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Journal of Physics A Mathematical and General

A piecewise deterministic process for homodyne measurement of an atom

Janusz Miśkiewicz

Published Dec 15, 1999• 1 citation
Publisher

Abstract

The piecewise deterministic process associated with event enhanced quantum theory is constructed in the case of homodyne detection of a two-level atom and the counting distribution is obtained. The non-Markovian property of the photocounting statistics is proved and the sub-Poissonian character of event registration established.
DOI
10.1088/0305-4470/33/1/305

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Quantum feedback can stabilize a two-level atom against decoherence (spontaneous emission), putting it into an arbitrary (specified) pure state. This requires perfect homodyne detection of the atomic emission, and instantaneous feedback. Inefficient detection was considered previously by two of us. Here we allow for a non-zero delay time tau\\tautau in the feedback circuit. Because a two-level atom is a nonlinear optical system, an analytical solution is not possible. However, quantum trajectories allow a simple numerical simulation of the resulting non-Markovian process. We find the effect of the time delay to be qualitatively similar to that of inefficient detection. The solution of the non-Markovian quantum trajectory will not remain fixed, so that the time-averaged state will be mixed, not pure. In the case where one tries to stabilize the atom in the excited state, an approximate analytical solution to the quantum trajectory is possible. The result, that the purity (P=2rmTr[rho2]−1P=2{\\rm Tr}[\\rho^{2}]-1P=2rmTr[rho2]−1) of the average state is given by P=1−4gammatauP=1-4\\gamma\\tauP=1−4gammatau (where gamma\\gammagamma is the spontaneous emission rate) is found to agree very well with the numerical results.
The statistics of the difference events in a balanced homodyne-detection scheme is studied without using the standard assumption of a strong, classical local oscillator. Starting from the quantum theory of photon counting, we derive the probability distribution for the difference events in the two detection channels. In the limit of the local oscillator being strong compared with the signal, the difference-number statistics tends to the statistics of the electric-field strength of the signal field. For weak signals, this limit may be obtained already far from a classical behavior of the local oscillator. While changing the local oscillator intensity, a transition of the observable concerning the signal field occurs from the photon-number difference towards the electric-field strength. Such a measurement scheme renders it possible to observe the quantum features of a coherent state or a single-photon state from the point of view of the field-strength statistics, a picture which is closely related to classical optics. The possibility to get some insight in the statistical properties of the phase difference of two microscopic fields is discussed.
The standard formalism of quantum theory is enhanced and definite meaning is given to the concepts of experiment, measurement and event. Within this approach one obtains a uniquely defined piecewise deterministic algorithm generating quantum jumps, classical events and histories of single quantum objects. The wave-function Monte Carlo method of Quantum Optics is generalized and promoted to the level of a fundamental process generating all the real events in Nature. The already worked out applications include SQUID-tank model and generalized cloud chamber model with GRW spontaneous localization as a particular case. Differences between the present approach and quantum measurement theories based on environment-induced master equations are stressed. Questions: what is classical, what is time, and what observers are addressed. Possible applications of the new approach are suggested, among them connection between the stochastic commutative geometry and Connes' noncommutative formulation of the Standard Model, as well as potential applications to the theory and practice of quantum computers.
We analyze a simultaneous continuous measurement of photon-counting and homodyne detection. The stochastic master equation or stochastic Schrödinger equation describing the measurement process includes both jump-type and diffusive-type stochastic increments. Analytic expressions of the wave function conditioned on homodyne and photon-counting records are obtained, yielding the probability density distributions and generating functions of the measurement records. Formula for the expectation values of the homodyne records conditioned on a photon-counting event is also derived which quantitatively describes the measurement backaction of photon-counting on the homodyne output. The obtained results are applied to typical initial states --- coherent, number, thermal, and squeezed states. Monte Carlo simulations of the measurement processes are also presented to demonstrate the dynamics of the combined measurement process.