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arXiv:1211.1701 · Physical Review A

Gaussification through decoherence

Paulina Marian, Iulia Ghiu, Tudor A. Marian

Published Jul 15, 2013• 23 citations
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Abstract

We investigate the loss of nonclassicality and non-Gaussianity of a single-mode state of the radiation field in contact with a thermal reservoir. The damped density matrix for a Fock-diagonal input is written using the Weyl expansion of the density operator. Analysis of the evolution of the quasiprobability densities reveals the existence of two successive characteristic times of the reservoir which are sufficient to assure the positivity of the Wigner function and, respectively, of the $P$ representation. We examine the time evolution of non-Gaussianity using three recently introduced distance-type measures. They are based on the Hilbert-Schmidt metric, the relative entropy, and the Bures metric. Specifically, for an $M$-photon-added thermal state, we obtain a compact analytic formula of the time-dependent density matrix that is used to evaluate and compare the three non-Gaussianity measures. We find a good consistency of these measures on the sets of damped states. The explicit damped quasiprobability densities are shown to support our general findings regarding the loss of negativities of Wigner and $P$ functions during decoherence. Finally, we point out that Gaussification of the attenuated field mode is accompanied by a nonmonotonic evolution of the von Neumann entropy of its state conditioned by the initial value of the mean photon number.

DOI
10.1103/physreva.88.012316

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We investigate the loss of nonclassicality and non-Gaussianity of a single-mode state of the radiation field in contact with a thermal reservoir. The damped density matrix for a Fock-diagonal input is written using the Weyl expansion of the density operator. Analysis of the evolution of the quasiprobability densities reveals the existence of two successive characteristic times of the reservoir which are sufficient to assure the positivity of the Wigner function and, respectively, of the PPP representation. We examine the time evolution of non-Gaussianity using three recently introduced distance-type measures. They are based on the Hilbert-Schmidt metric, the relative entropy, and the Bures metric. Specifically, for an MMM-photon-added thermal state, we obtain a compact analytic formula of the time-dependent density matrix that is used to evaluate and compare the three non-Gaussianity measures. We find a good consistency of these measures on the sets of damped states. The explicit damped quasiprobability densities are shown to support our general findings regarding the loss of negativities of Wigner and PPP functions during decoherence. Finally, we point out that Gaussification of the attenuated field mode is accompanied by a nonmonotonic evolution of the von Neumann entropy of its state conditioned by the initial value of the mean photon number.
The dynamical behavior of a photon-added thermal state (PATS) in a thermal reservoir is investigated by virtue of Wigner function (WF) and Wigner logarithmic negativity (WLN), where this propagation model is abstracted as an input–output problem in a thermal-loss channel. The density operator of the output optical field at arbitrary time can be expressed in the integration form of the characteristics function of the input optical field. The exact analytical expression of WF is given, which is closely related to the Laguerre polynomial and is dependent on the evolution time and other interaction parameters (related with the initial field and the reservoir). Based on the WLN, we observe the dynamical evolution of the PATS in the thermal reservoir. It is shown that the thermal noise will make the PATS lose the non-Gaussianity.
We investigate nonclassicality and decoherence of field states generated with the non-Gaussian operation of photon addition to a two-mode squeezed thermal state (TSTS) using the normally ordered density operator of the photon-added TSTS (PTSTS). The normalization factor of the PTSTS is just a Jacobi polynomial of the squeezing parameter γ and the average photon number n¯ of the thermal field. We show that the fields in such states exhibit remarkable quantum features, such as sub-Poissonian photon statistics, the anti-bunching effect, and photon-number distribution. The nonclassicality is discussed in phase space based on the partial negativity of the Wigner function (WF) with two-variable Hermite polynomials. The results show that the non-Gaussian WF always exhibits the quantum interference structure caused by the two-mode squeezing operation and has some negative regions for any value of n¯, γ, and the photon-addition numbers m, n. In addition, the effect of decoherence on the PTSTS in the thermal channel is studied by analytically deriving the time-evolution WF. We find that the WF becomes Gaussian (corresponding to a classical thermal state) at long times, losing its nonclassicality as a result of decoherence, and a larger m (or n) leads to a longer decoherence time κt.
The existence of positive well-defined quasiprobabilities is known to be related to the measure of nonclassicality for a given field state. We test the coincidence of the loss of well-known nonclassical properties with the appearance of the positive well-defined quasiprobabilities for various fields influenced by phase-insensitive reservoirs. For Gaussian fields the appearance of a positive Glauber-Sudarshan P function coincides with the loss of nonclassical properties in quadrature fluctuations and in photon statistics. We find that when an initial Fock state dissipates or is amplified the initial field loses its nonclassicality in photon statistics before the P function becomes positive, which is different from the Gaussian field. The Wigner function may be positive even when the decaying Fock state is sub-Poissonian. For the amplification of the Fock state, however, the Wigner function is positive only when the field is super-Poissonian. For the dissipation and amplification of the Fock state the point when a certain quasiprobability function becomes positive definite depends only on the average number of photons N of the heat bath. \textcopyright{} 1996 The American Physical Society.
We investigate non-Gaussianity properties for a set of classical one-mode states obtained by subtracting photons from a thermal state. Three distance-type degrees of non-Gaussianity used for these states are shown to have a monotonic behaviour with respect to their mean photon number. Decaying of their non-Gaussianity under damping is found to be consistently described by the distance-type measures considered here. We also compare the dissipative evolution of non-Gaussianity when starting from MMM-photon-subtracted and MMM-photon-added thermal states
We study the nonclassicality and decoherence in photon-loss channel of generalized squeezed Fock state, which is related to the squeezed operator with three real parameters. By using the Weyl ordering invariance under similarity transformations and the technique of integration within an ordered product of operators, we obtain the compact Wigner function of the generalized squeezed Fock state (Laguree-Gaussian function). According to the integration formula of the time evolution of the Wigner function, the Wigner expresssion in photon-loss channel is analytically deduced. In particular, we discussed the nonclassicality from the negativity of the Wigner function.
We analyze the polarization of a quantum radiation field under a de-Gaussification process. Specifically, we consider the addition of photons to a two-mode thermal state to get mixed non-Gaussian and nonclassical states which are still diagonal in the Fock basis. Stokes-operator-based degrees of polarization and two distance-type measures defined with Hilbert-Schmidt and Bures metrics are investigated. For a better insight we here introduce a polarization degree based on the relative entropy. Polarization of the thermal states is fully investigated and simple closed expressions are found for all the defined degrees. The evaluated degrees for photon-added states are then compared to the corresponding ones for the two-mode thermal states they originate from. We present interesting findings which tell us that some popular degrees of polarization are not fully consistent. However, the most solidly defined degrees, which are based on the Bures metric and relative entropy, clearly indicate an enhancement of polarization through de-Gaussification. This conclusion is supported by the behavior of the degrees of polarization of the Fock states, which are finally discussed as a limit case.