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Superfluid insulator transition
in a moving condensate
Anatoli Polkovnikov
Ehud Altman,
Eugene Demler,
Bertrand Halperin,
Misha Lukin
Harvard University
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.
No coupling to the environment.
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.
Interacting bosons in optical lattices.
Highly tunable periodic potentials with no defects.
Equilibrium system.
Interaction energy (two-body collisions):
U
Eint   N j ( N j  1)
j 2
Eint is minimized when Nj=N=const:
U
U
 2  2   6  0
2
2
Interaction suppresses number fluctuations
and leads to localization of atoms.
Equilibrium system.
Kinetic (tunneling) energy:
Etun   J  a †j ak  ak† a j
 jk 
aj  N j e
i j
Etun
2 JN  cos( j  k )
 jk 
Kinetic energy is minimized when the
phase is uniform throughout the system.
Classically the ground state will have
uniform density and a uniform phase.
However, number and phase are conjugate
variables. They do not commute:
 N ,   i

 N  1
There is a competition between the
interaction leading to localization and
tunneling leading to phase coherence.
Strong tunneling
Ground state is
a superfluid:
cos i   j   const,
i- j 
Weak tunneling
Ground state is
an insulator:
cos i   j   0,
i- j 
Superfluid-insulator quantum phase transition.
(M.P.A. Fisher, P. Weichman, G. Grinstein, D. Fisher, 1989)
M. Greiner et. al., Nature (02)
Adiabatic increase of lattice potential
Superfluid
Mott insulator
Measurement: time of flight imaging
Observe:
np   am† an eip ( nm)
m, n
R
pm
t
Nonequilibrium phase transitions
Fast sweep of the lattice potential
wait for time t
M. Greiner et. al. Nature (2002)
U
Eint ( N )  N ( N  1)
2
Explanation
  t  0   e a 0  0   1 
†
  t   0   e  iE
int (1) t
1 
2
Revival of the initial state at
N ( N  1)
Eint  N  t  2 n
2
2!
2
2!
2  ...
e  iEint (2)t 2  ...
2 n
t
U
Fast sweep of the lattice potential
wait for time t
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.
dN
 sin 
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.
Phase Coherence (arb. units)
Numerical results:
0.5
0.2
Semiclassical
approximation to
many-body dynamics:
0.1
A.P., PRA 68, 033609 (2003),
ibid. 68, 053604 (2003).
0.4
0.3
0.0
0.0
0.5
1.0
1.5 2.0
Time
2.5
3.0
Classical non-equlibrium phase transitions
Superfluids can support non-dissipative current.
accelarate the lattice
Theory: Wu and Niu
PRA (01); Smerzi et.
al. PRL (02).
Exp: Fallani et. al.,
(Florence) condmat/0404045
Theory: q   / 2 superfluid flow becomes unstable.
Based on the analysis of classical equations of motion
(number and phase commute).
Damping of a superfluid current in 1D
pmax   / 5   / 2
C.D. Fertig et. al. cond-mat/0410491
See: AP and D.-W. Wang, PRL 93, 070401 (2004).
What will happen if we have both
quantum fluctuations and non-zero
superfluid flow?
possible experimental sequence:
p
???
U/J
SF
p
MI
Unstable
/2
Stable
SF
???
MI
U/J
Simple intuitive explanation
Two-fluid model for Helium II
Landau (1941)
Viscosity of Helium II,
Andronikashvili (1946)
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
I  s p
SF current in free space
I  s sin p
SF current on a lattice
s – superfluid density, p – condensate momentum.
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!!!
Phase
Consider a fluctuation
dp
Site Position
1 2 E
2
E 
(

p
)
,
2
2 p
E
1 I
 I,   E 
( p) 2
p
2 p
1
E
2
no lattice: E   s p  I   s p 
2
p
If I decreases with p, there is a continuum of
resonant states smoothly connected with the
uniform one. Current cannot be stable.
Include quantum depletion.
In equilibrium  s   s ( J / U )
In a current state:

J  J eff  J cos p
p
So we expect:  s ( p)
s
(p)
sin(p)
With quantum
depletion the current
state is unstable at
I(p)
p
0.0
0.1
0.2
I  s ( p)sin p
0.3
*
0.4
0.5
Condensate momentum p/p
p  p*   / 2.
Quantum rotor model
Valid if N1:
U
2
H    2 JN cos( j  k )   2
2 j  j
j ,k
Deep in the superfluid regime (JNU) we can
use classical equations of motion:
d 2 j
dt
2
 2UJN sin  j 1   j   sin  j 1   j  
 j  pj   j
d 2 j
dt
2
 2UJN cos p  j 1   j 1  2 j 
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.
Need to coarse grain the system.
After coarse graining we get both amplitude and
phase fluctuations.
Time dependent Ginzburg-Landau:

       
2
2
2

S. Sachdev, Quantum
phase transitions;
Altman and Auerbach
(2002)
(  diverges at the transition)
Stability analysis around a current
Uc  U
carrying solution: pc ~ 1 
p
/2

pc ~ 1 
Superfluid
MI
U/J
Use time-dependent Gutzwiller approximation
to interpolate between these limits.
Time-dependent Gutzwiller approximation
p
/2
MI
Phase coherence (np)
Superfluid
U/J
1.0
p=/5
U=0.01t
Jz=1
N=1
0.8
0.6
2D
0.4
0.2
0.0
U/J
Meanfield (Gutzwiller ansatzt)
phase diagram
0.5
unstable
0.4
d=3
d=2
p/
0.3
d=1
0.2
stable
0.1
0.0
0.0
0.2
0.4
U/Uc
0.6
0.8
1.0
Is there current decay below the instability?
Role of fluctuations
Phase slip
E
p
Below the mean field transition superfluid
current can decay via quantum tunneling or
thermal decay .
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)
Current decay far from the
insulating transition
0.5
unstable
0.4
p/
0.3
0.2
stable
0.1
0.0
0.0
0.2
0.4
U/Uc
0.6
0.8
1.0
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
S
0
0
 1  dx 2

d 
 V ( x) 


 2m  d 



Asymptotical decay rate near the instability
 1  dx 2

2
3
S   d 
 ax  bx 
0
 2m  d 



a  ( pc  p)  0
0
Rescale the variables:
a
1
x  x,  =

b
2ma
1 a5/ 2
5/ 2
S
S0   pc  p 
2
2m b

  exp   S  exp C  pc  p 
S0  
0
0
5/ 2
 1  dx 2
 8 2
2
3
d     x  x  
 2  d 
 15



Many body system
1
S    d
2U
j
 j  pj   j
 d j 

  2 JN cos  j 1   j 
 d 
2
At p/2 we get
1  d j 
S    d


2
U
d

j


2
 JN cos p  j 1   j 
 j   j cos p,

2
3
JN

 j 1   j 

3

UNJ cos p
Many body system, 1D
5/ 2

JN  
 
  exp  7.1
  p 
U 2
 

7.1 – variational result
JN
– semiclassical parameter (plays
U . the role of 1/)
Large N~102-103
Small N~1
Higher dimensions.
Stiffness along the current is much smaller than
that in the transverse direction.
J||  J cos p, J   J
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
r 1

2
p
Phase slip tunneling is more expensive in
higher dimensions:
Sd  Cd
JN  

  p
U 2

6 d
2
d  exp  Sd 
Stability phase diagram
0.55
Sd  3
Unstable
0.50
0.40
p/
Stable
d=3
0.45
1  Sd  3
d=2
Crossover
0.35
0.30
0.25
0.20
0.0
Sd  1
d=1
Unstable
Stable
0.2
0.4
0.6
U/JN
0.8
1.0
Current decay in the vicinity of the
superfluid-insulator transition
0.5
unstable
0.4
p/
0.3
0.2
stable
0.1
0.0
0.0
0.2
0.4
U/Uc
0.6
0.8
1.0
Current decay in the vicinity of the
Mott transition.
In the limit of large  we can employ a different
effective coarse-grained theory (Altman and
Auerbach 2002):
C   d 3
Metastable current state:   1  p 2 2 eip x
This state becomes unstable at pc  1
 3
corresponding to the maximum of the current:
I  p   p 1  p2 2  .
2
Use the same steps as before to obtain
the asymptotics:
Sd 
Cd

3 d
1 
S1 
5.7
S2 
3.2

2

3 p
1 
1 

5
d
2
e
,
3 p

3 p

3
 Sd
2
1
2
S3  4.3
Discontinuous change of the decay rate
across the meanfield transition. Phase
diagram is well defined in 3D!
Large broadening in one and two dimensions.
Damping of a superfluid current
in one dimension
C.D. Fertig et. al. cond-mat/0410491
See also AP and D.-W. Wang, PRL, 93, 070401 (2004)
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).
Numerical simulation in the 1D case
Simulate thermal decay by adding weak
fluctuations to the initial conditions. Quantum
decay should be similar near the instability.
Scaled Current
1.0
p=2/5
p=/10
0.8
0.6
0.4
0.2
0.0
0
500
1000
Time (t)
1500
2000
The underdamped regime is realized in uniform
systems near the instability. This is also the case
in higher dimensions.
Effect of the parabolic trap
Expect that the motion becomes unstable first
near the edges, where N=1
Center of Mass Momentum
Gutzwiller ansatz simulations (2D)
0.00
0.17
0.34
0.52
0.69
0.86
N=1.5
N=3
0.2
0.1
0.0
-0.1
-0.2
0
100
200
300
Time
400
500
U=0.01 t
J=1/4
Exact simulations in small systems
Eight sites, two particles per site
p
U (t )  2 tanh 0.02t  tanh 0.02(200  t ) , J=1
U/J
SF
MI
8
p=0
np
6
p=/4
4
2
p=/2
0
0
50
100
Time
150
200
Semiclassical (Truncated Wigner) simulations of
damping of dipolar motion in a harmonic trap
D0
GP
N=500
1.0
Displacement D(t)
Displacement (D0)
10
1
0.5
D0

D1
0.0
-0.5
-1.0
ln(D0/D1)

Time
0.1
1
10
Inverse Tunneling (1/J)
AP and D.-W. Wang, PRL 93, 070401 (2004).
Detecting equilibrium superfluidinsulator transition boundary in 3D.
p
/2
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!
U/J
Summary
Smooth connection between the classical dynamical
instability and the quantum superfluid-insulator transition.
Quantum fluctuations
mean field
beyond mean field
Depletion of the condensate.
Reduction of the critical
current. All spatial dimensions.
p
Broadening of the
mean field transition.
Low dimensions
New scaling approach to
current decay rate:
asymptotical behavior
of the decay rate near
the mean-field
transition
/2
Superfluid
MI
U/J
Qualitative agreement with experiments and
numerical simulations.