrrdrdr
rVm
rdrH ext
''' '2
1
22
2
trtrtr ,'~
,,
trm
arV
mtr
ti ext ,
4
2,
2222
Parameters:Parameters:
x is measured in unit of x0= 1μm
t in unit of mx0/ h, t= 120
φ in unit of square root of n0
G= 4πn0ax02= 5-10
Gross-Pitaevskii equationGross-Pitaevskii equation
2 2 2
24
2 ext
ai V r
t m m
Long time solutionLong time solution
22
2 ( ) log(4 )12
2 2
( )( , ) ( log )
1( ) log(1 ( ) )
2
x xi i t
t t
xtx t e O t tt
k r kg
Theoretical explanationTheoretical explanation
1 2
''0 0
12 2 2
2n nk E V n V
Fringe positionFringe position
Central fringeCentral fringe
1 2"
0 1 1 0 04 2k k k V V
Experimental prediction:Experimental prediction:1. Energy level 2. Many wave packets1. Energy level 2. Many wave packets
Ratio of level width to level spacingRatio of level width to level spacing
22 n ng E w En n
n n
k Ee
k E
S. Inouye et al., Nature 392, 151 (1998).S. Inouye et al., Nature 392, 151 (1998).
4. BEC 4. BEC near Feshbach resonancenear Feshbach resonance-soliton-soliton
Z. X. Liang, Z. D. Zhang, W. M. Liu,Z. X. Liang, Z. D. Zhang, W. M. Liu,
Dynamics of a bright soliton in Bose-Einstein condensates
with time-dependent atomic scattering length in an expulsive parabolic potential,
Phys. Rev. Lett. 74, 050402 (2005).Phys. Rev. Lett. 74, 050402 (2005).
SupernovaSupernovaS.L. Cornish et al., Phys. Rev. Lett. 85, 1795 (2000).S.L. Cornish et al., Phys. Rev. Lett. 85, 1795 (2000).
L. Khaykovich et al., Science 296, 1290 (2002).
5. BEC in optical lattice–discrete soliton5. BEC in optical lattice–discrete soliton
K.E. Strecker et al., Nature 417, 150 (2002).
K.E. Strecker et al., Nature 417, 150 (2002).
Z.W. Xie, Z.X. Cao, E.I. Kats, W.M. LiuZ.W. Xie, Z.X. Cao, E.I. Kats, W.M. Liu,,
Nonlinear dynamics Nonlinear dynamics of of dipolardipolar Bose-Einstein condensate Bose-Einstein condensate
in optical lattice,in optical lattice,
Phys. Rev. A 71, 025601 (2005).Phys. Rev. A 71, 025601 (2005).
G.P. Zheng, J.Q. Liang, W.M. Liu,G.P. Zheng, J.Q. Liang, W.M. Liu,
Phase diagram of two-species Bose-Einstein conde
nsates in an optical lattice
Phys. Rev. A71, 053608 (2005)Phys. Rev. A71, 053608 (2005)
6. Two component BEC - soliton inelastic collision
Soliton filter and switchSoliton filter and switch
7. Spinor BEC - soliton 7. Spinor BEC - soliton J. Stenger, Nature 396, 345 (1998).
Z.W. Xie, W.P. Zhang, S.T. Chui, W.M. Liu,
Magnetic solitons of
spinor Bose-Einstein condensates
in optical lattice,
Phys. Rev. A69, 053609 (2004).
Z.D. Li, P.B. He, L.Li, J.Q. Liang, W.M. LiZ.D. Li, P.B. He, L.Li, J.Q. Liang, W.M. Liu,u,
Soliton collision of spinor Bose-Einstein condensates
in optical lattice,
Phys. Rev. A71, 053608 (2005).
L. Li, Z.D. Li, B. A. Malomed, D. Mihalache, W. M. Liu,L. Li, Z.D. Li, B. A. Malomed, D. Mihalache, W. M. Liu,
Exact soliton solutions and Exact soliton solutions and nonlinear modulation instability nonlinear modulation instability
in spinor Bose-Einstein condensates,in spinor Bose-Einstein condensates,
Phys. Rev. A 72, 03???? (2005).Phys. Rev. A 72, 03???? (2005).
BEC tunneling - instanton
BEC interference - long time solution
BEC near Feshbach resonance– solitonBEC near Feshbach resonance– soliton
BEC in optical lattice – discrete solitonBEC in optical lattice – discrete soliton
Two component BEC - soliton inelastic collision
Spinor BEC - soliton
8. Conclusion8. Conclusion
Bose-Einstein condensates become an
ultralow-temperature laboratory
for atom optics, collisional physics and many-body physics, superfluidity, quantized vortices, Josephson junctions and quantum phase transitions.