Entanglement and thermodynamics in non-equilibrium isolated quantum - PowerPoint PPT Presentation

Entanglement and thermodynamics in non-equilibrium isolated quantum systems Pasquale Calabrese SISSA-Trieste Rome, Feb 16th 2018 Joint work with Vincenzo Alba PNAS 114 , 7947 (2017) & more

Isolated systems out of equilibrium Quantum Quench 1) prepare a many-body quantum system in a pure state | Ψ 0 ⟩ that is not an eigenstate of the Hamiltonian 2) let it evolve according to quantum mechanics (no coupling to environment) | ⇧ ( t ) ⇤ = e − iHt | ⇧ 0 ⇤ , t • How can we describe the dynamics? Questions: • Does it exist a stationary state? • Can it be thermal? In which sense? Don’t forget: | Ψ ( t ) ⟩ is pure (zero entropy) for any t while the thermal mixed state has non-zero entropy

Quantum Newton cradle T. Kinoshita, T. Wenger and D.S. Weiss, Nature 440, 900 (2006) few hundreds 87 Rb atoms in a harmonic trap Essentially unitary time evolution

Quantum Newton cradle Kinoshita et al 2006 - 2D and 3D systems relax quickly and thermalize: 9 τ 0 τ 4 τ 2 τ - 1D system relaxes slowly in time, to a non-thermal distribution

Quantum Newton cradle Kinoshita et al 2006 - 2D and 3D systems relax quickly and thermalize: Local observables have the same values as if the entire system was in a thermal ensemble 9 τ 0 τ 4 τ 2 τ - 1D system relaxes slowly in time, to a non-thermal distribution Non-equilibrium new states of matter (with very unconventional features)

◉ ◉ Entanglement & thermodynamics Reduced density matrix: ρ A (t)=Tr B ρ ( t ) A B ρ A (t) corresponds to a mixed state The entanglement entropy Infinite system (A U B ) S A (t)= -Tr[ ρ A (t) ln ρ A (t)] measures A finite the bipartite entanglement between A & B ◉ Stationary state exists if for any finite subsystem A of an infinite system lim ρ A (t) = ρ A ( ∞ ) exists t →∞

Thermalization Consider the Gibbs ensemble for the entire system A U B ρ T = e - H/T /Z ⟨ Ψ 0 | H | Ψ 0 ⟩ = Tr[ ρ T H ] with T is fixed by the energy in the initial state: no free parameter!! Reduced density matrix for subsystem A: ρ A,T =Tr B ρ T The system thermalizes if for any finite subsystem A ρ A,T = ρ A ( ∞ ) In jargon: the infinite part B of the system acts as an heat bath for A

Generalized Gibbs Ensemble What about integrable systems? Proposal by Rigol et al 2007: The GGE density matrix ρ GGE = e - ∑ λ m Im /Z with λ m fixed by ⟨ Ψ 0 | I m | Ψ 0 ⟩ = Tr[ ρ GGE I m ] Again no free parameter!! I m are the integrals of motion of H, i.e. [ I m ,H ] = 0 Reduced density matrix for subsystem A: ρ A,GGE =Tr B ρ GGE The system is described by GGE if for any finite subsystem A of an infinite system [Barthel-Schollwock ’08] ρ A,GGE = ρ A ( ∞ ) [Cramer, Eisert, et al ’08] + ........ [PC, Essler, Fagotti ’12] But, which integral of motions must be included in the GGE? Too long and technical answer to be discussed here

Entanglement vs Thermodynamics The equivalence of reduced density matrices ρ A,TD = ρ A ( ∞ ) TD=Gibbs or GGE Implies that the subsystem’s entropies are the same: S A,TD = S A ( ∞ ) The TD entropy S TD =-Tr ρ TD ln ρ TD is extensive S A ( ∞ ) S A,TD S TD = ≃ L l l For large time the entanglement entropy becomes thermodynamic entropy The entropy of the stationary state is just the entanglement accumulated during time

http://science.sciencemag.org/ Quantum thermalization through Science 353, 794 (2016) Downloaded from entanglement in an isolated many-body system Adam M. Kaufman, M. Eric Tai, Alexander Lukin, Matthew Rispoli, Robert Schittko, Philipp M. Preiss, Markus Greiner * A quantum quench Mott insulator Many-body - Initialize Quench Even Odd interference 1.0 T � 0 - 0.8 pure state e 0.6 P(A) 680 nm 0.4 y - 0.2 45 Er x 0.0 e 6 Er 0 1 2 3 4 5 6 Observable A B Many-body purity On-site Statistics On-site Statistics 0 global 10 0 local - 1 1 unitary dynamics thermalization Entropy: -Log(Tr[ � 2 ]) - Locally thermal Locally pure Globally pure 0.8 0.8 Initial Purity: Tr[ � 2 ] u- t=0 ms t=16 ms state 1.0 T>0 10 -1 2.3 P(n) 0.6 P(n) 0.6 0.8 ge quench pure state 0.6 P(A) 0.4 0.4 s 0.4 0.2 0.2 0.2 Global thermal state purity s 10 -2 4.6 0.0 0 1 2 3 4 5 6 0 0 , Observable A 0 10 20 0 1 2 3 4 5 6 0 1 2 3 4 5 6 Particle number Particle number time after quench (ms) - FIG. 1. Schematic of thermalization dynamics in C Expand and Measure Local Occupation Number Expand and Measure Local and Global Purity closed systems . An isolated quantum system at zero tem- g perature can be described by a single pure wavefunction | Ψ i . 0 1 2 2 0 1 -1 -1 -1 1 -1 1 s ~ 50 Sites ~ 50 Sites Subsystems of the full quantum state appear pure, as long e as the entanglement (indicated by grey lines) between sub- c systems is negligible. If suddenly perturbed, the full system s evolves unitarily, developing significant entanglement between s all parts of the system. While the full system remains in a al pure, zero-entropy state, the entropy of entanglement causes - the subsystems to equilibrate, and local, thermal mixed states t appear to emerge within a globally pure quantum state. 2 0 1 0 1 2 1 1 -1 1 -1 1

http://science.sciencemag.org/ Quantum thermalization through Science 353, 794 (2016) Downloaded from entanglement in an isolated many-body system Adam M. Kaufman, M. Eric Tai, Alexander Lukin, Matthew Rispoli, Robert Schittko, Philipp M. Preiss, Markus Greiner * A quantum quench Mott insulator Many-body - Initialize Quench Even Odd interference 1.0 T � 0 - 0.8 pure state e 0.6 P(A) 680 nm 0.4 y - 0.2 45 Er x 0.0 It’s a paradigm shift about the origin of entropy e 6 Er 0 1 2 3 4 5 6 Observable A B Many-body purity On-site Statistics On-site Statistics 0 global 10 0 local - 1 1 unitary dynamics thermalization Entropy: -Log(Tr[ � 2 ]) - Locally thermal Locally pure Globally pure 0.8 0.8 Initial Purity: Tr[ � 2 ] u- t=0 ms t=16 ms state 1.0 T>0 10 -1 2.3 P(n) 0.6 P(n) 0.6 0.8 ge quench pure state 0.6 P(A) 0.4 0.4 s 0.4 0.2 0.2 0.2 Global thermal state purity s 10 -2 4.6 0.0 0 1 2 3 4 5 6 0 0 , Observable A 0 10 20 0 1 2 3 4 5 6 0 1 2 3 4 5 6 Particle number Particle number time after quench (ms) - FIG. 1. Schematic of thermalization dynamics in C Expand and Measure Local Occupation Number Expand and Measure Local and Global Purity closed systems . An isolated quantum system at zero tem- g perature can be described by a single pure wavefunction | Ψ i . 0 1 2 2 0 1 -1 -1 -1 1 -1 1 s ~ 50 Sites ~ 50 Sites Subsystems of the full quantum state appear pure, as long e as the entanglement (indicated by grey lines) between sub- The understanding of the quench dynamics cannot c systems is negligible. If suddenly perturbed, the full system s evolves unitarily, developing significant entanglement between prescind the characterisation of the entanglement s all parts of the system. While the full system remains in a al pure, zero-entropy state, the entropy of entanglement causes - the subsystems to equilibrate, and local, thermal mixed states t appear to emerge within a globally pure quantum state. 2 0 1 0 1 2 1 1 -1 1 -1 1

Light-cone spreading of entanglement entropy PC, J Cardy 2005 • After a global quench, the initial state | ψ 0 › has an extensive excess of energy • It acts as a source of quasi-particles at t =0. A particle of momentum p has energy E p and velocity v p = dE p /dp • For t > 0 the particles moves semiclassically with velocity v p • particles emitted from regions of size of the initial correlation length are entangled, particles from far points are incoherent • The point x ∈ A is entangled with a point x’ ∈ B if a left (right) moving particle arriving at x is entangled with a right (left) moving particle arriving at x ’. This can happen only if x ± v p t ∼ x ’ ∓ v p t 2t 2t B B A t 2t < l

Light-cone spreading of entanglement entropy B A B PC, J Cardy 2005 l • The entanglement entropy of an interval A of length l is proportional to the total number of pairs of particles emitted from arbitrary points such that at time t, x ∈ A and x’ ∈ B • Denoting with f(p) the rate of production of pairs of momenta ±p and their contribution to the entanglement entropy, this implies Z 1 Z Z Z x 0 � x � v p t x 00 � x + v p t � � � � dx 0 dx 00 S A ( t ) ⇡ dx f ( p ) dp � � x 0 2 A x 00 2 B �1 Z 1 Z 1 / t dpf ( p )2 v p ✓ ( ` � 2 v p t ) + ` dpf ( p ) ✓ (2 v p t � ` ) ( 0 0 • When v p is bounded (e.g. Lieb-Robinson bounds) | v p |< v max , the second term is vanishing for 2 v max t< l and the entanglement entropy grows linearly with time up to a value linear in l Note : This is only valid in the space-time scaling limit t, l→∞ , with t/ l constant

One example Transverse field Ising chain PC, J Cardy 2005 Analytically for t, l ⨠ 1 with t/ l constant M Fagotti, PC 2008 d ' d ' Z Z 2 ⇡ 2 | ✏ 0 | H (cos ∆ ' )+ ` S ( t ) = t 2 ⇡ H (cos ∆ ' ) 2 | ✏ 0 | t< ` 2 | ✏ 0 | t> ` (2) = 1 � cos ' ( h + h 0 ) + hh 0 H ( x ) = � 1 + x log 1 + x � 1 � x log 1 � x cos ∆ ' = ( . 2 2 2 2 ✏ ' ✏ 0 ' contains al