Multiple-Photon-Added and -Subtracted Two-Mode Binomial States:Nonclassicality and Entanglement *

We theoretically analyze the nonclassicality and entanglement of two new non-Gaussian entangled states generated by applying multiple-photon addition and subtraction to a two-mode binomial state. The nonclassical properties are investigated in terms of the partial negativity of the Wigner functions, whose results show that their nonclassicality can be enhanced via one-mode even-number photon operations and two-mode symmetrical operations for the initial two-mode binomial state. We also find that there exists some enhancement in the entanglement properties in certain parameter ranges via one-mode photon-addition and two-mode symmetrical operations.     

Quantum Phase Properties of Photon Added and Subtracted Displaced Fock States

Quantum phase properties of photon added and subtracted displaced Fock states (and their limiting cases) are investigated from a number of perspectives, and it is shown that the quantum phase properties are dependent on the quantum state engineering operations performed. Specifically, the analytic expressions for quantum phase distributions and angular Q distribution as well as measures of quantum phase fluctuation and phase dispersion are obtained. The uniform phase distribution of the initial Fock states is observed to be transformed by the unitary operation (i.e., displacement operator) into non‐Gaussian shape, except for the initial vacuum state. It is observed that the phase distribution is symmetric with respect to the phase of the displacement parameter and becomes progressively narrower as its amplitude increases. The non‐unitary (photon addition/subtraction) operations make it even narrower in contrast to the Fock parameter, which leads to broadness. The photon subtraction is observed to be a more powerful quantum state engineering tool in comparison to the photon addition. Further, one of the quantum phase fluctuation parameters is found to reveal the existence of antibunching in both the engineered quantum states under consideration. Finally, the relevance of the engineered quantum states in the quantum phase estimation is also discussed.     

Lower‐ and Higher‐Order Nonclassical Properties of Photon Added and Subtracted Displaced Fock States

Nonclassical properties of photon added and subtracted displaced Fock states are studied using various witnesses of lower‐ and higher‐order nonclassicality. Compact analytic expressions are obtained for the nonclassicality witnesses. Using those expressions, it is established that these states and the states that can be obtained as their limiting cases (except coherent states) are highly nonclassical as they show the existence of lower‐ and higher‐order antibunching and sub‐Poissonian photon statistics, in addition to the nonclassical features revealed through the Mandel $Q_M$ parameter, zeros of Q function, Klyshko's criterion, and Agarwal–Tara criterion. Further, some comparison between the nonclassicality of photon added and subtracted displaced Fock states have been performed using witnesses of nonclassicality. This has established that between the two types of non‐Gaussianity inducing operations (i.e., photon addition and subtraction) used here, photon addition influences the nonclassical properties more strongly. Further, optical designs for the generation of photon added and subtracted displaced Fock states from squeezed vacuum state have also been proposed.     

压缩真空态与压缩单光子态的相干叠加态的非经典性质

通过对量子比特态进行压缩操作,获得压缩真空态与压缩单光子态按一定的权重比例叠加而成的相干叠加态,此相干叠加态与压缩参数和权重因子有关,研究其非经典性和非高斯性.研究结果表明,通过压缩使得量子比特态的光子数分布范围变宽了.特别是从亚泊松分布和Wigner函数负定性情况分析了该量子态的非经典属性,从Wigner函数的边缘分布角度观察了该量子态的非高斯性.发现调节权重因子和压缩参数可以改善该态的非经典性和非高斯性.     

Manipulating Nonclassicality via Quantum State Engineering Processes: Vacuum Filtration and Single Photon Addition

The effect of two quantum state engineering processes that can be used to burn a hole at vacuum in the photon number distribution of quantum states of radiation field is compared using various witnesses of lower- and higher-order nonclassicality as well as a measure of nonclassicality. Specifically, the modification in nonclassical properties due to vacuum state filtration and a single photon addition on an even coherent state, binomial state, and Kerr state are investigated using the criteria of lower- and higher-order antibunching, squeezing, and sub-Poissonian photon statistics. Further, the amount of nonclassicality present in these engineered quantum states having enormous applications in continuous variable quantum communication is quantified and analyzed by using an linear entropy-based entanglement potential. It is observed that all the quantum states studied here are highly nonclassical, and the hole-burning processes can introduce/enhance nonclassical features. However, it is not true in general. A hole at vacuum implies a maximally nonclassical state (as far as Lee's nonclassical depth is concerned), but a particular process of hole burning at vacuum does not ensure the existence of any particular nonclassical feature. Specifically, lower- and higher-order squeezing are not observed for photon-added and vacuum filtered even coherent states.     

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.     

Nonclassicality and decoherence of the coherent superposition operation of photon subtraction and addition on the 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 are 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 the Wigner function(WF) of the M-SSVS,and find the negative region of the 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 the WF,the loss of nonclassicality is then discussed.     

Generalized Photon Added and Subtracted f-Deformed Displaced Fock States

In this paper, by making use of the nonlinear coherent states approach, the generalized photon added and subtracted f-deformed displaced Fock states are introduced. In other words, a natural link between photon added and subtracted displaced Fock states and nonlinear coherent states associated with nonlinear oscillator algebra is obtained. It is found that various kinds of nonclassical states can be generated by adopting appropriately controlling parameters in both linear and nonlinear regimes. Moreover, examining some of the most nonclassical properties such as Mandel's Q parameter, different types of squeezing, namely, quadrature, amplitude–squared and phase entropic squeezing, and Vogel's characteristic function, the nonclassicality features of the considered quantum states of interest are studied. Furthermore, to obtain the degree of quantum coherence, the relative entropy of coherence is investigated. Indeed, the nonclassicality aspects of the states obtained have been numerically studied to understand the roles of deformation functions, photons added and subtracted, and photon number occupied in the Fock state on physical properties. It is demonstrated that the depth and the domain of the nonclassicality features of the system can properly be controlled by selecting the suitable parameters.     

Higher-Order Nonclassicality in Photon Added and Subtracted Qudit States

Higher-order nonclassical properties of r photon added and t photon subtracted qudit states (referred to as rPAQS and tPSQS, respectively) are investigated here to answer: How addition and subtraction of photon can be used to engineer higher-order nonclassical properties of qudit states? To obtain the answer, higher-order moment of relevant bosonic field operators is first obtained and subsequently used to study the higher-order nonclassical properties (e.g., higher-order antibunching, higher-order squeezing, and higher-order sub-Poissonian photon statistics) of the corresponding states. These witnesses establish that rPAQS and tPSQS are highly nonclassical. To quantitatively establish this observation and to make a comparison between rPAQS and tPSQS, volumes of the negative part of Wigner function are computed. Finally, for the sake of verifiability of the obtained results, optical tomograms are also reported. Throughout the study, a particular type of qudit state named as a new generalized binomial state is used as an example.     

广义压缩粒子数态的非经典性质及其退相干

研究了三参数的压缩算符产生的广义压缩粒子数态的非经典性质及其在光子损失通道中的退相干问题.利用有序算符内的积分技术和Weyl编序算符在相似变换下的不变性,简洁地导出了广义压缩粒子数态的Wigner函数(Laguerre-Gaussian函数).基于Wigner函数的演化积分公式,解析地推导出了在耗散通道中的Wigner函数表达式.特别地,根据Wigner函数负部体积讨论了其非经典性.     

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.     

极化自由度对分束器出射光场的量子相干性影响的研究

我们在讨论粒子数态光场在分束器上干涉后得到的输出态的量子相干性时,考虑了入射场的极化自由度.利用campos[1]等人提出的量子分束器的SU(2)理论模型,计算得到了输出光场所处状态的表达式.进而讨论了光场在两个不同的入射空间模上极化方向对两个输出空间模上光场二阶量子干涉度的影响.     

Preparation of multi-photon Fock states and quantum entanglement properties in circuit QED

We demonstrate the controllable generation of multi-photon Fock states in circuit quantum electrodynamics （circuit QED）. The external bias flux regulated by a counter can effectively adjust the bias time on each superconducting flux qubit so that each flux qubit can pass in turn through the circuit cavity and thereby avoid the effect of decoherence. We further investigate the quantum correlation dynamics of coupling superconducting qubits in a Fock state. The results reveal that the lower the photon number of the light field in the number state, the stronger the interaction between qubits is, then the more beneficial to maintaining entanglement between qubits it will be.     

加光子双模SU(2)相干光场的制备及其与Λ型三能级原子相互作用的性质

提出了通过对分束器的条件输出测量制备加光子双模SU(2 )相干光场的方案 ,并研究了Λ型三能级原子与此双模光场相互作用系统中 ,加光子对原子布居概率及光场非经典性质的影响。结果表明 ,原子粒子布居概率发生改变 ,光场一模的光子数统计趋向于亚泊松分布 ,另一模趋向于超泊松分布 ,光场两模间相关度增大 ,非经典相关程度增强。     

A new optical field state as an output of diffusion channel when the input being number state

By analyzing theoretical scheme of quantum controlling through photon addition,we propose a new optical field whose density operator asρ=λ(1-λ)l:Ll(-λ2aa/1-λ)e-λaa:(here::denotes normal ordering symbol),which is named Laguerre-polynomialweighted chaotic state.We show that such state is the solution to the master equation d/dtρ=-κ(aaρ+ρaa-aρa-aρa),describing a diffusion channel,with the initial number state|l l|,andλ=1/(1+κt).This new state is characteristic of possessing photon number l+κt at time t,so the photon number by adjusting the diffusion parameterκcan be controlled.This master equation is solved using the summation method within ordered product of operators and the entangled state representation.The physical difference between the diffusion and the amplitude damping is noted.     

A Scheme of Controlled Quantum State Swapping

A scheme for controlled quantum state swapping is presented using maximally entangled five-qubit state,i.e.,Alice wants to transmit an entangled state of particle a to Bob and at the same time Bob wants to transmit an entangled state of particle b to Alice via the control of the supervisor Charlie.The operations used in this swapping process including C-not operation and a series of single-qubit measurements performed by Alice,Bob,and Charlie.     

A Scheme of Controlled Quantum State Swapping

A scheme for controlled quantum state swapping is presented using maximally entangled five-qubit state, i.e., Alice wants to transmit an entangled state of particle a to Bob and at the same time Bob wants to transmit an entangled state of particle b to Alice via the control of the supervisor Charlie. The operations used in this swapping process including C-not operation and a series of single-qubit measurements performed by Alice, Bob, and Charlie.     

Quantum entanglement of an entangled coherent state: Role of particle losses

We analyze entanglement properties of entangled coherent state （ECS）, ｜α,0） 1,2 ＋｜0,α） 1,2, with and without photon losses. By separating the coherent state into ]a） = co｜0） ＋ √-Co2｜α）, we derive exact results of the logarithmic negativity EN, which quantifies the degree of entanglement between the two bosonic modes. Without particle losses, E~ = 1 for the NOON state; while for the ECS, E jr increases from 0 to 1 as ｜α｜-→∞. In the presence of photon losses, we find that the ECS with large enough photon number is more robust than that of the NOON state. An optimal ECS is obtained by maximizing E~ with respect to l a 12.