Superfluid to insulator transition in
a moving system of interacting
bosons
Ehud Altman
Anatoli Polkovnikov
Bertrand Halperin
Mikhail Lukin
Eugene Demler
Physics Department,
Harvard University
References:
J. Superconductivity 17:577 (2004)
Phys. Rev. Lett. 95:20402 (2005)
Phys. Rev. A 71:63613 (2005)
Outline
Introduction. Cold atoms in optical lattices.
Superfluid to Mott transition. Dynamical instability
Mean-field analysis using Gutzwiller variational wavefunctions
Current decay by quantum tunneling
Current decay by thermal activation
Conclusions
Atoms in optical lattices. Bose Hubbard model
Theory: Jaksch et al. PRL 81:3108(1998)
Experiment: Kasevich et al., Science (2001)
Greiner et al., Nature (2001)
Cataliotti et al., Science (2001)
Phillips et al., J. Physics B (2002)
Esslinger et al., PRL (2004), …
Equilibrium superfluid to insulator transition
m
Theory: Fisher et al. PRB (89), Jaksch et al. PRL (98)
Experiment: Greiner et al. Nature (01)
U
Superfluid
Mott
insulator
n 1
t/U
Moving condensate in an optical lattice. Dynamical instability
Theory: Niu et al. PRA (01), Smerzi et al. PRL (02)
Experiment: Fallani et al. PRL (04)
v
Related experiments by
Eiermann et al, PRL (03)
This talk: How to connect
the dynamical instability (irreversible, classical)
to the superfluid to Mott transition (equilibrium, quantum)
p
p/2
Unstable
Stable
???
SF
This talk
MI
U/J
p
???
Possible experimental
U/t
sequence:
SF
MI
Superconductor to Insulator
transition in thin films
Bi films
d
Superconducting films
of different thickness
Marcovic et al., PRL 81:5217 (1998)
Dynamical instability
Classical limit of the Hubbard model.
Discreet Gross-Pitaevskii equation
Current carrying states
Linear stability analysis: States with p>p/2 are unstable
unstable
unstable
Amplification of
density fluctuations
r
Dynamical instability for integer filling
Order parameter for a current carrying state
Current
GP regime
. Maximum of the current for
When we include quantum fluctuations, the amplitude of the
order parameter is suppressed
decreases with increasing phase gradient
.
Dynamical instability for integer filling
s
(p)
sin(p)
p
p/2
I(p)
p
0.0
0.1
0.2
0.3
U/J
*
0.4
0.5
Condensate momentum p/
Vicinity of the SF-I quantum phase transition.
Classical description applies for
Dynamical instability occurs for
SF
MI
Dynamical instability. Gutzwiller approximation
Wavefunction
Time evolution
We look for stability against small fluctuations
0.5
unstable
0.4
d=3
Phase diagram. Integer filling
d=2
p/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
Order parameter suppression by the current.
Number state (Fock) representation
Integer filling
N-2
N-1
N
N+1
N+2
N-2
N-1
N
N+1
N+2
Order parameter suppression by the current.
Number state (Fock) representation
Integer filling
Fractional filling
N-2
N-1
N
N+1
N+2
N-2
N-1
N
N+1
N+2
N-3/2
N-3/2
N-1/2 N+1/2 N+3/2
N-1/2 N+1/2 N+3/2
Dynamical instability
Integer filling
Fractional filling
p
p
p/2
p/2
U/J
SF
MI
U/J
Center of Mass Momentum
Optical lattice and parabolic trap.
Gutzwiller approximation
0.00
0.17
0.34
0.52
0.69
0.86
N=1.5
N=3
0.2
0.1
The first instability
develops 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)
j
phase
j
phase
phase
Beyond semiclassical equations. Current decay by tunneling
Current carrying states are metastable.
They can decay by thermal or quantum tunneling
Thermal activation
Quantum tunneling
j
phase
phase
Decay of current by quantum tunneling
Quantum
phase slip
j
j
Escape from metastable state by quantum tunneling.
WKB approximation
S – classical action corresponding to the motion in an inverted potential.
Decay rate from a metastable state. Example
S
0
0
 1  dx 2

2
3
d 
  x  bx 


 2m  d 



  ( pc  p)  0
Weakly interacting systems. Quantum rotor model.
Decay of current by quantum tunneling
1  d j 
S    d

  2 JN cos  j 1   j 
2U  d 
j
2
 j  pj   j
At pp/2 we get
For the link on which the QPS takes place
2
3
1  d j 
JN
S    d
 j 1   j 


  JN cos p  j 1   j  
2U  d 
3
j
2
d=1. Phase slip on one link + response of the chain.
Phases on other links can be treated in a harmonic approximation
For d>1 we have to include transverse directions.
Need to excite many chains to create a phase slip
J ||  J cos p,
J  J
Longitudinal stiffness
is much smaller than
the transverse.
The transverse size of the phase slip diverges near a phase
slip. We can use continuum approximation to treat transverse
directions
Weakly interacting systems. Gross-Pitaevskii regime.
Decay of current by quantum tunneling
p
p/2
U/J
SF
MI
Fallani et al., PRL (04)
Quantum phase slips are
strongly suppressed
in the GP regime
Strongly interacting regime. Vicinity of the SF-Mott transition
p
p/2
Close to a SF-Mott transition
we can use an effective
relativistivc GL theory
(Altman, Auerbach, 2004)
U/J
SF
M
I
2 2 ip x


1

p
 e
Metastable current carrying state:
This state becomes unstable at pc  1 3 corresponding to the
maximum of the current: I  p   p 1  p 2 2  .
2
Strongly interacting regime. Vicinity of the SF-Mott transition
Decay of current by quantum tunneling
p
p/2
U/J
SF
Action of a quantum phase slip in d=1,2,3
MI
- correlation length
Strong broadening of the phase transition in d=1 and d=2
is discontinuous at the transition. Phase slips are not important.
Sharp phase transition
Decay of current by quantum tunneling
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
phase
phase
Decay of current by thermal activation
Thermal
phase slip
j
j
DE
Escape from metastable state by thermal activation
Thermally activated current decay. Weakly interacting regime
DE
Thermal
phase slip
Activation energy in d=1,2,3
Thermal fluctuations lead to rapid decay of currents
Crossover from thermal
to quantum tunneling
Decay of current by thermal fluctuations
Phys. Rev. Lett. (2004)
Conclusions
Dynamic instability is continuously connected to the
quantum SF-Mott transition
Quantum fluctuations lead to strong decay of current
in one and two dimensional systems
Thermal fluctuations lead to strong decay of current
in all dimensions