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Physica Scripta

Statistical properties of photon-added coherent state in a dissipative channel

Liyun Hu, Hong-Yi Fan

Published Mar 1, 2009• 43 citations
Publisher

Abstract

We adopt a new approach (using the entangled state representation) to study time evolution of the mixed state in a dissipative channel (DC) with decay rate κ. We then investigate the statistical properties of the m -photon-added coherent state (PACS) in the DC, by deriving the analytical expressions of the Wigner function (WF), which turns out to be the Laguerre–Gaussian function. The criterion time for the definite positivity of WF: , valid for arbitrary m , is revealed. Time evolution of the photocount distribution is also derived analytically, which is related to two-variable Hermite polynomials. The tomogram of PACS is also calculated through the intermediate coordinate–momentum representation.

DOI
10.1088/0031-8949/79/03/035004

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We adopt a new approach (using the entangled state representation) to study time evolution of the mixed state in a dissipative channel (DC) with decay rate κ. We then investigate the statistical properties of the m -photon-added coherent state (PACS) in the DC, by deriving the analytical expressions of the Wigner function (WF), which turns out to be the Laguerre–Gaussian function. The criterion time for the definite positivity of WF: , valid for arbitrary m , is revealed. Time evolution of the photocount distribution is also derived analytically, which is related to two-variable Hermite polynomials. The tomogram of PACS is also calculated through the intermediate coordinate–momentum representation.
We investigate the time evolution of a quantum state in the laser channel with cavity gain coefficient g and loss coefficient κ by virtue of the thermo entangled state representation. This is a new approach for solving master equations. We exactly calculate the evolving result of the photon-added coherent state (PACS) governed by the master equation describing the laser process, and the normal ordered form of the density operator in the laser channel is derived, based on which we calculate the Wigner function analytically and graphically. The corresponding law of the PACS evolution in the laser channel is evaluated.
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 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.
This paper investigates the decoherence of photo-subtracted squeezed vacuum state (PSSVS) in dissipative channel by describing its statistical properties with time evolution such as Wigner function, Husimi function, and tomogram. It first calculates the normalization factor of PSSVS related to Legendre polynomial. After deriving the normally ordered density operator of PSSVS in dissipative channel, one obtains the explicit analytical expressions of time evolution of PSSVS's statistical distribution function. It finds that these statistical distributions loss their non-Gaussian nature and become Gaussian at last in the dissipative environment as expected.
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.
In this paper, we propose a class of the generalized photon-added coherent states (GPACSs) obtained by repeatedly operating the combination of Bosonic creation and annihilation operatoes on the coherent state. The normalization factor of GPACS is related to Hermite polynomial. We also derive the explicit expressions of its statistical properties such as photocount distribution, Wigner function and tomogram and investigate their behaviour as the photon-added number varies graphically. It is found that GPACS is a kind of nonclassical state since Wigner function exhibits the negativity by increasing the photon-added number.