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Quantum Optics Journal of the European Optical Society Part B

Continuous measurements of quantum phase

V. P. Belavkin, C. Bendjaballah

Published Jun 1, 1994• 12 citations
Publisher

Abstract

The question of phase measurement for a quantum oscillator using the quantum non-demolition principle (NDP), which does not need the use of the wave packet reduction postulate, is studied. It is shown that this approach is appropriate to introduce in a proper way the relevant observables for unlocalized (unsharp) measurement of quantum phase. Our analysis suggests that in contrast to other quantum-mechanical problems (e.g. The position and the momentum of a particle), the operator to represent the phase of a quantum field cannot be separated from the measurement process. Thus, the quantum phase operator must be defined for a corresponding measurement scheme. Moreover, it allows us to describe the time continuous process of quantum phase observations and also to study the stochastic dynamics of the oscillator under such measurement.

DOI
10.1088/0954-8998/6/3/004

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The question of phase measurement for a quantum oscillator using the quantum non-demolition principle (NDP), which does not need the use of the wave packet reduction postulate, is studied. It is shown that this approach is appropriate to introduce in a proper way the relevant observables for unlocalized (unsharp) measurement of quantum phase. Our analysis suggests that in contrast to other quantum-mechanical problems (e.g. The position and the momentum of a particle), the operator to represent the phase of a quantum field cannot be separated from the measurement process. Thus, the quantum phase operator must be defined for a corresponding measurement scheme. Moreover, it allows us to describe the time continuous process of quantum phase observations and also to study the stochastic dynamics of the oscillator under such measurement.
Abstract. We give an explicit axiomatic formulation of the quantum measurement theory which is free of the projection postulate. It is based on the generalized nondemolition principle applicable also to the unsharp, continuous– spectrum and continuous-in-time observations. The “collapsed state–vector” after the “objectification ” is simply treated as a random vector of the a posteriori state given by the quantum filtering, i.e., the conditioning of the a priori induced state on the corresponding reduced algebra. The nonlinear phenomenological equation of “continuous spontaneous localization ” has been derived from the Schrödinger equation as a case of the quantum filtering equation for the diffusive nondemolition measurement. The quantum theory of measurement and filtering suggests also another type of the stochastic equation for the dynamical theory of continuous reduction, corresponding to the counting nondemolition measurement, which is more relevant for the quantum experiments. 1. The status of quantum measurement theory
A stochastic Schrödinger equation is presented to describe simultaneous continuous measurement of the position and momentum of a non-relativistic particle. The equation is solved to yield a state localised in position and momentum contingent on the uncertainty principle. The state is understood as that to which the particle tends after a period of continuous measurement. The solution takes the form of a wave packet of fixed Gaussian shape whose centre undergoes random motion in phase space. It is shown to be locally stable.
In the Action--Phase program, physical dynamics are formulated from the action SSS as the primary structure, with the phase ϕ=S/ℏ\phi = S/\hbarϕ=S/ℏ playing the role of the universal operational variable. Previous works have established the emergence of quantum dynamics, geometric structure, and gauge interactions as different modes of action upon the phase, as well as the closure of the relationship between spectrum, energy, and momentum. However, it remains to formulate intrinsically the notion of an observable and the measurement process, without the introduction of independent postulates. In this work, we show that this gap can be naturally filled through the spectral theory of self-adjoint operators. In particular, we demonstrate that physical observables are functionals of the spectral measure associated with the state and the relevant structural operator, while measurement corresponds to the effective restriction of this measure to dominant spectral regions. In this framework, the Born rule is recovered as a particular case in the discrete spectrum regime, while in more general situations, physical results are described by spectral regions with finite width. The proposed formulation eliminates the need for additional axioms for observables and measurement, unifying the dynamical and operational structure of quantum mechanics within the same fundamental principle based on action.
The concept of intrinsic and operational observables in quantum mechanics is introduced. In any realistic description of a quantum measurement that includes a macroscopic detecting device, it is possible to construct from the statistics of the recorded raw data a set of operational quantities that correspond to the intrinsic quantum-mechanical observable. This general approach is illustrated by the example of an operational measurement of the position and the momentum of a particle, as well as by an analysis of the operational detection of the phase of an optical field. For the latter we identify the intrinsic phase operator and report its explicit form.
Stochastic differential equations describing the Markovian evolution of state vectors in the quantum Hilbert space are studied as possible expressions of a universal dynamical principle. The general features of the considered class of equations as well as their dynamical consequences are investigated in detail. The stochastic evolution is proved to induce continuous dynamical reduction of the state vector onto mutually orthogonal subspaces. A specific choice, expressed in terms of creation and annihilation operators, of the operators defining the Markov process is then proved to be appropriate to describe continuous spontaneous localization of systems of identical particles. The dynamics obtained in such a way leaves practically unaffected the standard quantum evolution of microscopic systems and induces a very rapid suppression of coherence among macroscopically distinguishable states. The classical behavior of macroscopic objects as well as the reduction of the wave packet in a quantum measurement process can be consistently derived from the postulated universal dynamical principle.