Quasiprobability Distribution Functions Research Articles

Wigner distribution of elliptical quantum optical vortex

We calculate the Wigner quasiprobability distribution function of the quantum elliptical vortex in elliptical beam (EEV), produced by coupling squeezed coherent states of two modes. The coupling between the two modes is performed by using beam splitter (BS) or a dual channel directional coupler (DCDC). The quantum interference due to the coupling between the two modes promises the generation of controlled entanglement for quantum computation and quantum tomography.

Delayed creation of entanglement in superconducting qubits interacting with a microwave field

We explore the role played by the intrinsic decoherence in superconducting charge qubits in the presence of a microwave field applied as a magnetic flux. We study how the delayed creation of entanglement, which is opposite to the sudden death of entanglement, can be induced. We compute the time evolution of the population inversion, total correlation and entanglement, taking into account the junction mixed state and dissipation of the cavity field. We show that although decoherence destroys the correlation of the junction and field, information of the initial state may be obtained via quasi-probability distribution functions.

Wigner Functions for Klein-Gordon Oscillators in Non-commutative Space

As a quasi-probability distribution function in phase-space and as well as a special representation of the density matrix, the Wigner function is of great significance in Physics. This letter first makes a review of Wigner function and then provides three approaches of calculating it in non-commutative space. Finally, with the help of Moyal-Weyl multiplication and Bopp’s shift, the Wigner functions for Klein-Gordon oscillators in non-commutative space are deduced explicitly.

Perspectives for a superconducting charge qubit system interacting with bimodal cavity fields

We report on the observation of new features in a superconducting charge qubit system. The system we analyze comprises of a single Cooper-pair box sequentially coupled to two microwave cavity fields. Simulations of the full qubit–field dynamics show significant total correlation and coherence loss. By suitably choosing the system’s parameters and precisely controlling the dynamics, we demonstrate the generation of two-mode field states. We explore the nonclassical behavior of the system by studding the quasi-probability distribution function. Our scheme can be realized within the current experimental technology and may well be of use in quantum information processing applications.

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.

Quantum properties of a superposition of squeezed displaced two-mode vacuum and single-photon states

In this paper, we study some quantum properties of a superposition of displaced squeezed two-mode vacuum and single-photon states, such as the second-order correlation function, the Cauchy–Schwarz inequality, quadrature squeezing, quasiprobability distribution functions and purity. These type of states include two mechanisms, namely interference in phase space and entanglement. We show that these states can exhibit sub-Poissonian statistics, squeezing and deviate from the classical Cauchy–Schwarz inequality. Moreover, the amount of entanglement in the system can be increased by increasing the squeezing mechanism. In the framework of the quasiprobability distribution functions, we show that the single-mode state can tend to the thermal state based on the correlation mechanism. A generation scheme for such states is given.

Quasiprobability distribution functions for finite-dimensional discrete phase spaces: Spin-tunneling effects in a toy model

We show how quasiprobability distribution functions defined over ${N}^{2}$-dimensional discrete phase spaces can be used to treat physical systems described by a finite space of states which exhibit spin-tunneling effects. This particular approach is then applied to the Lipkin-Meshkov-Glick model in order to obtain the time evolution of the discrete Husimi function, and as a by-product the energy gap for a symmetric combination of ground and first excited states. Moreover, we also show how an angle-based potential approach can be efficiently employed to explain qualitatively certain features of the energy gap in terms of a spin tunneling. Entropy functionals are also discussed in this context. Such results reinforce not only the formalism per se but also the possibility of some future potential applications in other branches of physics.

Quasi-probability distribution functions for a single trapped ion interacting with a mixed laser field

We give a fully analytical description of the dynamics of the quasi-probability distribution (Q-PD) functions for any model describing a single two-level qubit interacting with a field. The Wigner function at time (t = 0) and Q function for a pure state as an initial field state are studied. But, in this paper, the temporal behavior of the Q-PD functions for a coherent superposition state (SS) and a statistical mixture of coherent states (SM) for the interaction between a single ion and laser field are investigated. It is found that, the temporal behavior of the Q-PD functions is in very good agreement with its counterpart for the entanglement. If the curve of the entanglement between the trapped ion (specially for the second red sideband) and laser field are known, we can expect the shape of the Q-PD functions.

Phase space description of spin dynamics

The spin system with Hamiltonian (ξ and σ are external and internal field parameters) is treated as an example of the phase space description of spin dynamics using a master equation for the quasiprobability distribution function of spin orientations in the representation (phase) space of the polar angles (analogous to the Wigner phase space distribution for translational motion). The master equation yields (via the Wigner–Stratonovich transformation of the density matrix) the solution as a Fourier series in the spherical harmonics with Fourier coefficients given by the statistical moments in a manner analogous to the classical distribution. The relaxation time of the longitudinal component of the spin can be estimated using a quantum generalization of the classical integral relaxation time via the stationary distribution and diffusion coefficient of the master equation.

Nonclassical properties of a nonlinear generalized geometric state

In this paper, we introduce and investigate some properties of a nonlinear generalized geometric state (the state that interpolates between the number state and a nonlinear pure thermal state). We mainly concentrate on the statistical properties. We have discussed the normal squeezing as well as the amplitude squared squeezing, further the Mandel's q-parameter is also considered. The investigation is also extended to include the quasi-probability distribution functions (W-Wigner and Q-functions). The quadrature distribution and the phase properties in the Pegg–Barnett formalism besides the phase variances are considered.

Transition state theory for spins: phase-space formulation

The quasi-probability distribution function for spins in the representation (phase) space of the polar and azimuthal angles (ϑ, φ) (analogous to the Wigner distribution for translational motion of a particle) is used to calculate the escape rate for a uniaxial spin system via quantum transition state theory (TST). The quantum corrections to the TST escape rate equation for classical magnetic dipoles appear both in the prefactor and in the exponential part of the escape rate unlike in the corresponding Wigner TST formula for particles and exhibit a marked dependence on the spin number.

Even and Odd Nonlinear Negative Binomial States

In this paper the even and odd nonlinear negative binomial states of the radiation field are introduced. These states interpolates between even (odd) number states and the even (odd) nonlinear coherent states. The Glauber second-order correlation function is calculated for these states. The squeezing phenomenon (normal and amplitude-squared squeezing), the quasi-probability distribution function Q-function, Wigner-function and the phase properties, are also discussed. Examination of the resonance fluorescence against the present state is given. It has been shown that the atomic inversion is sensitive to any variation in the even and odd nonlinear negative binomial number M and the nonlinearity parameter η.

Generation of squeezed vacuum states by PPKTP crystal and its Wigner quasi-probability distribution function reconstruction

We demonstrated experimentally the generation of squeezed vacuum by a periodically poled KTP crystal in a continuous-wave optical parametric oscillator，which was pumped with 532nm field. The squeezed vacuum of 3.41 dB at 1064 nm was detected by a homodyne detection system. Moreover，the Wigner quasi-probability distribution function of the squeezed vacuum was reconstructed using quantum tomography technique.

Phase-space formulation of the nonlinear longitudinal relaxation of the magnetization in quantum spin systems

Nonlinear longitudinal relaxation of a spin in a uniform external dc magnetic field is treated using a master equation for the quasiprobability distribution function of spin orientations in the configuration space of polar and azimuthal angles (analogous to the Wigner phase space distribution for translational motion). The solution of the corresponding classical problem of the rotational Brownian motion of a magnetic moment in an external magnetic field essentially carries over to the quantum regime yielding in closed form the dependence of the longitudinal spin relaxation on the spin size S as well as an expression for the integral relaxation time, which in linear response reduces to that previously given by D. A. Garanin [Phys. Rev. E 55, 2569 (1997)] using the density matrix approach. The nonlinear relaxation is dominated by a single exponential having as time constant the integral relaxation time. Thus a simple description in terms of a Bloch equation holds even for the nonlinear response of a giant spin.

Superposition of two coherent states π out of phase with average photon number as relative phase

This paper discusses nonclassical features of the superposition of two coherent states π out of phase with each other and parameterized solely by amplitude α, that is, when the relative phase has a relationship with average photon number ∣ α∣ 2. In particular, it discusses oscillatory and sub-Poissonian photon statistics, degree of quadrature squeezing, and quasiprobability distribution functions. We examine that some nonclassical properties of these states are different from those of the even and odd coherent states.

Statistical properties of the nonlinear negative binomial state

In this context, we introduce and investigate the properties of the nonlinear negative binomial state (the state which interpolates between the nonlinear coherent and the number states). Mainly we concentrate on the statistical properties for such state where we have discussed two different cases of squeezing phenomenon. The first case is the normal squeezing while the second is the amplitude squared squeezing, further the second order correlation function is also considered. Our discussion have been extended to include the quasi-probability distribution functions ( W-Wigner and Q-functions). The quadrature distribution and the phase properties in Pegg–Barnett formalism besides the phase variances are considered. Examination of the resonance fluorescence against the present state is given (single atom and thermodynamic limit). It has been shown that the atomic inversion is sensitive to any variation in the nonlinear negative binomial number m.

Generation and some statistical properties of nonlinear pair-coherent states

In this contribution a nonlinear type of the correlated two-mode states (nonlinear pair-coherent states (NLPCS)) is introduced. Based on the resonant ion–cavity interaction, we propose a scheme to generate these states revealing their connection with the pair-coherent state (PCS) and investigate their nonclassical properties. The practical feasibility of this method is also discussed. The s-parameterized characteristic function (CF) of the NLPCS is considered. The s-parameterized quasi-probability distribution function (QDF) is investigated.

Wigner function and transition amplitude of three mutually coupled oscillators

A full quantum mechanical treatment of three electromagnetic fields is considered. The proposed model consists of three different coupling parameters for which the rotating and counter-rotating terms are retained. An exact solution of the wave function in the Schrödinger picture is obtained and the connection with the coherent states wave function is given. The symmetrical ordered quasi-probability distribution function ( W-Wigner function) is calculated via the wave function in the coherent states representation. The squeezing phenomenon is also examined for both single mode and squared-amplitude, where the collapse and revival phenomena are observed. For the case in which λ 3 = 0 and ω 1 = ω 2 = ω 3 (exact resonances) we find that the late phenomenon is apparent but only after long period of the time considered. The transition amplitude between two different coherent states (a state in which all the coupling parameters are involved and a state when the coupling parameter λ 3 = 0 ) is calculated. It is shown that the probability amplitude is sensitive to the variation of the mean photon numbers, and the coupling parameters as well as to the field frequencies.

Statistical properties for a class of nonlinear squeezed states

In the present paper a class of nonlinear squeezed vacuum and first order excited squeezed states are introduced. Under conditions on the non-linearity function, the first order excited squeezed states are realizations of the SU(1,1) group. However, when the condition of the nonlinear function being unitary is removed, these states are defined as eigenstates of certain operators. Some of the properties of these states are investigated for the case of trapped ions. The normalized second-order correlation function, the phase properties, the quasiprobability distribution function and the position distribution of the nonlinear squeezed vacuum and first order excited squeezed states are discussed.

SCHEME FOR DIRECT MEASURING QUASIPROBABILITY DISTRIBUTIONS OF N TRAPPED IONS IN THE DISPERSIVE REGIME

In this letter, a procedure is proposed for reconstructing the motional quantum state of N trapped ions. Unlike other previous schemes, the present scheme works in the dispersive regime in the presence of dissipation. In our scheme, we show that it is possible to directly measure the quasiprobability distribution function corresponding to the motional coherent state in a trapped-ion system subjected to dissipation. We also provide an example of a motional Schrödinger cat state reconstruction to exemplify our procedure.