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Cold Atoms and Out of Equilibrium Quantum

Dynamics

Anatoli Polkovnikov, Boston University

Vladimir Gritsev Harvard Ehud Altman

- Weizmann Eugene Demler Harvard Bertrand

Halperin - Harvard Misha Lukin - Harvard

Texas AM, 11/29/2007

AFOSR

Two examples of complexity.

Single neuron relatively easy to characterize.

Is more fundamentally different or just more

complicated?

From single particle physics to many particle

physics.

Classical mechanics Need to solve Newtons

equation.

Single particle

Many particles

Instead of one differential equation need to

solve n differential equations, not a big deal!?

With modern computers we can simulate thousands

or even millions of particles.

M

Use specific numbers M200, n100.

n

Fermions

Bosons

A computer built from all known particles in

universe is not capable to exactly simulate even

such a small system.

Other issues.

Typical level spacing for our system

Gravitation field of the moon on electron

Typical time scales needed to resolve these

energy levels (e.g. to prepare the system in the

pure state)

Required accuracy of the theory (knowledge of

laws of nature) 10-80-10-50.

Many-body physics is fundamentally different from

single particle physics. It can not be derived

purely from microscopic description.

Experiments best simulators are real

systems. Problem with solid state (liquid)

systems Hamiltonians are two complex

Optical Lattices

I. Bloch, Nature Physics 1, 23 - 30 (2005)

OL are tunable (in real time) from weak

modulations to tight binding regime. Can change

dimensionality and study 1D, 2D, and 3D

physics. Both fermions and bosons.

Superfluid-insulator quantum phase transition in

interacting bosons (from particles to waves).

Repulsive Bose gas.

Lieb-Liniger, complete solution 1963.

M. Olshanii, 1998

interaction parameter

Experiments

T. Kinoshita, T. Wenger, D. S. Weiss ., Science

305, 1125, 2004

Also, B. Paredes1, A. Widera, V. Murg, O. Mandel,

S. Fölling, I. Cirac, G. V. Shlyapnikov, T.W.

Hänsch and I. Bloch, Nature 277 , 429 (2004)

Energy vs. interaction strength experiment and

theory.

No adjustable parameters!

Kinoshita et. Al., Science 305, 1125, 2004

Local pair correlations.

Kinoshita et. Al., Science 305, 1125, 2004

Cold atoms (controlled and tunable Hamiltonians,

isolation from environment)

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In the continuum this system is equivalent to an

integrable KdV equation. The solution splits into

non-thermalizing solitons Kruskal and Zabusky

(1965 ).

Qauntum Newton Craddle. (collisions in 1D

interecating Bose gas Lieb-Liniger model)

T. Kinoshita, T. R. Wenger and D. S. Weiss,

Nature 440, 900 903 (2006)

Cold atoms (controlled and tunable Hamiltonians,

isolation from environment)

3. 12 Nonequilibrium thermodynamics?

E.g. second law of thermodynamics.

Adiabatic process.

Assume no first order phase transitions.

Adiabatic theorem for integrable systems.

Density of excitations

Adiabatic theorem in quantum mechanics

Landau Zener process

In the limit ??0 transitions between different

energy levels are suppressed.

This, for example, implies reversibility (no work

done) in a cyclic process.

Adiabatic theorem in QM suggests adiabatic

theorem in thermodynamics

- Transitions are unavoidable in large gapless

systems. - Phase space available for these transitions

decreases with d. Hence expect

Is there anything wrong with this picture?

Hint low dimensions. Similar to Landau expansion

in the order parameter.

More specific reason.

- Equilibrium high density of low-energy states

-gt - strong quantum or thermal fluctuations,
- destruction of the long-range order,
- breakdown of mean-field descriptions,

Dynamics -gt population of the low-energy states

due to finite rate -gt breakdown of the adiabatic

approximation.

This talk three regimes of response to the slow

ramp

- Mean field (analytic) high dimensions
- Non-analytic low dimensions
- Non-adiabatic lower dimensions

Example crossing a QCP.

? ? ? t, ? ? 0

Gap vanishes at the transition. No true adiabatic

limit!

How does the number of excitations scale with ? ?

Transverse field Ising model.

Phase transition at g1.

Start at ggtgt1 and slowly ramp it to 0. Count the

number of domain walls

Critical exponents z?1 ? d?/(z? 1)1/2.

A.P. 2003, W. H. Zurek, U. Dorner, P. Zoller

2005, J. Dziarmaga 2005

Possible breakdown of the Fermi-Golden rule

(linear response) scaling due to bunching of

bosonic excitations.

Hamiltonian of Goldstone modes superfluids,

phonons in solids, (anti)ferromagnets,

In superfluids ? is determined by the

interactions.

Imagine an experiment, where we start from a

noninteracting superfluid and slowly turn on

interaction.

Zero temperature regime

Energy

Assuming the system thermalizes at a fixed energy

Finite Temperatures

d1,2

Non-adiabatic regime!

d3

Artifact of the quadratic approximation or the

real result?

Numerical verification (bosons on a lattice).

Nonintegrable model in all spatial dimensions,

expect thermalization.

Quantum expansion of dynamics

Leading order in ? start from random initial

conditions distributed according to the Wigner

transform of the density matrix and propagate

them classically (truncated Wigner approximation)

T0.02

Thermalization at long times.

2D, T0.2

Conclusions.

Three generic regimes of a system response to a

slow ramp

- Mean field (analytic)
- Non-analytic
- Non-adiabatic

Open questions general fate of linear response

at low dimensions, non-uniform perturbations,

M. Greiner et. al., Nature (02)

Adiabatic increase of lattice potential

What happens if there is a current in the

superfluid?

Together with E. Altman, E. Demler, M. Lukin, and

B. Halperin.

Drive a slowly moving superfluid towards MI.

Meanfield (Gutzwiller ansatzt) phase diagram

Is there current decay below the instability?

Role of fluctuations

Phase slip

Below the mean field transition superfluid

current can decay via quantum tunneling or

thermal decay .

1D System.

variational result

semiclassical parameter (plays the role of 1/ )

N1

Large N102-103

C.D. Fertig et. al., 2004

Fallani et. al., 2004

Higher dimensions.

Longitudinal stiffness is much smaller than the

transverse.

r

Need to excite many chains in order to create a

phase slip.

Phase slip tunneling is more expensive in higher

dimensions

Current decay in the vicinity of the

superfluid-insulator transition

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.

Detecting 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.

J. Mun, P. Medley, G. K. Campbell, L. G.

Marcassa, D. E. Pritchard, W. Ketterle, 2007

Conclusions.

Three generic regimes of a system response to a

slow ramp

- Mean field (analytic)
- Non-analytic
- Non-adiabatic

Smooth connection between the classical dynamical

instability and the quantum superfluid-insulator

transition.