Transcript Document

Superfluid insulator transition in a moving
condensate
Anatoli Polkovnikov
Ehud Altman,
Eugene Demler,
Bertrand Halperin,
Misha Lukin
Harvard University
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.
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 a  a a j
 jk 
aj  N j e
i j
Etun
†
j k
†
k
2 JN  cos( j  k )
 jk 
Kinetic energy is minimized when the phase is
uniform throughout the system.
Classically the ground state has a 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
Superfluid regime:
cos i   j   const,
i- j 
Weak tunneling
Insulating regime:
cos i   j   0,
i- j 
Superfluid-insulator quantum phase transition.
(M.P.A. Fisher, P. Weichman, G. Grinstein, D. Fisher, 1989)
Classical non-equlibrium phase transitions
Superfluids can support non-dissipative current.
Theory: Wu and Niu
PRA (01); Smerzi et.
al. PRL (02).
Exp: Fallani et. al.,
(Florence) condmat/0404045
Theory: p   / 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
Current damping below
classical instability.
No sharp transition.
See also : AP and D.-W. Wang, PRL 93,
070401 (2004).
What happens if we there are both quantum
fluctuations and superfluid flow?
possible experimental sequence:
p
~lattice
potential
???
U/J
p
/2
SF
MI
Unstable
Stable
SF
???
MI
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!!!
Include quantum depletion.
s  s ( J / U )
Equilibrium:

J  Jeff  J cos p
Current state:
p
s
(p)
sin(p)
p
0.1
0.2
 s ( p)
I  s ( p)sin p
With quantum
depletion the current
state is unstable at
I(p)
0.0

0.3
*
0.4
0.5
Condensate momentum p/
p  p*   / 2.
Quantum rotor model
U
2
H    2 JN cos( j  k )   2
2 j  j
j ,k
OK if N1:
Deep in the superfluid regime (JNU) use GP equations of motion:
d 2 j
dt
 j  pj   j
2
 2UJN sin  j 1   j   sin  j 1   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 carrying solution:
pc ~ 1 
p
Uc U
/2

pc ~ 1 
Superfluid
MI
U/J
Use time-dependent
Gutzwiller approximation
to interpolate between
these limits.
Phase coherence (np)
Time-dependent Gutzwiller approximation
p
/2
1.0
p=/5
U=0.01t
Jz=1
N=1
0.8
0.6
2D
0.4
0.2
0.0
U/J
Superfluid
MI
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
Below the mean
field transition
superfluid current
can decay via
quantum
tunneling or
p 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
2


0
1  dx 
S   d 
 V ( x) 


0
 2m  d 



Asymptotical decay rate near the instability
S
0
0
 1  dx 2

2
3
d 
  x  bx 


 2m  d 



Rescale the variables:
  ( pc  p )  0

1
x  x,  =

b
2m
1 
5/ 2
S
S0   pc  p 
2
2m b
5/ 2
  exp  S 
S0  
0
0
 1  dx 2
 8 2
2
3
d     x  x  
 2  d 
 15


Many body system, 1D

JN  

  exp  7.1

p


U 2


Large N~102-103
5/ 2



7.1 – variational result
JN
U
semiclassical
parameter (plays
the role of 1/)
Small N~1
Higher dimensions.
Longitudinal stiffness is much smaller than the transverse.
J||  J cos p,
J  J
r
r 1

2
Need to excite many chains in order to create a phase slip.
p
Phase slip tunneling is more expensive in higher dimensions:
S d  Cd



p


2

JN
U
6 d
2
d  exp  Sd 
Stability phase diagram
Sd  3
0.55
Unstable
Stable
0.50
d=3
0.45
p/
0.40
1  Sd  3 Crossover
d=2
0.35
0.30
0.25
0.20
0.0
Sd  1
d=1
Stable
0.2
0.4
0.6
U/JN
0.8
1.0
Unstable
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
Use the same steps as before to obtain the asymptotics:
Sd 
Cd

S1 
5.7
S2 
3.2

2

3 d
1
1 
1 
3 p
3 p

3 p

3

2
1
2
5
d
2
,
  exp  Sd 
Discontinuous change of
the decay rate across the
meanfield transition.
Phase diagram is well
defined in 3D!
S3  4.3
Large broadening in one and two dimensions.
Damping of a superfluid current in one dimension
C.D. Fertig et. al.
cond-mat/0410491
0.5
unstable
0.4
d=3
d=2
d=1
p/
0.3
0.2
stable
0.1
0.0
0.0
0.2
0.4
U/Uc
0.6
0.8
1.0
See also AP and D.-W. Wang,
PRL, 93, 070401 (2004)
Center of Mass Momentum
Effect of the parabolic trap
0.00
0.17
0.34
0.52
0.69
0.86
N=1.5
N=3
0.2
0.1
Expect that the motion
becomes unstable first
near the edges, where N=1
0.0
-0.1
U=0.01 t
J=1/4
-0.2
0
100
200
300
Time
400
500
Gutzwiller ansatz simulations (2D)
Exact simulations: 8 sites, 16 bosons
p U (t )  2tanh 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
GP
N=500
D0
1.0
Displacement D(t)
Displacement (D0)
10
1
0.5
D0

D1
0.0
-0.5
-1.0
ln(D0/D1)

Time
Smaller critical current
Broad transition
0.1
1
Quantum fluctuations:
10
Inverse Tunneling (1/J)
AP and D.-W. Wang, PRL 93, 070401 (2004).
Detecting equilibrium SF-IN transition boundary in 3D.
p
Easy to detect nonequilibrium
irreversible transition!!
/2
Superfluid
MI
U/J
Extrapolate
At nonzero current the SF-IN transition is irreversible: no restoration
of current and partial restoration of phase coherence in a cyclic ramp.
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
/2
Broadening of the mean field
transition. Low dimensions
asymptotical behavior
of the decay rate near
the mean-field
transition
Superfluid
MI
Qualitative agreement
with experiments and
U/J numerical simulations.