Superfluid insulator transition in a Superfluid insulator transition in a moving condensatemoving condensate

Anatoli Polkovnikov,Anatoli Polkovnikov,Boston UniversityBoston University

Collaboration:Ehud AltmanEhud Altman -- WeizmannWeizmannEugene Demler Eugene Demler - - HarvardHarvardBertrand HalperinBertrand Halperin - - HarvardHarvardMikhail LukinMikhail Lukin - - HarvardHarvard

Plan of the talk

1. Bosons in optical lattices. Equilibrium phase diagram.

2. Superfluid-insulator transition in a moving condensate.

• Mean field phase diagram.

• Role of quantum fluctuations.

3. Conclusions and experimental implications.

Interacting bosons in optical lattices. Interacting bosons in optical lattices.

Highly tunable periodic potentials with no defects.Highly tunable periodic potentials with no defects.

Equilibrium system. Equilibrium system.

Interaction energy (two-body collisions):

int ( 1)2 j j

j

UE N N

Eint is minimized when Nj=N=const:

2 2 6 02 2U U

Interaction suppresses number fluctuations and leads to Interaction suppresses number fluctuations and leads to localization of atoms.localization of atoms.

Equilibrium system. Equilibrium system.

Kinetic (tunneling) energy:Kinetic (tunneling) energy:

† †tun j k k j

jk

E J a a a a

e jij ja N

Kinetic energy is minimized when the phase is Kinetic energy is minimized when the phase is uniform throughout the system.uniform throughout the system.

2 cos( )tun j kjk

E JN

Classically the ground state has a uniform density and Classically the ground state has a uniform density and a uniform phase.a uniform phase.

However, number and phase are conjugate variables.

They do not commute: , 1N i N

There is a competition between the interaction leading to There is a competition between the interaction leading to localization and tunneling leading to phase coherence.localization and tunneling leading to phase coherence.

Superfluid regime: Superfluid regime:

(M.P.A. Fisher, P. Weichman, G. Grinstein, D. Fisher, 1989)

Superfluid-insulator quantum phase transition.

Strong tunnelingStrong tunneling

cos const,

-

i j

i j

Weak tunnelingWeak tunnelingInsulating regime: Insulating regime:

cos 0,

-

i j

i j

Classical non-equlibrium phase transitionsClassical non-equlibrium phase transitionsSuperfluids can support non-dissipative current.

Exp: Fallani et. al., (Florence) cond-mat/0404045

Theory: Wu and Niu PRA (01); Smerzi et. al. PRL (02).

Theory: superfluid flow becomes unstable.

/ 2p

Based on the analysis of classical equations of motion (number and phase commute).

Damping of a superfluid current in 1DDamping of a superfluid current in 1D

C.D. Fertig et. al. cond-mat/0410491

max / 5 / 2p

See also : AP and D.-W. Wang, PRL 93, 070401 (2004).

Current damping below Current damping below classical instability.classical instability.

No sharp transition.No sharp transition.

What happens if we there are both quantum What happens if we there are both quantum fluctuations and superfluid flow?fluctuations and superfluid flow?

???

p

U/J

Stable

Unstable

SF MI

p

SF MI

U/J???

possible experimental sequence: ~lattice potential

Physical Argument

sI p SF current in free space

sinsI p SF current on a lattice

Strong tunneling regime (weak quantum fluctuations): s = const. Current has a maximum at p=/2.

This is precisely the momentum corresponding to the onset of the instability within the classical picture.

Wu and Niu PRA (01); Smerzi et. al. PRL (02). Not a coincidence!!!

s – superfluid density, p – condensate momentum.

Include quantum depletion.Include quantum depletion.

Equilibrium: ( / )s s J U

Current state:coseffJ J J p

( )s p

0.0 0.1 0.2 0.3 0.4 0.5

p*

I(p)

s(p)

sin(p)

Condensate momentum p/

( )sinsI p p

With quantum depletion the current state is unstable at

* / 2.p p

p

OK if N1:

2

2,

2 cos( )2j k

j k j j

UH JN

Quantum rotor model

2

1 12 2 sin sinjj j j j

dUJN

dt

Deep in the superfluid regime (JN U) use GP equations of motion:

j jpj 2

1 12 2 cos 2jj j j

dUJN p

dt

Unstable motion for Unstable motion for p>p>/2/2

SF in the vicinity of the insulating transition: U JN.

Structure of the ground state:

It is not possible to define a local phase and a local phase gradient. Classical picture and equations of motion are not valid.

After coarse graining we get both amplitude and phase fluctuations.

Need to coarse grain the system.

Time dependent Ginzburg-Landau:

( diverges at the transition)

Stability analysis around a current carrying solution:

~ 1c cp U U

22 2

p

U/J

Superfluid MI

~ 1cp

S. Sachdev, Quantum phase transitions; Altman and Auerbach (2002)

Use time-dependent Gutzwiller approximation to interpolate between these limits.

0.0

0.1

0.2

0.3

0.4

0.5

0.0 0.2 0.4 0.6 0.8 1.0

d=3d=2

d=1

unstable

stable

U/Uc

p/

Meanfield (Gutzwiller ansatzt) phase diagram

Is there current decay below the instability?Is there current decay below the instability?

Role of fluctuations

Below the mean field transition superfluid current can decay via quantum tunneling or thermal decay .

E

p

Phase slip

Related questions in superconductivity

Reduction of TC and the critical current in superconducting wires

Webb and Warburton, PRL (1968)

Theory (thermal phase slips) in 1D:

Langer and Ambegaokar, Phys. Rev. (1967)McCumber and Halperin, Phys Rev. B (1970)

Theory in 3D at small currents:

Langer and Fisher, Phys. Rev. Lett. (1967)

0.0

0.1

0.2

0.3

0.4

0.5

0.0 0.2 0.4 0.6 0.8 1.0

unstable

stable

U/Uc

p/

Current decay far from the insulating transition

Decay due to quantum fluctuations

The particle can escape via tunneling:

exp S

S is the tunneling action, or the classical action of a particle moving in the inverted potential

02

0

1 ( )2

dxS d V xm d

Asymptotical decay rate near the instability

02

2 3

0

1 ( ) 02 c

dxS d x bx p pm d

Rescale the variables:1, =

2mx x

b

5/ 2

5/ 202

12 cS S p p

bm

02

2 30 0

1 8 22 15

dxS d x xd

exp S

Many body system, 1D

5/ 2

exp 7.12

JN pU

7.1 – variational result

JNU

semiclassical parameter (plays the role of 1/)

Small N~1Large N~102-103

Higher dimensions.

Longitudinal stiffness is much smaller than the transverse.

Need to excite many chains in order to create a phase slip.

12

r p

|| cos ,J J p

J J

r

62

2

d

d dJNS C pU

Phase slip tunneling is more expensive in higher dimensions:

expd dS

Stability phase diagram3dS

Crossover1 3dS

Stable

1dS Unstable

0.0

0.1

0.2

0.3

0.4

0.5

0.0 0.2 0.4 0.6 0.8 1.0

unstable

stable

U/Uc

p/

Current decay in the vicinity of the superfluid-insulator transition

Use the same steps as before to obtain the asymptotics:

52

3 1 3 , expd

dd dd

CS p S

32

1 2

12

2

3

5.7 1 3

3.2 1 3

4.3

S p

S p

S

Discontinuous change of the decay rate across the meanfield transition. Phase diagram is well defined in 3D!

Large broadening in one and two dimensions.Large broadening in one and two dimensions.

See also AP and D.-W. Wang, PRL, 93, 070401 (2004)

Damping of a superfluid current in one dimension

C.D. Fertig et. al. cond-mat/0410491

0.0

0.1

0.2

0.3

0.4

0.5

0.0 0.2 0.4 0.6 0.8 1.0

d=3d=2

d=1

unstable

stable

U/Uc

p/

Effect of the parabolic trap

Expect that the motion becomes unstable first near the edges, where N=1

0 100 200 300 400 500

-0.2

-0.1

0.0

0.1

0.2

0.00 0.17 0.34 0.52 0.69 0.86

Cen

ter o

f Mas

s M

omen

tum

Time

N=1.5 N=3

U=0.01 tJ=1/4

Gutzwiller ansatz simulations (2D)

Exact simulations: 8 sites, 16 bosons

0 50 100 150 2000

2

4

6

8

p=/2

p=/4

p=0

n p

Time

SF MI

( ) 2 tanh 0.02 tanh 0.02(200 ) , J=1U t t t p

U/J

Semiclassical (Truncated Wigner) simulations of damping of dipolar motion in a harmonic trap

1 100.1

1

10

-1.0

-0.5

0.0

0.5

1.0

GP N=500

Dis

plac

emen

t (D

0)

Inverse Tunneling (1/J)

ln(D0/D

1)

Time

D0

D1

Dis

plac

emen

t D(t)

AP and D.-W. Wang, PRL 93, 070401 (2004).

Quantum fluctuations:Quantum fluctuations:

Smaller critical Smaller critical currentcurrent

Broad transitionBroad transition

Detecting equilibrium SF-IN transition boundary in 3D.

p

U/J

Superfluid MI

ExtrapolateAt nonzero current the SF-IN transition is irreversible: no restoration of current and partial restoration of phase coherence in a cyclic ramp.

Easy to detect nonequilibrium Easy to detect nonequilibrium irreversible transition!!irreversible transition!!

Summary

asymptotical behavior of the decay rate near the mean-field transition

p

U/J

Superfluid MI

Quantum fluctuations

Depletion of the condensate. Reduction of the critical current. All spatial dimensions.

mean field beyond mean field

Broadening of the mean field transition. Low dimensions

Smooth connection between the classical dynamical instability and the quantum superfluid-insulator transition.

Qualitative agreement Qualitative agreement with experiments and with experiments and numerical simulations.numerical simulations.

p

U/J

Superfluid MI

Time-dependent Gutzwiller approximation

0.0

0.2

0.4

0.6

0.8

1.0

2D

p=/5U=0.01tJz=1N=1

Pha

se c

oher

ence

(np)

U/J