Title: Superfluid insulator transition in a moving condensate
1Superfluid insulator transition in a moving
condensate
Anatoli Polkovnikov (BU and
Harvard)
Ehud Altman, (Weizmann and Harvard) Eugene
Demler, Bertrand Halperin, Misha Lukin
(Harvard)
2Plan of the talk
- 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.
3Interacting bosons in optical lattices.
Highly tunable periodic potentials with no
defects.
Highly tunable periodic potentials with no
defects.
4Equilibrium system.
Interaction energy (two-body collisions)
Eint is minimized when NjNconst
Interaction suppresses number fluctuations and
leads to localization of atoms.
5Equilibrium system.
Kinetic (tunneling) energy
Kinetic energy is minimized when the phase is
uniform throughout the system.
6Classically the ground state has a 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.
7Strong tunneling
Superfluid regime
Weak tunneling
Insulating regime
8M. Greiner et. al., Nature (02)
Adiabatic increase of lattice potential
9Nonequilibrium phase transitions
Fast sweep of the lattice potential
wait for time t
M. Greiner et. al. Nature (2002)
IN
SF
SF
10Explanation
Revival of the initial state at
11Fast sweep of the lattice potential
wait for time t
- Tuchman et. al., 2001,
- cond-mat/0504762
Theory
A.P., S. Sachdev and S.M. Girvin, PRA 66, 053607
(2002), E. Altman and A. Auerbach, PRL 89,
250404 (2002)
12Classical non-equlibrium phase transitions
Superfluids can support non-dissipative current.
Theory superfluid flow becomes
unstable.
Theory Wu and Niu PRA (01) Smerzi et. al. PRL
(02).
Based on the analysis of classical equations of
motion (number and phase commute).
Exp Fallani et. al., (Florence) cond-mat/0404045
13Damping of a superfluid current in 1D
C.D. Fertig et. al. cond-mat/0410491
Current damping below classical instability. No
sharp transition.
See also AP and D.-W. Wang, PRL 93, 070401
(2004).
14What happens if we there are both quantum
fluctuations and superfluid flow?
???
15Simple intuitive explanation
Two-fluid model for Helium II
Landau (1941)
Viscosity of Helium II, Andronikashvili (1946)
Cold atoms quantum depletion at zero
temperature.
Friction between superfluid and normal components?
16Physical 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.
Not a coincidence!!!
Wu and Niu PRA (01) Smerzi et. al. PRL (02).
17Current state
Fluctuation
If I decreases with p, there is a continuum of
resonant states smoothly connected with the
uniform one. Current cannot be stable.
18Include quantum depletion.
Equilibrium
?
Current state
?
p
19SF 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.
20Time dependent Ginzburg-Landau
S. Sachdev, Quantum phase transitions Altman
and Auerbach (2002)
Use time-dependent Gutzwiller approximation to
interpolate between these limits.
21Meanfield (Gutzwiller ansatzt) phase diagram
Is there current decay below the instability?
22Role of fluctuations
Phase slip
Below the mean field transition superfluid
current can decay via quantum tunneling or
thermal decay .
23Related 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)
24Current decay far from the insulating transition
25Decay 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
26Asymptotical decay rate near the instability
Rescale the variables
27Many body system, 1D
variational result
Small N1
Large N102-103
28Higher dimensions.
Longitudinal stiffness is much smaller than the
transverse.
r
Need to excite many chains in order to create a
phase slip.
29Phase slip tunneling is more expensive in higher
dimensions
30Current decay in the vicinity of the
superfluid-insulator transition
31Use 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.
32Damping 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)
33Dynamics 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).
34Numerical 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.
35Effect of the parabolic trap
Expect that the motion becomes unstable first
near the edges, where N1
U0.01 t J1/4
Gutzwiller ansatz simulations (2D)
36Exact simulations 8 sites, 16 bosons
37Semiclassical (Truncated Wigner) simulations of
damping of dipolar motion in a harmonic trap
Quantum fluctuations Smaller critical
current Broad transition
AP and D.-W. Wang, PRL 93, 070401 (2004).
38Detecting equilibrium SF-IN transition boundary
in 3D.
p
Easy to detect nonequilibrium irreversible
transition!!
At nonzero current the SF-IN transition is
irreversible no restoration of current and
partial restoration of phase coherence in a
cyclic ramp.
39Summary
Smooth connection between the classical dynamical
instability and the quantum superfluid-insulator
transition.
Qualitative agreement with experiments and
numerical simulations.
40Quantum rotor model
OK if N?1
Deep in the superfluid regime (JN?U) use GP
equations of motion
Unstable motion for pgt?/2
41Time-dependent Gutzwiller approximation
42Many body system
At p??/2 we get
43Current 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
This state becomes unstable at
corresponding to the maximum of the
current