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Physical Review A

Quantum state reconstruction and detection of quantum coherences on different observation levels

Vladimír Bužek, G. Adam, G. Drobný

Published Jul 1, 1996• 42 citations
Publisher

Abstract

We apply the Jaynes principle of maximum entropy [Phys. Rev. 106, 620 (1957); 108, 171 (1957)] for a reconstruction of Wigner functions of quantum-mechanical states of light on different observation levels. We study how quantum interference between components of superpositions of coherent states, which is responsible for the appearance of nonclassical effects, can be detected on different observation levels. We analyze in detail the reconstruction of Wigner functions of squeezed states on different observation levels in the case of nonunit detection efficiency modeled as a decay of the state under consideration into a zero-temperature reservoir. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreva.54.804

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We apply the Jaynes principle of maximum entropy [Phys. Rev. 106, 620 (1957); 108, 171 (1957)] for a reconstruction of Wigner functions of quantum-mechanical states of light on different observation levels. We study how quantum interference between components of superpositions of coherent states, which is responsible for the appearance of nonclassical effects, can be detected on different observation levels. We analyze in detail the reconstruction of Wigner functions of squeezed states on different observation levels in the case of nonunit detection efficiency modeled as a decay of the state under consideration into a zero-temperature reservoir. \textcopyright{} 1996 The American Physical Society.
We use the operational phase-space distributions and sampling entropies developed in the preceding paper [V. Bu\ifmmode \check{z}\else \v{z}\fi{}ek, C. H. Keitel, and P. L. Knight, Phys. Rev. A 51, 2575 (1995)] to discuss the nature of quantum interference between components of superpositions of states. We show how the Wehrl entropy, a special case of the sampling entropy, is a useful discriminator between different kinds of superpositions and of statistical mixtures, and is determined essentially by the coherent-state content. Apart from interference terms, this content is given by the quantum uncertainty of a single coherent state and the classical contribution to the number of coherent states necessary to tile the dominant phase-space ``patch'' representing the quantum state of interest. We illustrate these ideas using nonclassical superpositions of coherent states, where interference modifies the phase-space distributions, and show how these features are sensitive to dissipation.
Using the Wigner function formalism in phase space, we analyze the decay of quantum coherences in phase-sensitive reservoirs. We show that the decay rate of quantum coherences in phase-sensitive reservoirs can be significantly modified compared to the decay rate in ordinary (phase-insensitive) thermal reservoirs. Depending on the phases of the quantum system (field mode) and the squeezed reservoir, the decay rate of the quantum coherence can be either enhanced or significantly suppressed, which is in agreement with the results obtained recently by other methods [T. A. B. Kennedy and D. F. Walls, Phys. Rev. A 37, 152 (1988)]. We show that in an ideally squeezed reservoir with a high degree of squeezing, the decay rate of the quantum coherence (i.e., the decay rate of off-diagonal terms of the density matrix in the coherent-state basis) can be equal to the decay rate of the energy of the system (i.e., the decay rate of diagonal terms of the density matrix). Suppression of the decay rate of the quantum coherence leads to preservation of nonclassical effects such as the oscillations in the photon number distribution. Moreover, we find that some initial superposition states of light exhibiting super-Poissonian photon statistics can be transformed into intermediate sub-Poissonian states under the influence of phase-sensitive reservoirs.
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.
Summary form only given.We show experimentally the reconstructions of Wigner functions for the output states of a laser. These states have classical Wigner functions with Gaussian cross-sections. In order to investigate the spectral properties of the output states, we normalize the Wigner functions in the inverse Radon transform algorithm to a coherent state. In this way, we are able to compare the full spectrum of Wigner functions obtained with different detection frequencies, paying particular attention to Wigner functions near the laser relaxation oscillation. The same normalized Wigner function is also used to measure squeezed optical states from an optical parametric oscillator. When large modulation is introduced to the laser beam, a dc component is introduced in the Wigner function. We show that Wigner functions of incoherent superposition states can be reconstructed using this modulation technique. In some cases, fluctuations can be large enough to produce highly non-Gaussian Wigner functions. We provide examples of such cases and show that optical homodyne tomography can provide essentially new information not detectable with standard quadrature variance measurements. Furthermore, we demonstrate that single sideband analysis used to date will sometimes no longer suffice in the analysis of such optical states. We include the higher order sideband correlations experimentally and discuss the necessity of these higher order sidebands for measuring non-Gaussian quantum states, e.g. Schrodinger's cat and Fock states.