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Chinese Physics B

Wigner function and density operator of the photon-subtracted squeezed thermal state

Liyun Hu, Fan Hong‐Yi

Published Nov 1, 2009• 30 citations
Publisher

Abstract

By using the technique of integration within an ordered product of operators, the normal ordered density operator of the photon-subtracted squeezed thermal state (PSSTS) is derived. Then the corresponding Wigner function is presented by using the coherent state representation of the Wigner operator. The nonclassical properties of the PSSTS are discussed based on the negativity of the Wigner function.

DOI
10.1088/1674-1056/18/11/008

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By using the technique of integration within an ordered product of operators, the normal ordered density operator of the photon-subtracted squeezed thermal state (PSSTS) is derived. Then the corresponding Wigner function is presented by using the coherent state representation of the Wigner operator. The nonclassical properties of the PSSTS are discussed based on the negativity of the Wigner function.
We investigate nonclassical properties of the field states generated by subtracting any number of photons from the squeezed thermal state (STS). It is found that the normalization factor of photon-subtracted STS (PSSTS) is a Legendre polynomial of squeezing parameter rrr and average photon number $\mathrm{n\ifmmode \bar{}\else \={}\fi{}}$ of the thermal state. Expressions of several quasiprobability distributions of PSSTS are derived analytically. Furthermore, the nonclassicality is discussed in terms of the negativity of the Wigner function (WF). It is shown that the WF of single PSSTS always has negative values if $\mathrm{n\ifmmode \bar{}\else \={}\fi{}}<\mathrm{sinh}{}^{2}r$ at the phase space center. The decoherence effect on PSSTS is then included by analytically deriving the time evolution of WF. The results show that the WF of single PSSTS has negative value if $2\ensuremath{\kappa}t<\mathrm{ln}{1\ensuremath{-}(2\mathrm{n\ifmmode \bar{}\else \={}\fi{}}+1)(\mathrm{n\ifmmode \bar{}\else \={}\fi{}}\ensuremath{-}\mathrm{sinh}{}^{2}r)/[(2\mathfrak{N}+1)(\mathrm{n\ifmmode \bar{}\else \={}\fi{}}\mathrm{cosh}2r+\mathrm{sinh}{}^{2}r)]}$, which is dependent not only on average number N\mathfrak{N}N of thermal photons in the environment, but also on $\mathrm{n\ifmmode \bar{}\else \={}\fi{}}$ and rrr.
We introduce the photon-subtracted squeezed coherent state (PSSCS), which is theoretically constructed by repeatedly subtracting photons from the squeezed coherent state (SCS) with squeezing parameter r and displacement amplitude α=|α|eiφ. Employing the normal ordering form of density operator of the SCS, we study the nonclassicality of the PSSCS by analyzing Mandel’s Q-parameter, quadratures squeezing, photon-number distribution (PND), and Wigner function (WF). We find that the PND in a PSSCS is a periodic function of φ with a period π and exhibits more remarkable oscillations than that of a SCS in the case of strong squeezing. The partial negative region of the WF is sensitive to r and |α|. The fidelity between PSSCS and SCS is analyzed, which manifests that larger photon subtraction number may result in lower fidelity. By virtue of the thermal entangled state representation the decoherence of the PSSCS in thermal environment is studied through the time evolution of the WF. The negative volume of the WF gradually diminishes with the increase of evolution time and thermal photon number, respectively. The study of the PSSCS shows that generating new photon-number-controllable nonclassical states from a weak coherent light may be realized by subtracting suitable photons from a SCS.
We investigate statistical properties of photon-subtracted squeezed vacuum (PSSV) in a thermal environment with average thermal photon number n¯ and dissipative coefficient κ by deriving the analytical expressions of the density operator and the Wigner function (WF). The normalization of m-PSSV is proved to be the m-order Lagendre polynomials. Time evolution of the photocount distribution is also related to Lagendre polynomials. The tomogram of PSSV is calculated by using the intermediate coordinate-momentum representation. Especially, the nonclassicality is discussed by virtue of the negativity of WF. It is found that the WF is always negative for all squeezing parameters λ if κt<½ ln(2n¯+2)/(2n¯+1) for the single-PSSV.
In this paper, photon-excited squeezed coherent states (PESCS) are studied theoretically by adding a photon to the squeezed coherent states repeatedly. By taking advantage of the normally ordered form of the density operator for PESCS, the analytical expressions of the normalization constant, Mandel Q -parameter, photon-number distribution, quadrature squeezing and Wigner function are derived and discussed in detail, and from these the sub-Poissonian photon statistics, photon-number oscillations, negative value of quadrature squeezing, and partial negativity and non-Gaussian distribution of a Wigner function are demonstrated, respectively. These criteria well exhibit the highly nonclassical properties and non-Gaussianity of PESCS. In addition, by deriving the normally ordered form of the density operator and Wigner function for PESCS in a thermal channel, the decoherence process of PESCS is analyzed both analytically and numerically.
We study nonclassical properties of the optical field when photon subtraction/addition coherent operation ta+ra† (|t|2+|r|2=1) acts on a squeezed thermal state (STS) by examining its quadrature squeezing, sub-Poissonian statistics, and negativity of the Wigner function. The degree of squeezing of the coherent operated squeezed thermal state (COSTS) becomes weaker. The Mandel Q parameter decreases with r and quickly becomes negative, which implies sub-Poissonian statistics. Both the negative dip and negative area of the Wigner function increase with r for small squeezing values of κ. Decoherence of the COSTS in an amplitude-damping channel is studied by the time evolution of the Wigner function. The length of time that this nonclassical field preserves its partial negativity of the Wigner function can be modulated by the coherent operation. All these results indicate that the nonclassicality of the STS is sensitive to the coherent operation.