NON-EQUILIBRIUM DYNAMICS IN

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1 NON-EQUILIBRIUM DYNAMICS IN ISOLATED QUANTUM SYSTEMS Masud Haque Maynooth University Dept. Mathematical Physics Maynooth, Ireland Max-Planck Institute for Physics of Complex Systems (MPI-PKS) Dresden, Germany

2 NON-EQUILIBRIUM DYNAMICS IN ISOLATED QUANTUM SYSTEMS Overview: motivations + spectral picture + selection of problems MULTI-MAGNON DYNAMICS IN SPIN CHAINS MANY-BODY BLOCH OSCILLATIONS IN 1D TILTED LATTICE

3 NON-EQUILIBRIUM DYNAMICS IN ISOLATED QUANTUM SYSTEMS Traditions & formalisms of condensed matter physics predominantly address ground state or thermal equilibrium Low-lying parts of the many-body spectrum explored. Typical issues in traditional Condensed Matter Physics... Time rarely appears on horizontal axis. X Time Gapped versus gapless spectra Universality Phase transitions Why? Symmetry breaking Electrons in solids relax fast... all involve low-lying part of spectrum

4 NON-EQUILIBRIUM DYNAMICS IN ISOLATED QUANTUM SYSTEMS THE NON-EQUILIBRIUM ERA Two classes of non-equilibrium situations: Ultracold trapped atoms STRONG ENVIRONMENT COUPLING GOOD ISOLATION FROM ENVIRONMENT Ultrafast pump-probe spectroscopy BATH, LEADS, DISSIPATION CHANNELS UNITARY, NON-DISSIPATIVE DYNAMICS Transport through quantum dots and molecules Condensation of cavity polaritons NMR quantum computing Isolated Quantum many-body systems Cold atoms some nano- & pump-probe experiments (upto some time scale) Some NMR experiments

5 EXPERIMENTAL CONTEXT - 1 Kinoshita et al, Nature (26) ULTRACOLD TRAPPED ATOMS A Quantum Newton s Cradle Excellent isolation from environment

6 EXPERIMENTAL CONTEXT - 2 ULTRAFAST PUMP-PROBE SPECTROSCOPY Dissipation, bath, may be negligible UP TO SOME TIME SCALE Data from van Loosdrecht group

7 EXPERIMENTAL CONTEXT - 3 Crystal of fluorapatite molecules NUCLEAR MAGNETIC RESONANCE Each molecule is a nuclear-spin chain Nuclear spins can serve as isolated spin lattice H = L 1 ij [ S z is z j 1 2 ( S x i S x j +S y i Sy j ) ] IN SOME WINDOW OF TIME SCALES Coupling to environment longer time scale External control through RF pulses: Cappellaro, Ramanathan, Cory, P.R. A (27) ] H rf = ω rf [S xi cosφ(t) + S yi sinφ(t) i

8 THE FULL-SPECTRUM PERSPECTIVE Isolated Quantum many-body systems Standard many-body quantum physics low-lying parts of the many-body spectrum. No tendency toward ground state Any part of spectrum can be important! real-time dynamics spectral structures Many new questions and new phenomena! Required: A fundamental re-thinking of the low-energy ideology.

9 NON-EQUILIBRIUM DYNAMICS IN ISOLATED QUANTUM SYSTEMS: NEW QUESTIONS & PHENOMENA Disclaimer!! (1) Following selection is biased & personal Selection is NOT comprehensive or complete (2) Not all relevant references are mentioned

10 NON-EQUILIBRIUM DYNAMICS IN ISOLATED QUANTUM SYSTEMS: NEW QUESTIONS & PHENOMENA [a] Thermalization in isolated systems [b] (Deviation from) adiabaticity in slow quenches [c] Propagation dynamics in lattice systems (bound clusters, localization, scattering...) [d] Collective dynamics in traps [e] Bloch oscillations [f] Oscillatory driving

11 NON-EQUILIBRIUM DYNAMICS IN ISOLATED QUANTUM SYSTEMS: NEW QUESTIONS & PHENOMENA [a] Thermalization in isolated systems Question Can an isolated system, driven out of equilibrium, thermalize without external bath? Answer No thermalization of full system Pure state ψ(t) remains pure under isolated evolution. Will never turn into a mixed thermal state ρ = tre β H = e βe α α α α

12 THERMALIZATION IN ISOLATED SYSTEMS: OBSERVABLES However: sub-regions of the isolated system could thermalize observables could thermalize Some observable time = observables relax to value dictated by thermal ensemble

13 THERMALIZATION OF OBSERVABLES If initial state is ψ() = c α α, α then observable A will relax to A(t) = ψ(t) A ψ(t) t α c α 2 α A α Some observable Thermalization: α c α 2 α A α = 1 e βe α Z(β) α A α α time α, E α eigenstates, eigenenergies β defined by energy In non-integrable systems, many observables thermalize.

14 THERMALIZATION OF OBSERVABLES Mechanism for thermalization: Eigenstate Thermalization Hypothesis α A α s are smooth functions of E α Deutsch, (1991); Srednicki, (1994) Rigol, Dunjko, Olshanii, Nature (28) Role of Integrability Generalized Gibbs Ensemble Kinoshita, Wenger Weiss, Nature (26) A Quantum Newton s Cradle Rigol et al, P.R.L. (27) Advertisement Beugeling, Moessner, Haque; P.R. E (214) Finite-Size Scaling of ETH

15 E.T.H. SCALING h i,j 1 2 (S+ i S j +S i S+ j )+ Sz is z j ; =.8 A αα = α Â α = α S z 2 α H ladder = H leg +λh rung E.T.H.: A αα s are smooth function of energy

16 E.T.H. SCALING Second example H = H XXZ +λ i (i i ) 2 S z j A αα = α Â α = α S z middle α Scaling of E.T.H. fluctuations: σ A D 1/2 e αl Beugeling, Moessner, Haque; P.R. E (214)

17 NON-EQUILIBRIUM DYNAMICS IN ISOLATED QUANTUM SYSTEMS: NEW QUESTIONS & PHENOMENA [a] Thermalization in isolated systems [b] (Deviation from) adiabaticity in slow quenches [c] Propagation dynamics in lattice systems (bound clusters, localization, scattering...) [d] Collective dynamics in traps [e] Bloch oscillations [f] Oscillatory driving

18 NON-EQUILIBRIUM DYNAMICS IN ISOLATED QUANTUM SYSTEMS: NEW QUESTIONS & PHENOMENA [b] (Deviation from) adiabaticity in finite-duration ramps λ(t) τ λ f Quantify non-adiabatic through excess excitation energy over final g.s. energy. Time t λ i τ =.1, 1, 4, 8 τ = sudden quench τ = adiabatic ramp Finite τ system doesn t reach final ground state. energy Time, t } `heating Q(τ) Q = E(t τ) E (λ f) g.s.

19 τ DEVIATIONS FROM ADIABATICITY: Q(τ) λ(t) λ f Time t λ i Adiabatic theorem: Q( ) = Asymptotic decrease of Q(τ) first correction to adiabaticity Q(τ) can decrease: Exponentially, as power-law; With/without oscillations or logarithms Q(τ) Q(τ) τ 1 1 τ Adiabaticity question [Q(τ)]: meaningful due to lack of dissipation energy δ

20 τ DEVIATIONS FROM ADIABATICITY: Q(τ) λ(t) λ f Time t λ i Ramps starting from excited eigenstate: energy spectrum c a λ b d fidelity, F c d.5 1 Ramp duration τ Realized in Bose-Hubbard ladders: Chen et al, Nat. Phys. (211) [Bloch group] Dóra, Haque, Zaránd; P.R.L. 211 Pollmann, Haque, Dóra; P.R.B 213 Venumadhav, Haque, Moessner; P.R.B 21 Tschischik, Haque, Moessner; P.R.A 212 Haque & Zimmer, P.R.A 213 Zimmer & Haque, arxiv: Luttinger Liquid Bose-Hubbard dimers, ladders Generic interacting trapped systems Non-adiabaticity measured using other quantities Many variants of the adiabaticity question have been studied Influence of phase transition (Kibble-Zurek physics) Effect of spectral gaps...

21 ASYMPTOTIC DECREASE OF Q(τ) : UNIVERSALITY IN TRAPPED SYSTEMS Haque & Zimmer, P.R.A (213) phase 1 QPT phase 2 λ Ramping across quantum phase transitions Kibble-Zurek scaling: Q(τ) τ α Uniform systems Most likely experimental realization: trapped atoms HARMONIC CONFINEMENT H = H system +V trap Trapping potential Density or filling INTERACTION RAMPS Asymptotic decay of Q(τ) UNIVERSAL FEATURES position position or site

22 NON-EQUILIBRIUM DYNAMICS IN ISOLATED QUANTUM SYSTEMS: NEW QUESTIONS & PHENOMENA [a] Thermalization in isolated systems [b] (Deviation from) adiabaticity in slow quenches [c] Propagation dynamics in lattice systems (bound clusters, localization, scattering...) [d] Collective dynamics in traps [e] Bloch oscillations [f] Oscillatory driving

23 NON-EQUILIBRIUM DYNAMICS IN ISOLATED QUANTUM SYSTEMS: NEW QUESTIONS & PHENOMENA [c] Propagation dynamics in lattice systems (Repulsively) Bound pairs & clusters In a lattice, interaction can bind particles. 2 4 Repulsive or attractive interactions! Spectrum, 2 particles in finite Bose-Hubbard lattice E ν 1 U = +1 H Bose.Hubbard = (a j a j+1 +a j+1 a j + U a j 2 a j a ja j ) E ν 1 U = 1 eigenvalue index ν

24 Two bosons start at site 1: U=.1 2 Two bosons start at site 1: U=1 2 < time < time position position

25 double occupancy D(t) 1.9 U = 12.2 Initial state: > Initial state 2... > > > Time t

26 COMPOSITE OBJECT DYNAMICS IN LATTICE SYSTEMS Magnetic systems, e.g. spin chains: Bound bi-magnons, tri-magnons, clusters Lattice bosons and lattice fermions: Repulsively bound pairs + bound triplets, clusters Dynamics of composite objects: PROPAGATION LOCALIZATION SCATTERING

27 BOUND COMPOSITES IN ITINERANT SYSTEMS Bose-Hubbard fermionic Hubbard Spin dependent hopping

28 ANISOTROPIC HEISENBERG (XXZ) CHAIN L 1 H = J x j=1 S x j Sx j+1 +Sy j Sy j+1 + Sz j Sz j+1 Isolated chain no dissipation Particle language: J x hopping J z = J x interaction Small parameter: 1/ Strong-coupling physics Ising-like regime All displayed data: = 1

29 LARGE PHENOMENA Clusters of s stick together and travel together as heavy object Clusters at very edge are locked to edge. Explanation: conservation of domain wall # OR AFM bond # Clusters near the edge can also be locked. A hierarchy of edge-locked configurations. NEW!

30 TRI-MAGNONS : PROPAGATION AND LOCKING OR OR propagates edge-locked (non-trivial) edge-locked (obvious) 5 "tri magnon" propagation.5 "tri magnon" propagation.5 < time < time position (site #) position (site #).5

31 XXZ CHAIN SPECTRUM (a) Periodic chain (b) Open chain Energy eigenvalues eigenvalue index N = 3 11 sites = 1 Bands marked with # domain walls

32 XXZ CHAIN SPECTRUM (a) Periodic chain (b) Open chain Energy eigenvalues eigenvalue index ± + small corrections N = 3 11 sites = 1

33 TRI-MAGNONS : PROPAGATION AND LOCKING OR OR propagates edge-locked (non-trivial) edge-locked (obvious) "tri magnon" propagation.5 "tri magnon" propagation.5 < time < time position (site #) position (site #).5

34 XXZ CHAIN SPECTRUM (a) Periodic chain (b) Open chain Energy eigenvalues N = 3 11 sites = eigenvalue index

35 THREE MAGNONS: PROPAGATION AND LOCKING Bi-magnon propagation?

36 THREE MAGNONS: PROPAGATION AND LOCKING Single magnon propagates Bi-magnon propagates "134" propagation.5 "145" propagation.5 < time < time position (site #) position (site #)

37 SPECTRAL EXPLANATAION Open chain 12 Energy eigenvalues N = 3 11 sites = 1 Sharma & Haque, P.R. A (214) eigenvalue index

38 SPECTRAL EXPLANATAION Open chain Energy eigenvalues N = 3 11 sites = 1 Sharma & Haque, P.R. A (214) eigenvalue index

39 SCATTERING OF BOUND OBJECTS ( QUANTUM BOWLING ) Spinless fermions with nearest-neighbor interactions: H = t (c j c j+1 + c j+1 c j ) + V n j n j+1 Strong coupling: V t Same results for spin chain (Heisenberg or XXZ chain; large anisotropy)

40 SCATTERING OF BOUND OBJECTS ( QUANTUM BOWLING ) With slightly modified interactions: Conjecture: Integrability suppresses reflection. Ganahl, Haque, Evertz, arxiv:

41 NON-EQUILIBRIUM DYNAMICS IN ISOLATED QUANTUM SYSTEMS Overview: motivations + spectral picture + selection of problems MULTI-MAGNON DYNAMICS IN SPIN CHAINS MANY-BODY BLOCH OSCILLATIONS IN 1D TILTED LATTICE