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Journal of the Physical Society of Japan

Nonclassicality of Photon-Added Displaced Thermal State via Quantum Phase-Space Distributions

Ran Zhang, Xiang‐Gao Meng, Chuan-Xun Du, Ji‐Suo Wang

Published Jan 22, 2018• 13 citations
Publisher

Abstract

We introduce a new kind of nonclassical mixed state generated by adding arbitrary photons to a displaced thermal state, i.e., the photon-added displaced thermal state (PADTS), and obtain the normalization factor, which is simply related to two-variable Hermite polynomials. We also discuss the nonclassicality of the PADTS by considering quantum phase-space distributions. The results indicate that the value of the photon count statistics is maximum when the number of detected photons is equal to the number of added photons, and that the photon-added operation has a similar modulation effect with increasing displacement. Moreover, the negative volume of the Wigner function for the PADTS takes a maximal value for a specific photon-added number.

DOI
10.7566/jpsj.87.024001

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We introduce a new kind of nonclassical mixed state generated by adding arbitrary photons to a displaced thermal state, i.e., the photon-added displaced thermal state (PADTS), and obtain the normalization factor, which is simply related to two-variable Hermite polynomials. We also discuss the nonclassicality of the PADTS by considering quantum phase-space distributions. The results indicate that the value of the photon count statistics is maximum when the number of detected photons is equal to the number of added photons, and that the photon-added operation has a similar modulation effect with increasing displacement. Moreover, the negative volume of the Wigner function for the PADTS takes a maximal value for a specific photon-added number.
In this paper, the decoherence features of the photon-added displaced thermal state (PADTS) in the amplitude decay channel were studied by analytically and numerically investigating the time evolution of the Wigner function. The normal product representation of density for the PADTS is presented. The results indicate that the nonclassicality of PADTS can revive after disappearing with time. This kind of revival can maintain for a period of time and inevitably disappear eventually. However, we find that the operation of adding photons has the effect of reducing the decoherence rate of PADTS in the amplitude decay channel, although it also can decrease the PADTS’s nonclassicality markedly.
Theoretical analysis is given of nonclassicality and decoherence of the field states generated by adding any number of photons to the squeezed thermal state (STS). Based on the fact that the squeezed number state can be considered as a single-variable Hermite polynomial excited state, the compact expression of the normalization factor is derived, a Legendre polynomial. The nonclassicality is investigated by exploring the sub-Poissonian and negative Wigner function (WF). The results show that the WF of single-photon–added STS (PASTS) always has negative values at the phase space center. The decoherence effect on PASTS is examined by the analytical expression of WF. It is found that a longer threshold value of decay time than in single-photon–subtraction STS is included in single PASTS.
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.
We investigate the nonclassical properties of multiple-photon-added displaced squeezed thermal states (PADSTSs). Particularly, we study the dependence of those nonclassical properties on the compound phase ϕ−θ/2 involved in squeezing and displacement parameters. We find that the Mandel Q parameter, the quadrature squeezing, the negative volume of the Wigner function, and the fidelity between PADSTSs and displaced squeezed thermal states (DSTSs) are all periodic functions of ϕ−θ/2 with a period π. Without considering the displacement parameter, these quantities are all phase independent. Finally, we investigate the non-Gaussianity of PADSTSs according to the fidelity between PADSTSs and DSTSs and find that non-Gaussianity and nonclassicality have similar behavior. Thus, the non-Gaussianity induced by the photon-addition operation is essentially nonclassical.
Using the normally ordered form of thermal state characteristic of average photon number nc, we introduce the photon-added thermal state (PATS) and investigate its statistical properties, such as Mandel's Q-parameter, photon number distribution (PND), and Wigner function (WF). We then study its decoherence in a thermal environment with average thermal photon number [Formula: see text] and dissipative coefficient κ by deriving analytical expressions of the WF. The nonclassicality is discussed in terms of the negativity of the WF. It is found that the WF is always positive when [Formula: see text] for any number PATS. The expression for time evolution of the PND and the tomogram of PATS are also derived analytically, which are related to hypergeometric function and single variable Hermite polynomials.
We studied the nonclassicality of photon-added squeezed vacuum (PASV) and its decoherence in thermal environment in terms of the sub-Poissonian statistics and the negativity of the Wigner function (WF). By converting the PASV to a squeezed Hermite polynomial excitation state, we derive a compact expression for the normalization factor of m-PASV, which is an m-order Legendre polynomial of squeezing parameter r. We also derive the explicit expression of WF of m-PASV and find the negative region of WF in phase space. We show that there is an upper bound value of r for this state to exhibit sub-Poissonian statistics increasing as m increases. Then we derive the explicit analytical expression of time evolution of WF of m-PASV in the thermal channel and discuss the loss of nonclassicality using the negativity of WF. The threshold value of decay time is presented for the single PASV.