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.     

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.     

Quantum statistical properties of photon-added spin coherent states

The photon-added spin coherent state as a new kind of coherent state has been defined by iterated actions of the proper raising operator on the ordinary spin coherent state. In this paper, the quantum statistical properties of photon-added spin coherent states such as photon number distribution, second-order correlation function and Wigner function are studied. It is found that the Wigner function shows the negativity in some regions and the second-order correlation function is less than unity. Therefore, the photon-added spin coherent state is a nonclassical state.     

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.     

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.     

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.     

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.     

Statistics of states generated by quantum-scissors device

Generating desired states is a prerequisite in quantum information. Some desired states can be generated by a quantum-scissors device(QSD). We present a detailed analysis of the properties of the generated states, including average photon numbers and intensity gains. The theoretical analysis shows that there is a nondeterministic amplification in terms of the average photon number under the condition that the average photon number of the input state is less than 1. In contrast to the input states, the generated states show the nonclassical property described by the negativity of the Wigner function. Furthermore, we generalize the QSD to truncate arbitrary photon number terms of the input states, which may be useful in high-dimensional quantum information processing.     

Quantum interference between a single-photon Fock state and a coherent state

We derive analytical expressions for the single mode quantum field state at the individual output ports of a beam splitter when a single-photon Fock state and a coherent state are incident on the input ports. The output states turn out to be a statistical mixture between a displaced Fock state and a coherent state. Consequently we are able to find an analytical expression for the corresponding Wigner function. Because of the generality of our calculations the obtained results are valid for all passive and lossless optical four port devices. We show further how the results can be adapted to the case of the Mach-Zehnder interferometer. In addition we consider the case for which the single-photon Fock state is replaced with a general input state: a coherent input state displaces each general quantum state at the output port of a beam splitter with the displacement parameter being the amplitude of the coherent state.     

Wigner function of an arbitrary-photon-catalyzed optical coherent state

In this paper, we give a detailed derivation process for the Wigner function (WF) of an arbitrary photon-catalyzed optical coherent state (APCOCS), which can be generated via interference between coherent and Fock states using quantum catalysis. We find that the WF is non-Gaussian and is related to the two-variable Hermite polynomial of the relative parameters, such as the coherent field strength, the number of catalyst photons and the control ratio of the beam splitter. It showed that the APCOCS created a wide range of nonclassical properties.     

Enhanced Phase Estimation in Parity-Detection-Based Mach–Zehnder Interferometer using Non-Gaussian Two-Mode Squeezed Thermal Input State

While the quantum metrological advantages of performing non-Gaussian operations on two-mode squeezed vacuum (TMSV) states have been extensively explored, similar studies in the context of two-mode squeezed thermal (TMST) states are severely lacking. This paper explores the potential advantages of performing non-Gaussian operations on TMST state for phase estimation using parity detection-based Mach–Zehnder interferometry and compares it with the TMSV case. To this end, a realistic photon subtraction, addition, and catalysis model is considered. A unified Wigner function of the photon subtracted, photon added, and photon catalyzed TMST state is derived, which is used to obtain the expression for the phase sensitivity. The results show that performing non-Gaussian operations on TMST states can enhance the phase sensitivity for significant squeezing and transmissivity parameter ranges. Because of the probabilistic nature of these operations, it is of utmost importance to consider their success probability. When the success probability is considered, the photon catalysis operation performed using a high transmissivity beam splitter is the optimal non-Gaussian operation. This contrasts with the TMSV case, where photon addition is observed as the most optimal. Further, the derived Wigner function of the non-Gaussian TMST states will be useful for state characterization and various quantum protocols.     

Experimental test of the quantum non-Gaussian character of a heralded single-photon state

We report on the experimental verification of quantum non-Gaussian character of a heralded single-photon state with a positive Wigner function. We unambiguously demonstrate that the generated state cannot be expressed as a mixture of Gaussian states. Sufficient information to witness the quantum non-Gaussian character is obtained from a standard photon anticorrelation measurement.     

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.     

Photon states and Wigner functions in parametric generation

The problem of preparation of photon states from two correlated modes of subharmonics generated in pulsed regime of optical parametric oscillator is investigated. Because of quantum correlation, detecting of n-photon Fock states in one mode transfers the other mode into also an n-photon state. Conditional Wigner functions are calculated whose negative parts describe the prepared photon states in cases of one-, two-, and three-photon schemes of measurements.     

Statistical properties of coherent photon-subtracted two-mode squeezed vacuum and its application in quantum teleportation

We introduce a kind of non-Gaussian entangled state, which can be obtained by operating a non-local coherent photon-subtraction operation on a two-mode squeezed vacuum. It is found that its normalization factor is only related to the Legendre polynomials, which is a compact expression. Its statistical properties are discussed by the negative region Wigner function with the analytical expression. As an application, the quantum teleportation for coherent states is considered by using the non-Gaussian state as an entangled channel. It is found that the teleportation fidelity can be enhanced by this non-Gaussian operation.     

Quantum state reconstruction of the single-photon Fock state

We have reconstructed the quantum state of optical pulses containing single photons using the method of phase-randomized pulsed optical homodyne tomography. The single-photon Fock state 1> was prepared using conditional measurements on photon pairs born in the process of parametric down-conversion. A probability distribution of the phase-averaged electric field amplitudes with a strongly non-Gaussian shape is obtained with the total detection efficiency of (55+/-1)%. The angle-averaged Wigner function reconstructed from this distribution shows a strong dip reaching classically impossible negative values around the origin of the phase space.     

Photon-added and photon-subtracted displaced Fock states and their non-classical properties

A new kind of quantum optical state, photon-added and -subtracted displaced Fock states, is introduced by applying the inverse of bosonic creation and annihilation operators to displaced Fock states. The quantum statistical properties of these states are investigated by numerical methods. Numerical results indicate that these states reveal some interesting non-classical properties, such as anti-bunching effects, sub-Poisson distributions and negativities of their Wigner functions.     

Two-Mode Excited Entangled Coherent State: Nonclassicality and Entanglement

Two-mode excited entangled coherent states (TME-ECSs) are introduced by operating repeatedly the photon-excited operator on the ECSs. It is shown that the normalization constant is related to the product of two Laguerre polynomials. The influence of the operation on nonclassical behaviour of the ECSs is investigated in terms of cross-correlation function, anti-bunching effect and the negativity of Wigner function, which show that nonclassical properties can be enhanced. In addition, inseparability properties of the TME-ECSs are discussed by using Bell inequality and concurrence. It is found that the degree of quantum entanglement of even ECSs increases with the increase of the total excited photon number, and the violation of Bell inequality can be present for both even and odd case only when the total excited photon numbers are even and odd, respectively.     

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.     

Nonclassical and non-Gaussian properties of states generated by the superposed photon added-and-subtracted operation on squeezed vacuum

A new kind of non-Gaussian quantum state is constructed by operating the superposed operator (SO) (cos θaa^† + sin θa^†a) on a squeezed vacuum state (SVS) S(r)|0〉. It is found that the SOSVS is just a superposition state between S(r)|0〉 and S(r)|2〉 with only even numbers of photons. The nonclassicality is investigated by exploring the negativity of Wigner function (WF) and the sub-Poissonian distribution of Mandel's Q-parameter. The non-Guassianity is exhibited via the fidelity between the SOSVS and the SVS and the marginal distribution of its Wigner function. It is found that such SO on the SVS can enhance the nonclassicality and change the non-Gaussianity of the SOSVS. This provides the possibility of generating quantum states with specific nonclassical and non-Gaussian properties.