Superfluid to insulator transition in a moving system of interacting bosons

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Superfluid to insulator transition in a moving
system of interacting bosons
Ehud Altman Anatoli Polkovnikov Bertrand
Halperin Mikhail Lukin Eugene Demler
References
J. Superconductivity 17577 (2004) Phys. Rev.
Lett. 9520402 (2005) Phys. Rev. A 7163613 (2005)
Physics Department, Harvard University
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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
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Atoms in optical lattices. Bose Hubbard model
Theory Jaksch et al. PRL 813108(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),
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Equilibrium superfluid to insulator transition
Theory Fisher et al. PRB (89), Jaksch et al. PRL
(98) Experiment Greiner et al. Nature (01)
Superfluid
Mott insulator
t/U
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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)
Related experiments by Eiermann et al, PRL (03)
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This talk How to connect the dynamical
instability (irreversible, classical) to the
superfluid to Mott transition (equilibrium,
quantum)
Possible experimental sequence
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Superconductor to Insulator transition in thin
films
d
Superconducting films of different thickness
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Dynamical instability
Classical limit of the Hubbard model.
Discreet Gross-Pitaevskii equation
Current carrying states
Linear stability analysis States with pgtp/2 are
unstable
Amplification of density fluctuations
unstable
unstable
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Dynamical instability for integer filling
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
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Dynamical instability for integer filling
Dynamical instability occurs for
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Dynamical instability. Gutzwiller approximation
Wavefunction
Time evolution
We look for stability against small fluctuations
Phase diagram. Integer filling
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Order parameter suppression by the current.
Number state (Fock) representation
Integer filling
N
N1
N-1
N2
N-2
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Order parameter suppression by the current.
Number state (Fock) representation
Integer filling
Fractional filling
N
N1
N-1
N2
N-2
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Dynamical instability
Integer filling
Fractional filling
p
p
p/2
p/2
U/J
U/J
SF
MI
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Optical lattice and parabolic trap.
Gutzwiller approximation
The first instability develops near the edges,
where N1
U0.01 t J1/4
Gutzwiller ansatz simulations (2D)
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Beyond semiclassical equations. Current decay by
tunneling
Current carrying states are metastable. They can
decay by thermal or quantum tunneling
Thermal activation
Quantum tunneling
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Decay of current by quantum tunneling
phase
j
Escape from metastable state by quantum
tunneling.
WKB approximation
S classical action corresponding to the motion
in an inverted potential.
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Decay rate from a metastable state. Example
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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
d1. Phase slip on one link response of the
chain. Phases on other links can be treated in a
harmonic approximation
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For dgt1 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
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Weakly interacting systems. Gross-Pitaevskii
regime. Decay of current by quantum tunneling
Fallani et al., PRL (04)
Quantum phase slips are strongly suppressed in
the GP regime
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Strongly interacting regime. Vicinity of the
SF-Mott transition
Close to a SF-Mott transition we can use an
effective relativistivc GL theory (Altman,
Auerbach, 2004)
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Strongly interacting regime. Vicinity of the
SF-Mott transition Decay of current by quantum
tunneling
Action of a quantum phase slip in d1,2,3
Strong broadening of the phase transition in d1
and d2
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Decay of current by quantum tunneling
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Decay of current by thermal activation
phase
j
DE
Escape from metastable state by thermal
activation
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Thermally activated current decay. Weakly
interacting regime
DE
Activation energy in d1,2,3
Thermal fluctuations lead to rapid decay of
currents
Crossover from thermal to quantum tunneling
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Decay of current by thermal fluctuations
Phys. Rev. Lett. (2004)
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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