Quantum chaos and complexity in decay, emission, and relaxation processes. Doron Cohen. Ben-Gurion University

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1 Quantum chaos and complexity in decay, emission, and relaxation processes Doron Cohen Credits: Ben-Gurion University In addition to the review of the literature, I will use material that is taken from works that have been done in collaborations with students and colleagues. In particular I would like to acknowledge: Itamar Sela (PhD, BGU) Alex Stotland (PhD, BGU) Yaron de Leeuw (PhD, BGU) Maya Chuchem (PhD, BGU) Amichay Vardi (Chemistry, BGU) Christine Khripkov (PhD, BGU) Igor Tikhonenkov (potdoc, BGU) James Anglin (Kaiserslautern) Tsampikos Kottos (Wesleyan) James Aisenberg (student, Wesleyan) Alex Elgart (Math, VirginiaTech) S time [Josephson periods] dcohen $DIP, $BSF, $ISF φ=π φ=4π/5 φ=3π/5 φ=2π/5 φ=π/5 φ=0

2 Simple decay process - The Friedriech model H = diag{e n } + matrix{v nm } V n,0 2 E n E 0 α 1 ω α 1 ρ LDOS (ε) ε 0 2 P (t) = FT [ρ LDOS(ε)] 2 For Ohmic (α = 1) continuum: P (t) = exp( Γt) Γ = 2π ϱ DOS V 2

3 Digression - Bandprofile, Sparsity and Texture [Feingold, Peres] V nm 2 (2πϱ) 1 Ccl (E n E m ) Hard Qchaos Taken from: C(ω) classical h=0.030 h=0.015 Cohen, Tsampikos 0.2 (PRE 2000) Weak Qchaos [median mean] Taken from: Stotland, Pecora, Cohen (EPL 2010, PRE 2011) X [scaled] classical QM - EW1 - mean QM - EW2 - mean QM - EW1 - median QM - EW2 - median ω [scaled]

4 Non-exponential decay due to non-ohmic bandprofile [Sela, Aisenberg, Kottos, Elgart, Cohen (JPA 2010, PRE 2010)] P (t) = FT [ρ LDOS(ε)] 2 H = diag{e n } + matrix{v nm } V n,m 2 E n E m α 1 Friedriech vs Wigner model Non-Ohmic continuum (α 1) P (t) exp [ (t/t 0 ) 2 α] P (t) 1 (t/t 0 ) 2 α [FM] The latter show up in the FM case, reflecting the singular mixing of nearby levels. α = 1.5 rho(w) rho(w) 10-2 WM b = 40 WM b = 80 WM b = 160 WM b = 320 FM N = 1600 FM theoretical Semicircle (fit) FM WM w WM b = 40 WM b = 80 WM b = 160 WM b = 320 FM b = 800 FM theoretical w

5 Non-exponential decay due to finite bandwidth One observes exponential decay if the dynamics is dominated by an isolated pole of the rezolvent. More generally we have the relation P (t) = FT [ρ LDOS(ω)] 2 0 d = mean level spacing cl /τ cl = semiclassical bandwidth Based on LDOS analysis, depending on the coupling (ν): First order regime (Γ(ν) < 0 ) - non decay FGR regime ( 0 < Γ(ν) < cl ) - Wigner type decay Non perturbative regime (Γ(ν) > cl ) - semiclassical decay Cohen (PRL 1999), Cohen, Heller (PRL 2000) Cohen, Izrailev, Kottos (PRL 2000), Cohen, Kottos ( ) Later the same idea of regime appeared in the context of quantum reversibility and fidelity-decay studies.

6 Semiclassical perspective - The Gamow formula H = p2 2m + V (q) P (t) exp( Γt) Γ = ν attempt g analogy with Landauer: G = e2 2π g Derivations: Approximating by the Friedriech model Hamiltonian Looking for outgoing stationary states of H(p, q) Looking for eigen-modes of the nonhermitian H reduced Stationary states of H with outgoing boundary conditions are associated with the complex poles of the resolvent 1 G(z) = (analytically continued from above) z H

7 Tunnel splitting Two level approximation: P (t) 1 2 [1 + cos(ωt)] Ω = 2ν attempt g For derivation of this formula as written above see [Sela and Cohen (PRB 78, 2008)] Note that Γ Ω 2 as implied by FGR.

8 Dynamical tunneling Abstract view of barrier in phase space [Davis and Heller (JCP 1981)] Pendulum example < n < Bosonic Josephson junction N/2 < n < N/2 H(n, ϕ) = Un 2 NK 2 cos(ϕ) n (population imbalance) is like angular momentum ϕ (conjugate phase) is like angular position U (on site interaction) is like inverse mass K (hopping coefficient) is like gravitation field U K [talk by Ofir Alon]

9 Island-Island coupling mediated by chaotic Sea [Lin and Ballentine (PRL 1990)] [Bohigas, Tomsovic, Ullmo (1993)] [..., Moiseyev,..., Raizen,...] Tunnel splitting: Chaos assisted tunneling Enhanced coupling due to multiple path options ( Interference ) The coupling fluctuates as a parameter is varied ( UCF [Lee and Stone]) Fingerprints of nearby degeneracies ( avoided crossings ) [*] [*] Mouchet, Miniatura, Kaiser, Grmaud, Delande (PRE 2001)

10 Decay processes: Resonance assisted tunneling Island decays to the sea, mediated by resonances [Brodier, Schlagheck, Ullmo, Lock, Backer, Ketzmerick,...] FGR based formula adds up multiple path options (stochastic addition) The coupling fluctuates as a parameter is varied Fingerprints of nearby exceptional points Relevant in the semiclassical regime, i.e. if A Γ = Γ [0 chaos] + Γ [0 resonances chaos] [talk by Ulrich Kuhl]

11 Not just decay processes... Consider active ( pumped ) medium. Within the framework of a semi-classical treatment the electromagnetic field is described by a non-hermitian Hamiltonian H The modes are stationary states of H They correspond to eigenvalues that fluctuate as a parameter is varied Fingerprints of nearby exceptional points [1] [2] [1] Pump-Induced Exceptional Points in Lasers [Liertzer, Ge, Cerjan, Stone, Tureci, Rotter (PRL 2012)] [2] Taming random lasers through active spatial control of the pump Bachelard, Andreasen, Gigan, Sebbah (PRL 2012)] [talk by Patrick Sebbah]

12 Under what conditions do we have a decay? The initial wave-packet should overlap a continuum of eigen-states, else quasi-periodic oscillation instead of decay. Some cases where there is no continuum contrary to semi-classical intuition: Anderson localization Small PN at the top of a barrier [with Chuchem, Khripkov, Vardi] u elliptic fixed point PN u log(n/u) saddle point N log(n/u) separatrix edge S φ=π φ=4π/5 φ=3π/5 φ=2π/5 φ=π/5 φ= time [Josephson periods] Scars at hyperbolic points [Kaplan, Heller] 1 PN 1 N hyperbolic fixed point cosh(λs) 2 s=

13 How to slow down the decay? H(n, ϕ) = Un 2 N 2 (K + f(t)) cos(ϕ) Kapitza effect: Periodic driving f(t) sin(ωt) Zeno effect: Noisy driving f(t)f(t ) = 2Dδ(t t ) The quantum Zeno effect is in fact classical... dρ dt = i[h 0 V eff, ρ] D[W, [W, ρ]] Kapitza effective potential term V eff = 1 4Ω 2 Noise induced radial diffusion coefficient S = exp { 1 } N [exp (8D wt) 1] Should be contrasted with S = exp D w = w2 J 8D { 1 } N 8D wt [W, [W, H]] S [Boukobza, Moore, Cohen, Vardi, Kapitza (PRL 2010)] [Khripkov, Vardi, Cohen, Zeno (PRA 2012)] time [Josephson periods]

14 Non-exponential decay due to weak quantum chaos Stotland, Kottos, Cohen (PRB 2010) de Leeuw, Cohen (PRE 2012) Weak quantum chaos implies sparsity. Resistor network modeling percolation VRH-type dynamics H total = diag{e n } f(t)v nm w n,m = ν V nm ResNet Spectral linear n c = P (t) 1 t d/2 (diffusion) D 10 5 D EXP d (1/s) e 1/s D X = 1 s

15 [Chirikov, Shepelyansky,..., Geisel,...] Non-exponential decay due to mixed phase space [Weiss, Hufnagel, Cristadoro, Ketzmerick] Mixed phase space with hierarchical structure Stickiness of trajectories close to the boundary P (t) 1 t γ 1 < γ < 3 This classical behavior possibly might be observed in the semi-classical regime.

16 Relaxation of mesoscopic subsystems [Tikhonenkov, Vardi, Anglin, Cohen (PRL 2012)] Minimal Fokker-Planck theory for the thermalization of mesoscopic subsystems. ρ = ( g(ε)d(ε) ( )) 1 t ε ε g(ε) ρ g(ε) = g 1 (ε) g 2 (E ε) Complexity of phase space might affect the thermalization. BEC trimer: long dwell times in sticky regions are reflected in ε(t)