Quantum Chaos  

In General [> s.a. 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 → 0!), but there are other qualitative effects.
* 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, cm/94-in]

lima → infty dX a(X + X1) ... a(X+Xn) = pairs J(Xi_1–Xi_2) ... J(Xi_{n–1}–Xi_n) ,

where the correlation function

J(X)  dP exp(i P · X/) (H(P, X)–Ua) ,      J(0) = 1 .

Decoherence, Entanglement, Relation Classical–Quantum > s.a. decoherence; semiclassical quantum mechanics.
* In general: Results from spin chains indicate that chaos reduces entanglement.
@ General references: Friedrich PW(92); Ball et al qp/99, PRE(00)qp/99 [model]; Cucchietti et al PhyA(00)cm [dynamical]; Zurek Nat(01)aug; Jordan & Srednicki qp/01 [sub-Planck physics]; Pattanayak & Sundaram qp/02 [parameter scaling].
@ Classical limit: Primack & Smilansky JPA(98) [semiclassical trace formula]; Greenbaum et al qp/06/PRE [semiclassics]; Kapulkin & Pattanayak qp/07 [non-monotonicity in quantum-classical transition]; Castagnino & Lombardi SHPMP(07) [and self-induced decoherence].
@ And classical chaos: Sengupta & Chattaraj PLA(96); Emerson qp/02-PhD; Huard et al qp/04-in; Lopaev et al PLA(05).
@ Large systems: Sugita & Shimizu JPSJ(05)qp/03 [correlations and entanglement].
@ Related topics: Vitali & Grigolini PLA(98) [celestial mechanics]; Wisniacki et al PRL(05)nl/04 [and homoclinic motion].

Specific Topics > s.a. Loschmidt Echo; lyapunov.
* 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].
@ 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, cm/94-in, cm/94, JPA(99); Dorfman PRP(98), 99.
@ Perturbations: Primack & Smilansky JPA(94); Ballentine & Zibin PRA(96); de Polavieja PRA(98).
@ Numerical aspects: Bäcker n.CD/02 [eigenvalues and eigenfunctions].
@ In Bohmian mechanics, pilot wave: 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].
@ And measurement: Dewdney & Malik 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 qp/00 [entropy production]; Chotorlishvili & Skrinnikov PLA(08) [and irreversibility].
@ Other: Srednicki & Stiernelof JPA(96) [fluctuations in eigenstates]; Zurek & Paz qp/96-in [correspondence principle]; Torres-Vega et al PRA(98) [separatrices]; Gallavotti Chaos(98) [fluctuations and non-equilibrium]; Kronz PhSc(98) [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].

References
@ 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 qp/04-in; Habib et al qp/05-in.
@ 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.

Specific Systems > s.a. Baker Map; bianchi models; foundations of quantum mechanics; quantum systems; spectral geometry; spin models.
@ Anharmonic oscillator: Adamyan et al PRE(01)qp; Caron et al PLA(01)qp [at finite T], PLA(04), JPA(04)qp [and classical].
@ Quantum billiard: Liboff PLA(00); Primack & Smilansky PRP(00) [semiclassical]; Bies et al qp/02 [2-point correlations].
@ In atomic physics: Monteiro CP(94).
@ In discrete systems: Smilansky JPA(07)-a0704 [on graphs]; > s.a. graphs [quantum].
@ And quantum computers: Kim & Mahler PLA(99)qp; Berman et al qp/01, PRE(01)qp.
@ Quantum field theory: Berg et al hl/00-in [gauge theories]; Kuvshinov & Kuzmin PLA(02)ht [criterion]; Beck 02.
@ Yang-Mills-Higgs: Salasnich MPLA(97)qp; Matrasulov et al EPJC(05)hp/03.
@ Other types of systems: Spiller & Ralph PLA(94) [open system]; Jona-Lasinio & Presilla PRL(96)cm [many-body, thermodynamic limit]; Roberts & Muzykantskii JPA(00) [mixing, spectral decompositions]; Dabaghian et al JETPL(01)qp [solvable spectra in 1D]; Dabaghian qp/04 [familiar example]; Dabaghian & jensen EJP(05) [in elementary quantum mechanics]; Ullmo RPP(08) [mesoscopic and nanoscopic systems].

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