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Physical Review A

Quantum measurement theory of optical heterodyne detection

Gerard J Milburn

Published Dec 1, 1987• 31 citations
Publisher

Abstract

An analysis is presented of optical heterodyne detection of an intracavity field within the context of quantum measurement theory. We find the intracavity density operator becomes diagonal in the basis which diagonalizes the measured quantity, a quadrature phase amplitude of the field. This representation is the ``pointer basis,'' and this result is the continuous-measurement equivalent of state reduction. The model illustrates a general feature of continuous measurement; one parameter given by the product of the system and measuring device coupling bandwidth and the fluctuations in the measuring device completely characterizes the measurement. This constant determines both the rate of diagonalization of the density operator and the rate of growth of fluctuations in the system quantity conjugate to the measured observable.

DOI
10.1103/physreva.36.5271

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quant-pharXiv:2512.07319•Dec 9, 2025•1 citation
An analysis is presented of optical heterodyne detection of an intracavity field within the context of quantum measurement theory. We find the intracavity density operator becomes diagonal in the basis which diagonalizes the measured quantity, a quadrature phase amplitude of the field. This representation is the ``pointer basis,'' and this result is the continuous-measurement equivalent of state reduction. The model illustrates a general feature of continuous measurement; one parameter given by the product of the system and measuring device coupling bandwidth and the fluctuations in the measuring device completely characterizes the measurement. This constant determines both the rate of diagonalization of the density operator and the rate of growth of fluctuations in the system quantity conjugate to the measured observable.
A specific measurement model based on quantum nondemolition monitoring of oscillator quanta is used to illustrate the essential elements of the quantum theory of measurement, viz., state reduction and existence of a ``pointer basis,'' recently discussed by Zurek. The form of the interaction between the meter and its environment determines the system observable recorded by the meter. In our model this quantity is the square of the oscillator's quanta. The corresponding pointer basis states are the meter coherent states. The more accurate the measurement the more excited (and classical) these states become. We show explicitly that the state reduction of the system during measurement is due to nonunitary evolution resulting from the meter-environment coupling. A particular realization of the environment as a photoelectron counter is discussed.
This paper presents a general theory for a continuous quantum-nondemolition measurement of photon number. This theory treats a time-distributed measurement as a sequence of measurements in which at most one photon can be detected in an infinitesimal time, and shows that the average number of photons remaining in the measured field increases when a photon is detected and decreases when no photon is detected. The state of the measured system evolves nonunitarily and reduces continuously to a nunber state whose eigenvalue is uniquely determined by the average rate of photodetection and whose probability distribution coincides with the initial photon-number distribution. Applying the general theory to typical quantum states---coherent, thermal, and squeezed states---shows that the continuous-state reduction towards a number state depends strongly on the initial photon statistics. Despite the nonunitarity of state evolution, an initially pure state keeps its purity: the initial density operator becomes diagonalized only if the readout information is discarded.
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.
We discuss a balanced homodyne detection scheme with imperfect detectors in the framework of the operational approach to quantum measurement. We show that a realistic homodyne measurement is described by a family of operational observables that depends on the experimental setup, rather than a single field quadrature operator. We find an explicit form of this family, which fully characterizes the experimental device and is independent of a specific state of the measured system. We also derive operational homodyne observables for the setup with a random phase, which has been recently applied in an ultrafast measurement of the photon statistics of a pulsed diode laser. The operational formulation directly gives the relation between the detected noise and the intrinsic quantum fluctuations of the measured field. We demonstrate this with two examples: the operational uncertainty relation for the field quadratures, and the homodyne detection of suppressed fluctuations in photon statistics.
The quantum limits on simultaneous phase and squared-amplitude measurements made via optical heterodyne detection on a single-mode radiation field are established. The analysis proceeds from a fully quantum mechanical treatment of heterodyning with ideal photon detectors. A high mean field uncertainty principle is proven for simultaneous phase and squared-amplitude observations under the condition that the signal and image band states are independent, and the image band has zero mean. Operator representations are developed which show that no such principle applies when arbitrary signal/image band dependence is permitted, although the mean observations are no longer functions of the signal field alone. A multimode two-photon coherent state illustrating this behavior at finite energy is exhibited. Potential applications for the resulting improved accuracy measurements are briefly described.
We present a high-speed continuous-variable quantum random number generator (QRNG) based on heterodyne detection of vacuum fluctuations. The scheme follows a source-device-independent (SDI) security model in which the entropy originates from quantum measurement uncertainty and no model of the source is required; security depends only on the trusted measurement device and the calibrated discretization, and thus remains valid even under adversarial state preparation. The optical field is split by a 90∘^\circ∘ optical hybrid and measured by two balanced photodiodes to obtain both quadratures of the vacuum state simultaneously. The analog outputs are digitized using a dual-channel 12-bit analog-to-digital converter operating at a sampling rate of 3.2 GS/s per channel, and processed in real time by an FPGA implementing Toeplitz hashing for randomness extraction. The quantum-to-classical noise ratio was verified through calibrated power spectral density measurements and cross-checked in the time domain, confirming vacuum-noise dominance within the 1.6 GHz detection bandwidth. After extraction, the system achieves a sustained generation rate of Rnet=33.92 Gbit/sR_{\rm net}= 33.92~\mathrm{Gbit/s}Rnet​=33.92 Gbit/s of uniformly distributed random bits, which pass all NIST and Dieharder statistical tests. The demonstrated platform provides a compact, FPGA-based realization of a practical heterodyne continuous-variable source-independent QRNG suitable for high-rate quantum communication and secure key distribution systems.