Cold Atoms and Out of Equilibrium Quantum Dynamics Anatoli Polkovnikov, Boston University AFOSR Vladimir Gritsev – Harvard Ehud Altman -Weizmann Eugene Demler – Harvard Bertrand Halperin - Harvard Misha Lukin -Harvard UMASS, Boston, 11/21/2007

Cold atoms: (controlled and tunable Hamiltonians, isolation from environment) 1. Equilibrium thermodynamics: Quantum simulations of equilibrium condensed matter systems Experimental examples:

Optical Lattices: 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. I. Bloch, Nature Physics 1, (2005)

Superfluid-insulator quantum phase transition in interacting bosons.

Repulsive Bose gas. 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) M. Olshanii, 1998 interaction parameter Lieb-Liniger, complete solution Experiments:

Energy vs. interaction strength: experiment and theory. Kinoshita et. Al., Science 305, 1125, 2004 No adjustable parameters!

Local pair correlations. Kinoshita et. Al., Science 305, 1125, 2004

Cold atoms: (controlled and tunable Hamiltonians, isolation from environment) 1. Equilibrium thermodynamics: Quantum simulations of equilibrium condensed matter systems 2. Quantum dynamics: Coherent and incoherent dynamics, integrability, quantum chaos, …

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) No thermalization in1D. Fast thermalization in 3D. Quantum analogue of the Fermi-Pasta- Ulam problem.

3. = 1+2 Nonequilibrium thermodynamics? Cold atoms: (controlled and tunable Hamiltonians, isolation from environment) 1. Equilibrium thermodynamics: Quantum simulations of equilibrium condensed matter systems 2. Quantum dynamics: Coherent and incoherent dynamics, integrability, quantum chaos, …

Adiabatic process. Assume no first order phase transitions. Adiabatic theorem: “Proof”: then

Adiabatic theorem for integrable systems. Density of excitations Energy density (good both for integrable and nonintegrable systems: E B (0) is the energy of the state adiabatically connected to the state A. For the cyclic process in isolated system this statement implies no work done at small .

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: Is there anything wrong with this picture? Hint: low dimensions. Similar to Landau expansion in the order parameter. 1.Transitions are unavoidable in large gapless systems. 2.Phase space available for these transitions decreases with  Hence expect

More specific reason. Equilibrium: high density of low-energy states  strong quantum or thermal fluctuations,strong quantum or thermal fluctuations, destruction of the long-range order,destruction of the long-range order, breakdown of mean-field descriptions,breakdown of mean-field descriptions, Dynamics  population of the low-energy states due to finite rate  breakdown of the adiabatic approximation.

This talk: three regimes of response to the slow ramp: A.Mean field (analytic) – high dimensions: B.Non-analytic – low dimensions C.Non-adiabatic – lower dimensions

Example: crossing a QCP. tuning parameter tuning parameter gap    t,   0   t,   0 Gap vanishes at the transition. No true adiabatic limit! How does the number of excitations scale with  ? A.P. 2003

Transverse field Ising model. Phase transition at g=1. Critical exponents: z= =1  d /(z +1)=1/2. A.P. 2003, W. H. Zurek, U. Dorner, P. Zoller 2005, J. Dziarmaga 2005 Start at g  and slowly ramp it to 0. Count the number of domain walls

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: Assuming the system thermalizes at a fixed energy Energy

Finite Temperatures d=1,2 d=1; d=2; Artifact of the quadratic approximation or the real result? Non-adiabatic regime! d=3

Numerical verification (bosons on a lattice). Use the fact that quantum fluctuations are weak in the SF phase and expand dynamics in the effective Planck’s constant: 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): Expectation value is substituted by the average over the initial conditions.Expectation value is substituted by the average over the initial conditions. Exact for harmonic theories!Exact for harmonic theories! Not limited by low temperatures!Not limited by low temperatures! Asymptotically exact at short times.Asymptotically exact at short times. Subsequent orders: quantum scattering events (quantum jumps)

T=0.02

Thermalization at long times.

2D, T=0.2

Conclusions. A.Mean field (analytic): B.Non-analytic C.Non-adiabatic Three generic regimes of a system response to a slow ramp: Open questions: general fate of linear response at low dimensions, non-uniform perturbations,…

SuperfluidMott insulator Adiabatic increase of lattice potential M. Greiner et. al., Nature (02) What happens if there is a current in the superfluid?

??? p U/J   Stable Unstable SF MI p SFMI U/J ??? possible experimental sequence: ~lattice potential Drive a slowly moving superfluid towards MI.

Meanfield (Gutzwiller ansatzt) phase diagram Is there current decay below the instability?

Role of fluctuations Below the mean field transition superfluid current can decay via quantum tunneling or thermal decay. E p Phase slip

1D System. – variational result N~1 Large N~ semiclassical parameter (plays the role of 1/ ) Fallani et. al., 2004 C.D. Fertig et. al., 2004

Higher dimensions. Longitudinal stiffness is much smaller than the transverse. Need to excite many chains in order to create a phase slip. r

Phase slip tunneling is more expensive in higher dimensions: Stability phase diagram Crossover Stable Unstable

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 U/J   SuperfluidMI 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 nonequilibrium irreversible transition!!

J. Mun, P. Medley, G. K. Campbell, L. G. Marcassa, D. E. Pritchard, W. Ketterle, 2007

Conclusions. A.Mean field (analytic): B.Non-analytic C.Non-adiabatic Three generic regimes of a system response to a slow ramp: Smooth connection between the classical dynamical instability and the quantum superfluid-insulator transition. Quantum fluctuations Depletion of the condensate. Reduction of the critical current. All spatial dimensions. mean field beyond mean field Broadening of the mean field transition. Low dimensions