Marginal Distribution of Wigner Function in Mesoscopic RLC Circuit at Finite Temperature and Its Application

By means of the Weyl correspondence and Wigner theorem the marginal distribution of Wigner function in mesoscopic RLC circuit at finite temperature was discussed. Here we endow the Wigner function with a new physical meaning, i.e., its marginal distributions’ statistical average for q ²/(2C) and p ²/(2L) are the temperature-related energy stored in capacity and in inductance of the mesoscopic RLC circuit, respectively.     

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     

Continuous Multipartite Entangled Representation and Its Wigner Operator and Marginal Distribution

By extending the EPR bipartite entanglement to multipartite case, we briefly introduce a continuous multipartite entangled representation and its canonical conjugate state in the multi-mode Fock space, analyze their Schmidt decompositions and give their entangling operators. Furthermore, based on the above analysis we also find the n-mode Wigner operator. In doing so we may identify the physical meaning of the marginal distribution of the Wigner function.     

Continuous Multipartite Entangled Representation and Its Wigner Operator and Marginal Distribution

By extending the EPR bipartite entanglement to multipartite case, we briefly introduce a continuous multipartite entangled representation and its canonical conjugate state in the multi-mode Fock space, analyze their Schmidt decompositions and give their entangling operators. Furthermore, based on the above analysis we also find the n-mode Wigner operator. In doing so we may identify the physical meaning of the marginal distribution of the Wigner function.     

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.     

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.     

New transformation of Wigner operator in phase space quantum mechanics for the two-mode entangled case

As a natural and important extension of Fan's paper (Fan Hong-Yi 2010 Chin. Phys. B 19 040305) by employing the formula of operators' Weyl ordering expansion and the bipartite entangled state representation this paper finds a new two-fold complex integration transformation about the Wigner operator Δ ( μ,v ) (in its entangled form) in phase space quantum mechanics, and its inverse transformation. In this way, some operator ordering problems regarding to ( a₁⁺-a₂) and (a₁+a₂⁺) can be solved and the contents of phase space quantum mechanics can be enriched, where a_i,a_i⁺ are bosonic creation and annihilation operators, respectively.     

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。     

Wigner函数的简单引入

通过结合坐标表象及动量表象完备性的纯高斯积分形式及Wigner函数的物理意义,在量子统计的意义下简单的引入了Wigner算符及Wigner函数     

The Wigner distribution function applied to laser radiation

The Wigner distribution function (WDF) is investigated analytically and then calculated numerically and represented graphically for Laguerre-Gaussian modes and for a laser with annular gain medium. The derivation of the normalized beam qualityM ² from the WDF on the basis of second order moments is discussed, andM ² is determined for the examples mentioned above.     

Two-variable Hermite Polynomial State and Its Wigner Function

In this paper we obtain the Wigner functions of two-variable Hermite polynomial states (THPS) and their marginal distribution using the entangled state |ξ〉 representation. Also we obtain tomogram of THPS by virtue of the Radon transformation between the Wigner operator and the projection operator of another entangled state |η,τ ₁,τ ₂〉.     

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.     

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.     

New Properties of the Wigner Function of the Tripartite Entangled State

For entangled three particles one should treat their wave function as a whole, there is no physical meaning talking about the wave function (or Wigner function) for any one of the tripartite, therefore thinking of the entangled Wigner function (Wigner operator) is of necessity, we introduce the entangled Wigner operator related to a pair of mutually conjugate tripartite entangled state representations and discuss some of its new properties, such as the trace product rule, the size of an entangled quantum state and the upper bound of the three-mode Wigner function. Deriving wave function from its corresponding tripartite entangled Wigner function is also presented. Those new properties of the tripartite entangled Wigner function play significant role in quantum physics because they provide us deeper insight into the shape of quantum states.     

Bivariate averaging and the Wigner distribution function

In order to gain insight into the nature of the Wigner and related distribution functions, bivariate averaging functions of real unbounded variables with absolutely continuous marginals that are ordinary probabilities are considered. Accordingly variables are chosen to be phase space variables that are respectively eigenvalues of position and momentum operators. The impact of the condition that the marginals are squared magnitudes of amplitudes that are Fourier transforms of one another is emphasized by the delay of the introduction of this Fourier transform condition until after the form for a bivariate distribution with the given marginals is obtained. When the respective amplitudes are fourier transforms of one another, special cases of the bivariate averaging function correspond to generalized Wigner functions characterized by a parameterα. Such anα-Wigner function can be used as the basis of a consistent averaging procedure if an appropriate corresponding representation for underlying operators to be averaged is specified. Properties of theα-Wigner functions are summarized.     

Wigner Surmise for Domain Systems

In random matrix theory the spacing distribution functions p ⁽ⁿ⁾(s) are well fitted by the Wigner surmise and its generalizations. In this approximation the spacing functions are completely described by the behavior of the exact functions in the limits s→0 and s→∞. Most non equilibrium systems do not have analytical solutions for the spacing distribution and correlation functions. Because of that, we explore the possibility to use the Wigner surmise approximation in these systems. We found that this approximation provides a first approach to the statistical behavior of complex systems, in particular we use it to find an analytical approximation to the nearest neighbor distribution of the annihilation random walk.     

Measurement of quantum states and the Wigner function

In quantum mechanics, the state of an individual particle (or system) is unobservable, i.e., it cannot be determined experimentally, even in principle. However, the notion of measuring a state is meaningful if it refers to anensemble of similarly prepared particles, i.e., the question may be addressed: Is it possible to determine experimentally the state operator (density matrix) into which a given preparation procedure puts particles. After reviewing the previous work on this problem, we give simple procedures, in the line of Lamb's operational interpretation of quantum mechanics, for measuring a translational state operator (whether pure or mixed), via its Wigner function. These procedures closely parallel methods that might be used in classical mechanics to determine a true phase space probability distribution; thus, the Wigner function simulates such a distribution not only formally, but operationally also.

E. P. Wigner⁽¹⁾

     

Quantum Statistics of the Toda Oscillator in the Wigner Function Formalism

Classical and quantum mechanical Toda systems (Toda molecules, Toda lattices, Toda quantum fields) recently found growing interest as nonlinear systems showing solitons and chaos. In this paper the statistical thermodynamics of a system of quantum mechanical Toda oscillators characterized by a potential energy V(q) = V_o cos h q is treated within the Wigner function formalism (phase space formalism of quantum statistics). The partition function is given as a Wigner- Kirkwood series expansion in terms of powers of h² (semiclassical expansion). The partition function and all thermodynamic functions are written, with considerable exactness, as simple closed expressions containing only the modified Hankel functions K_o and K₁ of the purely imaginary argument iζ with ζ = V_o/kT.     

Quantum Dynamical Behavior of the Morse Oscillator: the Wigner Function Approach

We explore the quantum dynamical behavior of the Morse oscillator in the phase space using the Wigner function. For an initial wave packet excited with Gaussian probability distribution, we calculate the associated Wigner function and compute its time evolution. By calculating the marginal probabilities, we study the formation of quantum carpets both in the position space and in the momentum space. In addition, in view of these probabilities, we present the time evolution of the position and momentum expectation values. The structure of quantum carpets and the time-evolved expectation values mimic the emergence of quantum revivals and fractional revivals.