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Physical Review A

Signal-to-noise ratio of preamplified homodyne detection in quantum tomography

Marcelo A. Marchiolli, Salomon Sylvain Mizrahi, V. V. Dodonov

Published May 1, 1998• 9 citations
Publisher

Abstract

The operational theory of homodyne detection by nonideal detectors, used in quantum tomography, was recently modified in order to incorporate the preamplification (before homodyne detection) of the input signal, thus enabling one to beat the handicap of the lower than 0.5 detector efficiency. In the present work we set expressions for the Mandel $Q$ parameter and the signal-to-noise ratio in terms of the operational (measured) moments of the preamplified homodyne detection formalism. These quantities furnish important information on the statistical properties of the input signal field and the photocounts at the output. We illustrate the theory by considering several kinds of fields (for the input signal) and determine the effects of the preamplification on the output signal-to-noise ratios. Here we essentially verify that (i) the preamplification shifts the statistics towards the super-Poissonian limit, without jeopardizing the capacity of reconstructing a sub-Poissionan input signal, and (ii) the preamplification is more effective, i.e., the rate of increase of the signal-to-noise ratio of the output photocount is larger for low-efficiency detectors than for ideal ones.

DOI
10.1103/physreva.57.3885

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The operational theory of homodyne detection by nonideal detectors, used in quantum tomography, was recently modified in order to incorporate the preamplification (before homodyne detection) of the input signal, thus enabling one to beat the handicap of the lower than 0.5 detector efficiency. In the present work we set expressions for the Mandel QQQ parameter and the signal-to-noise ratio in terms of the operational (measured) moments of the preamplified homodyne detection formalism. These quantities furnish important information on the statistical properties of the input signal field and the photocounts at the output. We illustrate the theory by considering several kinds of fields (for the input signal) and determine the effects of the preamplification on the output signal-to-noise ratios. Here we essentially verify that (i) the preamplification shifts the statistics towards the super-Poissonian limit, without jeopardizing the capacity of reconstructing a sub-Poissionan input signal, and (ii) the preamplification is more effective, i.e., the rate of increase of the signal-to-noise ratio of the output photocount is larger for low-efficiency detectors than for ideal ones.
We use the operational theory of homodyne detection, introduced by Banaszek and W\'odkiewicz [Phys. Rev. A 55, 3117 (1997)], to express a generic quantum phase space quasiprobability distribution (in the sense of Cahill and Glauber) of an electromagnetic field in terms of the photocount moments. We adapt the method of degenerate parametric (pre)amplification of a signal, within the operational theory, in order to overcome the drawback of nonunit efficiency of the detector. Simulations show that as the gain parameter of the amplification is increased, the quasiprobability distribution goes closer to the original one, even for an efficiency lower than 0.5. We also express the mean value of an arbitrary observable in terms of the same photocount moments.
A novel experimental scheme is proposed that allows us to avoid the deterioration of homodyne detection measurements due to nonideal detectors. The basic idea is to "preamplify" the signal by means of antisqueezing. Experimentally, we would employ a squeezer, e.g., a degenerate optical parametric amplifier, that squeezes just the nonobserved quadrature component of the electric field while antisqueezing the conjugate component which is measured. It is shown that for sufficiently strong antisqueezing one achieves the same measurement accuracy as with perfectly efficient detectors. In particular, in this way the actual Wigner function can be reconstructed in cptical homodyne tomography.
Recently, there has been some controversy concerning the role of signal-to-noise ratio (SNR) calculations in conjunction with optical amplifiers and phase-shift keying. Analysis has shown that the large-signal SNR at the output of an optical parametric amplifier (OPA) or optical parametric oscillator (OPO) can exceed the shot-noise limit when an injected signal beam is used to elicit small-signal gain saturation. Although no small-signal SNR gain is obtained in this way, the large-signal SNR improvement does translate into a communication performance improvement if binary phase-shift keying is applied to the resulting output beam. Analysis has also shown that the large-signal, shot-noise limited SNR can be exceeded when a binary phase-shift keyed coherent-state beam is applied to a preamplify-feedforward-homodyne receiver. In this case, however, there is no communication performance improvement. Indeed, the preamplify-feed-forward-homodyne receiver requires twice as many signal photons, on average, to equal the error probability of ordinary homodyne detection. In this talk we will show that there is no contradiction between these two examples.
Optical homodyne tomography consists in reconstructing the quantum state of an optical field from repeated measurements of its amplitude at different field phases (homodyne data). The experimental noise, which unavoidably affects the homodyne data, leads to a detection efficiency η<1\eta<1η<1 . The problem of reconstructing quantum states from noisy homodyne data sets prompted an intense scientific debate about the presence or absence of a lower homodyne efficiency bound ( η>0.5\eta> 0.5η>0.5 ) below which quantum features, like quantum interferences, cannot be retrieved. Here, by numerical experiments, we demonstrate that quantum interferences can be effectively reconstructed also for low homodyne detection efficiency. In particular, we address the challenging case of a Schrödinger cat state and test the minimax and adaptive Wigner function reconstruction technique by processing homodyne data distributed according to the chosen state but with an efficiency η<0.5\eta< 0.5η<0.5 . By numerically reproducing the Schrödinger’s cat interference pattern, we give evidence that quantum state reconstruction is actually possible in these conditions, and provide a guideline for handling optical tomography based on homodyne data collected by low efficiency detectors.
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.