Thermalization in Quantum Systems

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1 Thermalization in Quantum Systems Jonas Larson Stockholm University and Universität zu Köln Dresden 18/4-2014

2 Motivation Long time evolution of closed quantum systems not fully understood. Cold atom system Not only of academic interest. Open questions in closed system quantum dynamics: i. Criteria for equilibration/thermalization. ii. Mechanism behind thermalization. iii. Properties of equilibrated states. iv. Definition for quantum integrability. v. Many-body localization vi. Open systems 2

3 Outline 1. Quantum Thermalization. 2. Integrability. Problems. 3. Chaos. Problems. 4. Localization and absence of ETH. 3

4 Quantum Thermalization 4

5 Quantum Thermalization Characteristics of ρ for long times. 1) Weak coupling. 2) Infinite degrees of freedom (bath). 3) Delta correlated in time (bath): Markov approximation (no memory). 4) Factorizable system-bath state (Born approximation). Thermalization of system. Open system System ρ THERMAL BATH ρ Th (T) t ρ = i ρ, H sys + L ρ Breuer, Open Quantum Systems (OUP). 5

6 Quantum Thermalization Equilibration Characteristics of ρ s (t) for long times. Closed system No clear separation system/bath, no Born-Markov nor rotatingwave approximations. System ρ(t) System ρ(t) ρ A (t) ρ B (t) Srednicki, Phys. Rev. E (1994). 6

7 Quantum Thermalization Equilibration Characteristics of ρ(t) for long times. Equilibration: Thermalization: A = Tr Aρ(t), t independent as t A local observable. Closed system A = A Th at long times A Th = Tr Aρ Th, where ρ Th = thermal state. Temperature determined from H. No memory of initial state. Srednicki, Phys. Rev. E (1994). 7

8 Quantum Thermalization ETH Eigenstate Thermalization Hypotesis Ψ(t) = γ C γ e ieγt/ħ ψ γ A(t) = C δ C γ e i E δ E γ t/ħ γ,δ A δγ, A δγ = ψ δ A ψ γ If thermalization (long-time limit) A LT 1 = lim T T T 0 dt A(t) = C γ 2 γ A γγ. ETH: A γγ is approximately constant in the energy window of the state Ψ. ETH: For all γ, ρ A = Tr B ψ γ ψ γ is thermal. Rigol, Nature

9 Quantum Thermalization Thermalization Which systems thermalize? Possible candidates: 1) Quantum non-integrable systems. 2) Chaotic systems. 9

10 Integrability 10

11 Integrability 101 Quantum Integrability Classical systems: Definition: A system is integrable if the number of degrees of freedom N is smaller than or equal to the number K of independent constants of motion. Q n, H = 0, n = 1, 2,, K, Q n, Q m = 0 n, m Arnold, Mathematical Methods in Classical Mechanics 11

12 Integrability 101 Quantum Integrability Quantum systems: Definition 1: Replace, i, /ħ. Fails, take P γ = ψ γ ψ γ. Definition 2: Use definition 1, but consider relevant constants of motion - that is operators with classical counterparts. Fails, not all operators have any classical corresponding observable. Definition 3: Poissonian level statistics (P S integrability. Definition 4: Level crossings implies integrability. = e S ) implies Definition 64: A quantum system is integrable if it is exactly solvable. 12 Caux, J. Stat. Mech. 2011

13 Integrability vs thermalization Spin-orbit coupled particle Rabi Hamiltonian of quantum optics Z 2 -parity symmetry H R = ωa + a + Ω 2 σ z + v a + + a σ x. U p, H R = 0, U p = e iπ a+ a+ σ z 2. Drive term breaks Z 2 (total energy only preserved quantity) H dr = ωa + a + Ω 2 σ z + v a + + a σ x + γσ x. Scully & Zubary, Quantum Optics (CUP) 13

14 Integrability vs thermalization Spin-orbit coupled particle Is the driven Rabi model integrable? Definition 1-2: Two degrees of freedom, only one (relevant) constant of motion (energy) Non-integrable. Larson, J. Phys. B (2013) (JC issue) 14

15 Integrability vs thermalization Spin-orbit coupled particle Is the driven Rabi model integrable? Definition 3: Level-statistics. Two branches, neither Poissonian Non-integrable. Level statistics of the Rabi model. Larson, J. Phys. B (2013) (JC issue) 15

16 Integrability vs thermalization Spin-orbit coupled particle Is the driven Rabi model integrable? Definition 4: Avoided crossings. No vissible crossings Nonintegrable. Energies of the Rabi model. Larson, J. Phys. B (2013) (JC issue) 16

17 Integrability vs thermalization Spin-orbit coupled particle Is the driven Rabi model integrable? Definition 64: Solvable. Braak (PRL 2011) says it might be solvable but not integrable, others say it is quasi solvable integrable? Larson, J. Phys. B (2013) (JC issue) 17

18 Integrability vs thermalization Spin-orbit coupled particle Does the driven Rabi model thermalize? If quantum non-integrability implies thermalization a qualified guess would be yes. Scaled variance of n(t). Thermalization δ n = 0. No thermalization! Larson, J. Phys. B (2013) (JC issue) 18

19 Chaos 19

20 Classical chaos Butterfly effect Hamilton equations: dp j dt = H q j, dq j dt = H p j, j = 1,2,, n. A solution R 1 (t) = q 1 1 (t),, q n 1 (t), p 1 1 (t),, p n 1 (t) lives on a surface in 2n-dimensional phase space. Chaotic system exponential spreading: R 1 t R 2 (t) e λt, λ> 0. Lyapunov exponent λ Gutzwiller, Chaos in Classical and Quantum mechanics (Springer) 20

21 Classical chaos KAM Theory Regular motion: Any solution R 1 (t) = q 1 1 (t),, q n 1 (t), p 1 1 (t),, p n 1 (t) lives on a tori in the 2n-dimensional phase space. Add a perturbation V that beaks integrability. KAM describes how the tori is gradually deformed. Cranking up V: Going from regular to full blown chaos. Poincaré section Gutzwiller, Chaos in Classical and Quantum mechanics (Springer) 21

22 Quantum chaos Butterfly effect Schrödinger equation: dρ dt = i ρ, H. Trace distance T ρ 1 t, ρ 2 t 1 2 Tr ρ 1 t ρ 2 t 2 = 1 2 μ i eigenvalues of ρ 1 t ρ 2 (t) Quantum mechanics linear theory. No Butterfly effect! Or i μ i = const. Peres, Quantum Theory, Concepts and Methods (Kluwer) 22

23 Quantum chaos Butterfly effect Perturbation Γ: H 1 = H and H 2 = H + Γ. Evolution, dρ 1 dt = i ρ 1, H 1 and dρ 2 dt = i ρ 2, H 2. Trace distance T ρ 1 t, ρ 2 t e λt Quantum butterfly effect! Non-unitary evolution butterfly effect, dρ dt = i ρ, H + L ρ. 23 Zurek, Rev. Mod. Phys. (2003).

24 Quantum chaos Characteristics of quantum chaotic systems Spectrum E n. Energy separation s n = E n+1 E n. Normalized distribution P(S). Regular motion: P S = e S (Poisson distribution). Chaotic motion: P S = π 2 Sβ e πs2 /4 (Wigner distribution). Level repulsion Kicked top Haake, Quantum Signatures of Chaos (Springer). 24

25 Quantum chaos Characteristics of quantum chaotic systems Level repulsion varying time-scales. Level repulsion ergodicity. Level repuslion avoided crossings. Driven Rabi model Haake, Quantum Signatures of Chaos (Springer). 25

26 Chaos vs thermalization Spin-orbit coupled particle Mean-field for the bosons, parametrize the atom by θ = (1+Z) 2 (1 Z) 2 e iδ, Semi-classical Hamiltonian H cl = p2 2 + x2 2 + ω 2 Z + gx 2 + γ 1 Z2 cos δ. Poincaré section. This Hamiltonian is chaotic in a classical sense thermalization. Larson, J. Phys. B (2013) (JC issue) 26

27 Chaos vs thermalization Spin-orbit coupled particle 2D SO coupling. ω = 0 dispersions H SO = p2 2m mω2 r 2 + v x p xσ x + v y p yσ y. E ± p x, p y = 1 2m p x 2 + p y 2 ± v x p x 2 + v y p y 2. Larson, Phys. Rev. A (2009) 27

28 v x = v y U(1) symmetry J, H SO = 0. Chaos vs thermalization Spin-orbit coupled particle Larson, Phys. Rev. A (2009). 28

29 v x = v y U(1) symmetry J, H SO = 0. Chaos vs thermalization v x = v y and ω 0 H SO equals dual E ε Jahn-Teller model. Spin-orbit coupled particle Larson, Phys. Rev. A (2009). 29

30 Chaos vs thermalization Spin-orbit coupled particle v x = v y U(1) symmetry J, H SO = 0. v x = v y and ω 0 H SO equals dual E ε Jahn-Teller model. v x v y Z 2 symmetry J, H SO 0. H SO equals dual E β 1 + β 2 Jahn-Teller model. Larson, Phys. Rev. A (2009). 30

31 Chaos vs thermalization Classical dynamics Poincaré sections (v x v y ). y = 0 p y = 0 Classical chaos x x Larson, Phys. Rev. A (2013). 31

32 Chaos vs thermalization Quantum dynamics Distributions (v x v y ). Quantum Scars Remnants of periodic classical solutions. Full Quantum Heller, Phys. Rev. Lett. (1984). Truncated Wigner (Semi-classical) Larson, Phys. Rev. A (2013). 32

33 Chaos vs thermalization KAM theory Islands may survive large integrability breaking perturbations. Poincaré sections Larson, Phys. Rev. A (2013). 33

34 Chaos vs thermalization KAM theory Initiate a state in one island. Distribution after long time when initial state in a regular island. No thermalization: Not all eigenstates obey ETH. Larson, Phys. Rev. A (2013). 34

35 Localization 35

36 ETH revisited Ergodicity Thermalization ergodicity. Quantum information spreads over the whole accessible phase space. The information about a subsystem A is shared in the whole system S: ρ A (t) diagonal/mixed. ρ A (t) obeys a volume law. Can ergodicity be lost in quantum non-integrable/chaotic systems? 36

37 Anderson localization Quantum interference Ψ Add disorder to your system. Time inversion symmetry. Enhance probability to scatter into the same state (factor 2) than an arbritary state. Quantum interference effect. Ψ Ψ No counterpart in classical systems (particles). Anderson, Phys. Rev. (1058). 37

38 Localization vs thermalization Spin models good for studying many-body localization Lieb-Robinson H = (J x σ x iσ x i+1 + J y σ y iσ y i+1 i +Jz σ z iσ z i+1 + h i σ z i) Clean XXX and XXY solvable, XYZ + h non-integrable. Localization with strong enough disorder h i W, +W. Localized eigenstates are not thermal, no thermalization! Larson, In progress. 38

39 Thanks! Summary Classical vs. Quantum choas. Possibilities to study closed quantum dynamics. Equilibration and thermalization. Not well understood: i. Criteria for equilibration/thermalization. ii. Mechanism behind thermalization. iii. Definition for Quantum integrability. iv. Open systems 39

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