Photon-number distribution of two-mode squeezed thermal states by entangled state representation

Using the entangled state representation, we convert a two-mode squeezed number state to a Hermite polynomial excited squeezed vacuum state. We first analytically derive the photon number distribution of the two-mode squeezed thermal states. It is found that it is a Jacobi polynomial; a remarkable result. This result can be directly applied to obtaining the photon number distribution of non-Gaussian states generated by subtracting from (adding to) two-mode squeezed thermal states.     

Single-mode photon number measurement for the squeezed two-mode number state

Based on the fact that a two-mode squeezed number state is a two-variable Hermite polynomial excitation of the two-mode squeezed vacuum state, the result of one-mode l-photon measurement for the two-mode squeezed number state $S_2|m,n\rangle$ is discussed. It is found that a remaining field-mode simultaneously collapses into a number state $|n-m+l\rangle$ with the coefficient being a Jacobi polynomial of n, m and l, which manifestly exhibits the entanglement between the two modes, i.e. it depends on the number-difference between the two modes. The second mode collapses into an excited coherent state when the first mode is measured as a coherent state.     

Excited Two-Mode Generalized Squeezed Vacuum State as a Squeezed Two-Variable Hermite Polynomial Excitation State

A new class of excited two-mode generalized squeezed vacuum states denoted by |r,s,m,n〉 are presented, which are obtained by repeatedly applying creation operators a ^† and b ^† on the two-mode generalized squeezed vacuum state. We find that it is just regarded as a generalized squeezed two-variable Hermite polynomial excitation on the vacuum state and its normalization constant is just a Jacobi polynomial. Their statistical properties are investigated such as squeezing properties, photon number distribution and the violations of Cauchy-Schwartz inequality. Especially, the Wigner function for |r,s,m,n〉 depending on the excitation photon numbers is discussed graphically.     

Photon number cumulant expansion and generating function for photon added- and subtracted-two-mode squeezed states

For the first time, we derive the photon number cumulant for two-mode squeezed state and show that its cumulant expansion leads to normalization of two-mode photon subtracted-squeezed states and photon added- squeezed states. We show that the normalization is related to Jacobi polynomial, so the cumulant expansion in turn represents the new generating function of Jacobi polynomial.     

Nonclassicality and decoherence of two-mode photon-added squeezed vacuum

We investigate the nonclassicality of two-mode photon-added squeezed vacuum state (TPASV) and its decoherence in a thermal environment according to the sub-Poissonian statistics and the negative region of Wigner function (WF). Based on the fact that the TPASV can be seen as a squeezed Hermite polynomial excitation state, we further obtain a compact expression for the normalization factor of TPASV, i.e., a Jacobi polynomial of the squeezing parameter. We also derive the explicit expression of WF of TPASV and then the decoherence effect on this state is also discussed by deriving the time evolution of the WF and its negative region.     

Entanglement of Photon-Subtracted Two-Mode Squeezed Thermal State and Its Decoherence in Thermal Environments

Using a non-Gaussian operation—photon subtraction from two-mode squeezed thermal state (PS-TMSTS), we construct a kind of entangled state. A Jacobi polynomial is found to be related to the normalization factor. The negativity of Wigner function (WF) is used to discuss its nonclassicality. The investigated entanglement properties turn out that the symmetrical PS-TMSTS may be more effective than the non-symmetric for quantum teleportation. Then the time evolution of WF is used to examine the decoherence effect, which indicates that the characteristic time of single PS-TMSTS depends not only on the average photon number of environment, but also on the average photon number of thermal state and the squeezing parameter.     

New formulas for normalizing photon-added（-subtracted） two-mode squeezed thermal states

For the first time,we derive the compact forms of normalization factors for photon-added(-subtracted) two-mode squeezed thermal states by using the P-representation and the integration within an ordered product of operators(IWOP) technique.It is found that these two factors are related to the Jacobi polynomials.In addition,some new relationships for Jacobi polynomials are presented.     

非简并双光子Jaynes-Cummings模型中贝尔非定域性的演化

基于赝自旋算符的关联所对应的贝尔算符期待值,研究了各种双模非经典态(纠缠相干态、对相干态以及双模压缩真空态)在非简并双光子Jaynes-Cummings模型中贝尔非定域性的动力学特性.结果表明:对于纠缠相干态,贝尔非定域性的演化与双模场平均光子数的大小息息相关;对于对相干态和双模压缩真空态,贝尔非定域性会在有限的时间内完全消失,之后又基本上复原到初始值,呈现出周期性振荡现象.     

量子光学理论中双模厄米多项式的新应用

强调双模厄米多项式在量子光学理论中的地位,认为它是研究连续变量纠缠态和压缩态的必要函数,具有明确的物理意义。利用双模厄米多项式,结合有序算符内的积分技术,给出了若干新的算符恒等式和互逆的积分变换公式,证明了压缩双模粒子数态恰好是双变量厄米多项式激发压缩真空态。     

Evolution of a two-mode squeezed vacuum for amplitude decay via continuous-variable entangled state approach

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.     

Quantum disentanglement and phase measurements

A 5050 beam splitter disentangles a two-mode squeezed vacuum state into two single-mode squeezed vacuum states. With the proper choice of parameters these two single-mode states will be identical. If one is passed through a device which shifts its phase, then the phases of the shifted and reference (unshifted) modes can be determined by the Vogel-Schleich technique [Phys. Rev. A44 (1991) 7642]. In this way the phase difference, i.e. the phase shift, can be measured to an accuracy of 1/N, whereN is the total number of photons coming into the beam splitter. We also propose an improved scheme involving the disentanglement of a shifted two-mode squeezed vacuum state. This leads to two shifted squeezed vacaum states at the output of the beam splitter. If one of these is passed through the phase shifter, then by performing homodyne measurements on the shifted and unshifted modes the phase shift can again be determined to an accuracy of 1/N.This research was supported by the National Science Foundation under grants INT-9221716 and PHY-9201912.     

Nonclassicality and decoherence of coherent superposition operation of photon subtraction and photon addition on squeezed state

The statistical properties of m-coherent superposition operation (μa+νa⁺)^m on the single-mode squeezed vacuum state (M-SSVS) and its decoherence in a thermal environment have been studied. Converting the M-SSVS to a squeezed Hermite polynomial excitation state, we obtain a compact expression for the normalization factor of M-SSVS, which is the Legendre polynomial of the squeezing parameter. We also derive the explicit expression of Wigner function (WF) of M-SSVS, and find the negative region of WF in phase space. The decoherence effect on this state is then discussed by deriving the time evolution of the WF. Using the negativity of WF, the loss of nonclassicality has been discussed.     

Laguerre-Polynomial-Weighted Two-Mode Squeezed State

We propose a new optical field named Laguerre-polynomial-weighted two-mode squeezed state. We find that such a state can be generated by passing the l-photon excited two-mode squeezed vacuum state C_la^?lS₂|00〉 through an single-mode amplitude damping channel. Physically, this paper actually is concerned what happens when both excitation and damping of photons co-exist for a two-mode squeezed state, e.g., dessipation of photon-added two-mode squeezed vacuum state. We employ the summation method within ordered product of operators and a new generating function formula about two-variable Hermite polynomials to proceed our discussion.     

Simple approach to deriving operator identities and mathematical formulas regarding to two-variable Hermite polynomials

By virtue of operator ordering technique and the generating function of polynomials, we provide a simple and neat approach to studying operator identities and mathematical formulas regarding to two-variable Hermite polynomials, which differs from the existing mathematical ways. We not only derive some new integration formulas and summation relations about two-variable Hermite polynomial, but also draw a conclusion that two-variable Hermite polynomial excitation of two-mode squeezed vacuum state is a squeezed two-mode number state. This may open a new route of developing mathematics by virtue of the quantum mechanical representations and operator ordering technique.     

The dependence of fidelity on the squeezing parameter in teleportation of the squeezed coherent statesvglue4pt

This paper investigates an analytical expression of teleportation fidelity in the teleportation scheme of a single mode of electromagnetic field. The fidelity between the original squeezed coherent state and the teleported one is expressed in terms of the squeezing parameter r and the quantum channel parameter (two-mode squeezed state) p. The results of analysis show that the fidelity increases with the increase of the quantum channel parameter p, while the fidelity decreases with the increase of the squeezing parameter r of the squeezed state. Thus the coherent state (r=0) is the best quantum signal for continuous variable quantum teleportation once the quantum channel is built.     

Nonclassicality of Coherent Photon-Subtracted Two Single-Modes Squeezed Vacuum State

We introduce a two-mode non-Gaussian state, generated by nonlocal coherent photon-subtraction (CPS) from two single-mode squeezed vacuum states (TSSV). Its normalized factor is turned out to be related with a Legendre polynomial. We further investigate the nonclassical properties of the CPS-TSSV through cross-correlation function, antibunching effect, photon number distribution, wave function and Wigner function. It is shown that the CPS operation can produce the vortex state and can exhibit the highly nonclassical properties.     

The atomic Wehrl entropy of a <Emphasis Type="Italic">V</Emphasis>-type three-level atom interacting with two-mode squeezed vacuum state

We study the interaction of a V-type three-level atom with a two-mode field through the two-photon interaction when the atom is initially in the upper state and the initial field is in the two-mode squeezed vacuum state. The influence of the atomic motion on the evolution of the atomic Q-function and atomic Wehrl entropy is examined. The results show that the atomic motion and the mode structure play important roles in the evolution of the atomic Q-function, atomic Wehrl entropy, and marginal atomic Wehrl entropy.     

Optical enhanced interferometry with two-mode squeezed twin-Fock states and parity detection

We theoretically investigate the quantum enhanced metrology using two-mode squeezed twin-Fock states and parity detection. Our results indicate that, for a given initial squeezing parameter, compared with the two-mode squeezed vacuum state, both phase sensitivity and resolution can be enhanced when the two-mode squeezed twin-Fock state is considered as an input state of a Mach–Zehnder interferometer. Within a constraint on the total photon number, although the two-mode squeezed vacuum state gives the better phase sensitivity when the phase shift φ to be estimated approaches to zero, the phase sensitivity offered by these non-Gaussian entangled Gaussian states is relatively stable with respect to the phase shift itself. When the phase shift slightly deviates from φ= 0, the phase sensitivity can be still enhanced by the two-mode squeezed twin-Fock state over a broad range of the total mean photon number where the phase uncertainty is still below the quantum standard noise limit. Finally, we numerically prove that the quantum Cramer–Rao bound can be approached with the parity detection.     

单-双模组合压缩热态的纠缠性质及在量子隐形传态中的应用

基于单-双模组合压缩真空态一定范围内能够获得压缩增强的效果,引入单-双模组合压缩热态(DSMST),讨论其纠缠性质.利用Weyl编序算符在相似变换下的不变性,简洁方便地导出了DSMST的纠缠度-负对数值,并给出了当热效应存在时保持纠缠的条件.研究表明:与通常的双模压缩态相比,随着参数的增加,DSMST的纠缠度增加.作为DSMST的应用,利用其实现相干态的量子隐形传输.结果表明:不同于纠缠度随压缩参数增加,保真度获得改善是有条件的,该条件恰好就是一正交分量涨落出现压缩增强的参数区域.此外,解析推导了有效隐形传输保真度(1/2)的条件.     

Nonclassical Properties of Squeezed Finite-Dimensional Pair Coherent State

We study the effect of the squeeze operator on the finite pair-coherent state. The state is a type of a correlated two-mode state in finite dimension based on the resonant ion-cavity interaction. We have discussed some statistical properties for such state, especially the quadrature variances as well as the second-order correlation function. Furthermore we have also considered the variance in the photon number sum and difference in addition to the linear correlation function. It is shown that the strong correlation occurs for a large value of the photon number difference. Our discussion is extended to include the quasiprobability distribution functions (W-Wigner and Q-functions). In the meantime we have considered the quadrature distribution function and phase distribution. It has been shown that the squeezed finite pair coherent state is sensitive to the variation in the state parameter, ξ and the squeeze parameter, r, as well as the q parameter which in fact plays a crucial role of controlling the behavior of the system.