Chirikov standard map Dima Shepelyansky

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1 Chirikov standard map Dima Shepelyansky (959) Chirikov criterion (969) Classical map: p = p + K sin x x = x + p (979) Quantum map (kicked rotator): ψ = e iˆp2 /2 e ik/ cos ˆx ψ (959-28) Hamiltonian classical/quantum chaos: H(ˆp, ˆx) = ˆp 2 /2 + K cos ˆx m [ˆp, ˆx] = i (2) Quantum computations δ(t m) (28) Ongoing experiments with cold atoms and Bose-Einstein condensates (Quantware group, CNRS, Toulouse) Chirikov Memorial Seminar BINP (28) / 2

2 Examples of quantum/classical Poincaré sections: initial coherent state at p = x = π/5, K =., t = 2 4, = 2π/N, N = 2 nq, nq = 2 (left), nq = 6 (middle), classical (right) A quantum computer with about n q qubits performs one map iteration in O(n 3 q) quantum gates for a vector of N = 2 nq states while a usual computer needs O(2 nq ) operations (B.Georgeot, DS, PRL 86, 289 (2)) Random matrix theory for quantum errors (K.Frahm, R.R.Fleckinger, DS, EPJD 29, 39 (24)) Quantum map implementation on a 3-qubit NMR-based quantum computer at MIT with cos x x 2 (M.K.Henry, J.Emerson, R.Martinez, D.G.Cory, PRA 74, 6237 (26)) (Quantware group, CNRS, Toulouse) Chirikov Memorial Seminar BINP (28) 2 / 2

3 Various faces of the Chirikov standard map Frenkel-Kontorova model (938) - atomic chain in a periodic potential Veksler (944) - particle dynamics in a microtron Chirikov ( ) - properties of chaos, universality, applications Casati, Chirikov, Ford, Izrailev (979) - quantum map (kicked rotator) Koch et al. (988) - hydrogen atoms in a microwave field Chirikov, Vecheslavov (989) - comet Halley Raizen et al. (995) - dynamical localization with cold atoms Phillips et al. (26) - Bose-Einstein condensates in kicked optical lattices ************************************************************ fractal Weyl law Chirikov typical map Boltzmann - Loschmidt dispute on time reversibility, time reversal of Bose-Einstein condensates (BEC) ************************************************************ Other faces & links: B.Chirikov, DS, Scholarpedia, 3(3):355 (28) (Quantware group, CNRS, Toulouse) Chirikov Memorial Seminar BINP (28) 3 / 2

4 Fractal Weyl law Gamow states in the kicked rotator with absorption: ψ = ˆPe iˆp2 /4 e ik/ cos ˆx e iˆp2 /4 ψ = e iλ γ/2 ψ projection on N/2 < n = p/ < N/2, N/K = a = 2, K = 7 (Fig: Husimi function of quantum fractal eigenstates with minimal escape rate γ; N = 25, 497, 6349, classical; fractal dimension of strange repeller d =.723) N γ N ν (d ) : number of states with < γ < γ b = 8/a 2..8 ν γ c/ Λ d DS, PRE 77, 522(R) (28) (Quantware group, CNRS, Toulouse) Chirikov Memorial Seminar BINP (28) 4 / 2

5 Chirikov typical map (969) Standard map with random, periodically repeated phases φ m: p = p + K sin(x + φ m), x = x + p, φ m+t = φ m chaos border: T 3/2 < K Kolmogorov-Sinai entropy: h K 2/3, diffusion rate per period T : D = K 2 T/2, => continuous time flow (Fig: Husimi function at K =., T =, t = 2 4, l = 2π/N, N = 2 6, initial coherent state at p =, x = π) l 2D/ 2 : dynamical localization (Fig:. K, T, = 2π/7.68) D h 2.. K.Frahm, DS, in preparation (28) (Quantware group, CNRS, Toulouse) Chirikov Memorial Seminar BINP (28) 5 / 2

6 Boltzmann - Loschmidt dispute on time reversibility (876) * irreversible kinetic theory from reversible equations Sitzungsberichte der Akademie der Wissebschaften, Wien, II 73, 28 (876); 75, 67 (877) (Quantware group, CNRS, Toulouse) Chirikov Memorial Seminar BINP (28) 6 / 2

7 Time reversal for the Chirikov standard map BESM-6 computation, rescaled energy or squared momentum vs. time t: K = 5, = (left), = /4 (right) DS, Physica D 8, 28 (983) (Quantware group, CNRS, Toulouse) Chirikov Memorial Seminar BINP (28) 7 / 2

8 * Experimental realization of time reversal: spin echo (E.L.Hahn (95)); acoustic waves (M.Fink (995)); electromagnetic waves (M.Fink (24)) * Loschmidt cooling by time reversal of atomic matter waves Wβ p p β.5..5 proposal of time reversal in kicked optical lattices: k = K/, = 4π + ǫ (forward), = 4π ǫ (back) and k k; Fig: k = 4.5, ǫ = 2, t r =, k B T o/e r = 2 4 (red), k B T o/e r = 2 6 (blue); momentum β and energy E r are give in recoil units J.Martin, B.Georgeot, DS, PRL, 446 (28) (Quantware group, CNRS, Toulouse) Chirikov Memorial Seminar BINP (28) 8 / 2

9 * Time reversal of Bose-Einstein condensates p p 5 3 Tf/T. Wp p k 6 8 The Gross-Pitaevskii equation with kicks: ( ) i t ψ = 2 2 2m g ψ 2 + k cos x δ x 2 T (t) ψ Left: same as in previous Fig. for g =, 5, (insets), 5, 2 (top to bottom), (t = ); Right: cooling ratio T f /T for g = (blue curve), g =.5 (green), g = (red) J.Martin, B.Georgeot, DS, arxiv:84.354[cond-mat] (28) (Quantware group, CNRS, Toulouse) Chirikov Memorial Seminar BINP (28) 9 / 2

10 * Loschmidt paradox for Bose-Einstein condensates 5 6 ln δ 7 5 x 5 5 x t r Soliton initial condition (Zakharov, Shabat (973)): ψ(x, t) = g 2 exp(ip (x x p t/2)+ig 2 t/8) cosh( g 2 (x x p t)) Left: time resersal of soliton at g =, k =, T = = 2, K = kt = 2, t r = 4 inside chaotic (left inset) and regular (right inset) domains; line shows divergence given by the Kolmogorov-Sinai entropy h =.45. Right: Poincaré section at K = 2 But the real BEC is quantum and should return back since the Ehrenfest time t E ln eff /h ln N/2h 3 for BEC with N = 5 atoms J.Martin, B.Georgeot, DS, arxiv:84.354[cond-mat] (28) (Quantware group, CNRS, Toulouse) Chirikov Memorial Seminar BINP (28) / 2

11 959 - Hamiltonian chaos - Chirikov, dissipative chaos - Lorenz,... (Quantware group, CNRS, Toulouse) Chirikov Memorial Seminar BINP (28) / 2

12 Boris Chirikov - Sputnik of Chaos (Quantware group, CNRS, Toulouse) Chirikov Memorial Seminar BINP (28) 2 / 2