Quantum State Evolution  

In General > s.a. Physical Process; quantum states; quantum-state collapse; quantum systems; time in quantum mechanics.
* Idea: For a pure state, time evolution is usually taken to be given by the Schrödinger equation, in which the time derivative of the state vector corresponds to the action of the Hamiltonian operator; For a mixed state, time evolution is usually taken to be given by the Liouville-von Neumann equation, in which the time derivative of the density matrix corresponds to the action of the Liouvillian operator.
@ General references: Aharonov & Albert PRD(84) [relativistic]; Styer AJP(90)aug, Weigert PRL(00)qp/99 [in terms of expectation values and uncertainties]; Mohrhoff FP(04)qp/03 [and Pondicherry interpretation]; Oppenheim & Reznik PRA(04)qp/03 [and probability/info]; Mizel PRA(04) [ground state]; D'Alessandro & Romano JMP(06)qp [and entanglement]; García Quijas & Arévalo Aguilar PS(07)qp/06 [factorization]; Vaidman qp/06/JPA [backward]; Schuch & Moshinsky PRA(06) [Ermakov invariant]; Mohseni & Lidar PRL(06)qp [direct characterization of quantum dynamics]; Bernatska & Messina PS(12)-a1006 [reconstruction of Hamiltonian]; McClean et al PNAS(13)-a1301 [discrete, and Feynman's clock]; > s.a. quantum-classical relationship [evolution to classicality].
@ Geometric: Kryukov FP(07)-a0704 [as geodesic motion on space of states]; Rezakhani et al PRA(10)-a1004 [geometry of adiabatic evolution]; Laba & Tkachuk CondMP(17)-a1006 [curvature and torsion of evolution].
@ Specific types of systems: Andrews AJP(08)dec-a0801 [free wave packets]; Evnin FdP(08)-a0805-proc [with singularities]; Fröhlich & Pizzo a2101 [isolated, open systems, replacement for the Schrödinger equation]; > s.a. Friedrichs Model.
@ Quantum jumps, transitions: Macomber AJP(77)jun; Greenstein & Zajonc AJP(95)aug [time]; Belavkin & Melsheimer QSO(96)qp/05 [stochastic Hamiltonian model]; Brun PRL(97) [decoherence]; Stenholm & Wilkens CP(97); Wiseman & Gambetta PRL(12)-a1110 + news physorg(12)jun [objective or observer-dependent?]; Dick SHPMP-a1703 [superpositions and the need for quantum field theory]; Minev a1902-PhD [catching and reversing jumps].
@ From pure to mixed states: Horwitz et al qp/96-proc [and Lax-Phillips theory]; Svec a0708 [in pion creation]; Fortin & Lombardi FP(14) [interpretation of partial traces and reduced states for subsystems]; > s.a. arrow of time; decoherence; mixed states [including thermalization]; Semigroup.
@ Related topics: Hiller et al PRL(04) [reversal of evolution, and echo]; Gammelmark et al PRL(13)-a1305 [past quantum states]; Aharonov et al ch(14)-a1305 [time evolution as correlations between universes]; Lorek a1908-PhD [influence of acceleration]; > s.a. Ehrenfest Theorem; Recurrence; Reversibility; Steering.

Evolution of Semiclassical States > s.a. hamilton-jacobi theory; semiclassical states; zeno effect.
@ Quantum vs classical evolution: Benet et al JPA(03) [fluctuations around classical average]; Habib qp/04 [Gaussian approximation]; Bojowald & Skirzewski RVMP(06)mp/05 [effective equations of motion and corrections to symplectic structure]; Gosselin et al EPJB(07)ht/06 [Berry phase corrections]; Schubert et al JPA(12)-a1112 [evolution of localized states]; Tsang & Caves PRX(12) [quantum-uncertainty-free subsystems]; Bojowald et al PRD(12)-a1208 [effective dynamics]; González-Arroyo & Nuevo PRD(12) [Wigner-function approach]; Lochan et al GRG(15)-a1404 [spontaneous evolution to classicality]; Brizuela PRD(14)-a1410 [for a generic probability distribution]; Pal et al PRE(16)-a1510 [generalized Gaussian wave packet dynamics].
@ Specific types of systems: Emerson & Ballentine PRA(01) [interacting spins]; Katz et al PRL(07) [driven non-linear Duffing resonator]; Gat JPA(07) [non-linear oscillator];
@ Related topics: Jacquod & Amiet JPA(97) [ergodic behavior]; Roncadelli & Schulman PRL(07) [density of paths around a semiclassical trajectory]; Mittal et al PRE(20)-a1911 [evolution of expectation values]; Lasser & Lubich ANum(20)-a2002 [numerical approaches].
@ Spreading: Ketzmerick et al PRL(97); Killip et al m.SP/01 [bounds on rate]; Damanik & Tcheremchantsev JAMS-mp/05 [bounds, application to Fibonacci operator]; Schubert et al JPA(12) [from localized coherent states to delocalized Lagrangian states, in Ehrenfest time]; > s.a. Ehrenfest Time.
@ State diffusion: Gisin & Percival JPA(92), JPA(93), JPA(93); Percival JPA(94); Breuer & Petruccione JPA(98) [relativistic], comment Diósi JPA(98), reply JPA(98); Percival & Strunz JPA(98); Burić JPA(07) [geometry and dynamics]; > s.a. histories formulations.
@ Wave-packet revival: Nauenberg et al SA(94)jun; Robinett PRP(04); Wang & Heller JPA(09) [1D]; Strange PRL(10) [relativistic]; Romera & de los Santos PRA(08)-a1409, de los Santos et al PhyE(10)-a1409 [and Rényi uncertainty relations]; Romera & de los Santos PLA(13)-a1409 [and Fisher information]; Dowker JPA(17)-a1605 [in free field CFT]; > s.a. classical-quantum relationship; quantum systems; random walk; types of coherent states.
> State revivals, specific types of systems: see Caldeira-Leggett Model; Infinite Well; Quantum Carpet.

Decay of Unstable States > s.a. quantum systems; types of quantum states / Leggett-Garg Inequality; spin-statistics [effect of exclusion principle].
* In general: The time evolution of quantum states for unstable particles can be divided into three domains, the very short time where Zeno's paradox is relevant, the intermediate interval where the exponential decay holds more or less, and the very long time where the decay is governed by a power law.
* Exponential decay: An explanation was given by Gamov in terms of complex "eigenvalues" and exponentially growing "eigenfunctions"; > s.a. resonances.
* Non-exponential decay: 1957, Khalfin proved that for large values of t the amplitude tends to zero more slowly than any exponential function of t; 1997, observed in tunneling of trapped Na ions.
@ General references: Fonda et al RPP(78); Jakobovits et al AJP(95)may [exponential]; Benatti & Floreanini PLB(98)ht [of open systems]; Rotter a0710 [and resonance, Feshbach projector description]; Courbage & Fathi PhyA(08) [decay probability distribution]; Dürr et al EJP(11)-a1011 [Gamow's exponential decay]; Bohm RPMP(11) [formalism]; Marchetti & Wreszinski 13, RVMP(13) [asymptotic, rev]; Giacosa & Pagliara PRA(14)-a1405 [effect of a detector]; Zloshchastiev EPJD(15)-a1505 [quantum fluctuations and instability of pure states]; Burgarth et al OSID(17)-a1609 [not due to classical noise]; Anastopoulos IJTP-a1808 [pedagogical]; Ramírez & Kelkar JPA(19)-a1809 [approaches].
@ In quantum field theory: Giacosa & Pagliara MPLA(11)-a1005 [non-exponential]; Boyanovsky AP(19)-a1810 [quasiparticle formation, Zeno and anti-Zeno effects].
@ (Non)-Exponential time dependence: Chiu et al PRD(77) [and Zeno's paradox]; Peres AP(80); Nakazato & Pascazio MPLA(95); Wilkinson et al Nat(97)jun; Jittoh et al PRA(05)qp/04; Martorell et al PRA(08)-a0709; Exner RPMP(07); Urbanowski a0806-conf [and cosmology]; Matinian & Perel'man PLA(08)-a0809 [exponential]; Semkow et al PLB(09) [explaining away oscillations in radioactive decay]; Urbanowski a1007 [astrophysical and cosmological consequences]; Aglietti & Santini a1010 [analysis of model]; Giacosa FP(12)-a1110 [more than one decay channels]; Aston EPL(12)-a1204 [and radioactivity]; del Campo NJP(16)-a1504 [many-body system]; Urbanowski EPJD(17)-a1606; Burgarth & Facchi PRA(17)-a1705 [exponential decay from Hamiltonians bounded below]; Beau PRL(17) + news PhysOrg(17)oct [non-exponential decay in open quantum systems].
@ Bohm-Gadella theory controversy: de la Madrid JPA(06)qp; Gadella & Wickramasekara JPA(07), a0707; de la Madrid JPA(07)-a0704; Baumgärtel a0704; de la Madrid a0705.
@ Related topics: Gurvitz & Marinov PLA(90) [interference effects]; Belavkin & Staszewski JMP(00)mp/05 [stochastic semiclassical model]; Kofman & Kurizki PRL(01)qp [dynamical control]; Castagnino et al JPA(02) [Gamov vectors]; Klein & Nystrand PLA(03) [wave-function-collapse test]; Agudov et al JPA(04) [noise and delay]; Luo & Zhang LMP(05) [survival amplitudes]; García-Calderón & Mendoza-Luna PRA(11)-a1106 [decay of two identical particles]; Civitarese & Gadella PhyA(14) [complex-energy states, entropy]; Sonnleiter et al PRL(17) + news pw(17)feb [friction force on decaying atom].
> Related topics: see arrow of time; atomic physics [effect of quantum coherence]; particle effects; relativistic quantum mechanics; zeno effect.

Speed of Evolution > s.a. Margolus-Levitin Theorem.
* Motivation: The speed at which a quantum system can change state is important for quantum computation.
* Result: One version of the energy-time uncertainty principle states that the minimum time for a quantum system to evolve from a given state with energy uncertainty ΔE to any orthogonal state is h/4ΔE.
@ General references: Bender Sigma(07)-a0712-proc, Bender & Brody a0808-in [faster in non-hermitian quantum mechanics]; Kupferman & Reznik PRA(08)-a0806 [for mixed states]; Masuda & Nakamura PRA(08) [speedup]; Mostafazadeh PRA(09) [optimal]; Castagnoli a0912/PRA [explanation], PRA(10)-a1005 [speedup]; Caneva et al PRL(09) [optimal control]; Margolus a1109; Taddei et al PRL(13)-a1209; Zander et al JPA(13) [and entanglement, for two interacting qubits]; Cianciaruso et al a1704 [role of non-Markovianity and backflow of information]; Gessner & Smerzi a1712 [statistical speed].
@ Speed limit: Margolus & Levitin PhyD(98)qp/97; Giovannetti et al PRA(03)qp/02 [and subsystem entanglement]; Andrecut & Ali IJTP(04); Zieliński & Zych PRA(06)qp [generalized bound]; Borrás et al PRA(06)qp [and entanglement]; Bender et al PRL(07)qp/06 [no limit in PT-symmetric quantum mechanics]; Giri IJTP(08)-a0706; Levitin & Toffoli PRL(09)-a0905; Yurtsever PS(10)-a0912; Jones & Kok PRA(10)-a1004, comment Zwierz PRA(12)-a1207 [derivation]; Poggi et al EPL(13)-a1308 [for 2-level systems]; Zhang et al sRep(14)-a1312 [arbitrary initial states]; Hegerfeldt PRL(13)-a1305; Brouzos et al PRA(15)-a1412 [complex interacting many-body systems]; Paiva Pires et al PRX(16); Epstein & Whaley PRA(17)-a1612; Jordan PRA(17)-a1701 [and computational speed]; Deffner NJP(17)-a1704 [in Wigner space]; Okuyama & Ohzeki PRL(18)-a1710 [in classical mechanics]; > s.a. open systems; uncertainty relations [time-energy].
@ Speed limit, for mixed states: Mondal et al PLA(16)-a1506 [set by quantum coherence]; Campaioli et al PRL(18)-a1709 [better bound].
@ Speed limit, tests: del Campo a2007 [proposal, with ultracold gases].

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