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arXiv:quant-ph/9709002 · Journal of Physics A Mathematical and General

Classical and quantum measurements of position

Carlo Presilla, Roberto Onofrio, Marco Patriarca

Published Nov 7, 1997• 22 citations
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Abstract

We study the dynamics of classical and quantum systems undergoing a continuous measurement of position by schematizing the measurement apparatus with an infinite set of harmonic oscillators at finite temperature linearly coupled to the measured system. Selective and non-selective measurement processes are then introduced according to a selection of or an average over all possible initial configurations of the measurement apparatus. At quantum level, the selective processes are described by a nonlinear stochastic Schrödinger equation whose solutions evolve into properly defined coherent states in the case of linear systems. For arbitrary measured systems, classical behaviour is always recovered in the macroscopic limit.

DOI
10.1088/0305-4470/30/21/014

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We study the dynamics of classical and quantum systems undergoing a continuous measurement of position by schematizing the measurement apparatus with an infinite set of harmonic oscillators at finite temperature linearly coupled to the measured system. Selective and non-selective measurement processes are then introduced according to a selection of or an average over all possible initial configurations of the measurement apparatus. At quantum level, the selective processes are described by a nonlinear stochastic Schrödinger equation whose solutions evolve into properly defined coherent states in the case of linear systems. For arbitrary measured systems, classical behaviour is always recovered in the macroscopic limit.
We study the dynamics of classical and quantum systems undergoing a continuous measurement of position by schematizing the measurement apparatus with an infinite set of harmonic oscillators at finite temperature linearly coupled to the measured system. Selective and nonselective measurement processes are then introduced according to a selection of or an average over all possible initial configurations of the measurement apparatus. At quantum level, the selective processes are described by a nonlinear stochastic Schrödinger equation whose solutions evolve into properly defined coherent states in the case of linear systems. For arbitrary measured systems, classical behavior is always recovered in the macroscopic limit.
We develop a statistical model of microscopic stochastic deviation from classical mechanics based on a stochastic process with a transition probability that is assumed to be given by an exponential distribution of infinitesimal stationary action. We apply the statistical model to stochastically modify a classical mechanical model for the measurement of physical quantities reproducing the prediction of quantum mechanics. The system+apparatus always has a definite configuration at all times, as in classical mechanics, fluctuating randomly following a continuous trajectory. On the other hand, the wavefunction and quantum mechanical Hermitian operator corresponding to the physical quantity arise formally as artificial mathematical constructs. During a single measurement, the wavefunction of the whole system+apparatus evolves according to a Schrödinger equation and the configuration of the apparatus acts as the pointer of the measurement so that there is no wavefunction collapse. We will also show that while the outcome of each single measurement event does not reveal the actual value of the physical quantity prior to measurement, its average in an ensemble of identical measurements is equal to the average of the actual value of the physical quantity prior to measurement over the distribution of the configuration of the system.
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.
Quantum and classical stochastic evolution equations are derived for the statistical state of a system subject to continuous observation of a position variable. In the classical case this results in a stochastic Fokker - Planck equation for the conditional state of the system conditioned on observation records. In the quantum case a stochastic master equation results, or a stochastic Schrödinger equation if the system starts in a pure state. In both cases the resulting equations are nonlinear. A simple modification of the model results in a linear stochastic Schrödinger equation which can also be interpreted as a conditional evolution equation for observations which give no information about the measured system but indicate the size of the random perturbations induced by the environment coupling.
One of the fundamental conceptual difficulties in quantum mechanics arises from the fact that two of the axioms of the theory concerning the evolution of a microscopic system are contradictory. The first axiom assumes that the state of a quantum system evolving undisturbed by any observer obeys a linear deterministic law expressed by the SchrSdinger equation. The second one (wave-packet reduction principle) states that, when the quantum system interacts with an apparatus devised to measure some of its dynamical variables, its state changes abruptly and stochastically, jumping into a normalized eigenstate of the measured variable. This evolution is not deterministic nor linear and cannot be accounted for by the SchrSdinger equation. As a consequence the internal coherence of the theory becomes questionable. On one side the influence of the apparatus on the quantum system cannot be described as an interaction of the quantum constituents of the measuring macroscopic apparatus with the measured quantum system. On the other side, the impossibility of considering a macroscopic object as made by quantum constituents, i .e . the difficulty of quantum mechanics in describing the behaviour of macroscopic systems, is even more dramatic if we consider the evolution of the closed system consisting of the macroscopic apparatus and of the microscopic system. In such a case the appearance of superpositions of macr0scopically distinguishable states related to the apparatus cannot be avoided. These superpositions contradict both our classical experience and wave-packet reduction principle, which provides for statistical mixtures.