1. Slow dynamics in gapless low-dimensional systems Anatoli Polkovnikov, Boston University AFOSR Vladimir Gritsev – Harvard Ehud Altman -Weizmann Eugene Demler – Harvard Bertrand Halperin - Harvard Misha Lukin -Harvard

2. 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, … Problem: huge Hilbert space. 3. = 1+2 Nonequilibrium thermodynamics?

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

4. 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 .

5. 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.

6. 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

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

8. 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

9. 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  ? KZ scaling, (A.P. 2003)

10. Transverse field Ising model. Phase transition at g=1. Critical exponents: z= =1  d /(z +1)=1/2. Linear Response: A. P., 2003 Correct result (J. Dziarmaga 2005): Interpretation as the Kibble-Zurek mechanism: W. H. Zurek, U. Dorner, Peter Zoller, 2005

11. Possible breakdown of the Fermi-Golden rule (linear response) scaling due to bunching of bosonic excitations. Zero temperature. We can view the response of the system on a slow ramp as parametric amplification of quantum or thermal fluctuations.

12. Most divergent regime:    Agrees with the scaling result. Assuming the system thermalizes at long times:

13. Finite temperatures. Instead of wave function use density matrix (Wigner form). d=1,2 Non-adiabatic regime! Non-analytic regime! Real result or the artifact of the harmonic approximation?

14. Numerical verification (bosons on a lattice).  

15. T=0.02

16. Thermalization at long times.

17. 2D, T=0.2

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

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

20. Include quantum depletion. Equilibrium: Current state: With quantum depletion the current state is unstable at  p 

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

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

23. 1D System. – variational result N~1 Large N~10 2 -10 3 semiclassical parameter (plays the role of 1/ ) Fallani et. al., 2004 C.D. Fertig et. al., 2004

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

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

26. Current decay in the vicinity of the superfluid-insulator transition

27. 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.

28. 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!!

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

30. 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