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We analytically investigate the Wigner function (WF) and the optical tomography for the two-variable Hermite polynomial state (THPS) and the effect of decoherence on the THPS via the entangled-state representations. The nonclassicality of the THPS is investigated in terms of the partial negativity of the WF, which depends much on the polynomial orders m, n and the squeezing parameter r. We also extend recent theoretical studies of optical tomography and introduce the radon transformation between the Wigner operator and the projection operator of the entangled state |η,τ1,τ2〉 to derive the tomogram of the THPS. Furthermore, we investigate how the WF for the THPS evolves undergoing the thermal channel. The results show that quantum dissipation in the decoherence channel can thoroughly deteriorate the nonclassicality of the THPS, and thermal noise leads to much quicker decoherence than amplitude damping.

Extending the recent work completed by Biswas and Agarwal [Phys. Rev. A75, 032104 (2007)] to the case of m-photon-subtracted squeezed vacuum states (m-PSSVSs), we focus our study on nonclassicality and decoherence of the m-PSSVSs. The nonclassical properties are investigated in terms of the squeezing character, the oscillation of photon-number distribution, and the partial negativity of Wigner function (WF).We then study the effect of decoherence on the m-PSSVSs in two different channels, viz., thermal process and phase damping. In each case, the time-evolution of density operators and WFs of such states is derived analytically. After undergoing the thermal channel, the initial pure m-PSSVSs evolve into a mixed state that turns out to be a Laguerre polynomial of combination of creation and annihilation operators within normal ordering; however, they become another mixed state with the exponential decay due to phase damping. At long times, these fields decay to a highly classical thermal field as a result of thermal noise, but they still keep nonclassicality in the phase-damping channel.

We investigate how the squeezed thermal state (STS) and squeezed number state (SNS) evolve undergoing decoherence in the laser process. Remarkably, the initial SNS, an example of a pure state, evolves into a mixed state, which turns out to be a Laguerre polynomial of combination of creation and annihilation operators within normal ordering; however, the STS, a mixed state, still keeps squeezed and thermal. At long times, these fields lose their nonclassical nature and decay to a highly classical thermal field. The normally ordered density operators of such states in the laser channel lead to deriving the analytical time-evolution expressions of the Wigner functions (WFs). Their nonclassicality is investigated in reference to the time-evolution WFs, which indicates that both of the WFs decay to the same thermal WF as a result of decoherence when the decay time t or the cavity loss κ of the laser is large enough.

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.