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Physical Review A

Homodyne statistics

Samuel L. Braunstein

Published Jul 1, 1990• 80 citations
Publisher

Abstract

It is usually assumed that a balanced homodyne detector measures a variable called the quadrature phase amplitude of a signal field. We examine this assumption by obtaining analytic expressions for the probability statistics for the output of such a detector for a single-mode signal with both signal and local oscillator treated quantum mechanically. We investigate the conditions under which the statistics of the quadrature phase are reproduced in the actual output of the detector. We show that the most obvious condition---that the number of quanta in the local oscillator be much larger than the number in the signal---is not sufficient to ensure that a balanced homodyne detector acts like an ideal detector of quadrature phases. Furthermore, we obtain the explicit conditions that are necessary and sufficient to reproduce the interference features in the homodyne statistics of a superposition of coherent states.

DOI
10.1103/physreva.42.474

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It is usually assumed that a balanced homodyne detector measures a variable called the quadrature phase amplitude of a signal field. We examine this assumption by obtaining analytic expressions for the probability statistics for the output of such a detector for a single-mode signal with both signal and local oscillator treated quantum mechanically. We investigate the conditions under which the statistics of the quadrature phase are reproduced in the actual output of the detector. We show that the most obvious condition---that the number of quanta in the local oscillator be much larger than the number in the signal---is not sufficient to ensure that a balanced homodyne detector acts like an ideal detector of quadrature phases. Furthermore, we obtain the explicit conditions that are necessary and sufficient to reproduce the interference features in the homodyne statistics of a superposition of coherent states.
The balanced homodyne detector is the paradigm of phase-sensitive detection schemes in quantum optics. We study the effects of a quantum local oscillator in a coherent state for this detector when it is coupled to a single signal mode. We calculate explicit general expressions for the output statistics of the difference current of this detector (applicable even to non-classical states), and compare them to the statistics of the so-called quadrature-phase amplitude. We show that the naive condition that the number of quanta in the local oscillator be much larger than the number in the signal is not sufficient to ensure that the homodyne detector approximates an ideal detectorof the quadrature-phase amplitude, further, we derive explicit conditions for this approximation to be achieved when the signal is a coherent state or a superposition of coherent states.
We show that the full statistics of the outputs of two detectors in a balanced homodyne detection setup involving a local oscillator in an ideal coherent state are experimentally accessible despite the excess noise existing in actual laser sources. This possibility is illustrated using phase randomized coherent states’ signals from which the statistics of Fock states can accurately be obtained. The experimental verification of the recently predicted [Phys. Rev. A98, 013832 (2018)PLRAAN1050-294710.1103/PhysRevA.98.013832] two-detector correlation probability for Fock states |1⟩ and |2⟩ is presented.
We consider the moment operators of the observable (i.e. a semispectral measure or POM) associated with the balanced homodyne detection statistics, whilst paying attention to the correct domains of these unbounded operators. We show that the high amplitude limit, when performed on the moment operators, actually determines uniquely the entire statistics of a rotated quadrature amplitude of the signal field, thereby verifying the usual assumption that the homodyne detection achieves a measurement of that observable. We also consider, in a general setting, the possibility of constructing a measurement of a single quantum observable from a sequence of observables by taking the limit on the level of moment operators of these observables. In this context, we show that under some natural conditions (each of which is satisfied by the homodyne detector example), the existence of the moment limits ensures that the underlying probability measures converge weakly to the probability measures of the limiting observable. The moment approach naturally requires that the observables be determined by their moment operator sequences (which does not automatically happen), and it turns out, in particular, that this is the case for the balanced homodyne detector.
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.
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.
Using only elementary calculus we prove that the balanced homodyne detector measures the quadrature phase amplitude of a signal. More precisely, after the measurement of photon numbers lll, the collapse of the composite state of a strong laser beam and a signal approximates the collapse of the signal to the eigen-state of quadrature phase amplitude with eigen-value rrr.