Quantum stochastic equation for a test particle interacting with a dilute Bose gas

Pechen, Alexander

doi:10.1063/1.1626806

Cited by 15 publications

(18 citation statements)

References 26 publications

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“…Here we make a connection between the objects defined in section II and the model of a test particle interacting with a dilute Bose gas (see Ref. 9 for details). The one particle Hilbert space for this model has the form H ≡ L 2 (R 3 ), where R 3 is the 3-dimensional coordinate or momentum space.…”

Section: Remark 4 the Limiting Correlation Functions Could Be Represementioning

confidence: 99%

The multitime correlation functions, free white noise, and the generalized Poisson statistics in the low density limit

Pechen

2006

Journal of Mathematical Physics

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In the present paper the low density limit of the non-chronological multitime correlation functions of boson number type operators is investigated. We prove that the limiting truncated non-chronological correlation can be computed using only a sub-class of diagrams associated to non-crossing pair partitions and thus coincide with the non-truncated correlation functions of suitable free number operators. The independent in the limit subalgebras are found and the limiting statistics is investigated. In particular, it is found that the cumulants of certain elements coincide in the limit with the cumulants of the Poisson distribution. An explicit representation of the limiting correlation functions and thus of the limiting algebra is constructed in a special case through suitably defined quantum white noise operators.

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Section: Remark 4 the Limiting Correlation Functions Could Be Represementioning

confidence: 99%

The multitime correlation functions, free white noise, and the generalized Poisson statistics in the low density limit

Pechen

2006

Journal of Mathematical Physics

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“…can be obtained in the weak-coupling limit [18][19][20][21], the singular-coupling limit [22,23], the stochastic limit [24,25], the low-density limit for gas environment [26][27][28][29][30][31][32][33], the stroboscopic limit in the collision model [34][35][36][37], and monitoring approach to derivation of linear Boltzmann equation [38][39][40][41]. In all these approximations, the particular form of L is expressed through the system-environment interaction Hamiltonian and the reservoir equilibrium state.…”

Section: Introductionmentioning

confidence: 99%

Quantum master equations for a system interacting with a quantum gas in the low-density limit and for the semiclassical collision model

2020

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A quantum system interacting with a diluted gas experiences the irreversible dynamics. The corresponding master equation can be derived within two different approaches: the fully quantum description in the low-density limit and the semiclassical collision model, where the motion of gas particles is classical whereas their internal degrees of freedom are quantum. The two approaches have been extensively studied in the literature but their predictions have not been compared. This is mainly due to a fact that the low-density limit is extensively studied for mathematical physics analysis, whereas the collision models have been essentially developed for quantum information tasks as a tractable description of the open quantum dynamics. Here we develop and for the first time compare both approaches for a spin system interacting with a gas of spin particles. Using some approximations, we explicitly find the corresponding master equations including the Lamb shifts and the dissipators. The low density limit in the Born approximation for fast particles is shown to be equivalent to the semiclassical collision model in the stroboscopic approximation. We reveal that the both approaches give exactly the same master equation if the gas temperature is high enough. This allows to interchangeably use complicated calculations in the low density limit and rather simple calculations in the collision model.

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“…This consideration was extended in Ref. [10] to the case of arbitrary formfactors and arbitrary, not necessarily equilibrium, quasifree low density states of the gas. In Ref.…”

Section: Introductionmentioning

confidence: 99%

“…White Noise Approach without Fock-antiFock RepresentationThe approach to derivation of the quantum white noise equations directly in terms of the correlation functions, without use of the Fock-antiFock representation, was developed in Ref [10]…”

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confidence: 99%

White Noise Approach to the Low Density Limit

Pechen

2005

Quantum Probability and Infinite Dimensional Analysis

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The white noise approach to the investigation of the dynamics of a quantum particle interacting with a dilute and in general non-equilibrium gaseous environment in the low density limit is outlined. The low density limit is the kinetic Markovian regime when only pair collisions (i.e., collisions of the test particle with one particle of the gas at one time moment) contribute to the dynamics. In the white noise approach one first proves that the appropriate operators describing the gas converge in the sense of appropriate matrix elements to certain operators of quantum white noise. Then these white noise operators are used to derive quantum white noise and quantum stochastic equations describing the approximate dynamics of the total system consisting of the particle and the gas. The derivation is given ab initio, starting from the exact microscopic quantum dynamics. The limiting dynamics is described by a quantum stochastic equation driven by a quantum Poisson process. This equation then applied to the derivation of quantum Langevin equation and linear Boltzmann equation for the reduced density matrix of the test particle. The first part of the paper describes the approach which was developed by L. Accardi, I.V. Volovich and the author and uses the Fock-antiFock (or GNS) representation for the CCR algebra of the gas. The second part presents the approach to the derivation of the limiting equations directly in terms of the correlation functions, without use of the Fock-antiFock representation. This approach simplifies the derivation and allows to express the strength of the quantum number process directly in terms of the one-particle S-matrix.

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Quantum stochastic equation for a test particle interacting with a dilute Bose gas

Cited by 15 publications

References 26 publications

The multitime correlation functions, free white noise, and the generalized Poisson statistics in the low density limit

The multitime correlation functions, free white noise, and the generalized Poisson statistics in the low density limit

Quantum master equations for a system interacting with a quantum gas in the low-density limit and for the semiclassical collision model

White Noise Approach to the Low Density Limit

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