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Quantum Optics Journal of the European Optical Society Part B

Direct and heterodyne detection and other applications of quantum stochastic calculus to quantum optics

Alberto Barchielli

Published Dec 1, 1990• 97 citations
Publisher

Abstract

Quantum stochastic calculus is an operator analogue of classical Ito's stochastic calculus which was originally developed for treating quantum noise. However, it turned out to be also useful in other cases of open systems, as in the case of measurement theory in quantum mechanics and in the treatment of quantum input and output channels. In the present paper, this calculus is used for developing a theory of direct, homodyne and heterodyne detection in quantum optics.

DOI
10.1088/0954-8998/2/6/002

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Quantum stochastic calculus is an operator analogue of classical Ito's stochastic calculus which was originally developed for treating quantum noise. However, it turned out to be also useful in other cases of open systems, as in the case of measurement theory in quantum mechanics and in the treatment of quantum input and output channels. In the present paper, this calculus is used for developing a theory of direct, homodyne and heterodyne detection in quantum optics.
Within the quantum theory of measurements continuous in time, a photon detection theory was formulated by using quantum stochastic calculus; this is a purely quantum formulation, where the usual notions of quantum mechanics appear: the dynamics is given by unitary operators and the observables are represented by commuting self-adjoint operators. In this paper we show how this theory can be equivalently developed by means of classical stochastic differential equations for vectors in a Hilbert space and for trace-class operators. This second formulation is linked to Belavkin's equation for a posteriori states and to quantum trajectory theory. A great part of this paper is dedicated to proving this equivalence between the purely quantum formulation and the stochastic one. The theory of direct and heterodyne detection is developed, with emphasis on the fact that the theory of heterodyne detection can be obtained as a limiting case (strong local oscillator) from the theory of direct detection. We discuss also the connections between the quantum Monte Carlo wavefunction method and some of the stochastic equations which appear in the theory of direct detection.
Quantum Open System Theory is nowadays a highly developed field which involves, on the mathematical side, many mathematical concepts such as operator theory, semigroups of operators on certain Banach spaces (algebras), classical and quantum stochastic differential equations. Inside Quantum Open System Theory there is the theory of quantum continual measurements: a quantum system is monitored with continuity in time. This is surely an open system because, apart from possible interactions with the external environment like a thermal bath, the system is interacting with the measuring apparatus. The typical field in which a theory of continual measurements is needed is quantum optics: when light emitted by some system is observed (by counting of photons - direct detection - or through some interference mechanism - heterodyne and homodyne detection) a continual measurement is performed.
Quantum noise in optical heterodyning and homodyning, usually analyzed under the assumption that the photodetector output is conditionally Poisson,1,2 is frequently supposed to arise from local oscillator (LO) shot noise. However, a fully quantum mechanical analysis from Yuen and Shapiro3 has shown that it actually arises from the signal quantum fluctuation; in particular homodyning and heterodyning are realizations of abstract quantum measurements of the field quadratures. We analyze the effect of LO noise, both quantum and excess noise, on homodyne and heterodyne detection. For conventional input light, i.e., fields in coherent states or their random superpositions, we demonstrate how a conditionally Poisson calculation would yield the correct answer but incorrect interpretation for the detection quantum noise.
Abstract The stochastic Schrödinger equation, of classical or quantum type, allows to describe open quantum systems under measurement in continuous time. In this paper we review the link between these two descriptions and we study the properties of the output of the measurement. For simplicity we deal only with the diffusive case. Firstly, we discuss the quantum stochastic Schrödinger equation, which is based on quantum stochastic calculus, and we show how to transform it into the classical stochastic Schrödinger equation by diagonalization of suitable commuting quantum observables. Then, we give the a posteriori state, the conditional system state at time t given the output up to that time, and we link its evolution to the classical stochastic Schrödinger equation. Moreover, the relation with quantum filtering theory is shortly discussed. Finally, we study the output of the continuous measurement, which is a stochastic process with probability distribution given by the rules of quantum mechanics. When the output process is stationary, at least in the long run, the spectrum of the process can be introduced and its properties studied. In particular we show how the Heisenberg uncertainty relations give rise to characteristic bounds on the possible spectra and we discuss how this is related to the typical quantum phenomenon of squeezing. We use a simple quantum system, a two-level atom stimulated by a laser, to discuss the differences between homodyne and heterodyne detection and to explicitly show squeezing and anti-squeezing of the homodyne spectrum and the Mollow triplet in the fluorescence spectrum.
Quantum-mechanical calculations of the mean-square fluctuation spectra in optical homodyning and heterodyning are made for arbitrary input and local-oscillator quantum states. In addition to the unavoidable quantum fluctuations, it is shown that excess noise from the local oscillator always affects homodyning and, when it is broadband, also heterodyning. Both the quantum and the excess noise of the local oscillator can be eliminated by coherent subtraction of the two outputs of a 50-50 beam splitter. This result also demonstrates the fact that the basic quantum noise in homodyning and heterodyning is signal quantum fluctuation, not local-oscillator shot noise.