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Physical Review A

Phase-sensitive reservoir modeled by beam splitters

Myung Shik Kim, Nobuyuki Imoto

Published Sep 1, 1995• 59 citations
Publisher

Abstract

The superposition of input fields in a lossless beam splitter is studied in the Schr\"odinger picture by using the convolution of the positive P representations, and the convolution law for these representations is extended to other quasiprobability functions such as the Wigner and Q functions. We show that the reservoir can be modeled by an infinite array of beam splitters, and we use the convolution law and this model to derive the Fokker-Planck equation for a system coupled with a phase-sensitive reservoir. Solving this equation shows that a phase-sensitive attenuation and amplification can be described by the superposition of two independent quantum fields, one of which is the initial signal field and the other the squeezed thermal noise field representing the reservoir.

DOI
10.1103/physreva.52.2401

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The superposition of input fields in a lossless beam splitter is studied in the Schr"odinger picture by using the convolution of the positive P representations, and the convolution law for these representations is extended to other quasiprobability functions such as the Wigner and Q functions. We show that the reservoir can be modeled by an infinite array of beam splitters, and we use the convolution law and this model to derive the Fokker-Planck equation for a system coupled with a phase-sensitive reservoir. Solving this equation shows that a phase-sensitive attenuation and amplification can be described by the superposition of two independent quantum fields, one of which is the initial signal field and the other the squeezed thermal noise field representing the reservoir.
We show that the reservoir influence can be modeled by an infinite array of beam splitters. The superposition of the input fields in the beam splitter is discussed with the convolution laws for their quasiprobabilities. We derive the Fokker-Planck equation for the cavity field coupled with a phase-sensitive reservoir using the convolution law. We also analyze the amplification in the phase-sensitive reservoir with use of the modified beam splitter model. We show the similarities and differences between the dissipation and amplification models. We show that a super-Poissonian input field cannot become sub-Poissonian by the phase-sensitive amplification.
A lossless beam splitter (a dielectric interface, a passive interferometer, or a linear coupler) changes the quantum state of two incident modes by an SU(2) transformation. Apart from phase shifting, the argument of the quadrature wave function of the system undergoes a rotation. Quasiprobabilities are changed by the inverse mode transformation. The use of balanced beam splitting allows the simultaneous measurement of conjugate quadrature components via homodyning the emerging beams with two strong coherent reference fields that differ in their phases by \ensuremath{\pi}/2. The measured probability distribution is given by a generalized Q function. It depends on the state of the field entering the second beam-splitter port. For a vacuum, the Q function will be obtained. The use of unbalanced beam splitting allows the measurement of a squeezed Q function without using squeezed states. Dissipation in Gaussian reservoirs corresponds exactly to a heuristic beam-splitter model. As a mathematical tool, the Fokker-Planck equation of damping in phase-sensitive reservoirs and the corresponding quantum master equation were solved. The dissipative decay of a Schr\"odinger-cat state was studied as an example. The sensitivity of quantum coherence with respect to damping can be interpreted geometrically.
We make a comparative study of various quasiprobability distributions in phase-sensitive quantum-optical systems. Starting from a general, linear master equation for the field, which emerges in different models of correlated-emission lasers, we derive the Fokker-Planck equations in the Glauber-Sudarshan P, the antinormal ordering Q, the Wigner W, the complex P, and the positive P representations and find the steady-state solutions for the five distributions. Simple relations between the complex and positive P functions are discovered for the first time. Various moments calculated by using these distributions are found to be identical, as expected. An application of these distributions to the two-photon correlated-emission laser shows that the intracavity field can be near-perfectly squeezed in the phase quadrature and the maximum quadrature squeezing is reached when the mean laser amplitude vanishes.
We study phase properties of quantum mechanical superposition states decaying into phase-sensitive reservoirs. We show that the Wigner phase quasiprobability distribution reflects the quantum interference effects very well while the Pegg–Barnett and the Husimi phase probability distributions are quite insensitive with respect to the character of the quantum interference between component states. We study in detail phase properties of the even coherent state which decays into the thermal and the squeezed reservoirs. In thermal reservoirs the phase of the superposition state becomes randomized under the influence of thermal fluctuations. The higher the number of thermal photons the faster the randomization process is. On the other hand we show that depending on the relative phase between the superposition state and the phase-sensitive reservoir the process of phase randomization can be either enhanced or completely suppressed.
Wigner-positive quantum states have the peculiarity to admit a Wigner function that is a genuine probability distribution over phase space. The Shannon differential entropy of the Wigner function of such states—called Wigner entropy for brevity—emerges as a fundamental information-theoretic measure in phase space and is subject to a conjectured lower bound, reflecting the uncertainty principle. In this work, we prove that this Wigner entropy conjecture holds true for a broad class of Wigner-positive states known as beam-splitter states, which are obtained by evolving a separable state through a balanced beam splitter and then discarding one mode. Our proof relies on known bounds on the p -norms of cross Wigner functions and on the interference formula, which relates the convolution of Wigner functions to the squared modulus of a cross Wigner function. Originally discussed in the context of signal analysis, the interference formula is not commonly used in quantum optics although it unveils a strong symmetry under convolution exhibited by Wigner functions of pure states. We provide here a simple proof of the formula and highlight some of its implications. Finally, we prove an extended conjecture on the Wigner-Rényi entropy of beam-splitter states, albeit in a restricted range for the Rényi parameter α ≥ 1 / 2 .