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Physical Review A

Quantum-mechanical model for continuous position measurements

Carlton M. Caves, Gerard J Milburn

Published Dec 1, 1987• 343 citations
Publisher

Abstract

We present an idealized model for a sequence of position measurements, and we then take an appropriate limit in which the measurements become continuous. The measurements lead to fluctuations without systematic dissipation, and they rapidly destroy off-diagonal terms in the position basis; thus the pointer basis is position. A modification of the model incorporates systematic dissipation via a feedback mechanism; in the modified model there is no decay of off-diagonal coherence in the position basis.

DOI
10.1103/physreva.36.5543

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arXiv:1407.5007•New Journal of Physics•Nov 11, 2014•1 citation
We present an idealized model for a sequence of position measurements, and we then take an appropriate limit in which the measurements become continuous. The measurements lead to fluctuations without systematic dissipation, and they rapidly destroy off-diagonal terms in the position basis; thus the pointer basis is position. A modification of the model incorporates systematic dissipation via a feedback mechanism; in the modified model there is no decay of off-diagonal coherence in the position basis.
We present a model of continuous (in time) position measurements on a quantum system using a single pseudoclassical meter. The nonselective evolution of the system is described by a master equation which is identical to that obtained from previous models. The selective evolution is described by a stochastic nonlinear Schr\"odinger equation. The significance of this equation is that the stochastic term has a physical interpretaion. By carefully choosing the parameters which define the meter and the system-meter coupling, we obtain a meter pointer with well-defined position which undergoes fluctuations. This ``jitter'' in the pointer position gives rise to the stochastic dynamical collapse of the system wave function. By the inclusion of feedback on the meter, the pointer is made to relax towards an appropriate readout. We apply this model to the selective measurement of the position of a particle in a double-well potential. In contrast to a recent claim [H. Fearn and W. E. Lamb, Jr., Phys. Rev. A 46, 1199 (1992)] we show that truly continuous position measurements lead to a quantum Zeno effect in certain parameter regimes. This is manifest by the changing of the particle dynamics from coherent tunneling between the well minima to incoherent flipping, as in a random telegraph. As the measurement strength increases, the average length of time the particle remains stuck in one well increases proportionally.
A systematic construction of exactly solvable models of interactions between the measuring apparatus and the particle whose position is to be measured is shown. The two types of root-mean-square errors are calculated for these models and many new models with better statistics than the standard von Neumann model are continuously found. The best among them is the model which has been shown to circumvent the standard quantum limit for the position monitoring. A feature of the models similar to Bondurant's improvement of the interferometric position measurement is also pointed out.
This paper is concerned with two questions in the decoherent histories approach to quantum mechanics: the emergence of approximate classical predictability, and the fluctuations about it necessitated by the uncertainty principle. It is in part a continuation of an earlier paper on the uncertainty principle for quantum-mechanical histories. We consider histories characterized by position samplings at n moments of time. This includes the case of position sampling continuous in time in the limit n\ensuremath{\rightarrow}\ensuremath{\infty}. We use this to construct a probability distribution on the value of (discrete approximations to) the field equations, F=mx\ifmmode\ddot\else\textasciidieresis\fi{}+V'(x), at n-2 times. We find that it is peaked around F=0; thus classical correlations are exhibited. We show that the width of the peak \ensuremath{\Delta}F is largely independent of the initial state. We show that the uncertainty principle takes the form 2${\mathrm{\ensuremath{\sigma}}}^{2}$(\ensuremath{\Delta}F)2{)}^{2})2\ensuremath{\ge}${\mathrm{\ensuremath{\Elzxh}}}^{2}$/t2{\mathit{t}}^{2}t2, where \ensuremath{\sigma} is the width of the position samplings, and t is the time scale between projections. We determine the modifications to this result when the system is coupled to a thermal environment, thus obtaining a measure of the comparative sizes of quantum and thermal fluctuations. We show that the thermal fluctuations become comparable to the quantum fluctuations under the same conditions that decoherence effects come into play, in agreement with earlier work. We also study an alternative measure of classical correlations, namely, the conditional probability of finding a sequence of position samplings, given that particular initial phase space data have occurred.We use these results to address the issue of the formal interpretation of the probabilities for sequences of position samplings in the decoherent histories approach to quantum mechanics. Under appropriate conditions, which we describe, they admit an interpretation as a statistical ensemble of classical solutions, with the probability of each individual classical solution given by a smeared Wigner function of its initial data. We study the decoherence properties of histories characterized by the value of the field equations, F, at a sequence of times. We argue that they will be decoherent if their initial data are decoherent.
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 present and discuss a master equation blueprint for a generic class of quantum measurement-feedback-based models of friction. A desired velocity-dependent friction force is realized on average by random repeated applications of unsharp momentum measurements followed by immediate outcome-dependent momentum displacements. The master equations can describe arbitrarily strong measurement-feedback processes as well as the weak continuous limit resembling diffusion master equations of Caldeira-Leggett type. We show that the special case of linear friction can be equivalently represented by an average over random position measurements with squeezing and position displacements as feedback. In fact, the dissipative continuous spontaneous localization model of objective wave-function collapse realizes this representation for a single quantum particle. We reformulate a consistent many-particle generalization of this model and highlight the possibility of feedback-induced correlations between otherwise noninteracting particles.
The dynamics for an open quantum system can be 'unravelled' in infinitely many ways, depending on how the environment is monitored, yielding different sorts of conditioned states, evolving stochastically. In the case of ideal monitoring these states are pure, and the set of states for a given monitoring forms a basis (which is overcomplete in general) for the system. It has been argued elsewhere (Atkins et al 2005 Europhys. Lett. 69 163 ) that the 'pointer basis' as introduced by Zurek et al (1993 Phys. Rev. Lett. 70 1187 ), should be identified with the unravelling -induced basis which decoheres most slowly. Here we show the applicability of this concept of pointer basis to the problem of state stabilization for quantum systems. In particular we prove that for linear Gaussian quantum systems, if the feedback control is assumed to be strong compared to the decoherence of the pointer basis, then the system can be stabilized in one of the pointer basis states with a fidelity close to one (the infidelity varies inversely with the control strength). Moreover, if the aim of the feedback is to maximize the fidelity of the unconditioned system state with a pure state that is one of its conditioned states, then the optimal unravelling for stabilizing the system in this way is that which induces the pointer basis for the conditioned states. We illustrate these results with a model system: quantum Brownian motion. We show that even if the feedback control strength is comparable to the decoherence, the optimal unravelling still induces a basis very close to the pointer basis. However if the feedback control is weak compared to the decoherence, this is not the case.