Superfluid insulator transition in a moving condensate

Anatoli Polkovnikov

Harvard University

Ehud Altman,

Eugene Demler,

Bertrand Halperin,

Misha Lukin

Plan of the talk

1. General motivation and overview.

2. Bosons in optical lattices. Equilibrium phase diagram. Examples of quantum dynamics.

3. Superfluid-insulator transition in a moving condensate.

• Qualitative picture

• Non-equilibrium phase diagram.

• Role of quantum fluctuations

4. Conclusions and experimental implications.

Why is the physics of cold atoms interesting?

It is possible to realize strongly interacting systems, both fermionic and bosonic.

Parameters of the Hamiltonian are well known and well controlled.

One can address not only conventional thermodynamic questions but also problems of quantum dynamics far from equilibrium.

No coupling to the environment.

Interacting bosons in optical lattices.

Highly tunable periodic potentials with no defects.

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 2

U U

Interaction suppresses number fluctuations and leads to localization of atoms.

Equilibrium system.

Kinetic (tunneling) energy:

† †tun j k k j

jk

E J a a a a

e ji

j ja N

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

2 cos( )tun j kjk

E JN

Classically the ground state will have uniform density and 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 localization and tunneling leading to phase coherence.

Ground state is a superfluid:

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

Superfluid-insulator quantum phase transition.

Strong tunneling

cos const,

-

i j

i j

Weak tunneling

Ground state is an insulator:

cos 0,

-

i j

i j

M. Greiner et. al., Nature (02)

Superfluid Mott insulator

Adiabatic increase of lattice potential

Observe:† ( )

,

ip n mp m n

m n

n a a e

Measurement: time of flight imaging

Rp m

t

Nonequilibrium phase transitions

wait for time t

Fast sweep of the lattice potential

M. Greiner et. al. Nature (2002)

int ( ) ( 1) 2

UE N N N

Revival of the initial state at 2 n

tU

Explanation

†

int int

2

2(1) (2)

0 0 0 1 2 ...2!

0 1 2 ...2!

a

iE t iE t

t e

t e e

int

( 1)2

2

N NE N t n

wait for time t

Fast sweep of the lattice potential

A. Tuchman et. al., (2001)

A.P., S. Sachdev and S.M. Girvin, PRA 66, 053607 (2002), E. Altman and A. Auerbach, PRL 89, 250404 (2002)

Two coupled sites. Semiclassical limit.

sindN

dt

The phase is not defined in the initial insulting phase. Start from the ensemble of trajectories.

Interference of multiple classical trajectories results in oscillations and damping of the phase coherence.

0.0 0.5 1.0 1.5 2.0 2.5 3.00.0

0.1

0.2

0.3

0.4

0.5

Pha

se C

oher

ence

(ar

b. u

nits

)

Time

Numerical results:

Semiclassical approximation to many-body dynamics:A.P., PRA 68, 033609 (2003), ibid. 68, 053604 (2003).

Classical non-equlibrium phase transitions

Superfluids can support non-dissipative current.

accelarate the lattice

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

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

Theory: superfluid flow becomes unstable. / 2q

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

Damping of a superfluid current in 1D

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

max / 5 / 2p

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

What will happen if we have both quantum fluctuations and non-zero superfluid flow?

p

SF MI

U/J???

possible experimental sequence:

???

p

U/J

Stable

Unstable

SF MI

Simple intuitive explanation

Viscosity of Helium II, Andronikashvili (1946)

Two-fluid model for Helium II

Landau (1941)

Cold atoms: quantum depletion at zero temperature.

The normal current is easily damped by the lattice. Friction between superfluid and normal components would lead to strong current damping at large U/J.

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.

dp

Pha

se

Site Position

Consider a fluctuation

22 2

2

1 1( ) , , ( )

2 2

E E IE p I E p

p p p

If I decreases with p, there is a continuum of resonant states smoothly connected with the uniform one. Current cannot be stable.

no lattice: 21

2 s s

EE p I p

p

Include quantum depletion.

In equilibrium ( / )s s J U

In a current state:

coseffJ J J p

So we expect: ( )s p

0.0 0.1 0.2 0.3 0.4 0.5

p*

I(p)

s(p)

sin(p)

Condensate momentum p/p

( )sinsI p p

With quantum depletion the current state is unstable at

* / 2.p p

p

Valid if N1:

2

2,

2 cos( )2j k

j k j j

UH JN

Quantum rotor model

2

1 122 sin sinj

j j j j

dUJN

dt

Deep in the superfluid regime (JNU) we can use classical equations of motion:

j jpj

2

1 122 cos 2j

j j j

dUJN p

dt

Unstable motion for p>/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.

p

U/J

Superfluid MI

0.0

0.2

0.4

0.6

0.8

1.0

2D

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

Pha

se c

oher

ence

(n p)

U/J

Time-dependent Gutzwiller approximation

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=3

d=2

d=1

unstable

stable

U/U

c

p/

Meanfield (Gutzwiller ansatzt) phase diagram

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/U

c

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

0

2

0

1( )

2

dxS d V x

m d

Asymptotical decay rate near the instability

0

22 3

0

1

2

( ) 0c

dxS d ax bx

m d

a p p

Rescale the variables:

1, =

2ma

ax x

b

5/ 25/ 2

02

5/ 2

1

2

exp exp

c

c

aS S p p

bm

S C p p

0

22 3

0 0

1 8 2

2 15

dxS d x x

d

Many body system

2

1

1 2 cos

2j

j jj

dS d JN

U d

At p/2 we get

2

2 3

1 1

1

2

cos3

j

j

j j j j

dS d

U d

JNJN p

j jpj

cos , cos

j j pUNJ p

Many body system, 1D

5/ 2

exp 7.12

JNp

U

7.1 – variational result

JNU

– semiclassical parameter (plays . the role of 1/)

Small N~1Large N~102-103

Higher dimensions.

Stiffness along the current is much smaller than that in the transverse direction.

We need to excite many chains in order to create a phase slip. The effective size of the phase slip in d-dimensional space time is

12

r p

|| cos , J J p J J

6

2

2

d

d d

JNS C p

U

Phase slip tunneling is more expensive in higher dimensions:

expd dS

Stability phase diagram

3dS

Crossover

1 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/U

c

p/

Current decay in the vicinity of the superfluid-insulator transition

In the limit of large we can employ a different effective coarse-grained theory (Altman and Auerbach 2002):

3dC

Metastable current state: 2 21 ip xp e

This state becomes unstable at

corresponding to the maximum of the current:

13cp

2 2 21 .I p p p

Current decay in the vicinity of the Mott transition.

Use the same steps as before to obtain the asymptotics:

5

23

1 3 , dd

Sdd d

CS p e

32

1 2

12

2

3

5.71 3

3.21 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.

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

Dynamics of the current decay.

Underdamped regime Overdamped regime

Single phase slip triggers full current decay

Single phase slip reduces a current by one step

Which of the two regimes is realized is determined entirely by the dynamics of the system (no external bath).

0 500 1000 1500 2000

0.0

0.2

0.4

0.6

0.8

1.0 p=2/5 p=/10

Scal

ed C

urre

nt

Time (t)

Numerical simulation in the 1D case

The underdamped regime is realized in uniform systems near the instability. This is also the case in higher dimensions.

Simulate thermal decay by adding weak fluctuations to the initial conditions. Quantum decay should be similar near the instability.

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

of M

ass

Mom

entu

m

Time

N=1.5 N=3

U=0.01 tJ=1/4

Gutzwiller ansatz simulations (2D)

Exact simulations in small systems

0 50 100 150 2000

2

4

6

8

p=/2

p=/4

p=0

n p

Time

Eight sites, two particles per site

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).

Detecting equilibrium superfluid-insulator transition boundary in 3D.

p

U/J

Superfluid MI

Extrapolate

At 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!

Summary

New scaling approach to

current decay rate:

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 with experiments and numerical simulations.