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: wmliu@aphy.iphy.ac.cnPhone: 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).

HamiltonianHamiltonian2

2 2( , )sin ( )2 lpH U x y z mgzm

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 xm

V x d V xx ikx

x d x

2

0

( )2

( ) ( )

2

W

l

B

PH V x Fxm

V x d V xE 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

2 ImE

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( ( , ) )

/ 2lU 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 dd

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 ( )2

1 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 01( , ) cos ( , )( )sin3l lV U x y U x y 0 arcsin( )

4 ( , )l

mgU x y

Euler-Lagrange equation

21 ( ) ( ( ))2

dzm V z Ed

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) ( )

lk 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

w ek

z zw w

is energy dependent frequencyis energy dependent frequency

20

0 2

16 ( , ) coslU x ywm

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 cot6

nn

l

Vn w

V V U x y

Harmonic approximationHarmonic approximation

Metastable ground stateMetastable ground state

max

0

365max

0 00

3122

VwVw e

w

Tunneling rate of Landau-Zener regimeTunneling rate of Landau-Zener regime

2

2

4

8

cgg

LZ

c

mg e

g

Atoms:Atoms:

Yale experimental Yale experimental

parametersparameters

87Rb

850( , ) 2.1

50l R

LZ

nmU x y E

ms

3 10

1

12.26 10

12.3788

LZ

LZ

s

sms

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

nmU x y E

s

TheoryTheory

3 10

1

2.63 10

2.600.39

LZ

LZ

s

ss

At high temperature:At high temperature:Arrhenius lawArrhenius lawmax /0

2BV k T

AR e

Temperature dependenceTemperature dependence

0max0

0

432

0( ) (1 )

wk TB

B

Vwe

wk TT e e

Crossover temperatureCrossover temperature

0

2257

( , ) 2.1

crB

cr

l R

hwT

kT nKU 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 eZ

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

''' '21

22

2

trtrtr ,'~,,

trm

arVm

trt

i ext ,42

, 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

242 ext

ai V rt m m

Long time solutionLong time solution

222 ( ) log(4 )

12

2 2

( )( , ) ( log )

1( ) log(1 ( ) )2

x xi i tt t

xtx t e O t tt

k r kg

Theoretical explanationTheoretical explanation

1 2''

0 012 2 22n 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 E ek 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 condensates in an optical latticePhys. 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.