A semi-classical treatment of dissipative processes based on Feynman's influence functional method

Möhring, K.; Smilansky, Uzy

doi:10.1016/0375-9474(80)90131-1

Cited by 59 publications

(34 citation statements)

References 11 publications

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“…In the quasi-Ohmic regime [174,175,176,197,121,122], the dissipative response at low frequencies is proportional to ω, as obvious for a friction-dominated harmonic oscillator. We thus take (for ω > 0)…”

Section: Equilibrium State Of the Bathmentioning

confidence: 94%

“…Depending on the model, their large number reflects the large size of the pointer or that of some environment. The situation is the same as for a large set of coupled harmonic oscillators [174,175,176,197,121,122], which in practice present damping although some exceptional quantities involving a single mode or a few modes oscillate. In § 6.1.2 we have studied a generic situation where the frequencies of the modes are random.…”

Section: Dutch Expressionmentioning

confidence: 97%

“…They explicitly adopt the statistical interpretation of quantum mechanics. General statements are illustrated via the Caldeira-Leggett model [173,174,175,176]: a two-level system coupled to a bath of harmonic oscillators. This model undergoes a second-order phase transition with relatively weak decay of off-diagonal terms in the thermodynamic limit, provided that the coupling of the two-level system to the bath is sufficiently strong.…”

Section: Spontaneous Symmetry Breakingmentioning

confidence: 99%

“…and choose forK(ω) the simplest expression having the required properties, namely the quasi-Ohmic form [174,175,197,121,122]…”

Section: Equilibrium State Of the Bathmentioning

confidence: 99%

“…The registration device is schematized as a set M of N Ising spins (the magnet). The size of the dot is supposed to be much smaller than the range of the interactions, both among the N spins and between them and the tested spin S. We further simplify by taking into account only interactions between the z-components of the spins of M and S. Finally, as in a real magnetic dot, phonons (with a quasi-ohmic behavior [121,122,174,175,197]) behave as a thermal bath B which ensures equilibrium in the final state (Fig. 3.1).…”

Section: Features Of the Curie-weiss Modelmentioning

confidence: 99%

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Understanding quantum measurement from the solution of dynamical models

Allahverdyan¹,

Balian²,

2013

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The quantum measurement problem, to wit, understanding why a unique outcome is obtained in each individual experiment, is currently tackled by solving models. After an introduction we review the many dynamical models proposed over the years for elucidating quantum measurements. The approaches range from standard quantum theory, relying for instance on quantum statistical mechanics or on decoherence, to quantum-classical methods, to consistent histories and to modifications of the theory. Next, a flexible and rather realistic quantum model is introduced, describing the measurement of the z-component of a spin through interaction with a magnetic memory simulated by a Curie-Weiss magnet, including N 1 spins weakly coupled to a phonon bath. Initially prepared in a metastable paramagnetic state, it may transit to its up or down ferromagnetic state, triggered by its coupling with the tested spin, so that its magnetization acts as a pointer. A detailed solution of the dynamical equations is worked out, exhibiting several time scales. Conditions on the parameters of the model are found, which ensure that the process satisfies all the features of ideal measurements. Various imperfections of the measurement are discussed, as well as attempts of incompatible measurements. The first steps consist in the solution of the Hamiltonian dynamics for the spin-apparatus density matrixD(t). Its off-diagonal blocks in a basis selected by the spin-pointer coupling, rapidly decay owing to the many degrees of freedom of the pointer. Recurrences are ruled out either by some randomness of that coupling, or by the interaction with the bath. On a longer time scale, the trend towards equilibrium of the magnet produces a final stateD(t f ) that involves correlations between the system and the indications of the pointer, thus ensuring registration. AlthoughD(t f ) has the form expected for ideal measurements, it only describes a large set of runs. Individual runs are approached by analyzing the final states associated with all possible subensembles of runs, within a specified version of the statistical interpretation. There the difficulty lies in a quantum ambiguity: There exist many incompatible decompositions of the density matrixD(t f ) into a sum of sub-matrices, so that one cannot infer from its sole determination the states that would describe small subsets of runs. This difficulty is overcome by dynamics due to suitable interactions within the apparatus, which produce a special combination of relaxation and decoherence associated with the broken invariance of the pointer. Any subset of runs thus reaches over a brief delay a stable state which satisfies the same hierarchic property as in classical probability theory; the reduction of the state for each individual run follows. Standard quantum statistical mechanics alone appears sufficient to explain the occurrence of a unique answer in each run and the emergence of classicality in a measurement process. Finally, pedagogical exercises are proposed and lessons for future works on m...

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Section: Equilibrium State Of the Bathmentioning

confidence: 94%

Section: Dutch Expressionmentioning

confidence: 97%

Section: Spontaneous Symmetry Breakingmentioning

confidence: 99%

“…and choose forK(ω) the simplest expression having the required properties, namely the quasi-Ohmic form [174,175,197,121,122]…”

Section: Equilibrium State Of the Bathmentioning

confidence: 99%

Section: Features Of the Curie-weiss Modelmentioning

confidence: 99%

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Understanding quantum measurement from the solution of dynamical models

Allahverdyan¹,

Balian²,

2013

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Periodic Orbit Approach to the Quantum‐Kramers‐Rate

Hänggi

Hontscha

1991

Ber Bunsenges Phys Chem

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Chaos and Energy Spreading for Time-Dependent Hamiltonians, and the Various Regimes in the Theory of Quantum Dissipation

Cohen

2000

Annals of Physics

134

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We make the first steps toward a generic theory for energy spreading and quantum dissipation. The Wall formula for the calculation of friction in nuclear physics and the Drude formula for the calculation of conductivity in mesoscopic physics can be regarded as two special results of the general formulation. We assume a time-dependent Hamiltonian H(Q, P; x(t)) with x(t)=Vt, where V is slow in a classical sense. The rate-of-change V is not necessarily slow in the quantum-mechanical sense. The dynamical variables (Q, P) may represent some``bath'' which is being parametrically driven by x. This bath may consist of just a few degrees of freedom, but it is assumed to be classically chaotic. In the case of either the Wall or Drude formula, the dynamical variables (Q, P) may represent a single particle. In any case, dissipation means an irreversible systematic growth of the (average) energy. It is associated with the stochastic spreading of energy across levels. The latter can be characterized by a transition probability kernel P t (n | m), where n and m are level indices. This kernel is the main object of the present study. In the classical limit, due to the (assumed) chaotic nature of the dynamics, the second moment of P t (n | m) exhibits a crossover from ballistic to diffusive behavior. In order to capture this crossover within quantum mechanics, a proper theory for the quantal P t (n | m) should be constructed. We define the V regimes where either perturbation theory or semiclassical considerations are applicable in order to establish this crossover. In the limit Ä 0 perturbation theory does not apply but semiclassical considerations can be used in order to argue that there is detailed correspondence, during the crossover time, between the quantal and the classical P t (n | m). In the perturbative regime there is a lack of such correspondence. Namely, P t (n | m) is characterized by a perturbative core-tail structure that persists during the crossover time. In spite of this lack of (detailed) correspondence there may be still a restricted correspondence as far as the second moment is concerned. Such restricted correspondence is essential in order to establish the universal fluctuation-dissipation relation.

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A semi-classical treatment of dissipative processes based on Feynman's influence functional method

Cited by 59 publications

References 11 publications

Understanding quantum measurement from the solution of dynamical models

Understanding quantum measurement from the solution of dynamical models

Periodic Orbit Approach to the Quantum‐Kramers‐Rate

Chaos and Energy Spreading for Time-Dependent Hamiltonians, and the Various Regimes in the Theory of Quantum Dissipation

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