Obtaining Multimode Entangled State Representation by Generalized Radon Transformation of the Wigner Operator

In a preceding paper (Fan and Lv in J. Math. Phys. 50:102108, 2009), the phase-space integration corresponding to the straight line characteristic of two different real parameters λ,τ over the Wigner operator (i.e. the Radon transformation) leads to pure-state density operator |u〉 _{λ,τ  λ,τ} 〈u|, where |u〉 _λ,τ is just the coordinate-momentum intermediate representation. In this work we show that generalized Radon transformation of the Wigner operator yields multimode density operator of continuum variables. This provides us with a new approach for obtaining multimode entangled state representation. The Weyl ordering of the Wigner operator is used in our discussions.     

S-ordered Expansion of the Intermediate Representation Projector and Its Applications

We first deduce the s-ordered expansion of the Wigner operator. Since Radon transformation of Wigner operator is just the intermediate representation |x〉 _λ,ν projector, we naturally obtain the s-ordered product of |x〉 _λ,νλ,ν〈x|. Accordingly, the completeness relation is still preserved under the s-ordering. Finally, based on it, we obtain the s-ordered expansion of some useful operator in quantum optics, and some new operator identities are revealed accordingly.     

A New Kind of Bipartite Entangled State and Some of Its Applications

Using the technique of integration within an ordered product (IWOP) of operators we derive a new kind of bipartite entangled state |β ₁,β ₂〉, which is perfectly different from the preceding entangled states in construction. We also present main properties, generation scheme of |β ₁,β ₂〉, superposition of |β ₁,β ₂〉, Wigner function and some applications of |β ₁,β ₂〉 in quantum optics.     

Research on the Dynamic Problems of 3D Cross Coupling Quantum Harmonic Oscillator by Virtue of Intermediate Representation |<Emphasis Type="Italic">x</Emphasis>〉<Subscript><Emphasis Type="Italic">λ</Emphasis>,<Emphasis Type="Italic">ν</Emphasis></Subscript>

The intermediate representation (namely intermediate coordinate-momentum representation) |x〉 _λ,ν are introduced and employed to research the expression of the operator in intermediate representation |x〉 _λ,ν . The systematic Hamilton operator of 3D cross coupling quantum harmonic oscillator was diagonalized by virtue of quadratic form theory. The quantity of λ,ν,τand σ were figured out. The dynamic problems of 3D cross coupling quantum harmonic oscillator are researched by virtue of intermediate representation. The energy eigen-value and eigenwave function of 3D cross coupling quantum harmonic oscillator were obtained in intermediate representation. The importance of intermediate representation was discussed. The results show that the Radon transformation of Wigner operator is just the projectional operator |x〉 _{λ,ν λ,ν} 〈x|, and the Radon transformation of Wigner function is just a margin distribution.     

Generalized Two-Mode Squeezed States and Some Nonclassical Characteristics

A class of generalized two-mode squeezed states |φ〉 is presented, which are generated from the generalized two-mode squeezing operator U(γ,λ) acting on the two-mode coherent state |α ₁,α ₂〉. We first investigate some mathematical properties of U(γ,λ) including the squeezing transformation under U(γ,λ), ket-bra integral form in the coordinate representation, normally ordered form. Then we evaluate some nonclassical characteristics of the state |φ〉 such as higher-order squeezing behavior, entanglement analysis and analytical expression of the Wigner function.     

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.     

A Kind Tripartite Entangled State Representation and?Its Application in Quantum Teleportation

We discuss entanglement for a tripartite system by setting up a new state vector representation |p,χ ₁,χ ₂〉 in three-mode Fock space. The Schmidt decomposition of |p,χ ₁,χ ₂〉 is presented and its application in teleporting a bipartite entangled state or a two-mode squeezed state to a pair of receivers is analyzed.     

The Wigner Function and the Husimi Function of the One- and Two-Mode Combination Squeezed State

In this paper we study the character of the Wigner function and Husimi function of the one- and two mode combining squeezed state (OTCSS) on the basis of plotting the three dimensional graphics of the Wigner function and Husimi function. It is easy to calculate the Husimi function of the OTCSS in entangled two-mode state by virtue of the formula of entangled two-mode Husimi operator: Δ _h (σ,γ;κ)=| σ,γ〉 _{κ κ} 〈σ,γ | (Fan, H.-Y., Guo, Q. in Phys. Lett. A 358:203–210, 2006). It is clearly found that the evolution law of Husimi function of OTCSS is different from the Wigner function. Work supported by the specialized research fund for the doctoral progress of higher education in China.     

A new finite-dimensional pair coherent state studied by virtue of the entangled state representation and its statistical behavior

In this paper we construct a new type of finite-dimensional pair coherent states |ξ, q〉 as realizations of SU(2) Lie algebra. Using the technique of integration within an ordered product of operator, the nonorthogonality and completeness properties of the state |ξ, q〉 are investigated. Based on the Wigner operator in the entangled state |τ〉 representation, the Wigner function of |ξ, q〉 is obtained. The properties of |ξ, q〉 are discussed in terms of the negativity of its Wigner function. The tomogram of |ξ, q〉 is calculated with the aid of the Radon transform between the Wigner operator and the projection operator of the entangled state |η, κ₁, κ₂〉. In addition, using the entangled state |τ〉 representation of |ξ, q〉 to show that the states |ξ, q〉 are just a set of energy eigenstates of time-independent two coupled oscillators.     

A Generalized Hadamard-Fresnel Complementary Transformation Derived by Virtue of New Coherent-Entangled State

The new coherent-entangled state |z,x;θ〉 is proposed in the two-mode Fock space, which exhibits both the properties of coherent and entangled states. The completeness relation of |z,x;θ〉 is proved by virtue of the technique of integral within an ordered product of operators. A generalized Hadamard-Fresnel complementary transformation derived by virtue of the coherent-entangled state |z,x;θ〉, which is unitary. The new unitary operator plays the role of both Hadamard transformation for ([^(a)]₁sinq-[^(a)]₂cosq)(\hat{a}_{1}\sin\theta -\hat{a}_{2}\cos\theta) and Fresnel transformation for ([^(a)]₁cosq+[^(a)]₂sinq)(\hat{a}_{1}\cos\theta +\hat{a}_{2}\sin\theta), respectively.     

The Controlled Teleportation of an Arbitrary Two-Atom Entangled State in Driven Cavity QED

In this paper, we propose a scheme for the controlled teleportation of an arbitrary two-atom entangled state |φ〉₁₂=a|gg〉₁₂+b|ge〉₁₂+c|eg〉₁₂+d|ee〉₁₂ in driven cavity QED. An arbitrary two-atom entangled state can be teleported perfectly with the help of the cooperation of the third side by constructing a three-atom GHZ entangled state as the controlled channel. This scheme does not involve apparent (or direct) Bell-state measurement and is insensitive to the cavity decay and the thermal field. The probability of the success in our scheme is 1.0.     

Entanglement and Mixedness of Locally Cloned Non-Maximal W-State

In this work we describe a protocol by which two of three parties generate two bipartite entangled state among themselves without involving third party, from a non maximal W-state or W-type state |X〉=α|001〉₁₂₃+β|010〉₁₂₃+γ|100〉₁₂₃,α ²+β ²+γ ²=1 shared by three distant partners. Also we have considered the case β=γ, to obtain a range for α ², for which the local output states are separable and non local output states are inseparable. We also find out the dependence of the mixedness of inseparable states with their amount of inseparability, for that range of α ².     

Correlations of polarizations and entangled states in the two-photon system

The correlations of the linear and circular polarizations in the system of two photons have been theoretically investigated. The polarization of a two-photon state is described by the one-photon Stokes parameters and by the components of the correlation “tensor” in the Stokes space. It is shown that in the case of two-photon decays π⁰ → 2γ, η → 2γ, K _L ⁰ → 2γ, K _S ⁰ → 2γ and the cascade process |0〉 → |1〉 + γ → |0〉 + 2γ(|0〉 and |1〉 are states with the spin 0 and 1, respectively) the final two-photon state represents a characteristic example of the entangled (nonfactorizable) state, and the correlations between the Stokes parameters in all these decays have the purely quantum character: the incoherence inequalities of the Bell type for the components of the correlation “tensor”, established previously for the case of classical “mixtures”, are violated. The general analysis of the registration procedure for two correlated photons by two one-photon detectors is performed.     

Husimi Operator for Describing Probability Distribution of Electron States in Uniform Magnetic Field Studied by Virtue of Entangled State Representation

For the first time we introduce an operator Δ _h (γ,ε;κ) for studying Husimi distribution function in phase space (γ,ε) for electron’s states in uniform magnetic field, where κ is the Gaussian spatial width parameter. The marginal distributions of the Husimi function are Gaussian-broadened version of the Wigner marginal distributions. Using the Wigner operator in the entangled state 〈λ | representation we find that Δ _h (γ,ε;κ) is just a pure squeezed coherent state density operator | γ,ε〉 _{κ κ} 〈γ,ε |, which brings much convenience for studying Husimi distribution, so we name Δ _h (γ,ε;κ) the Husimi operator. We then derive Husimi operator’s normally ordered form that provides us with an operator version to examine various properties of the Husimi distribution. Work supported by the National Natural Science Foundation under the grant: 10775097.     

Application of Generalized EPR Entangled State in Quantum Teleportation

We show how the generalized EPR entangled state |η,θ〉 generated by an asymmetric beamsplitter can be directly applied to quantum teleportation to make detailed analysis and calculation. In the whole process of teleportation, the entanglement source is more flexible with the change of transmission and reflection coefficients of the beam splitter. When the squeezed state of |η,θ〉 is used as quantum channel, the fidelity depends on the degrees of squeezing for entanglement and the reflection coefficient. Our calculation has been greatly simplified by using the Schmidt decomposition of |η,θ〉.     

Kraus Operator-Sum Representation and Time Evolution of Distribution Functions in Phase-Sensitive Reservoirs

Using the thermal entangled state representation 〈η|, we examine the master equation (ME) describing phase-sensitive reservoirs. We present the analytical expression of solution to the ME, i.e., the Kraus operator-sum representation of density operator ρ is given, and its normalization is also proved by using the IWOP technique. Further, by converting the characteristic function χ(λ) into an overlap between two “pure states” in enlarged Fock space, i.e., χ(λ)=〈η _=−λ |ρ|η ₌₀〉, we consider time evolution of distribution functions, such as Wigner, Q- and P-function. As applications, the photon-count distribution and the evolution of Wigner function of photon-added coherent state are examined in phase-sensitive reservoirs. It is shown that the Wigner function has a negative value when kt\leqslant\frac 12ln( 1+m_￥) \kappa t\leqslant\frac {1}{2}\ln ( 1+\mu_{\infty}) is satisfied, where μ _∞ depends on the squeezing parameter |M|² of environment, and increases as the increase of |M|.     

Complex Wavelet Transform of the Bell States

By employing the technique of an integral within an ordered product (IWOP) of operators we recast the complex wavelet transform to a matrix element of the two-mode squeezing-displacing operator U ₂(μ,σ) between the mother wavelet vector 〈ψ| and the two-mode quantum state vector |f〉 to be transformed, i.e., we propose that 〈ψ|U ₂(μ,σ)|f〉 can be considered as a new kind of spectra for analyzing |f〉, this may have some potential applications in quantum information and calculation. As an example, we numerically calculate wavelet-transform spectrum for the Bell states, which may play a role of distinguishing them one from another.     

Wavelet Transform of Even- and Odd-Coherent States

By employing the technique of integral within an ordered product (IWOP) of operators we recast classical wavelet transform to a matrix element of the squeezing-displacing operator U(μ,s) between the mother wavelet vector 〈ψ| and the state vector |f〉 to be transformed, i.e., we propose that 〈ψ|U(μ,s)|f〉 can be considered as a new kind of spectrum for analyzing the quantum state |f〉. In this way we do numerical calculation of wavelet-transform spectrum for the even- and odd-coherent states and then plot their figures, respectively. Thus this kind of spectrum can be used to recognize a variety of quantum optical states.     

Anisotropic impurity scattering in palladium silver alloys

Measurements have been made of the Hall coefficientR of some alloys of silver in palladium over the temperature range 1°K to 120°K. The alloys contain between ∼1 and ∼10 at.-% silver. Values ofR were also obtained at room temperature and these were in good agreement with earlier published work. The values ofR are negative in all the alloys, and |R| increases both on reducing the temperature and increasing the silver concentration,c. Below ∼10°K, |R| becomes independent of temperature but shows a linear dependence onc, increasing by a factor of 2.5 over the concentration range measured. This increase is too great to be accounted for in terms of band structure changes alone, so we have examined the effects of anisotropic impurity scattering. To a first approximation it can be shown thatR is proportional to an anisotropy parameterA, defined asA=〈τ ²(k)〉/〈τ(k)〉², whereτ(k) represents the relaxation time of an electron in a statek, and 〈〉 is an average over the Fermi surface. In palladium we assume that the majority of the current is carried by the s-electrons. In the presence of silver impurities these electrons can be scattered into s-states or d-states with relaxation times given byτ _ss α1/c(1−c) andτ _sd α1/c ²(1−c) respectively. FollowingPlate we have assumed thatτ _ss is isotropic and thatτ _sd is anisotropic, leading to an overall anisotropic relaxation time for impurity scattering. We then find the parameterA increases approximately linearly with silver content, in accordance with our experimental results.