Wigner function and tomogram of the Hermite polynomial state

Using the coherent state representation of Wigner operator and the technique of integration within an ordered product (IWOP) of operators, this paper derives the Wigner function for the Hermite polynomial state (HPS). The tomogram of the HPS is calculated with the aid of newly introduced intermediate coordinate-momentum representation in quantum optics.     

Generalized photon-added coherent state and its quantum statistical properties

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.     

Quasiprobability Distribution Functions of Squeezed Pair Coherent States

Using the entangled state representation of Wigner operator and some formulae related to the two-variable Hermite polynomials, the Wigner function of the squeezed pair coherent state (SPCS) and its two marginal distributions are derived. Based on the entangled Husimi operator introduced by Fan et al. (Phys. Lett. A 358:203, 2006) and the Weyl ordering invariance under similar transformations, we also obtain the Husimi function of the SPCS and its marginal distribution functions. The comparison between the two quasibability functions shows that, for the same amount of information included in two functions, the solving process of the Husimi function is simpler than that of the Wigner function. Work supported by the Natural Science Foundation of Shandong Province of China under Grant Y2008A23 and the Natural Science Foundation of Liaocheng University under Grant X071049.     

A new type of photon-added squeezed coherent state and its statistical properties

We construct a new type of photon-added squeezed coherent state generated by repeatedly operating the bosonic creation operator on a new type of squeezed coherent state [Fan H Y and Xiao M 1996 Phys. Lett. A 220 81]. We find that its normalization factor is related to single-variable Hermite polynomials. Furthermore, we investigate its statistical properties, such as Mandel’s Q parameter, photon-number distribution, and Wigner function. The nonclassicality is displayed in terms of the intense oscillation of photon-number distribution and the negativity of the Wigner function.     

Statistical Properties of the Generalized Photon-Added Pair Coherent State

We present a class of generalized photon-added pair coherent states (GPAPCS) and analyze some prominent nonclassical properties such as sub-Poissonian distribution and violations of Cauchy-Schwarz inequalities. In addition, we derive that the Wigner function of GPAPCS involves correlation of two two-variable Hermite polynomials and its Husimi function is related to a two-variable Hermite polynomial. Their behaviors varying with the phase space parameters are also graphically discussed. We find that the nonclassical effects of GPAPCS exhibits more with increasing of excitation photon numbers.     

奇偶对相干态的维格纳函数和层析图函数

利用纠缠态η〉表象下的维格纳算符,重构了奇偶对相干态的维格纳函数.根据维格纳函数在相空间中随变量ρ和γ的变化规律,讨论了奇偶对相干态的非经典性质和量子干涉效应.研究发现,奇偶对相干态总呈现非经典性质,并且当q取奇数时,奇偶对相干态更容易出现非经典性质.奇偶对相干态的量子干涉效应的显著程度与q取值有关,但对于q的同一取值,奇对相干态的量子干涉效应更为显著.利用纠缠态η〉表象下的维格纳算符Δ1,2(ρ,γ)和纠缠态η,τ1,τ2〉的投影算符之间满足的拉东变换,获得了奇偶对相干态的量子层析图函数.     

增光子奇偶相干态的Wigner函数

利用相干态表象下的Wigner算符, 重构了增光子奇偶相干态的Wigner函数.根据此Wigner函数在相空间中随复变量α的变化关系, 讨论了增光子奇偶相干态的非经典性质. 结果表明, 增光子奇偶相干态总可呈现非经典性质, 且在m取奇(或偶)数时, 增光子偶(或奇)相干态更容易出现非经典性质. 根据增光子奇偶相干态的Wigner函数的边缘分布, 阐明了此Wigner函数的物理意义. 同时, 利用中介表象理论获得了增光子奇偶相干态的量子tomogram函数. 关键词： 增光子奇偶相干态 Wigner函数 中介表象 tomogram函数     

Nonclassical Properties of Multiple-Photon-Added Two-Mode Squeezed Coherent States

We investigate how the photon addition operation affects the nonclassical properties of the non-Gaussian squeezed state generated by adding photons to each mode of the two-mode squeezed coherent state (TMSCS). By the generating function of two-variable Hermite polynomials, the compact expression of normalization factor is derived. We show that the fields in such states exhibit remarkable sub-Poissonian photon statistics. The photon addition operation can enhance the cross-correlation for appropriate combinations of several parameters involved in the TMSCS. Compared with that of TMSCS, the Wigner function of the photon–added TMSCS (PA-TMSCS) is modulated by a factor which is also related with two-variable Hermite polynomials. Such Wigner functions have some negativity regions and show a strong quantum mechanical interference. In addition, the normalization factor, Mandel’s Q parameter, cross-correlation function and Wigner functions are all sensitive to the compound phase involved in TMSCS.     

Wigner function for the generalized excited pair coherent state

This paper introduces the generalized excited pair coherent state （GEPCS）. Using the entangled state 〈η〉 representation of Wigner operator, it obtains the Wigner function for the GEPCS. In the ρ-γ phase space, the variations of the Wigner function distributions with the parameters q, α, k and l are discussed. The tomogram of the GEPCS is calculated with the help of the Radon transform between the Wigner operator and the projection operator of the entangled state ｜η1, η2, τ1, τ2｜. The entangled states ｜η〉 and ｜η1, η2, τ1, τ2〉 provide two good representative space for studying the Wigner functions and tomograms of various two-mode correlated quantum states.     

Comparison of nonclassicality between photon-added and photon-subtracted squeezed vacuum states

In this paper, we introduce photon-added and photon-subtracted squeezed vacuum state (PASV and PSSV) and obtain their normalized factors, which have the similar forms involved in Lengendre polynomials. Moreover, we give the compact expressions of Wigner function, which are related to single-variable Hermite polynomials. Especially, we compare their nonclassicality in terms of Mandel Q-factor and the negativity of Wigner function.     

Photon Statistics of Generic Two-Mode Squeezed Coherent Light

for two-mode squeezed coherent states, the photon distribution function is expressed in terms of both four-variable and two-variable Hermite polynomials depending on two squeezing parameters, the relative phase between the two oscillators, their “spatial” orientation, and a four-dimensional shift in phase space of the electromagnetic field oscillator. The oscillations of the photon distribution function are discussed. The Wigner function and Q-function of the generic two-mode squeezed vacuum and squeezed coherent state are calculated explicitly.     

Non-Gaussianity and decoherence of generalized photon-added coherent state as a Hermite-excited coherent state

Generalized photon-added coherent state (GPACS) is obtained by repeatedly acting the combination of Bose creation and annihilation operations on the coherent state. It is found that GPACS can be regarded as a Hermite-excited coherent state due to its normalization factor related to a Hermite polynomial. In addition, we adopt the Hilbert-Schmidt distance to quantify the non-Gaussian character of GPACS and discuss the decoherence of GPACS in dissipative channel by studying the loss of nonclassicality in reference of the negativity of Wigner function.     

Nonclassicality Generated by Applying Hermite–Polynomials Photon-Added Operator on the Even/Odd Coherent States

We examine nonclassical properties of the quantum state generated by applying Hermite polynomials photon-added operator on the even/odd coherent state (HPECS/HPOCS). Explicit expressions for its nonclassical properties, such as quantum statistical properties and squeezing phenomenon, are obtained. It is interesting to find that the HPECS/HPOCS exhibits sub-Poissonian distribution, anti-bunching effects and negative values of the Wigner function. Thus, we confirm the HPPECS/HPPOCS is a new nonclassical state. Finally, we reveal that the HPPECS/HPPOCS is a novel intelligent state by its squeezing effects in position distribution and quadrature squeezing.     

Statistical Properties of Photon-Added Two-Mode Squeezed Coherent States

The nonclassical and non-Gaussian quantum states—photon-added two-mode squeezed coherent states have been theoretically introduced by adding multiple photons to each mode of the two-mode squeezed coherent states. Starting from the new expression of two-mode squeezing operator in entangled states representation, the normalization factor is obtained, which is directly related to bivariate Hermite polynomials. The sub-Poissonian photon statistics, cross-correlation between two modes, partial negative Wigner function are observed, which fully reflect the nonclassicality of the target states. The negative Wigner function often display non-Gaussian distribution meanwhile. The investigations may provide experimentalists with some better references in quantum engineering.     

Continuous-Variable Entanglement and Wigner-Function Negativity via Adding or Subtracting Photons

The performance of adding or subtracting photons on two-mode squeezed thermal states via examining the Einstein–Podolsky–Rosen (EPR) correlation, the Hillery–Zubairy (HZ) correlation, the fidelity of teleportation, and the negativity of Wigner function is theoretically investigated. The normalization factors and the teleportation fidelity are related to Jacobi polynomials, and the (evolved) Wigner functions are simply associated with two-variable Hermite polynomials. Compared with the original squeezed thermal states, the EPR correlation and the teleportation fidelity can be enhanced by photon subtraction and basically weakened by photon addition symmetric operations, but they cannot be enhanced for both photon addition and subtraction asymmetric cases. Also, HZ correlation can provide a better option relative to the EPR correlation in detecting the entanglement, and the fidelity for teleporting a squeezed state with a large squeezing can also be enhanced via photon addition symmetric operations, in contrast to teleporting a coherent state. Additionally, the nonclassicality is discussed in terms of the negativity of the (evolved) Wigner functions, which shows that photon addition and subtraction and the squeezing cannot restrain the deteriorate of nonclassicality, and the evolved Wigner functions become Gaussian (corresponding to vacuum) with long decay times as a result of amplitude decay.     

Nonclassicality of photon-modulated spin coherent states in the Holstein—Primakoff realization

Two new photon-modulated spin coherent states (SCSs) are introduced by operating the spin ladder operators J_± on the ordinary SCS in the Holstein-Primakoff realization and the nonclassicality is exhibited via their photon number distribution, second-order correlation function, photocount distribution and negativity of Wigner distribution. Analytical results show that the photocount distribution is a Bernoulli distribution and the Wigner functions are only associated with two-variable Hermite polynomials. Compared with the ordinary SCS, the photon-modulated SCSs exhibit more stronger nonclassicality in certain regions of the photon modulated number k and spin number j, which means that the nonclassicality can be enhanced by selecting suitable parameters.     

Decoherence and Fidelity in Teleportation of Coherent Photon-Added Two-Mode Squeezed Thermal States

We theoretically introduce a kind of non-Gaussian entangled resources, i.e., coherent photon-added two-mode squeezed thermal states (CPA-TMSTS), by successively performing coherent photon addition operation to the two-mode squeezed thermal states. The normalization factor related to bivariate Hermite polynomials is obtained. Based upon it, the nonclassicality and decoherence process are analyzed by virtue of the Wigner function. It is shown that the coherent photon addition operation is an effective way in generating partial negative values of Wigner function, which clearly manifests the nonclassicality and non-Gaussianity of the target states. Additionally, the fidelity in teleporting coherent states using CPA-TMSTS as entangled resource is quantified both analytically and numerically. It is found that the CPA-TMSTS is an entangled resource of high-efficiency and high-fidelity in quantum teleportation.     

Two-particle Wigner functions in a one-dimensional Calogero-Sutherland potential

We calculate the Wigner distribution function for the Calogero-Sutherland system which consists of harmonic and inverse-square interactions. The Wigner distribution function is separated out into two parts corresponding to the relative and center-of-mass motions. A general expression for the relative Wigner function is obtained in terms of the Laguerre polynomials by introducing a new identity between Hermite and Laguerre polynomials.     

Wigner Function, Husimi Function and Tomogram of?Atomic?Coherent State as Energy Eigenstate of?Two?Coupled?Oscillators

In this work, we find that a set of energy eigenstates of the Hamiltonian of two coupled oscillators can be classified as the atomic coherent state in the Schwinger Bosonic realization. The statistical properties of these states are also discussed by analytically calculating their Wigner function, Husimi function and tomogram.     

Wigner function and Bell’s inequalities for even and odd coherent states

The Wigner function and the symplectic tomogram of an entangled quantum state, which is a superposition of the photon’s coherent states (even and odd coherent states), is studied. Photon statistics and violation of Bell’s inequality for the photon state are discussed.