Transcript Superfluid to insulator transition in a moving system of

Non-equilibrium dynamics of cold
atoms in optical lattices
Vladimir Gritsev
Anatoli Polkovnikov
Ehud Altman
Bertrand Halperin
Mikhail Lukin
Eugene Demler
Harvard
Harvard/Boston University
Harvard/Weizmann
Harvard
Harvard
Harvard
Harvard-MIT CUA
Motivation:
understanding transport phenomena
in correlated electron systems
e.g. transport near quantum phase transition
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 81:5217 (1998)
Scaling near the superconductor to insulator
transition
Yazdani and Kapitulnik
Phys.Rev.Lett. 74:3037 (1995)
Breakdown of scaling near the
superconductor to insulator transition
Mason and Kapitulnik
Phys. Rev. Lett. 82:5341 (1999)
Outline
v
Current decay for interacting atoms in
optical lattices. Connecting classical dynamical
instability with quantum superfluid to Mott transition
Phase dynamics of coupled 1d condensates.
Competition of quantum fluctuations and tunneling.
Application of the exact solution of quantum
sine Gordon model
Conclusions
J
Current decay for interacting atoms
in optical lattices
Connecting classical dynamical instability
with quantum superfluid to Mott transition
References:
J. Superconductivity 17:577 (2004)
Phys. Rev. Lett. 95:20402 (2005)
Phys. Rev. A 71:63613 (2005)
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), …
Equilibrium superfluid to insulator transition
m
Theory: Fisher et al. PRB (89), Jaksch et al. PRL (98)
Experiment: Greiner et al. Nature (01)
U
Superfluid
Mott
insulator
n 1
t/U
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)
This talk: How to connect
the dynamical instability (irreversible, classical)
to the superfluid to Mott transition (equilibrium, quantum)
p
p/2
Unstable
Stable
???
SF
This talk
MI
U/J
p
???
Possible experimental
U/t
sequence:
SF
MI
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
unstable
unstable
Amplification of
density fluctuations
r
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
.
Dynamical instability for integer filling
s
(p)
sin(p)
p
p/2
I(p)
p
0.0
0.1
0.2
0.3
U/J
*
0.4
0.5
Condensate momentum p/
Vicinity of the SF-I quantum phase transition.
Classical description applies for
Dynamical instability occurs for
SF
MI
Dynamical instability. Gutzwiller approximation
Wavefunction
Time evolution
We look for stability against small fluctuations
0.5
unstable
0.4
d=3
Phase diagram. Integer filling
d=2
p/p
0.3
d=1
0.2
stable
0.1
0.0
0.0
0.2
0.4
U/Uc
0.6
0.8
1.0
Order parameter suppression by the current.
Number state (Fock) representation
Integer filling
N-2
N-1
N
N+1
N+2
N-2
N-1
N
N+1
N+2
Order parameter suppression by the current.
Number state (Fock) representation
Integer filling
Fractional filling
N-2
N-1
N
N+1
N+2
N-2
N-1
N
N+1
N+2
N-3/2
N-3/2
N-1/2 N+1/2 N+3/2
N-1/2 N+1/2 N+3/2
Dynamical instability
Integer filling
Fractional filling
p
p
p/2
p/2
U/J
SF
MI
U/J
Center of Mass Momentum
Optical lattice and parabolic trap.
Gutzwiller approximation
0.00
0.17
0.34
0.52
0.69
0.86
N=1.5
N=3
0.2
0.1
The first instability
develops near the edges,
where N=1
0.0
-0.1
U=0.01 t
J=1/4
-0.2
0
100
200
300
Time
400
500
Gutzwiller ansatz simulations (2D)
j
phase
j
phase
phase
Beyond semiclassical equations. Current decay by tunneling
Current carrying states are metastable.
They can decay by thermal or quantum tunneling
Thermal activation
Quantum tunneling
j
phase
phase
Decay of current by quantum tunneling
Quantum
phase slip
j
j
Escape from metastable state by quantum tunneling.
WKB approximation
S – classical action corresponding to the motion in an inverted potential.
Decay rate from a metastable state. Example
S
0
0
 1  dx 2

2
3
d 
  x  bx 


 2m  d 



  ( pc  p )  0
Weakly interacting systems. Quantum rotor model.
Decay of current by quantum tunneling
1  d j 
S    d

  2 JN cos  j 1   j 
2U  d 
j
2
 j  pj   j
At pp/2 we get
For the link on which the QPS takes place
2
3
1  d j 
JN
S    d
 j 1   j 


  JN cos p  j 1   j  
2U  d 
3
j
2
d=1. Phase slip on one link + response of the chain.
Phases on other links can be treated in a harmonic approximation
For d>1 we have to include transverse directions.
Need to excite many chains to create a phase slip
J||  J cos p,
J  J
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
Weakly interacting systems. Gross-Pitaevskii regime.
Decay of current by quantum tunneling
p
p/2
U/J
SF
MI
Fallani et al., PRL (04)
Quantum phase slips are
strongly suppressed
in the GP regime
Strongly interacting regime. Vicinity of the SF-Mott transition
p
p/2
Close to a SF-Mott transition
we can use an effective
relativistivc GL theory
(Altman, Auerbach, 2004)
U/J
SF
M
I
2 2 ip x


1

p
 e
Metastable current carrying state:
This state becomes unstable at pc  1 3 corresponding to the
maximum of the current: I  p   p 1  p 2 2  .
2
Strongly interacting regime. Vicinity of the SF-Mott transition
Decay of current by quantum tunneling
p
p/2
U/J
SF
Action of a quantum phase slip in d=1,2,3
MI
- 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
Decay of current by quantum tunneling
0.5
unstable
0.4
d=3
d=2
d=1
p/
0.3
0.2
stable
0.1
0.0
0.0
0.2
0.4
U/Uc
0.6
0.8
1.0
phase
phase
Decay of current by thermal activation
Thermal
phase slip
j
j
DE
Escape from metastable state by thermal activation
Thermally activated current decay. Weakly interacting regime
DE
Thermal
phase slip
Activation energy in d=1,2,3
Thermal fluctuations lead to rapid decay of currents
Crossover from thermal
to quantum tunneling
Decay of current by thermal fluctuations
Phys. Rev. Lett. (2004)
Dynamics of interacting bosonic systems
probed in interference experiments
Interference of two independent condensates
Andrews et al., Science 275:637 (1997)
Interference experiments with low d condensates
1D condensates: Schmiedmayer et al., Nature Physics (2005,2006)
Transverse imaging
Longitudial imaging
trans.
imaging
long. imaging
2D condensates: Hadzibabic et al., Nature 441:1118 (2006)
z
Time of
flight
x
Studying dynamics using interference experiments
Motivated by experiments and discussions with
Bloch, Schmiedmayer, Oberthaler, Ketterle, Porto, Thywissen
J
Prepare a system by
splitting one condensate
Take to the regime of finite
or zero tunneling
Measure time evolution
of fringe amplitudes
Studying coherent dynamics
of strongly interacting systems
in interference experiments
Coupled 1d systems
J
Interactions lead to phase fluctuations within individual condensates
Tunneling favors aligning of the two phases
Interference experiments measure only the relative phase
Coupled 1d systems
Conjugate variables
J
Relative phase
Particle number
imbalance
Small K corresponds to strong quantum fluctuations
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
Dynamics of quantum sine-Gordon model
Hamiltonian formalism
Initial state
Quantum action in space-time
Initial state provides a boundary condition at t=0
Solve as a boundary sine-Gordon model
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
Sine-Gordon
+ boundary condition in space
Boundary
Sine-Gordon
Model
Sine-Gordon
+ boundary condition in time
two coupled 1d BEC
quantum impurity problem
space and time
enter equivalently
Boundary sine-Gordon model
Initial state is a generalized squeezed state
creates solitons, breathers with rapidity q
creates even breathers only
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)
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 Mathieu’s functions
Time evolution
Coherence
Dynamics of quantum Josephson Junction
Power spectrum
power
spectrum
w
E2-E0
Main peak
“Higher harmonics”
Smaller peaks
E4-E0
E6-E0
Dynamics of quantum sine-Gordon model
Coherence
Main peak
“Higher harmonics”
Smaller peaks
Sharp peaks
Dynamics of quantum sine-Gordon model
power
spectrum
w
main peak
“higher harmonics”
smaller peaks
sharp peaks (oscillations without decay)
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