Quantum Chaos  

In General > s.a. quantum statistical mechanics.
* Idea: The study of quantum systems whose classical counterparts are chaotic.
* History: 2001, The field is now coming of age; Key people have been M Gutzwiller (trace formula), M Berry.
* Issues: Some classically chaotic systems are not chaotic when quantized; How general is this?
* Picture: There seems to be no sensitive dependence on initial conditions, although some authors disagree; Motion becomes stable after some time (that goes to ∞ as \(\hbar\) → 0!), but there are other qualitative effects, such as repulsion of the energy levels for systems whose classical counterparts are chaotic.
* Gutzwiller's trace formula: Expresses the energy level density in terms of a sum over classical unstable orbits; > s.a. particle creation.
* Berry's conjecture: The energy eigenfunctions of a bounded, isolated, quantum chaotic system appear to be Gaussian random variables, in the sense that [@ Srednicki PRE(94)cm, ANYAS(95)cm/94]

lima → ∞ dX ψa(X + X1) ... ψa(X + Xn) = pairs J(Xi1Xi2) ... J(Xin−1Xin) ,

where the correlation function

J(X) ~  dP exp(i P · X/\(\hbar\)) δ(H(P, X) − Ua) ,      J(0) = 1 .

Decoherence, Entanglement, Relation Classical–Quantum > s.a. decoherence; semiclassical quantum mechanics.
* Issue: There is an apparent paradox in the fact that classically chaotic systems are regular in the quantum regime, with no signature of classical chaos whatsoever apparent in the corresponding quantum dynamics; In optomechanics at least, transient chaos provides a resolution.
* In general: Results from spin chains indicate that chaos reduces entanglement.
@ General references: Friedrich PW(92)apr; Ball et al JOB(99)qp, PRE(00)qp/99 [model]; Cucchietti et al PhyA(00)cm [dynamical]; Zurek Nat(01)aug; Jordan & Srednicki qp/01 [sub-Planck physics]; Pattanayak et al PRL(03)qp/02 [parameter scaling].
@ Classical limit: Primack & Smilansky JPA(98) [semiclassical trace formula]; Greenbaum et al PRE(07)qp/06 [semiclassics]; Kapulkin & Pattanayak PRL(08)qp/07 [non-monotonicity in quantum-classical transition]; Castagnino & Lombardi SHPMP(07) [and self-induced decoherence].
@ And classical chaos: Sengupta & Chattaraj PLA(96); Emerson PhD(01)qp/02; Huard et al qp/04-conf; Lopaev et al PLA(05); Wang et al SRep(16)-a1701 [optomechanics and transient chaos].
@ Large systems: Sugita & Shimizu JPSJ(05)qp/03 [correlations and entanglement]; Maldacena et al JHEP(16)-a1503 [bound on the rate of growth of chaos].
@ Related topics: Vitali & Grigolini PLA(98) [celestial mechanics]; Wisniacki et al PRL(05)nl/04 [and homoclinic motion].

Specific Topics > s.a. chaotic systems; complexity; ergodic theory [ergodic hierarchy]; Loschmidt Echo; lyapunov exponents.
* Scarring: An anomalous localization of quantum probability density along unstable periodic orbits.
@ Criteria: Bunakov et al PLA(98); Benatti & Fannes JPA(98) [variables]; Jirari et al PLA(01) [and quantum instantons]; Lahiri qp/03 [entropy in subsystem]; Inoue et al qp/04 [entropic chaos degree]; Madhok a1212 [and dynamical generation of entanglement]; Ali et al PRD-a1905 [time evolution of complexity].
@ Chaotic observables: van Winter JMP(99) [free particle].
@ Gutzwiller trace formula: Gutzwiller JMP(70), JMP(71); Muratore-Ginanneschi PRP(03).
@ Approach to equilibrium: Srednicki PRE(94)cm, ANYAS(95)cm/94, cm/94, JPA(99); Dorfman PRP(98), 99; Altland & Haake PRL(12)-a1110.
@ Perturbations: Primack & Smilansky JPA(94); Ballentine & Zibin PRA(96); de Polavieja PRA(98).
@ Eigenfunctions and eigenvalues: Bäcker n.CD/02 [numerical]; Nonnenmacher a1005-in [analytical approaches]; Revuelta et al PRE(13)-a1303 [basis sets of scar functions].
@ In Bohmian mechanics, pilot-wave theory: Dürr et al JSP(92); Parmenter & Valentine PLA(95); Konkel & Makowski PLA(98); Iacomelli & Pettini PLA(96); Wu & Sprung PLA(99); Makowski & Frackowiak APPB(01)qp [model]; Wisniacki & Pujals EPL(05)qp; > s.a. foundations.
@ Edge of chaos: Weinstein et al PRL(02)cm, qp/03-in [border regular/chaotic dynamics]; Bogomolny et al JPA(09) [near-integrable system].
@ And measurement: Dewdney & Malik PLA(96)qp/95 [quantum pinball]; Habib et al PRL(06)qp/04 [continuous observation].
@ And entropy: Benatti et al CMP(98) [dynamical entropy], JPA(04) [and discretization]; Monteoliva & Paz PRL(00)qp [entropy production]; Chotorlishvili & Skrinnikov PLA(08) [and irreversibility].
@ Quantum butterfly effect: Campisi & Goold PRE(17)-a1609; Cotler et al AP(18)-a1704 [and out-of-time-order operators].
@ Other: Srednicki & Stiernelof JPA(96) [fluctuations in eigenstates]; Zurek & Paz qp/96-proc [correspondence principle]; Torres-Vega et al PRA(98) [separatrices]; Gallavotti Chaos(98) [fluctuations and non-equilibrium]; Kronz PhSc(98)mar [and non-separability]; Shigehara et al IEICE(98)qp [chaos upon quantization]; Caron et al PLA(01)qp [T > 0]; Demikhovskii et al PRL(02)qp/01 [Arnold diffusion analog]; Romanelli et al PLA(03)qp/02 [master equation]; Gruebele & Wolynes PRL(07) [control]; Rivas JPA(07) [scar function, semiclassical]; Kowalewska-Kudlaszyk et al PRE(08)-a0905 [and Wigner-function non-classicality]; Flom et al a1507 [and tunneling]; Zisling et al a2012 [number of particles in many-body system for strong chaos]; Blake & Liu a2102 [maximally chaotic systems].

References > s.a. non-equilibrium statistical mechanics.
@ I, II: NS(87)nov19; Gutzwiller SA(92)jan; Stehle 94; Stone PT(05)aug.
@ General: Berry JPA(77), in(83), in(91), & Keating JPA(90); Bohigas et al PRL(84) [random matrix theory and spectra]; Sieber & Steiner PRL(91), Aurich et al PRL(92); Casati Chaos(96) [rev]; Albrecht Nat(01)aug; Huard et al in(03)qp/04; Habib et al ANYAS(05)qp; Yan & Chemissany a2004 [sensitivity to initial conditions]; Leone et al a2102 [quantum chaos cannot be simulated on a classical computer].
@ Books: Ozorio de Almeida 88; Gutzwiller 90; Lichtenberg & Lieberman 92; Reichl 92; Haake 06.
@ Books, III: Nakamura 93; Hurt 97 [mathematical]; Stöckmann 99; Kamenev & Berman 00 [perturbed harmonic oscillator].
@ Existence?: Majewski qp/98.

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