Reversing chaos Boris Fine Skolkovo Institute of Science and Technology University of Heidelberg

2 Postdoctoral and Ph.D. positions are available in the group of B.F.

What is quantum chaos? No fully consistent answer. Fundamentally all quantum systems are nonchaotic at the level of the Schroedinger equation for many-particle wave functions. Yet, the averaged behavior of quantum systems is often strongly reminiscent of the behavior of classical chaotic systems. Best empirical indication of quantum chaos is the Wigner-Dyson statistics of the spacings between quantum energy levels. D. Poilblanc et al., EPL 22, 537 (1993) P(Δ)P(Δ) Distribution of energy level spacings Integrable Chaotic Δ Δ Δ Problem: In many-particle systems Δ is exponentially small. The corresponding timescale is unphysically long.

Nuclear Magnetic Resonance (NMR) nucleus Spin 1/2 Larmor precession

NMR free induction decay magnetic dipolar interaction: t = 0t ~ T 2 T = ∞

Generic long-time behavior of nuclear spin decays: [B. F., Int. J. Mod. Phys. B 18, 1119 (2004)] where Markovian behavior on non-Markovian time scale is a manifestation of chaos. Chaotic eigenmodes of the time evolution operator: Pollicott-Ruelle resonances [D. Ruelle, PRL 56, 405 (1986)] Spins 1/2, experiment B. Meier et al., PRL 108, (2012).

Investigations of Lyapunov instabilities in classical spin systems A. de Wijn, B. Hess, and B. F., Phys. Rev. Lett. 109, (2012) …. J. Phys. A 46, (2013)

Definition of Lyapunov exponents Equations of motion: Local fields: Phase space vectors: Small deviations: Largest Lyapunov exponent: Lesson learned: is an intensive quantity for spin lattices with short-range interactions

How to extract Lyapunov exponent from the observable behavior of a many-particle system? Difficulties:  We cannot experimentally observe phase space coordinates of every particle  We cannot controllably prepare two close initial conditions in a many-particle phase space  Statistical behavior of many-particle systems masks Lyapunov instabilities Relaxation path in many-body phase space MxMx Solution: time-reversal of relaxation! (Loschmidt echo) MxMx

Sensitivity of Loschmidt echoes to small perturbations Physica A 283, 166 (2000)

Time reversal of magnetization noise B.F., T.A. Elsayed, C. M. Kropf, and A.S. de Wijn, Phys. Rev. E 89, (2014)

Observable exponential sensitivity of classical spin systems to small perturbations B.F., T.A. Elsayed, C. M. Kropf, and A.S. de Wijn, Phys. Rev. E 89, (2014)

Each Ψ ν is obtained from Ψ - by complete flipping of a small fraction of spins 5x5 lattice of spins 1/2 No exponential growth in the regime 1-F( τ ) << 1 Absence of exponential sensitivity to small perturbations in nonintegrable systems of spins 1/2 B.F., T.A. Elsayed, C. M. Kropf, and A.S. de Wijn, Phys. Rev. E 89, (2014)

Larger quantum spins T.A. Elsayed, B. F., Phys. Scripta (2015) Solid black: 6 spins 15/2 Blue: 6 classical spins Dashed: 26 spins 1/2

Summary: 1. NMR experiments indicate that relaxation in quantum spin systems exhibits generic similarities with the relaxation in chaotic classical spin systems. 2. Loschmidt echoes in chaotic systems of classical spins exhibit exponential sensitivity to small perturbations controlled by twice the value of the largest Lyapunov exponent. 3. Loschmidt echoes in non-integrable systems of spins 1/2 exhibit power- law sensitivity to small perturbations. - Implication for generic many-particle quantum systems: ergodicity ≠ exponential sensitivity to small perturbations - Good news for the efforts to create quantum simulators!

Investigations of Lyapunov spectra in classical spin systems [A. de Wijn, B. Hess, and B. F., J. Phys. A 46, (2013)] Lyapunov spectra for four different lattices Size dependence of the entire spectra (Heisenberg chains) Size dependence of (Heisenberg Hamiltonian) Lattices considered