Nonadiabatic dynamics in closed Hamiltonian systems. Anatoli Polkovnikov, Boston University University of Utah, Condensed Matter Seminar, 10/27/2009 R. Barankov BU Claudia De Grandi BU Vladimir Gritsev U. of Fribourg

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 phase transition in an optical lattice Greiner et. al. 2003

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:

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, …

Thermalization in classical systems (origin of ergodicity): chaotic many-body dynamics implies exponential in time sensitivity to initial fluctuations. Thermalization in quantum systems (EoM are linear in time – no chaos?) Consider the time average of a certain observable A in an isolated system after a quench. Eignestate thermalization hypothesis (M. Rigol, V. Dunjko & M. Olshanii, Nature 452, 854, 2008.): A n,n ~ const (n) so there is no dependence on  nn. Eignestate thermalization hypothesis (Srednicki 1994; M. Rigol, V. Dunjko & M. Olshanii, Nature 452, 854, 2008.): A n,n ~ const (n) so there is no dependence on  nn. Information about equilibrium is fully contained in diagonal elements of the density matrix.

This is true for all thermodynamic observables: energy, pressure, magnetization, …. (pick your favorite). They all are linear in . This is not true about von Neumann entropy! Off-diagonal elements do not average to zero. The usual way around: coarse-grain density matrix (remove by hand fast oscillating off-diagonal elements of . Problem: not a unique procedure, explicitly violates time reversibility and Hamiltonian dynamics.

Von Neumann entropy: always conserved in time (in isolated systems). More generally it is invariant under arbitrary unitary transfomations Thermodynamics: entropy is conserved only for adiabatic (slow, reversible) processes. Otherwise it increases. Quantum mechanics: for adiabatic processes there are no transitions between energy levels: If these two adiabatic theorems are related then the entropy should only depend on  nn. Simple resolution: the sum is taken in the instantaneous energy basis. ???

Connection between two adiabatic theorems allows us to define heat ( A.P., Phys. Rev. Lett. 101, , 2008 ). Consider an arbitrary dynamical process and work in the instantaneous energy basis (adiabatic basis). Adiabatic energy is the function of the state.Adiabatic energy is the function of the state. Heat is the function of the process.Heat is the function of the process. Heat vanishes in the adiabatic limit. Now this is not the postulate, this is a consequence of the Hamiltonian dynamics!Heat vanishes in the adiabatic limit. Now this is not the postulate, this is a consequence of the Hamiltonian dynamics!

Isolated systems. Initial stationary state. Unitarity of the evolution: In general there is no detailed balance even for cyclic processes (but within the Fremi-Golden rule there is).

yields If there is a detailed balance then Heat is non-negative for cyclic processes if the initial density matrix is passive. Second law of thermodynamics in Thompson (Kelvin’s form). The statement is also true without the detailed balance but the proof is more complicated (Thirring, Quantum Mathematical Physics, Springer 1999).

Properties of d-entropy ( A. Polkovnikov, arXiv: ). Jensen’s inequality: Therefore if the initial density matrix is stationary (diagonal) then Now assume that the initial state is thermal equilibrium Let us consider an infinitesimal change of the system and compute energy and entropy change.

Recover the first law of thermodynamics (Fundamental Relation). If stands for the volume the we find

Classic example: freely expanding gas Suddenly remove the wall by Liouville theorem double number of occupied states result of Hamiltonian dynamics!

Thermodynamic adiabatic theorem. General expectation: In a cyclic adiabatic process the energy of the system does not change: no work done on the system, no heating, and no entropy is generated. - is the rate of change of external parameter.

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: Breakdown of Taylor expansion in low dimensions, especially near singularities (phase transitions). 1.Transitions are unavoidable in large gapless systems. 2.Phase space available for these transitions decreases with the rate  Hence expect Low dimensions: high density of low energy states, breakdown of mean-field approaches in equilibrium

Three regimes of response to the slow ramp: A.P. and V.Gritsev, Nature Physics 4, 477 (2008) A.Mean field (analytic) – high dimensions: B.Non-analytic – low dimensions, near singularities like QCP C.Non-adiabatic – low dimensions, bosonic excitations In all three situations quantum and thermodynamic adiabatic theorem are smoothly connected. The adiabatic theorem in thermodynamics does follow from the adiabatic theorem in quantum mechanics.

Origin of quadratic scaling: adiabatic perturbation theory Can we say anything about system response in the limit  0? Assume that we start in the ground state. Need to solve Convenient to work in the adiabatic (co-moving, instantaneous) basis

Schrödinger equation in the adiabatic basis Assume the proximity to the instantaneous ground state (small excitation probability)

Landau-Zener problem Now start t i  - , t f – finite (or vice versa) For finite interval of excitation the transition probability scales as the second power of the rate (not exponentially).

Gapless systems with quasi-particle excitations Ramping in generic gapless regime (low energy contribution) E E* Uniform system: Density of quasi-particles (entropy): Absorbed energy density (heating):

High dimensions: high energies dominate dissipation, low- dimensions – low energies dominate dissipation. Low energy contribution: High energy contribution. E E* Similarly

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, Zurek, Dorner, Zoller 2005

Another Example: loading 1D condensate into an optical lattice or merging two 1D condensates (C. De Grandi, R. Barankov, AP, PRL 2008 ) Relevant sine Gordon model: K – Luttinger liquid parameter

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

Sudden and slow quenches starting from the QCP

Probing quasi-particle statistics in nonlinear dynamical probes. K 0 1 massive bosons massive fermions (hard core bosons) T=0 T>0 More adiabatic Less adiabatic T bosonic-like fermionic-like transition?

Conclusions 1.Adiabatic theorems in quantum mechanics and thermodynamics are directly connected. 2.Diagonal entropy satisfies laws of thermodynamics from microscopics. Heat and entropy change result from the nonadiabatic transitions between microscopic energy levels. 3.Maximum entropy state with  nn =const is the natural attractor of the Hamiltonian dynamics. 4.Universal scaling of density of excitations, heat, entropy for sudden and slow dynamics near QCP. These scaling laws are closely related to scaling of fidelity and other susceptibilities.

M. Rigol, V. Dunjko & M. Olshanii, Nature 452, 854 (2008) a, Two-dimensional lattice on which five hard-core bosons propagate in time. b, The corresponding relaxation dynamics of the central component n(k x = 0) of the marginal momentum distribution, compared with the predictions of the three ensembles c, Full momentum distribution function in the initial state, after relaxation, and in the different ensembles.

Non-adiabatic regime (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.

T=0.02

2D, T=0.2

Classical systems. probability to occupy an orbit with energy E. Instead of energy levels we have orbits. describes the motion on this orbits. Classical d-entropy The entropy “knows” only about conserved quantities, everything else is irrelevant for thermodynamics! S d satisfies laws of thermodynamics, unlike the usually defined