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Chinese Physics B

Wigner function for squeezed negative binomial state and evolution of density operator for amplitude decay*

Heng-Yun Lv, Ji‐Suo Wang, Xiaoyan Zhang, Meng-Yan Wu, Bao‐Long Liang, Xiang‐Guo Meng

Published Aug 6, 2019• 3 citations
Publisher

Abstract

Using the thermal-entangled state representation and the operator-ordering method, we investigate Wigner function (WF) for the squeezed negative binomial state (SNBS) and the analytical evolution law of density operator in the amplitude decay channel. The results show that the analytical WF is related to the square of the module of single-variable Hermite polynomials, which leads to a new two-variable special function and its generating function, and the parameters s and γ play opposite roles in the WF distributions. Besides, after undergoing this channel, the initial pure SNBS evolves into a new mixed state related to two operator Hermite polynomials within normal ordering, and fully loses its nonclassicality and decays to vacuum at long decay time.

DOI
10.1088/1674-1056/ab38a6

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Using the thermal-entangled state representation and the operator-ordering method, we investigate Wigner function (WF) for the squeezed negative binomial state (SNBS) and the analytical evolution law of density operator in the amplitude decay channel. The results show that the analytical WF is related to the square of the module of single-variable Hermite polynomials, which leads to a new two-variable special function and its generating function, and the parameters s and γ play opposite roles in the WF distributions. Besides, after undergoing this channel, the initial pure SNBS evolves into a new mixed state related to two operator Hermite polynomials within normal ordering, and fully loses its nonclassicality and decays to vacuum at long decay time.
Abstract We theoretically investigate the nonclassicality of the squeezed Hermite polynomial state (SHPS) by examining the squeezing feature, the oscillation of photon-number distribution (PND) and the negativity of the Wigner function (WF). Our results indicate that the squeezing operation leads to a large squeezing degree and negative volume of the WF, but the Hermite polynomial creation operation has less impact on them, and the two operations have opposite modulations of the PND. In addition, we study the evolution law of the SHPS in the amplitude decay channel (ADC). It is found that the density-operator evolution is related to two conjugated operator Hermite polynomials within normal ordering, which brings about a new two-variable special function and its generating function. We also find that the analytical WF evolution is represented as the compact form of this new special function, and that the large squeezing is more robust against the decoherence of the SHPS in the ADC.
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.
Extending the recent work completed by Fan et al. [ Front. Phys. 9(1), 74 (2014)] to a two-mode case, we investigate how a two-mode squeezed vacuum evolves when it undergoes a two-mode amplitude dissipative channel, with the same decay rate κ, using the continuous-variable entangled state approach. Our analytical results show that the initial pure-squeezed vacuum state evolves into a definite mixed state with entanglement and squeezing, decaying over time as a result of amplitude decay. We also investigate the time evolutions of the photon number distribution, the Wigner function, and the optical tomogram in this channel. Our results indicate that the evolved photon number distribution is related to Jacobi polynomials, the Wigner function has a standard Gaussian distribution (corresponding to the vacuum) at long periods, losing its nonclassicality due to amplitude decay, and a larger squeezing leads to a longer decay time.
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.
We theoretically put forward the Hermite-excited squeezed thermal states (HESTS) by applying operator Hermite polynomials on squeezed thermal states. Starting from the normally ordered density operator of squeezed thermal states and operator Hermite polynomials, the normalization factor is obtained, which is related to the Legendre polynomials. Several phase-space distribution functions, i.e., the Q function, P function, Wigner function (WF), and R function, are analytically derived. And the non-Gaussianity and nonclassicality are mainly reflected by the negativity of non-Gaussian WF and the existence of nonclassical depth. In addition, by deriving the normally ordered density operator and WF of HESTS in the laser channel, the decoherence effect is studied and discussed. Finally, the quantity in measuring non-Gaussianity is calculated to further quantitatively measure the non-Gaussianity of the resulting states.
By introducing the thermo entangled state representation, we convert the calculation of Wigner function (WF) of density operator to an overlap between "two pure" states in a two-mode enlarged Fock space. Furthermore, we derive a new WF evolution formula of any initial state in self-Kerr Medium with photon loss and find that the photon number distribution for any initial state is independent of the coupling factor with Kerr Medium, where the number state is not affected by the Kerr nonlinearity and evolves into a density operator of binomial distribution.