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Journal of the Optical Society of America B

Wigner function, optical tomography of two-variable Hermite polynomial state, and its decoherence effects studied by the entangled-state representations

Xiang‐Guo Meng, Zhen Wang, Ji‐Suo Wang, Hong-Yi Fan

Published May 21, 2013• 26 citations
Publisher

Abstract

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.

DOI
10.1364/josab.30.001614

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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.
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 the statistical properties of the photon-subtractions from the two-mode squeezed vacuum state and its decoherence in a thermal environment. It is found that the state can be considered as a squeezed two-variable Hermite polynomial excitation vacuum and the normalization of this state is the Jacobi polynomial of the squeezing parameter. The compact expression for Wigner function (WF) is also derived analytically by using the Weyl ordered operators' invariance under similar transformations. Especially, the nonclassicality is discussed in terms of the negativity of WF. The effect of decoherence on this state is then discussed by deriving the analytical time evolution results of WF. It is shown that the WF is always positive for any squeezing parameter and photon-subtraction number if the decay time exceeds an upper bound (}$κt>{1/2}\ln \frac{2\bar{n}+2}{2\bar{n}+1}).
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.
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.
We investigate the statistical properties of photon subtractions from a two-mode squeezed vacuum state and its decoherence in a thermal environment. It is found that the state can be considered as a squeezed two-variable Hermite polynomial excitation vacuum, and the normalization of this state is the Jacobi polynomial of the squeezing parameter. A compact expression for the Wigner function (WF) is also derived analytically by using the Weyl-ordered operator invariance under similar transformations. Especially, the nonclassicality is discussed in terms of the negativity of the WF. The decoherence effect on this state is then discussed by deriving the time evolution of the WF. It is shown that the WF is always positive for any squeezing parameter and any photon-subtraction number if the decay time exceeds an upper bound (κt>½ ln 2n¯+2/2n¯+1).
The nonclassicality of the two-variable Hermite polynomial state is investigated. It is found that the two-variable Hermite polynomial state can be considered as a two-mode photon subtracted squeezed vacuum state. A compact expression for the Wigner function is also derived analytically by using the Weyl-ordered operator invariance under similar transformations. Especially, the nonclassicality is discussed in terms of the negativity of the Wigner function. Then violations of Bell’s inequality for the two-variable Hermite polynomial state are studied.