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Journal of Physics Conference Series

Quantum measurements without Schrödinger cat states

Dominique Spehner, Fritz Haake

Published Oct 1, 2007• 1 citation
Publisher

Abstract

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.

DOI
10.1088/1742-6596/84/1/012018

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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 study a class of quantum measurement models. A microscopic object is entangled with a macroscopic pointer such that a distinct pointer position is tied to each eigenvalue of the measured object observable. Those different pointer positions mutually decohere under the influence of an environment. Overcoming limitations of previous approaches we (i) cope with initial correlations between pointer and environment by considering them initially in a metastable local thermal equilibrium, (ii) allow for object–pointer entanglement and environment-induced decoherence of distinct pointer readouts to proceed simultaneously, such that mixtures of macroscopically distinct object–pointer product states arise without intervening macroscopic superpositions, and (iii) go beyond the Markovian treatment of decoherence.
We consider the interaction of a quantum system (spin-12)\frac{1}{2})21​) with a macroscopic quantum apparatus (harmonic oscillator) which in turn is coupled to a bath of harmonic oscillators. Exact solutions of the Markovian master equation show that the reduced density matrix of the system-apparatus combination decoheres to a statistical mixture where up and down spins eventually correlate with pointer states of the apparatus (harmonic oscillator), with associated probabilities in accordance with quantum principles. For the zero-temperature bath these pointer states turn out to be coherent states of the harmonic oscillator (apparatus) for arbitrary initial states of the apparatus. Further, we see that the decoherence time is inversely proportional to the square of the separation between the two coherent states with which the spins correlate. For a high-temperature bath, pointer states no longer remain coherent states but are Gaussian distributions (generalized coherent states). Spin up and down states of the system now correlate with nearly diagonal distributions in position of these generalized coherent states. The diagonalization in position increases with the temperature of the bath. The off-diagonal elements in spin space decohere over a time scale which goes inversely as the square of the separation between the peaks of the two position distributions that correlate with the spin states. Zurek's earlier approximate result for the decoherence time is consistent with our exact results. Our analysis brings out the importance of looking at a measurementlike scenario where definite correlations are established between the system and apparatus to determine the nature of the pointer basis of the apparatus. Further, our exact results demonstrate in an unambiguous way that the pointer states in this measurement model emerge independent of the initial state of the apparatus.
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