Quantum and Semiclassical Optics Journal of the European Optical Society Part B

Detection theory in quantum optics: stochastic representation

Alberto Barchielli, Anna Maria Paganoni

Published Feb 1, 1996• 27 citations

Abstract

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.

DOI: 10.1088/1355-5111/8/1/011

Related papers

Quantum Stochastic Calculus with Unbounded Coefficient

Ricardo Castro Santis

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.

Aug 12, 20101 citation

Quantum trajectories and quantum measurement theory

Howard M. Wiseman

Beyond their use as numerical tools, quantum trajectories can be ascribed a degree of reality in terms of quantum measurement theory. In fact, they arise naturally from considering continuous observation of a damped quantum system. A particularly useful form of quantum trajectories is as linear (but non-unitary) stochastic Schrödinger equations. In the limit where a strong local oscillator is used in the detection, and where the system is not driven, these quantum trajectories can be solved. This gives an alternative derivation of the probability distributions for completed homodyne and heterodyne detection schemes. It also allows the previously intractable problem of real-time adaptive measurements to be treated. The results for an analytically soluble example of adaptive phase measurements are presented, and future developments discussed.

arXiv:quant-ph/0302080Quantum and Semiclassical Optics Journal of the European Optical Society Part BFeb 1, 1996251 citations

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

Alberto Barchielli

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.

Quantum Optics Journal of the European Optical Society Part BDec 1, 199097 citations

Quantum Trajectories and Quantum Measurement Theory

Howard M. Wiseman

Beyond their use as numerical tools, quantum trajectories can be ascribed a degree of reality in terms of quantum measurement theory. In fact, they arise naturally from considering continuous observation of a damped quantum system. A particularly useful form of quantum trajectories is as linear (but non-unitary) stochastic Schrodinger equations. In the limit where a strong local oscillator is used in the detection, and where the system is not driven, these quantum trajectories can be solved. This gives an alternate derivation of the probability distributions for completed homodyne and heterodyne detection schemes. It also allows the previously intractable problem of real-time adaptive measurements to be treated. The results for an analytically soluble example of adaptive phase measurements are presented, and future developments discussed.

Feb 11, 2003127 citations

Quantum continuous measurements: The stochastic Schrödinger equations and the spectrum of the output

Alberto Barchielli, Matteo Gregoratti

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.

arXiv:1301.3626Quantum Measurements and Quantum MetrologyAug 13, 201319 citations

Stochastic Dynamics In Hilbert Space

Heinz‐Peter Breuer, Francesco Petruccione

Abstract This chapter contains the foundations of the stochastic formulation of continuous measurements carried out on an open quantum system. It shows that the time evolution of the state vector, conditioned on the measurement record, is given by a piecewise deterministic process in Hilbert space involving continuous time-evolution periods which are broken by randomly occurring, sudden quantum jumps. This so-called unravelling of the quantum master equation in the form of a stochastic process is based on a close relation between quantum dynamical semigroups and piecewise deterministic processes. The general theory is illustrated by means of a number of examples, such as direct, homodyne, and heterodyne photodetection. The chapter also treats the corresponding description of continuous measurements through stochastic Schr ö dinger equations and stochastic differential equations for the density matrix, as well as the stochastic representation of multitime correlations functions.

Jan 25, 20070 citations