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Physical Review A

Generalized model of the quantum measurement process in an environment

Masashi Ban, Sachiko Kitajima, Fumiaki Shibata

Published Mar 25, 2009• 2 citations
Publisher

Abstract

A generalized model of the quantum measurement process is considered, in which the relaxation of the detector system interacting with an environment can be treated. When the effect of the environment on the detector system is described by the quantum Markovian process, the model can be solved exactly. The time evolution of the particle and detector is investigated in detail. It is shown that the decay of quantum coherence of the particle becomes negligible when the relaxation time of the detector system is sufficiently short. Furthermore the stochastic model is found, which is equivalent to the proposed model with respect to the particle state.

DOI
10.1103/physreva.79.032113

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A generalized model of the quantum measurement process is considered, in which the relaxation of the detector system interacting with an environment can be treated. When the effect of the environment on the detector system is described by the quantum Markovian process, the model can be solved exactly. The time evolution of the particle and detector is investigated in detail. It is shown that the decay of quantum coherence of the particle becomes negligible when the relaxation time of the detector system is sufficiently short. Furthermore the stochastic model is found, which is equivalent to the proposed model with respect to the particle state.
The dissipation and decoherence (for example, the effects of noise in quantum computations), interaction with thermostat or in general with physical vacuum, measurement and many other complicated problems of open quantum systems are a consequence of interaction of quantum system with the environment. These problems are described mathematically in terms of complex probabilistic process (CPP). Particularly, treating the environment as a Markovian process we derive an Langevin-Schroedinger type stochastic differential equation (SDE) for describing the quantum system interacting with environment. For the 1D randomly quantum harmonic oscillator (QHO) model L-Sh SDE is a solution in the form of orthogonal CPP. On the basis of orthogonal CPP the stochastic density matrix (SDM) method is developed and in its framework relaxation processes in the uncountable dimension closed system of "QHO+environment" is investigated. With the help of SDM method the thermodynamical potentials, like nonequilibrium entropy and the energy of ground state are exactly constructed. The dispersion for different operators are calculated. In particular, the expression for uncertain relations depending on parameter of interaction with environment is obtained. The Weyl transformation for stochastic operators is specified. Ground state Winger function is developed in detail.
The measuring process of an object system and a detector are studied in the von Neumann model and one of its variants, in which the detector consists of a single particle. If the states of the particle after the interaction with the object system are described partly by coherent states, then the particle can play the role of a detector under certain conditions. The mass (or free Hamiltonian) of the detector does not affect the measuring process at all, while the mass of the object system plays an important role. In the von Neumann model, if the mass of the object system is quite large, the position measurement can be considered, and under certain conditions the statistical operator \ensuremath{\rho} tends to just its mixed-state part, i.e., quantum interference disappears. However, when the time after the interaction becomes very large, interference emerges again. On the other hand, if the mass of the object system is small, we can even consider the momentum measurement and show that \ensuremath{\rho} approaches its mixed-state part when the time becomes very large, i.e., \ensuremath{\rho} is equivalent to its mixed-state part. The momentum can also be measured almost completely in the variant model; the operator \ensuremath{\rho} becomes equivalent to its mixed-state part. The different quantum states distinguish the detectors from the object systems.
Quantum mechanics must be regarded as open systems. On one hand, this is due to the fact that, like in classical physics, any realistic system is subjected to a coupling to an uncontrollable environment which influences it in a non-negligible way. The theory of open quantum systems thus plays a major role in many applications of quantum physics since perfect isolation of quantum system is not possible and since a complete microscopic description or control of the environment degrees of freedom is not feasible or only partially so [1]. Practical considerations therefore force one to seek for a simpler, effectively probabilistic description in terms of an open system. There is a close physical and mathematical connection between the evolution of an open system, the state changes induced by quantum measurements, and the classical notion of a stochastic process. The paper provides a bibliographic review of this interrelations, it shows the mathematical equivalence between markovian master equation and generalized piecewise deterministic processes [1] and it introduces the open system in an open observed environment model.
Abstract We study the evolution of a two-state system that is monitored continuously but with interactions with the detector tuned so as to avoid the Zeno affect. The system is allowed to interact with a sequence of prepared probes. The post-interaction probe states are measured and this leads to a stochastic evolution of the system’s state vector, which can be described by a single angle variable. The system’s effective evolution consists of a deterministic drift and a stochastic resetting to a fixed state at a rate that depends on the instantaneous state vector. The detector readout is a counting process. We obtain analytic results for the distribution of number of detector events and the time-evolution of the probability distribution. Earlier work on this model found transitions in the form of the steady state on increasing the measurement rate. Here we study transitions seen in the dynamics. As a spin-off we obtain, for a general stochastic resetting process with diffusion, drift and position dependent jump rates, an exact and general solution for the evolution of the probability distribution.
Quantum mechanical decoherence process is studied with the use of a boson detector model. Generalizing the previous method, thermal effect is taken into account: Coherence of an incident particle is destroyed in interaction with its surroundings at finite temperature. Vanishing of off-diagonal elements of a reduced density matrix is rigorously shown when the detector (surroundings) is composed of a large number of constituents and/or its temperature is very high. The temperature effect reveals itself through a renormalization of interaction strength between an incident particle and the detector. Time evolution of the von Neumann entropy is explicitly determined.
Time-evolutions of two-level and multi-level systems in phase relaxation processes are investigated in a systematic way. A general form of the decoherence function which determines phase relaxation of a quantum state is obtained by making use of a microscopic interaction Hamiltonian between the relevant system and reservoir. Furthermore the loss of the phase coherence is described in terms of the Pegg–Barnett phase operator. The decoherence function is explicitly calculated for the stochastic dephasing, the boson detector model and the Coleman–Hepp model.