Time-dependent projection-operator approach to master equations for coupled systems

Willis, C. R.; Picard, R. H.

doi:10.1103/physreva.9.1343

Cited by 192 publications

(111 citation statements)

References 17 publications

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“…Following suggestions of Bohr that the proper quantum measurement should imply a classical apparatus [83,84,85], there were several attempts to work out interaction between a quantum and an explicitly classical system [95,96,97,98,99,100,101,102,103,104,105,106,107,108,109]. (Neither Bohr [83,84], nor Landau and Lifshitz [85] who present Bohr's opinion in quite detail, consider the proper interaction processes.)…”

Section: Dutch Proverbmentioning

confidence: 99%

“…This subject is referred to as hybrid (quantum-classical) dynamics. Besides the measurement theory it is supposed to apply in quantum chemistry [95,96,103] (where the full modeling of quantum degrees of freedom is difficult) and in quantum 21 What is allowed for Jupiter, is not allowed for the rind 22 Don't throw away old shoes before you have new ones gravity [110], where the proper quantum dynamics of the gravitational field is not known. There are several versions of the hybrid dynamics.…”

Section: Dutch Proverbmentioning

confidence: 99%

“…There are several versions of the hybrid dynamics. The situation, where the classical degree of freedom is of a mean-field type is especially well-known [95,96]. In that case the hybrid dynamics can be derived variationally from a simple combination of quantum and classical Lagrangian.…”

Section: Dutch Proverbmentioning

confidence: 99%

“…This conception of probabilities, the current one in the XVII th and XVIII th centuries, has been given a mathematical foundation, on which we will return in § 11.2.2, by Venn [309] and von Mises [310]. On the other hand, in the "logical Bayesian" approach, initiated by Bayes and Laplace, and later on formalized by Cox [311] and advocated by Jaynes [289], probabilities are defined as a mathematical measure of likelihood of events; they are not inherent to the considered object alone, but are tools for making reasonable predictions about this object through consistent inference 95 . Both interpretations are relevant to quantum theory, and their equivalence has been established [313], in the context of assigning a quantum probability distribution to a system ( § 10.2.2).…”

Section: Interpretation Of Probabilities and Statistical Ensembles Ofmentioning

confidence: 99%

“…With a single Universe at hand, this leads to the unsolvable cosmic variance problem. The same ideas hold whenever probabilities are applied to a single system or event [311], and this is the subject of standard and thorough developments in books of probabilities, including already Laplace's 94 Kolmogorov's axioms, the starting point of many mathematical treatises, do not prejudge how probabilities may be interpreted in applications 95 We keep aside the "subjective Bayesian" interpretation, developed by de Finetti [312], and suited more to ordinary life or economy than to science. There, probabilities are associated with the state of mind of an agent, and help him to take rational decisions.…”

Section: Interpretation Of Probabilities and Statistical Ensembles Ofmentioning

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: Dutch Proverbmentioning

confidence: 99%

Section: Dutch Proverbmentioning

confidence: 99%

Section: Dutch Proverbmentioning

confidence: 99%

Section: Interpretation Of Probabilities and Statistical Ensembles Ofmentioning

confidence: 99%

Section: Interpretation Of Probabilities and Statistical Ensembles Ofmentioning

confidence: 99%

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Understanding quantum measurement from the solution of dynamical models

Allahverdyan¹,

Balian²,

2013

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Vibrational and Rotational Collision Processes

DePristo

Rabitz

1980

Advances in Chemical Physics

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Verallgemeinerte Mastergleichungen und Fokker‐Planck‐Gleichungen für Prozesse in inhomogenen Systemen weitab vom Gleichgewicht

Haberlandt¹

1986

Annalen der Physik

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Prof. Dr. Robert Rompe zum 80. Ceburtstag zugeeignetInhaltsiibersicht. Unter Verwendung verallgemeinerter kanonischer Gesamtheiten werden verallgemeinerte Mastergleichungen und Fokker-Planck-Gleichungen fur irreversible Prozesse in inhomogenen, raumlich und zeitlich nichtlokalen Systemen, die weit vom Gleichgewicht entfernt sein diirfen, hergeleitet. Ihr Zusammenhang mit Spezialfallen wird diskutiert.Abstract. Using generalized canonical ensembles we derive generalized masterequations and Fokker-Planck-Equations describing irreversible processes in inhomogeneous systems far from equilibrium and nonlocal in space and time.The connection of these equations to special cases is discusses.

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Time-dependent projection-operator approach to master equations for coupled systems

Cited by 192 publications

References 17 publications

Understanding quantum measurement from the solution of dynamical models

Understanding quantum measurement from the solution of dynamical models

Vibrational and Rotational Collision Processes

Verallgemeinerte Mastergleichungen und Fokker‐Planck‐Gleichungen für Prozesse in inhomogenen Systemen weitab vom Gleichgewicht

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