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Classical and quantum measurements of position

Carlo Presilla A, Roberto Onofrio, Marco Patriarca

Published Jan 1, 1997• 12 citations
OpenAlex

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 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.

Related papers

Classical and quantum measurements of position

Carlo Presilla, Roberto Onofrio, Marco Patriarca

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.

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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.

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Toward the Collapse of State

Slobodan Prvanović, Z. Marić, Belgrade +1 more

The basic concepts of classical mechanics are given in the operator form. Then, the hybrid systems approach, with the operator formulation of both quantum and classical sector, is applied to the case of an ideal nonselective measurement. It is found that the dynamical equation, consisting of the Schrödinger and Liouville dynamics, produces noncausal evolution when the initial state of measured system and measuring apparatus is chosen to be as it is demanded in discussions regarding the problem of measurement. Nonuniqueness of possible realizations of transition from pure noncorrelated to mixed correlated state is analyzed in details. It is concluded that collapse of state is the only possible way of evolution of physical systems in this case.

arXiv:quant-ph/9910020•Oct 5, 1999•0 citations
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 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 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.
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.
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.
The basic concepts of classical mechanics are given in the operator form. Then, the hybrid systems approach, with the operator formulation of both quantum and classical sector, is applied to the case of an ideal nonselective measurement. It is found that the dynamical equation, consisting of the Schrödinger and Liouville dynamics, produces noncausal evolution when the initial state of measured system and measuring apparatus is chosen to be as it is demanded in discussions regarding the problem of measurement. Nonuniqueness of possible realizations of transition from pure noncorrelated to mixed correlated state is analyzed in details. It is concluded that collapse of state is the only possible way of evolution of physical systems in this case.