1. Quantum dynamics in low dimensional isolated systems. Anatoli Polkovnikov, Boston University AFOSR Condensed Matter Colloquium, 04/03/2008 Roman Barankov Claudia De Grandi Vladimir Gritsev

2. Cold atoms: (controlled and tunable Hamiltonians, isolation from environment) 1. Equilibrium thermodynamics: Quantum simulations of equilibrium condensed matter systems Superfluid insulator phase transition in an optical lattice Greiner et. al. 2003

3. 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, …

5. In the continuum this system is equivalent to an integrable KdV equation. The solution splits into non-thermalizing solitons Kruskal and Zabusky (1965 ).

6. 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.

7. 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, …

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

9. Adiabatic theorem for isolated systems. Integrable systems: density of excitations Alternative (microcanonical) definition: In a cyclic adiabatic process the energy of the system does not change. This implies absence of work done on the system and hence absence of heating. E B (0) is the energy of the state adiabatically connected to the state A. General expectation:

10. 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.

11. 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

12. 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.

13. 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

14. 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

15. Example: optimal crossing of a QCP. (work in progress with Roman Barankov) tuning parameter tuning parameter gap  =(  t) r,   0 =(  t) r,   0 Gap vanishes at the transition. No true adiabatic limit! power corresponding to an optical adiabatic passage through a critical point.

16. Possible breakdown of the Fermi-Golden rule (linear response) scaling due to bunching of bosonic excitations. Bogoliubov Hamiltonian: In cold atoms: start from free Bose gas and slowly turn on interactions. Hamiltonian of Goldstone modes: superfluids, phonons in solids, (anti)ferromagnets, …

17. Zero temperature regime: Assuming the system thermalizes at a fixed energy Energy

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

19. 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.

20. T=0.02

21. Thermalization at long times.

22. 2D, T=0.2

23. Another Example: loading 1D condensate into an optical lattice or merging two 1D condensates ( work in progress with R. Barankov and C. De Grandi ) Relevant sine Gordon model: K – Luttinger liquid parameter

24. Results: K=2 corresponds to a SF-IN transition in an infinitesimal lattice (H.P. Büchler, et.al. 2003)

25. Expansion of quantum dynamics around classical limit. Classical (saddle point) limit: (i) Newtonian equations for particles, (ii) Gross-Pitaevskii equations for matter waves, (iii) Maxwell equations for classical e/m waves and charged particles, (iv) Bloch equations for classical rotators, etc. Questions: What shall we do with equations of motion? What shall we do with initial conditions? Challenge : How to reconcile exponential complexity of quantum many body systems and power law complexity of classical systems?

26. Partial answers. Leading order in  : equations of motion do not change. Initial conditions are described by a Wigner “probability’’ distribution: G.S. of a harmonic oscillator: Quantum-classical correspondence: ;

27. Semiclassical (truncated Wigner approximation): Exact for harmonic theories!Exact for harmonic theories! Not limited to low temperatures and to 1D!Not limited to low temperatures and to 1D! Asymptotically exact at short times.Asymptotically exact at short times.Summary: Expectation value is substituted by the average over the initial conditions.

28. Beyond the semiclassical approximation. Quantum jump. Each jump carries an extra factor of  2. Recover sign problem = exponential complexity in exact formulation of quantum dynamics.

29. Example (back to FPU problem). m = 10,  = 1, = 0.2, L = 100 Choose initial state corresponding to initial displacement at wave vector k = 2  /L (first excited mode). Follow the energy in the first excited mode as a function of time.

30. Classical simulation

31. Classical + semiclassical simulations

33. Similar problem with bosons in an optical lattice. Prepare and release a system of bosons from a single site. Little evidence of thermalization in the classical limit. Strong evidence of thermalization in the quantum and semiclassical limits.

34. Many-site generalization 60 sites, populate each 10 th site.

35. Conclusions. A.Mean field (analytic): B.Non-analytic C.Non-adiabatic Three generic regimes of a system response to a slow ramp: Many open challenging questions on nonequilibrium quantum dynamics. Cold atoms should be able to provide unique valuable experiments.