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arXiv:cond-mat/9903403

Quantum Measurement: A Solvable Model

Dmitry Mozyrsky, Vladimir Privman A

Published Mar 27, 1999• 0 citations
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Abstract

We present an exact derivation of a process in which a microscopic measured system interacts with “heat bath ” and pointer modes of a measuring device, via a linear coupling involving Hermitian operator Λ of the system. In the limit of strong interaction with these modes, over a small time interval, we show that the measured system and the “pointer ” part of the measuring device evolve into a statistical mixture of direct-product states such that the system is in each eigenstate of Λ with the correct quantummechanical probability, whereas the expectation values of pointer-space operators retain amplified information of the system’s eigenstate. PACS: 03.65.Bz, 03.65.Ca – 1 – The problem of quantum measurement has fascinated scientists for a long time [1]. It has been argued that a large “bath ” is an essential ingredient of the measurement process. Interaction with the bath, which might be a “heat bath ” in thermal equilibrium, causes decoherence which is

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We present an exact derivation of a process in which a microscopic measured system interacts with “heat bath ” and pointer modes of a measuring device, via a linear coupling involving Hermitian operator Λ of the system. In the limit of strong interaction with these modes, over a small time interval, we show that the measured system and the “pointer ” part of the measuring device evolve into a statistical mixture of direct-product states such that the system is in each eigenstate of Λ with the correct quantummechanical probability, whereas the expectation values of pointer-space operators retain amplified information of the system’s eigenstate. PACS: 03.65.Bz, 03.65.Ca – 1 – The problem of quantum measurement has fascinated scientists for a long time [1]. It has been argued that a large “bath ” is an essential ingredient of the measurement process. Interaction with the bath, which might be a “heat bath ” in thermal equilibrium, causes decoherence which is
We present an exact derivation of a process in which a microscopic measured system interacts with the heat-bath and pointer modes of a measuring device, via a coupling involving a general Hermitian operator Λ of the system. In the limit of strong interaction with these modes, over a small time interval, we derive the exact effective many-body density matrix of the measured system plus pointer. We then discuss the interpretation of the dynamics considered as the first stage in the process of quantum measurement, eventually involving the wave-function collapse due to interactions with "the rest of the universe". We establish that the effective density matrix represents the required framework for the measured system and the pointer part of the measuring device to evolve into a statistical mixture described by direct-product states such that the system is in each eigenstate of Λ with the correct quantum-mechanical probability, whereas the expectation values of the pointer-space operators retain amplified information of the system's eigenstate.
We review results of a recently developed model of a microscopic quantum system interacting with the macroscopic world components which are modeled by collections of bosonic modes. The interaction is via a general operator (Lambda) of the system, coupled to the creation and annihilation operators of the environment modes. We assume that in the process of a nearly instantaneous quantum measurement, the function of the environment involves two distinct parts: the pointer and the bath. Interaction of the system with the bath leads to decoherence such that the system and the pointer both evolve into a statistical mixture state described by the density matrix such that the system is in one of the eigenstates of (Lambda) with the correct quantum mechanical probability, whereas the expectation values of pointer operators retain amplified information on that eigenstate. We argue that this process represents the initial step of a quantum measurement.
We report and give an alternative derivation of some results on a model for a quantum measurement studied in [1]. The measured microscopic system is coupled to the position of a macroscopic pointer, which itself interacts with its environment via its momentum. The entanglement between the system and the pointer produced by their mutual interaction is simultaneous with the decoherence of distinct pointer readings resulting from leakage of information to the environment. After a discussion on the various time scales in the model we calculate the matrix elements of the system-pointer density operator between eigenstates of the measured observable with distinct eigenvalues. In general, the decay with time of these coherences is neither exponential nor gaussian. We determine the decoherence (decay) time in terms of the strength of the system-pointer and pointer-environment couplings. This decoherence time does not depend upon the details of the pointer-bath coupling as soon as it is smaller than the bath correlation time (non-Markov regime). In contrast, in the Markov regime it depends strongly on whether this coupling is Ohmic or super-Ohmic.
We study a class of quantum measurement models. A microscopic object is entangled with a macroscopic pointer such that each eigenvalue of the measured object observable is tied up with a specific pointer deflection. Different pointer positions mutually decohere under the influence of a bath. Object-pointer entanglement and decoherence of distinct pointer readouts proceed simultaneously. Mixtures of macroscopically distinct object-pointer states may then arise without intervening macroscopic superpositions. Initially, object and apparatus are statistically independent while the latter has pointer and bath correlated according to a metastable local thermal equilibrium. We obtain explicit results for the object-pointer dynamics with temporal coherence decay in general neither exponential nor Gaussian. The decoherence time does not depend on details of the pointer-bath coupling if it is smaller than the bath correlation time, whereas in the opposite Markov regime the decay depends strongly on whether that coupling is Ohmic or super-Ohmic.
This is the first of two papers showing that the quantum problem of measurement remains unsolved even when the initial state of the apparatus is described by a statistical operator and when the results of measurement have a small probability of being erroneous. A realistic treatment of the measurement of observables of microscopic objects (e.g., the position or the spin of an electron) by means of observables of macroscopic apparatus (e.g., the position of a spot on a photographic plate) requires the consideration of errors. The first paper considers measurement procedures of the following type: An initial eigenstate of the object observable leads to a final statistical operator of the object plus apparatus which describes a mixture of “approximate” eigenstates of the apparatus observable. It is proved that each of a large class of initial states leads to a final statistical operator which does not describe any mixture containing even one “approximate” eigenstate of the apparatus observable. INTRODUCTION Several writers have tried to solve the quantum-mechanical problem of measurement through one or both of the following proposals: (a) describthe initial state of the measuring apparatus by a projection onto a subspace of the associated Hilbert space or by a statistical operator, thus taking into account the practical impossibility of knowing the exact quantum state of a macroscopic object; (b) recognizing that there may be some physical inaccuracy, such as a small error in the position of a pointer needle, in the final registration of the outcome of the measurement by the apparatus.
We study the extent to which the outcomes of a quantum measurement can be manipulated by changing the state of the measurement apparatus. The measurement process is modeled as decoherence induced by the experimenter, to gain knowledge about a particular system. The measurement apparatus is assumed to contain a large number of degrees of freedom, and the measurement outcomes are obtained by looking at the long interaction time limit. We study two cases which show contrasting behaviour. With a fixed axis coupling, the measurement is performed along the pointer basis with a high degree of robustness, for a wide variety of bath states. In a second model with Heisenberg interactions, the measurement outcomes can be altered considerably by changing the state of the bath.