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International Journal of Modern Physics A

QUANTUM NONDEMOLITION MEASUREMENTS FROM PATH INTEGRAL

G. A. Golubtsova, Michael B. Mensky

Published Jul 10, 1989• 21 citations
Publisher

Abstract

The path integration method is applied to continuous quantum nondemolition (QND) measurements. Direct and indirect QND measurements are considered. It is shown that an observable is QND variable then and only then if the equation of motion for this observable due to a special “effective Hamiltonian” coincides with the classical equation of motion for this observable. The method is developed for finding a class of QND variables in the case of quadratic Hamiltonian.

DOI
10.1142/s0217751x89001059

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The path integration method is applied to continuous quantum nondemolition (QND) measurements. Direct and indirect QND measurements are considered. It is shown that an observable is QND variable then and only then if the equation of motion for this observable due to a special “effective Hamiltonian” coincides with the classical equation of motion for this observable. The method is developed for finding a class of QND variables in the case of quadratic Hamiltonian.
This chapter discusses the type of measurement in the framework of the path-integral method. It begins from the simplest case of measurement of the linear momentum of a free particle, which allows one to see all the typical features characteristic of quantum nondemolition (QND) measurements. The idea of QND measurement is to measure a variable such that the unavoidable disturbance of the complementary one does not disturb the evolution of the chosen variable. The weight factor describes monitoring of the momentum, which for a free particle turns out to be a QND variable. This can be seen from the formula for the probability distribution of the measurement results. Usual relation between linear momentum and velocity is violated in the modified dynamics. From the physical point of view this is a consequence of measurement, of the influence of a measuring device.
In this work a family of quantum nondemolition variables for the case of a particle caught in a Paul trap is obtained. Afterwards, in the context of the so called restricted path integral formalism, a continuous measuring process for this family of parameters is considered, and then the corresponding propagators are calculated. In other words, the time evolution of a particle in a Paul trap, when the corresponding quantum nondemolition parameter is being continuously monitored, is deduced. The probabilities associated with the possible measurement outputs are also obtained, and in this way new theoretical results emerge, which could allow us to confront the predictions of this restricted path integral formalism with the readouts of some future experiments. 1
Preface INTRODUCTION TO CONTINUOUS QUANTUM MEASUREMENTS Quantum and Classical Systems Amplitudes and Alternatives Paths and Continuous Measurements The Action Uncertainty Principle INSTANTANEOUS AND SEQUENTIAL MEASUREMENTS Measurement of a Quantum System Quantum Zeno Paradox Approximate and Sequential Measurements TECHNIQUE OF PATH INTEGRALS Propagators and Path Integrals Definition of a Path Integral The Path Integral for an Oscillator Gaussian Path Integrals CONTINUOUS MEASUREMENT AND EVOLUTION OF THE MEASURED SYSTEM The Measurement Amplitude Effective Lagrangian Evolution of a Quantum System Subject to Continuous Measurement Scattering by the Measuring Medium CONTINUOUS MEASUREMENTS OF OSCILLATORS Position Monitoring (Path measurement) Estimation of Force Acting on an Oscillator Spectral Measurements of an Oscillator Evolution of a Harmonic Oscillator Subject to Spectral Measurement Measurement of a Path of a Nonlinear Oscillator CONTINUOUS QUANTUM NONDEMOLITION MEASUREMENTS Quantum Nondemolition Measurements Monitoring of Linear Momentum Velocity versus Momentum QND Measurements of Oscillators MEASUREMENT OF AN ELECTROMAGNETIC FIELD Exposition of the Method Measurement Amplitude Analysis of the Results Another Definition of the Field Strength Explicit Model of a Meter TIME IN QUANTUM COSMOLOGY Statement of the Problem Measurements in Quantum Cosmology Minisuperspace without Measurement Minisuperspace under Measurement Analysis of the results Concluding Remarks THE ACTION UNCERTAINTY PRINCIPLE Uncertainties in Continuous Measurements Classical Regime of Measurements Quantum Regime and AUP AUP and Equations of Motion Examples from Quantum Mechanics Quantum Nondemolition Measurements AUP in Quantum Gravity Concluding Remarks GROUP-THEORETICAL STRUCTURE OF QUANTUM CONTINUOUS MEASUREMENTS Evolution under Measurement Transverse Group for Alternatives Longitudinal Semigroup for Evolution Unification of Transverse and Longitudinal Structures Concluding Remarks PATHS AND MEASUREMENTS: FURTHER DEVELOPMENT Quantum and Classical Features in Continuous Measurement The Two-Slit Experiment Wave-Particle Dualism Conceptual Problems References Index
In the scheme of a quantum nondemolition (QND) measurement, an observable is measured without perturbing its evolution. In the context of studies of decoherence in quantum computing, we examine the 'open' quantum system of a two-level atom or, equivalently, a spin-1/2 system, in interaction with quantum reservoirs of either oscillators or spins, under the QND condition of the Hamiltonian of the system commuting with the system–reservoir interaction. For completeness, we also examine the well-known non-QND spin-Bose problem. For all these many-body systems, we use the methods of functional integration to work out the propagators. The propagators for the QND Hamiltonians are shown to be analogous to the squeezing and rotation operators, respectively, for the two kinds of baths considered. Squeezing and rotation being both phase-space area-preserving canonical transformations, this brings out an interesting connection between the energy-preserving QND Hamiltonians and the homogeneous linear canonical transformations.
In the scheme of a quantum nondemolition (QND) measurement, an observable is measured without perturbing its evolution. In the context of studies of decoherence in quantum computing, we examine the `open' quantum system of a two-level atom, or equivalently, a spin-1/2 system, in interaction with quantum reservoirs of either oscillators or spins, under the QND condition of the Hamiltonian of the system commuting with the system-reservoir interaction. For completeness, we also examine the well-known non-QND spin-Bose problem. For all these many-body systems, we use the methods of functional integration to work out the propagators. The propagators for the QND Hamiltonians are shown to be analogous to the squeezing and rotation operators, respectively, for the two kinds of baths considered. Squeezing and rotation being both phase space area-preserving canonical transformations, this brings out an interesting connection between the energy-preserving QND Hamiltonians and the homogeneous linear canonical transformations.