Quench, Hydro and Floquet dynamics in integrable systems RAQIS, - PowerPoint PPT Presentation

Quench, Hydro and Floquet dynamics in integrable systems RAQIS, Annecy, 12 September 2018 Jean-Sébastien Caux Universiteit van Amsterdam

Plan of the talk Equilibrium dynamics Out-of-equilibrium dynamics Quenches Quasisolitons and Generalized Hydrodynamics Floquet dynamics

Applications of integrability Quantum magnetism Ultracold atoms Quantum dots, Atomic nuclei NV centers

The general idea, simply stated: Start with your favourite quantum state (expressed in terms of Bethe states) |{ λ } i O Apply some operator on it Reexpress the result in the basis of Bethe states: X F O O|{ λ } i = { µ } , { λ } |{ µ } i { µ } F O using ‘matrix elements’ { µ } , { λ } = h { µ }|O|{ λ } i

Equilibrium Dynamics from Integrability

Heisenberg spin chain S ( k, ω ) , ∆ = 1 , h = 0

Integrability ‘probed’ in the lab Quantum magnetism Ultracold atoms H a L 3 2nK < p 2 > H arb. u. L ‡ · 2 ‡ · ‡ · ‡ · ‡ · ‡ · ‡ · ‡ · · ‡ 4nK ‡ · ‡ · ‡ · · ‡ · ‡ · ‡ · ‡ 1 · ‡ · ‡ ‡ · ‡ · · ‡ · ‡ ‡ · · ‡ · ‡ · ‡ · ‡ · ‡ · ‡ ‡ · 0 · ‡ · ‡ · ‡ 2 ‡ · · ‡ ‡ · 1nK ‡ · · ‡ · ‡ ‡ · ‡ · H b L 5nK Ï Ì 0.0 0.5 < p 2 > H arb. u. L 1.0 1.5 2.0 2.5 Ï Ì Ì Ï Ï Ì Ï Ì Ï Ì Ï Ì Bragg detuning H kHz L Ï Ì 1 Ï Ì 10nK Ï Ì 2 Ï Ì Ì Ï Ï Ì Ï Ì H c L 1nK � Ï Ì ��� Ï Ì < p 2 > H arb. u. L 10nK Ï Ì � Ï Ì Á Ê Ï Ì Ê Á Ï Ì Á Ê Ï Ì ��� Ï Ì Ï Ì ��� Ì Ï Ï Ì Ê Á Ì Ï Ï Ì Ï Ì Ì Ï Ï Ì 0 Ï Ì 3 Ï Ì Ì Ï Ï Ì 3 Á Ê Ï Ì Ï Ì Ï Ì b 10 Ï Ì Ì Ï a T = 0nK / τ = 0 T = 25nK / τ = 0.44 Ì Ï Á Ê Ì Ï 1 Ê Á Á Ê Á Ê Ê Á Ê Á � � � � � ������ �� ������ � � Á Ê χ 2 /DOF Ê Á 15nK ��� ��������������������� T = 50nK / τ = 0.88 T = 75nK / τ = 1.19 0.0 0.5 1.0 1.5 2.0 2.5 ��� Ê Á 2 2 Ê Á Ê Á H d L 1 Á Ê Bragg detuning H kHz L Á Ê Á Ê Δ E [a.u.] 0 20 40 60 80 10 1nK Ê Á Ê Á ��� Á Ê Á Ê T (nK) < p 2 > H arb. u. L Á Ê Ê Á 10nK Ê Á Ê Á Á Ê Á Ê 1 Á Ê 2 Ê Á Á Ê 0 Ê Á Á Ê Ê Á Ê Á Á Ê χ 2 /DOF Á Ê Á Ê ��� Ê Á Ê Á Û Ú Ú Û 0.0 0.5 1.0 1.5 2.0 2.5 1 Ú Û Ú Û Û Ú 0 Û Ú Ú Û Ú Û Δ E [a.u.] Û Ú Ú Û 20nK Ú Û Û Ú ��� Û Ú 3 Bragg detuning H kHz L c Ú Û 10 Ú Û Ú Û Û Ú Ú Û Ú Û χ 2 /DOF Ú Û Û Ú 1 ��� Ú Û 1 Û Ú Ú Û Ú Û 0 20 40 60 Ú Û 80 2 Û Ú Ú Û Ú Û Û Ú Û Ú Ú Û T (nK) Ú Û Ú Û Û Ú Û Ú Ú Û Ú Û 1 0 Û Ú Ú Û Ú Û Ú Û Ú Û Δ E [a.u.] Û Ú Ú Û Û Ú Û Ú Û Ú Û Ú Û Ú 0 20 40 60 80 Û Ú T (nK) 0.0 0.5 1.0 1 1.5 2.0 2.5 ��� 0 Bragg detuning H kHz L 0 ���������� ������ 0 5 10 15 20 0 5 10 15 20 � � ω /2 π [kHz] ω /2 π [kHz] � ��� ��� ��� � ������������������� � ����������� Hope: also Quantum dots, NV centers, Atomic nuclei

Out-of- Equilibrium Dynamics from Integrability

Out-of-equilibrium using integrability The super Tonks-Girardeau gas Split Fermi seas (Moses states) Pulsed: Spin echo in quantum dots Quasisolitons Interaction quench in Richardson Domain wall release in Heisenberg Geometric quench Interaction cutoff in Lieb-Liniger Quenched: Release of trapped Lieb-Liniger BEC to Lieb-Liniger quench Quantum Newton’s Cradle in TG Néel to XXZ quench Generalized hydrodynamics Floquet driving central spin Driven: Floquet driving spin chains

Quantum Quenches

Progress on quenches

Quasisoliton Dynamics in Spin Chains

In memoriam Ludvig Dmitrievich Faddeev 23/3/1934 - 26/2/2017

Spinon dynamics in real space/time Vlijm, Caux, PRB 2016

Spinon dynamics in real space/time Vlijm, Caux, PRB 2016

Solitons (classical) John Scott Russell : solitary wave of translation (1834) (Herriot-Watt University)

Solitons (classical) ( Boussinesq ) Korteweg-de Vries equation ∂ t u + u ∂ x u + δ 2 ∂ 3 x u = 0 First simulations: Fermi-Pasta-Ulam- (Tsingou) absence of ergodicity Further simulations: Zabusky & Kruskal 1965 concept of a soliton Classical inverse scattering

Quasisoliton scattering (quantum) Vlijm, Ganahl, Fioretto, Brockmann, Haque, Evertz and Caux, 2015 1str-1str 2str-2str 0 0 0.5 ‘Worldlines’ 50 of colliding 0.4 j ( t ) i 20 100 wavepackets: tJ h S z 0.3 150 40 ∆ = 2 200 0.2 25 50 75 25 50 75 j j 0 6 Single magnons Bound magnons Displacement as a Displacement − χ (1 , 1) 4 − χ (2 , 2) 100 function of anisotropy tJ 2 (fixed incoming momenta) 200 Measured av. Linear fits 0 20 40 60 80 0 1 2 3 4 ∆ j

Generalized Hydrodynamics

Generalized Hydrodynamics (GHD) B. Bertini, M. Collura, J. De Nardis and M. Fagotti, PRL 117, 207201 (2016) O. A. Castro-Alvaredo, B. Doyon and T. Yoshimura, PRX 6, 041065 (2016) B. Doyon and T. Yoshimura, SciPost Phys. 2, 014 (2017) 
 Quench from spatially inhomogeneous state gr After initial dephasings: v eff ‘hydrodynamic’ evolution v described by local GGE Fig. 3 from B. Bertini, M. Collura, J. De Nardis and Two-equation summary: M. Fagotti, PRL 117, 207201 (2016) Local continuity equation ∂ t ρ ( λ ) + ∂ x ( v eff ( λ ) ρ ( λ )) = 0 (in terms of Bethe root densities) d λ 0 ϕ ( λ , λ 0 ) Z Local effective velocity p 0 ( λ ) ρ ( λ 0 ) ( v eff ( λ 0 ) − v eff ( λ ) = v gr ( λ ) + eff ( λ )) v

GHD as ‘molecular dynamics’: the flea gas B. Doyon, T. Yoshimura and JSC, PRL 2018 Encode initial state as a gas of quasisolitons Loop: evolve, collide and scatter (as if quasisolitons were classical particles, using displacement calculated from quantum phase shifts) Attractive interactions Repulsive interactions As simple as it gets in the integrability business!

David Weiss’s quantum Newton’s cradle experiment Ergodicity (or lack thereof) in interacting quantum systems close to an integrable model

The flea gas in a force field: simulating the quantum Newton’s cradle JSC, B. Doyon, J Dubail, R. Konik and T. Yoshimura, arXiv1711.00873 ‘Oscillation’-like dynamics at short time scales

The flea gas in a force field: simulating the quantum Newton’s cradle JSC, B. Doyon, J Dubail, R. Konik, and T. Yoshimura, arXiv:1711.00873 ‘Relaxation’-like dynamics at long time scales

Floquet Dynamics

The simple pendulum on its head Pyotr L. Kapitza (8/7/1894-8/4/1984)

Floquet basics Unitary time evolution operator under H ( t ) = ˆ ˆ action of periodic Hamiltonian H ( t + T ) R t P ( t ) e − i ˆ ˆ 0 dt 0 H ( t 0 ) ) = ˆ U ( t ) = T ( e − i H F t “fast motion” operator Floquet P ( t + T ) = ˆ ˆ Hamiltonian P ( t ) ˆ P ( nT ) = 1 ( n ∈ Z ) Stroboscopic Floquet operator U ( T ) = e − i ˆ U F ≡ ˆ ˆ H F T Diag’n: X X ˆ ˆ e − i θ n | φ n ih φ n | ✏ n | � n ih � n | H F = U F = n n t evolution: | φ n ( t ) i = ˆ | ψ n ( t ) i = e − i ✏ n t | φ n ( t ) i , P ( t ) | φ n i

Floquet’ing integrable models P . Claeys and JSC, arXiv:1708.07324 Idea: take a one-parameter family of integrable models do quench/dequench sequences on this manifold T { H 2 (1 − η ) T H 1 η T t Integrable manifold η = 0 Time-avg H 2 η = 1 Hamiltonian H avg = η H 1 + (1 − η ) H 2 also integrable H 1

“Quench-dequench” Floquet protocol Let’s consider the simple case of periodically “switching” between two Hamiltonians: ( ˆ for 0 < t < η T, H 1 ˆ H ( t ) = ˆ for H 2 η T < t < T, so U F ≡ e − i ˆ H F T = e − i (1 − η ) T ˆ H 2 e − i η T ˆ ˆ H 1 One then finds the nice identities ✏ n = ✓ n ∂θ n T = h � n | ˆ ∂ T = h φ n | ˆ H F | � n i H avg | φ n i H Avg = η ˆ ˆ H 1 + (1 − η ) ˆ with t-avg Hamiltonian H 2

Floquet Dynamics: XXZ chain

Floquet’ing integrable models P . Claeys and JSC, arXiv:1708.07324 XXZ Hamiltonian: ˆ X i +1 + S y i S y S x i S x i +1 + ∆ ( t ) S z i S z ⇥ ⇤ H ( t ) = − J i +1 i and Floquet protocol: binary switch between ∆ 1 ∆ 2 U F = e − i (1 − η ) T ˆ H 2 e − i η T ˆ Stroboscopic Floquet operator: ˆ H 1 No analytical solution ☹ , must rely on numerics Focus on sector with k = 0 and m z = 1 / 3

Floquet’ing integrable models P . Claeys and JSC, arXiv:1708.07324 ∆ 1 , 2 = − 2 , − 3 ∆ 1 , 2 = − 2 , 3 π π Floquet phases, θ n 0 0 quasienergies and avg quasienergies − π − π as a function of T 4 2 2 1 θ n /T 0 0 All crossings − 2 − 1 are avoided − 4 − 2 4 2 2 1 Possible to achieve ∂ T θ n 0 0 targeted state − 2 − 1 preparation − 4 − 2 0 1 2 3 0 2 4 6 Period T Period T