1. Superfluid to insulator transition in a moving system of interacting bosons
Ehud Altman
Anatoli Polkovnikov
Bertrand Halperin
Mikhail Lukin
Eugene Demler
References:
J. Superconductivity 17:577 (2004)
Phys. Rev. Lett. 95:20402 (2005)
Phys. Rev. A 71:63613 (2005)
Physics Department,
Harvard University

2. 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

3. 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), …

4. Equilibrium superfluid to insulator transition
Theory: Fisher et al. PRB (89), Jaksch et al. PRL (98)
Experiment: Greiner et al. Nature (01) U m
Superfluid
Mott insulator
What if the ground state cannot be described by single particle matter waves? Current experiments are reaching such regimes. A nice example is the experiment by Markus and collaborators in Munich where cold bosons on an optical lattice were tuned across the SF-Mott transition.
Explain experiment.
SF well described by a wavefunction in which the zero momentum state is macroscopically occupied.
Mott state is rather well described by definite RS occupations.
t/U

5. 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)

6. ??? This talk: How to connect
the dynamical instability (irreversible, classical)
to the superfluid to Mott transition (equilibrium, quantum)
Unstable
??? p
U/J
p/2
Stable SF MI
This talk
U/t p SF MI
???
Possible experimental
sequence:

7. Superconductor to Insulator transition in thin films
Marcovic et al., PRL 81:5217 (1998)
Bi films d
Superconducting films
of different thickness

8. 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
Amplification of
density fluctuations
unstable
unstable r

9. 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

10. Dynamical instability for integer fillingSF MI p
U/J
p/2
Vicinity of the SF-I quantum phase transition.
Classical description applies for
The amplitude and the phase of the order parameter can only be defined if we coarse grain the system on the scale ksi.
Applying the same argument as before we find the condition that p ksi should be of the order of pi over two.
Near the SF-I transition ksi diverges. So the critical phase gradient goes to zero.
Dynamical instability occurs for

11. Dynamical instability. Gutzwiller approximation
Wavefunction
Time evolution
We look for stability against small fluctuations
Phase diagram. Integer filling

12. Order parameter suppression by the current.
Number state (Fock) representation
Integer filling N
N+1
N-1
N+2
N-2
Distribution of the occupation numbers is determined by the competition of the kinetic and interaction
energies. Kinetic energy favors wide distribution. Interaction energy favors distribution peaked at a single
filling. The wider is the distribution, the larger is the order parameter. When there is a phase gradient,
the effect of the kinetic energy is reduced and the distribution gets squeezed. This reduces the order parameter.
N-2
N-1 N
N+1
N+2

13. Order parameter suppression by the current.
Number state (Fock) representation
Integer filling
Fractional filling N
N+1
N-1
N+2
N-2
N-1/2
N+1/2
N-3/2
N+3/2
N-1/2
N+1/2
N-3/2
N+3/2
Distribution of the occupation numbers is determined by the competition of the kinetic and interaction
energies. Kinetic energy favors wide distribution. Interaction energy favors distribution peaked at a single
filling. The wider is the distribution, the larger is the order parameter. When there is a phase gradient,
the effect of the kinetic energy is reduced and the distribution gets squeezed. This reduces the order parameter.
N-2
N-1 N
N+1
N+2

14. Dynamical instability
Integer filling
Fractional filling p p
p/2
p/2
U/J
U/J SF MI

15. Optical lattice and parabolic trap. Gutzwiller approximation
The first instability develops near the edges, where N=1
U=0.01 t
J=1/4
Gutzwiller ansatz simulations (2D)

16. Beyond semiclassical equations. Current decay by tunneling
phase j
phase j
phase j
Current carrying states are metastable.
They can decay by thermal or quantum tunneling
Thermal activation
Quantum tunneling

17. Decay of current by quantum tunneling
phase j
phase
Quantum
phase slip j
Escape from metastable state by quantum tunneling.
WKB approximation
S – classical action corresponding to the motion in an inverted potential.

18. Decay rate from a metastable state. Example

19. Weakly interacting systems. Quantum rotor model.
Decay of current by quantum tunneling
At p/2 we get
For the link on which the QPS takes place
d=1. Phase slip on one link + response of the chain.
Phases on other links can be treated in a harmonic approximation

20. For d>1 we have to include transverse directions.
Need to excite many chains to create a phase slip
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

21. Weakly interacting systems. Gross-Pitaevskii regime.
Decay of current by quantum tunneling SF MI p
U/J
p/2
Fallani et al., PRL (04)
Quantum phase slips are
strongly suppressed
in the GP regime

22. Strongly interacting regime. Vicinity of the SF-Mott transitionMI p
U/J
p/2
Close to a SF-Mott transition
we can use an effective
relativistivc GL theory
(Altman, Auerbach, 2004)
Metastable current carrying state:
This state becomes unstable at corresponding to the maximum of the current:

23. Strongly interacting regime. Vicinity of the SF-Mott transition
Decay of current by quantum tunneling SF MI p
U/J
p/2
Action of a quantum phase slip in d=1,2,3
- 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

24. Decay of current by quantum tunneling

26. Decay of current by thermal activation
phase j
phase
Thermal
phase slip j DE
Escape from metastable state by thermal activation

27. Thermally activated current decay. Weakly interacting regime
phase slip DE
Activation energy in d=1,2,3
Thermal fluctuations lead to rapid decay of currents
Crossover from thermal
to quantum tunneling

28. Decay of current by thermal fluctuations
Phys. Rev. Lett. (2004)

29. 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