The one-dimensional Hulthén potential in the quantum phase space representation

The analytic expression of the Wigner function for bound eigenstates of the Hulthén potential in quantum phase space is obtained and presented by plotting this function for a few quantum states. In addition, the correct marginal distributions of the Wigner function in spherical coordinates are determined analytically.     

用热场动力学理论研究介观电路的Wigner函数

利用热场动力学及相干热态表象理论,重构了有限温度下介观RLC电路的Wigner函数,研究了有限温度下介观RLC电路的量子涨落.借助于Weyl-Wigner理论讨论了有限温度下介观RLC电路Wigner函数的边缘分布,并进一步阐明了Wigner函数边缘分布统计平均的物理意义.结果表明:有限温度下介观RLC电路中电荷和电流的量子涨落随着温度和电阻值的增加而增加,回路中的电荷和电流之间存在着压缩效应,这种量子效应是由于系统零点振动的涨落而引起的;有限温度下介观RLC电路Wigner函数边缘分布的统计平均正好是储存在介观RLC电路中电容和电感上的能量.     

Quantum distributions for localized quantum states

The principle of ergodicity of the quantum theory has been used for elaboration of a new technique for numerical simulation of the Wigner function of open dissipative quantum systems. With this purpose the density matrix of a quantum system is represented via averaging over the ensemble of quantum states in time intervals instead of averaging over the ensemble of stochastic variables. It is shown that this approach leads to new approximate expressions for quantum distributions in the phase space, in particular, Wigner functions for systems localized in the region of classical phase trajectories. As an application, the Wigner functions are calculated for the process of intracavity second harmonic generation in the region of Hopf bifurcations.     

双模激发压缩真空态的维格纳函数

利用纠缠态表象下的维格纳(Wigner)算符,构造了双模激发压缩真空态的维格纳函数,并根据该函数在相空间ρ-γ中随参量m,n和r的变化关系,讨论了双模激发压缩真空态的量子干涉特性和压缩效应。结果表明,对于参量m,n不同的取值,双模激发压缩真空态的量子干涉效应的强弱不同;而对于不同的压缩参量r,双模激发压缩真空态呈现出不同程度的压缩效应。最后,根据双模激发压缩真空态的维格纳函数的边缘分布,阐明了此维格纳函数的物理意义。     

Factorization of the Wigner distribution in prime power dimensions

There are different techniques that allow us to gain complete knowledge about an unknown quantum state, e.g., to perform full tomography of this state. For instance, quasi-distributions such as the Weyl or Wigner distributions provide complete information about a quantum state which is equivalent to the information contained in the density matrix. In the case of composite systems, of which the subsystems are not necessarily located at the same place, the experimental feasibility of the tomographic process is considerably simplified whenever it can be realized through local operations and classical communications between local observers. This brings us naturally to study the possibility to factorize the (discrete) Wigner distribution of a composite system into the product of local Wigner distributions, which is the subject of the present paper. The discrete Heisenberg-Weyl group is an essential ingredient of our approach.     

Wigner functions and tomograms of the even and odd binomial states

Using the coherent state representation of Wigner operator and the technique of integration within an ordered product (IWOP) of operators, the Wigner functions of the even and odd binomial states (EOBSs) are obtained. The physical meaning of the Wigner functions for the EOBSs is given by means of their marginal distributions. Moreover, the tomograms of the EOBSs are calculated by virtue of intermediate coordinate-momentum representation in quantum optics.     

Probability Representation of Quantum States

The review of new formulation of conventional quantum mechanics where the quantum states are identified with probability distributions is presented. The invertible map of density operators and wave functions onto the probability distributions describing the quantum states in quantum mechanics is constructed both for systems with continuous variables and systems with discrete variables by using the Born’s rule and recently suggested method of dequantizer–quantizer operators. Examples of discussed probability representations of qubits (spin-1/2, two-level atoms), harmonic oscillator and free particle are studied in detail. Schrödinger and von Neumann equations, as well as equations for the evolution of open systems, are written in the form of linear classical–like equations for the probability distributions determining the quantum system states. Relations to phase–space representation of quantum states (Wigner functions) with quantum tomography and classical mechanics are elucidated.     

Wigner Measures in Noncommutative Quantum Mechanics

We study the properties of quasi-distributions or Wigner measures in the context of noncommutative quantum mechanics. In particular, we obtain necessary and sufficient conditions for a phase-space function to be a noncommutative Wigner measure, for a Gaussian to be a noncommutative Wigner measure, and derive certain properties of the marginal distributions which are not shared by ordinary Wigner measures. Moreover, we derive the Robertson-Schrödinger uncertainty principle. Finally, we show explicitly how the set of noncommutative Wigner measures relates to the sets of Liouville and (commutative) Wigner measures.     

Quasidistributions and coherent states for finite-dimensional quantum systems

In recent years, an approach to discrete quantum phase spaces which comprehends all the main quasiprobability distributions known has been developed. It is the research that started with the pioneering work of Galetti and Piza, where the idea of operator bases constructed of discrete Fourier transforms of unitary displacement operators was first introduced. Subsequently, the discrete coherent states were introduced, and finally, the s-parametrized distributions, that include the Wigner, Husimi, and Glauber–Sudarshan distribution functions as particular cases. In the present work, we adapt its formulation to encompass some additional discrete symmetries, achieving an elegant yet physically sound formalism.     

On the Moyal Formulation of Quantum Identical Particles

Wigner functions of permutation operators are obtained and are used as a basis for a phase space formulation of quantum identical particles. Non-spin systems as well as 1/2 spin systems are considered. The general results are applied to a couple of examples.     

Quantum coherent versus classical coherent light

The paper shows that the Wigner distribution function of quantum optical coherent states, or of a superposition of such states, can be produced and measured with a classical optical set-up using classical coherent light fields. This measurement cannot be done directly in quantum optics since the quantum phase space variables correspond to non-commuting operators. As an example, the Wigner distribution function of Schrödinger cat states of light has been measured. It is also shown that the possibility of measuring the Wigner distribution function of quantum coherent states with classical coherent fields is unique in the sense that it cannot be extended to other quantum states, not even to the incoherent limit of the superposition of coherent states.     

Evolution Equation for a Joint Tomographic Probability Distribution of Spin-1 Particles

The nine-component positive vector optical tomographic probability portrait of quantum state of spin-1 particles containing full spatial and spin information about the state without redundancy is constructed. Also the suggested approach is expanded to symplectic tomography representation and to representations with quasidistributions like Wigner function, Husimi Q?function, and Glauber-Sudarshan P?function. The evolution equations for constructed vector optical and symplectic tomograms and vector quasidistributions for arbitrary Hamiltonian are found. The evolution equations are also obtained in special case of the quantum system of charged spin-1 particle in arbitrary electro-magnetic field, which are analogs of non-relativistic Proca equation in appropriate representations. The generalization of proposed approach to the cases of arbitrary spin is discussed. The possibility of formulation of quantum mechanics of the systems with spins in terms of joint probability distributions without the use of wave functions or density matrices is explicitly demonstrated.     

Quantum States and Hardy’s Formulation of the Uncertainty Principle: a Symplectic Approach

We express the condition for a phase space Gaussian to be the Wigner distribution of a mixed quantum state in terms of the symplectic capacity of the associated Wigner ellipsoid. Our results are motivated by Hardy’s formulation of the uncertainty principle for a function and its Fourier transform. As a consequence we are able to state a more general form of Hardy’s theorem. The authors were supported under the EU-project MEXT-CT-2004-51715.     

Path sums for Brownian motion and quantum mechanics

A treatment is given of classical Brownian motion in phase space based on path summation. It treats efficiently the usual exactly solvable cases when the external force is linear in momentum or position. The method might be useful for generating approximations for more complicated external forces. A path sum formalism is given to generate the Wigner propagator in the Wigner-Weyl phase space formulation of quantum mechanics. The short-time Brownian and Wigner propagators bear a generic similarity.     

增光子奇偶相干态的Wigner函数

利用相干态表象下的Wigner算符, 重构了增光子奇偶相干态的Wigner函数.根据此Wigner函数在相空间中随复变量α的变化关系, 讨论了增光子奇偶相干态的非经典性质. 结果表明, 增光子奇偶相干态总可呈现非经典性质, 且在m取奇(或偶)数时, 增光子偶(或奇)相干态更容易出现非经典性质. 根据增光子奇偶相干态的Wigner函数的边缘分布, 阐明了此Wigner函数的物理意义. 同时, 利用中介表象理论获得了增光子奇偶相干态的量子tomogram函数. 关键词： 增光子奇偶相干态 Wigner函数 中介表象 tomogram函数     

Quantum chaos in the two-center shell model

Within an axially symmetric two-center shell model single-particle levels with Ω=1/2 are analyzed with respect to their level-spacing distributions and avoided level crossings as functions of the shape parameters. Only for shapes sufficiently far from any additional symmetry, ideal Wigner distributions are found as signature for quantum chaos.     

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.     

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.     

SORE APPLICATIONS OF THE COHERENT STATE FORMULATION OF THE WIGNER OPERATOR

On the basis of the preceding paper^[1]we present some new applications of both the normal rroduct form and the coherent state form of the Wigner operator,which involve deriving some new quantum operator formulas, giving the coherent state generalization of the Moyal theorem, evaluating some quantum operators which correspond to the given classical functions in the weyl manner and vice versa. Were it not for the Wigner operator's coherent state formulation given by us the above-mentioned calculations would be hard to perform.     

Wigner distribution,nonclassicality and decoherence of generalized and reciprocal binomial states

There are quantum states of light that can be expressed as finite superpositions of Fock states (FSFS). We demonstrate the nonclassicality of an arbitrary FSFS by means of its phase space distributions such as the Wigner function and the Q-function. The decoherence of the FSFS is studied by considering the time evolution of its Wigner function in amplitude decay and phase damping channels. As examples, we determine the nonclassicality and decoherence of generalized and reciprocal binomial states.