Superfluid to insulator transition in a moving system of interacting bosons

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Non-equilibrium dynamics of cold atoms in optical
lattices
Vladimir Gritsev
Harvard Anatoli Polkovnikov
Harvard/Boston University Ehud Altman
Harvard/Weizmann Bertrand Halperin
Harvard Mikhail Lukin
Harvard Eugene Demler
Harvard
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Motivation understanding transport phenomena
in correlated electron systems e.g. transport
near quantum phase transition
3
Superconductor to Insulator transition in thin
films
Tuned by film thickness
Tuned by magnetic field
V.F. Gantmakher et al., Physica B 284-288, 649
(2000)
Marcovic et al., PRL 815217 (1998)
4
Scaling near the superconductor to insulator
transition
Yazdani and Kapitulnik Phys.Rev.Lett. 743037
(1995)
5
Breakdown of scaling near the superconductor to
insulator transition
Mason and Kapitulnik Phys. Rev. Lett. 825341
(1999)
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Outline
Current decay for interacting atoms in optical
lattices. Connecting classical dynamical
instability with quantum superfluid to Mott
transition
Phase dynamics of coupled 1d condensates. Competit
ion of quantum fluctuations and
tunneling. Application of the exact solution of
quantum sine Gordon model
Conclusions
7
Current decay for interacting atoms in optical
lattices Connecting classical dynamical
instability with quantum superfluid to Mott
transition
References
J. Superconductivity 17577 (2004) Phys. Rev.
Lett. 9520402 (2005) Phys. Rev. A 7163613 (2005)
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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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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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Dynamics of interacting bosonic systems probed in
interference experiments
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Interference of two independent condensates
Andrews et al., Science 275637 (1997)
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Interference experiments with low d condensates
1D condensates Schmiedmayer et al., Nature
Physics (2005,2006)
Longitudial imaging
Transverse imaging
2D condensates Hadzibabic et al., Nature
4411118 (2006)
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Studying dynamics using interference experiments
Motivated by experiments and discussions with
Bloch, Schmiedmayer, Oberthaler, Ketterle, Porto,
Thywissen
Prepare a system by splitting one condensate
Take to the regime of finite or zero tunneling
Measure time evolution of fringe amplitudes
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Studying coherent dynamics of strongly
interacting systems in interference experiments
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Coupled 1d systems
Interactions lead to phase fluctuations within
individual condensates
Tunneling favors aligning of the two phases
Interference experiments measure only the
relative phase
39
Coupled 1d systems
Conjugate variables
Relative phase
Particle number imbalance
Small K corresponds to strong quantum
fluctuations
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Quantum Sine-Gordon model
Hamiltonian
Imaginary time action
Quantum Sine-Gordon model is exactly integrable
Excitations of the quantum Sine-Gordon model
soliton
antisoliton
many types of breathers
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Dynamics of quantum sine-Gordon model
Hamiltonian formalism
Initial state
Quantum action in space-time
Initial state provides a boundary condition at t0
Solve as a boundary sine-Gordon model
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Boundary sine-Gordon model
Exact solution due to
Ghoshal and Zamolodchikov (93) Applications to
quantum impurity problem Fendley, Saleur,
Zamolodchikov, Lukyanov,
Limit enforces boundary
condition
Boundary Sine-Gordon Model
space and time enter equivalently
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Boundary sine-Gordon model
Initial state is a generalized squeezed state
Matrix and are known
from the exact solution of the boundary
sine-Gordon model
Time evolution
Coherence
Matrix elements can be computed using form factor
approach Smirnov (1992), Lukyanov (1997)
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Quantum Josephson Junction
Limit of quantum sine-Gordon model when spatial
gradients are forbidden
Initial state
Eigenstates of the quantum Jos. junction
Hamiltonian are given by Mathieus functions
Time evolution
Coherence
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Dynamics of quantum Josephson Junction
power spectrum
w
E6-E0
E2-E0
E4-E0
Main peak
Higher harmonics
Smaller peaks
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Dynamics of quantum sine-Gordon model
Coherence
Main peak
Higher harmonics
Smaller peaks
Sharp peaks
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Dynamics of quantum sine-Gordon model
power spectrum
w
main peak
higher harmonics
smaller peaks
sharp peaks (oscillations without decay)
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Conclusions
Dynamic instability is continuously connected to
the quantum SF-Mott transition. Quantum and
thermal fluctuations are important
Interference experiments can be used to do
spectroscopy of the quantum sine-Gordon model