Time-extended measurement of the position of a driven harmonic oscillator

Roig, Francesc S.

doi:10.48550/arxiv.1209.2445

Cited by 2 publications

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“…Next we will obtain and alternative expression for the decoherent functional and we will assume that the initial state is a pure state of the form Ψ 0 (x, X) = ϕ 0 (x)Φ 0 (X). In reference [2] the following expression was obtained for the propagator of a driven harmonic oscillator interacting with a von Neumann apparatus:…”

Section: Formalismmentioning

confidence: 99%

“…In this paper we consider the system of a particle interacting with a von Neumann measuring apparatus, which was studied in detail in references [1,2]. Standard quantum mechanics was applied to this system to study quantum measurements of finite duration of the position of the particle.…”

Section: Introductionmentioning

confidence: 99%

“…For a driven oscillator acted on by a driving force f D (t) symmetric about the midpoint of the interval [0, T ], the expressions for s(x, x ′ ), M eff and φ(x, x ′ ) in (1.15), are given by equations ( 29), ( 31) and (34) in Ref. [2] respectively. In particular in Ref.…”

Section: Introductionmentioning

confidence: 99%

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Decoherent histories for a particle coupled to a von Neumann apparatus

Roig

2014

Preprint

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Using the Gell-Mann and Hartle formalism of generalized quantum mechanics of closed systems, we study coarse-grained decoherent histories. The system under consideration is one-dimensional and consists of a particle coupled to a von Neumann apparatus that measures its position. The particle moves in a quadratic potential; in particular we consider a driven harmonic oscillator. The real line is divided into intervals of the same length, and coarse-grained histories are defined by the arithmetic average of the initial and final position of the particle to be within one of these intervals. The position of the pointer correlates with this arithmetic average. In addition a constant term is added to this average, which is a result of the presence of a driving force on the oscillator. We investigate decoherence for such histories via the decoherence functional for the particle-apparatus. An exact expression for this functional has been derived. If the particle or the pointer of the apparatus is in an exact position state initially, then exact decoherence ensues, and the relative probability for such histories is obtained. If the initial state of the particle, which is centered at some arbitrary point in the x-axis, is narrow compared to the width of the intervals on the x-axis, then we obtain approximate decoherence. The same result follows if the initial state of the pointer is narrow. In these two situations, we evaluate expressions for the decoherence functional qualitatively and quantitatively. In the latter case we obtain the first two terms in an expansion in powers of the width of the initial state of the particle or the pointer respectively. Approximate relative probabilities can be assigned in this case, and we have obtained an expansion up to second order in powers of the width of the initial state for an expression for an upper bound of these probabilities.

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Section: Formalismmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Decoherent histories for a particle coupled to a von Neumann apparatus

Roig

2014

Preprint

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“…As done in Ref. [5] the sum over pointer paths can easily be carried out by inserting two complete sets of eigenstates of the momentum P of the pointer and normalized according to P | P ′ = δ(P − P ′ ). The propagator (1.5) can be rewritten in terms of a reduced propagator as…”

Section: General Potential and General Coupling Functionmentioning

confidence: 99%

“…(3.7), were already obtained in Refs. [5,10]. The expressions for the lengths ℓ and Z in the exponential in the integrantion over k are given by…”

Section: A the Class Operatorsmentioning

confidence: 99%

Decoherence in the classical limit of histories of a particle coupled to a von Neumann apparatus

Roig¹

2016

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Using the Gell-Mann and Hartle formalism of generalized quantum mechanics of closed systems, we study the classical limit of coarse-grained spacetime histories and their decoherence. The system under consideration is one-dimensional and consists of a particle coupled to a von Neumann apparatus that performs a measurement of the position of the particle during the finite time interval during which the histories of this system take place. We consider two cases: a free particle and a harmonic oscillator. The real line is divided into intervals of the same length, and coarse-grained histories are defined by the time average of the position of the particle on a given Feynman path to be within one of these intervals. The position of the pointer in each Feynman path correlates with this time average. The class operators for this system have been evaluated, and the decoherence functional shows that these coarse-grained histories do not decohere, not even when initially either the particle or the pointer is in an eigenstate of position. Decoherence is obtained only when the classical limit is taken. Qualitative arguments for decoherence in the classical limit are presented for the case of a general particle potential.

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Time-extended measurement of the position of a driven harmonic oscillator

Cited by 2 publications

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Decoherent histories for a particle coupled to a von Neumann apparatus

Decoherent histories for a particle coupled to a von Neumann apparatus

Decoherence in the classical limit of histories of a particle coupled to a von Neumann apparatus

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