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Physical Review A

Operational approach for reconstruction of quantum distributions in a preamplified homodyne-detection scheme

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

Published Nov 1, 1997• 13 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.56.4278

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Physical Review A•Oct 1, 1997•19 citations
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.
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.
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.
We propose the reconstruction of the quantum state of light from the homodyne photocounting statistics of a single, realistic photodetector. Contrary to the development of homodyning over the last decade, our approach is based on unbalanced detection with a weak local oscillator. Representing the quantum state in terms of s-parametrized quasiprobability distributions, the method even allows one to determine the Wigner function provided the quantum efficiency of the detector is sufficiently large. We show that perturbing effects due to classical noise of the local oscillator are small. \textcopyright{} 1996 The American Physical Society.
The experimental schemes for measuring quantum-mechanical phase properties of light suggested and partly also realized thus far, namely (i) amplification, with the help of a quantum amplifier, of the microscopic field before phase measurement, (ii) heterodyning the field with a strong local oscillator, and (iii) performing two separate homodyne measurements on the field after beam splitting, are compared from a theoretical point of view.They share the common feature that undesired noise enters the experimental setup, which makes the measurement fuzzy.It will be pointed out that all three schemes amount to measuring the Q function of the original field, and hence are fully equivalent.Since the Q function can be interpreted as a smoothed Wigner function, one may associate with the introduced noise a smoothing process in which intrinsically quantum-mechanical features displayed by finer details of the Wigner function -especially by the occurrence of negative values -are lost.As a consequence, the measured phase distribution will be broader than the "true" one based on the concept of a quantum-mechanical phase operator.In realistic experiments, the nonunit detection efficiency further deteriorates the measuring results.It will be shown that also this echect can be properly described by an (additional) smoothing process leading to a certain s-parametrized quasiprobability distribution, with a parameter s that is connected with the detection efficiency in a simple way, as the distribution that is actually measurable.
We provide a simple analytic relation that connects the density operator \ensuremath{\rho}^ of the electromagnetic field with the tomographic homodyne probabilities for generic quantum efficiency \ensuremath{\eta} of detectors. The problem of experimentally ``sampling'' a general matrix element 〈\ensuremath{\psi}\ensuremath{\Vert}\ensuremath{\rho}^\ensuremath{\Vert}\ensuremath{\Phi}〉 is addressed in the statistically rigorous sense of the central-limit theorem. We show that experimental sampling is possible also for nonunit efficiency, provided that \ensuremath{\eta} satisfies a lower bound related to the ``resolutions'' of vectors \ensuremath{\Vert}\ensuremath{\psi}〉 and \ensuremath{\Vert}\ensuremath{\Phi}〉 in the quadrature representations. For coherent and number states the bound is \ensuremath{\eta}\ensuremath{\gtrsim}1/2. On the basis of computer-simulated experiments we show the feasibility of detecting delicate quantum probability oscillations, which otherwise would be smeared out by inefficient detection.
A quantum state can be described by its quasiprobability functions or by its density matrix. The quasiprobability functions and density matrix for nonclassical fields have been measured by making use of homodyne detection schemes. We suggest direct photodetection of the amplified signal fields for the quasiprobability function measurements. Linear amplification is described in phase space by a convolution of the signal field to the idler field, which normally determines added noise during amplification. Taking the coherent idler field and using the convolution relation, we show that the direct photodetection of the amplified field results in a quasiprobability function for the signal field. In particular, the probability of there being no photons in the amplified field is the scaled QQQ function for the signal field.