Superfluid insulator transition in a moving condensate

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Superfluid insulator transition in a moving
condensate
Anatoli Polkovnikov
Ehud Altman, Eugene Demler, Bertrand
Halperin, Misha Lukin
Harvard University
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Plan of the talk

• General motivation and overview.
• Bosons in optical lattices. Equilibrium phase
  diagram. Examples of quantum dynamics.
• Superfluid-insulator transition in a moving
  condensate.
• Qualitative picture
• Non-equilibrium phase diagram.
• Role of quantum fluctuations
• Conclusions and experimental implications.

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Why is the physics of cold atoms interesting?
It is possible to realize strongly interacting
systems, both fermionic and bosonic.
No coupling to the environment.
Parameters of the Hamiltonian are well known and
well controlled.
One can address not only conventional
thermodynamic questions but also problems of
quantum dynamics far from equilibrium.
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Interacting bosons in optical lattices.
Highly tunable periodic potentials with no
defects.
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Equilibrium system.
Interaction energy (two-body collisions)
Eint is minimized when NjNconst
Interaction suppresses number fluctuations and
leads to localization of atoms.
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Equilibrium system.
Kinetic (tunneling) energy
Kinetic energy is minimized when the phase is
uniform throughout the system.
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Classically the ground state will have uniform
density and a uniform phase.
However, number and phase are conjugate
variables. They do not commute
There is a competition between the interaction
leading to localization and tunneling leading to
phase coherence.
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Strong tunneling
Ground state is a superfluid
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M. Greiner et. al., Nature (02)
Adiabatic increase of lattice potential
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Nonequilibrium phase transitions
Fast sweep of the lattice potential
wait for time t
M. Greiner et. al. Nature (2002)
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Explanation
Revival of the initial state at
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Fast sweep of the lattice potential
A. Tuchman et. al., (2001)
A.P., S. Sachdev and S.M. Girvin, PRA 66, 053607
(2002), E. Altman and A. Auerbach, PRL 89, 250404
(2002)
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Two coupled sites. Semiclassical limit.
The phase is not defined in the initial insulting
phase. Start from the ensemble of trajectories.
Interference of multiple classical trajectories
results in oscillations and damping of the phase
coherence.
Numerical results
Semiclassical approximation to many-body
dynamics A.P., PRA 68, 033609 (2003), ibid. 68,
053604 (2003).
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Classical non-equlibrium phase transitions
Superfluids can support non-dissipative current.
accelarate the lattice
Theory Wu and Niu PRA (01) Smerzi et. al. PRL
(02).
Exp Fallani et. al., (Florence) cond-mat/0404045
Theory superfluid flow becomes
unstable.
Based on the analysis of classical equations of
motion (number and phase commute).
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Damping of a superfluid current in 1D
C.D. Fertig et. al. cond-mat/0410491
See AP and D.-W. Wang, PRL 93, 070401 (2004).
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What will happen if we have both quantum
fluctuations and non-zero superfluid flow?
???
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Simple intuitive explanation
Two-fluid model for Helium II
Landau (1941)
Viscosity of Helium II, Andronikashvili (1946)
Cold atoms quantum depletion at zero
temperature.
The normal current is easily damped by the
lattice. Friction between superfluid and normal
components would lead to strong current damping
at large U/J.
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Physical Argument
SF current in free space
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!!!
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Consider a fluctuation
no lattice
If I decreases with p, there is a continuum of
resonant states smoothly connected with the
uniform one. Current cannot be stable.
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Include quantum depletion.
In equilibrium
In a current state
?
p
So we expect
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Quantum rotor model
Valid if N?1
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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.
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Time dependent Ginzburg-Landau
S. Sachdev, Quantum phase transitions Altman and
Auerbach (2002)
Use time-dependent Gutzwiller approximation to
interpolate between these limits.
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Time-dependent Gutzwiller approximation
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Meanfield (Gutzwiller ansatzt) phase diagram
Is there current decay below the instability?
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Role of fluctuations
Phase slip
Below the mean field transition superfluid
current can decay via quantum tunneling or
thermal decay .
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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)
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Current decay far from the insulating transition
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Decay due to quantum fluctuations
The particle can escape via tunneling
S is the tunneling action, or the classical
action of a particle moving in the inverted
potential
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Asymptotical decay rate near the instability
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Many body system
At p??/2 we get
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Many body system, 1D
variational result
semiclassical parameter (plays . the role of
1/?)
Small N1
Large N102-103
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Higher dimensions.
Stiffness along the current is much smaller than
that in the transverse direction.
We need to excite many chains in order to create
a phase slip. The effective size of the phase
slip in d-dimensional space time is
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Phase slip tunneling is more expensive in higher
dimensions
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Current decay in the vicinity of the
superfluid-insulator transition
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Current decay in the vicinity of the Mott
transition.
In the limit of large ? we can employ a different
effective coarse-grained theory (Altman and
Auerbach 2002)
Metastable current state
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Use the same steps as before to obtain the
asymptotics
Discontinuous change of the decay rate across the
meanfield transition. Phase diagram is well
defined in 3D!
Large broadening in one and two dimensions.
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Damping of a superfluid current in one dimension
C.D. Fertig et. al. cond-mat/0410491
See also AP and D.-W. Wang, PRL, 93, 070401 (2004)
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Dynamics of the current decay.
Underdamped regime
Overdamped regime
Single phase slip triggers full current decay
Single phase slip reduces a current by one step
Which of the two regimes is realized is
determined entirely by the dynamics of the system
(no external bath).
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Numerical simulation in the 1D case
Simulate thermal decay by adding weak
fluctuations to the initial conditions. Quantum
decay should be similar near the instability.
The underdamped regime is realized in uniform
systems near the instability. This is also the
case in higher dimensions.
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Effect of the parabolic trap
Expect that the motion becomes unstable first
near the edges, where N1
Gutzwiller ansatz simulations (2D)
U0.01 t J1/4
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Exact simulations in small systems
Eight sites, two particles per site
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Semiclassical (Truncated Wigner) simulations of
damping of dipolar motion in a harmonic trap
AP and D.-W. Wang, PRL 93, 070401 (2004).
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Detecting equilibrium superfluid-insulator
transition boundary in 3D.
p
p/2
U/J
Superfluid
MI
Extrapolate
At nonzero current the SF-IN transition is
irreversible no restoration of current and
partial restoration of phase coherence in a
cyclic ramp.
Easy to detect!
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Summary
Smooth connection between the classical dynamical
instability and the quantum superfluid-insulator
transition.
Qualitative agreement with experiments and
numerical simulations.