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Journal of Modern Optics

Correlations and Excess Noise in One-port and Balanced Two-port Homodyne Detection

A. Bandilla

Published Dec 1, 1990• 1 citation
Publisher

Abstract

The fluctuations in the processes of one-port and balanced two-port homodyning are calculated in the operator picture for a weak signal and a strong local oscillator beam where the last is contaminated with Gaussian excess noise. For such a mixture of coherent and chaotic light, we construct a physical model with the help of a linear laser amplifier. To be able to measure phase-dependent quantum fluctuations like squeezing with one-port homodyning, demands a Poisson-like photon distribution of the local oscillator and stringent requirements on its intensity. The Poisson-like photon distribution can also be achieved by a very low reflectivity which is equivalent to strong attenuation. These conditions are relaxed for balanced two-port homodyning but certain less stringent consequences of the excess noise are derived to occur.

DOI
10.1080/09500349014552151

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The fluctuations in the processes of one-port and balanced two-port homodyning are calculated in the operator picture for a weak signal and a strong local oscillator beam where the last is contaminated with Gaussian excess noise. For such a mixture of coherent and chaotic light, we construct a physical model with the help of a linear laser amplifier. To be able to measure phase-dependent quantum fluctuations like squeezing with one-port homodyning, demands a Poisson-like photon distribution of the local oscillator and stringent requirements on its intensity. The Poisson-like photon distribution can also be achieved by a very low reflectivity which is equivalent to strong attenuation. These conditions are relaxed for balanced two-port homodyning but certain less stringent consequences of the excess noise are derived to occur.
The process of two-port homodyning is calculated for a coherent signal and the subtraction of the photocurrents is modelled by double-mode two-photon absorption until the stationary state is attained. This allows a simple physical interpretation and can be treated analytically for a coherent local oscillator field. In addition, it is shown how the excess noise of a chaotic local oscillator is balanced out. The shot-noise level is given by the square root of the local oscillator photon number. This follows explicitly for a vanishing signal but is also derived for a present signal field.
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.
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.
A simple but rigorous analysis of the important sources of noise in homodyne detection is presented. Output noise and signal-to-noise ratios are compared for direct detection, conventional (one-port) homodyning, and two-port homodyning, in which one monitors both output ports of a 50-50 beam splitter. It is shown that two-port homodyning is insensitive to local-oscillator quadrature-phase noise and hence provides (1) a means of detecting reduced quadrature-phase fluctuations (squeezing) that is perhaps more practical than one-port homodyning and (2) an output signal-to-noise ratio that can be a modest to significant improvement over that of one-port homodyning and direct detection.
The eight-port homodyne detector is an optical circuit designed to perform the monitoring of two quadratures of an optical field: the signal. By using quantum Bose fields and quantum stochastic calculus, we give a complete quantum description of this apparatus, when used as quadrature detector in continuous time. We can treat either the traveling waves in the optical circuit or the observables involved in the detection part: two couples of photodiodes, postprocessing of the output currents\dots{}. The analysis includes imperfections, such as not perfectly balanced beam splitters, detector efficiency, electronic noise, phase and intensity noise in the laser acting as local oscillator; this last noise is modeled by using mixtures of field coherent states as statistical operator of the laser component. Due to the monitoring in continuous time, the output is a stochastic process and its full probability distribution is obtained. When the output process is sampled at discrete times, the quantum description can be reduced to discrete mode operators, but at the price of having random operators, which contain also the noise of the local oscillator. Consequently, the local oscillator noise has a very different effect on the detection results with respect to an additive noise, such as the noise in the electronic components. As an application, the problem of secure random-number generation is considered, based on the local oscillator shot noise. The rate of random bits that can be generated is quantified by the min-entropy; the possibility of classical and quantum side information is taken into account by suitable conditional min-entropies. The final rate depends on which parts of the apparatus are considered to be secure and on which ones are considered to be exposed to the intervention of an intruder. In some experimentally realistic situations, the entropy losses are computed, depending on the values of the parameters quantifying the imperfections.
The theory of balanced homodyne and heterodyne detection is developed for inputs in which the signal field is in an arbitrary quantum state and the local-oscillator field is in a highly excited coherent state. Exact expressions are derived for the photocount moment-generating functions in the special case of a coherent signal. For more general signals, the first two moments of the photocount probability distribution are determined. The moments are evaluated for the examples of a coherent signal with a chaotic noise component, and for squeezed light derived from a degenerate and from a non-degenerate parametric amplifier. The corresponding moments for direct detection are obtained so that comparisons can be made. The Kelley-Kleiner photocount distribution formula is adapted to balanced detection schemes. Light beams are characterized throughout by their energy fluxes, and the theory accordingly describes steady-state experiments.