Covariant Wigner function approach to the relativistic quantum electron gas in a strong magnetic field

This paper is concerned with a unified approach to some equilibrium properties of the relativistic quantum electron plasma embedded in a strong external magnetic field. This unified approach rests on the systematic use of a covariant Wigner function. The equilibrium Wigner function of the noninteracting gas is derived and its main properties are studied. In particular, it satisfies equations that are the complete analog of the usual Liouville equation and thus can be termed “relativistic quantum Liouville equation” whose properties are considered. The equations of state are rederived in this formalism and the results obtained earlier by Canuto and Chiu are found anew. Also, the covariant Wigner funetion of the magnetized vacuum is derived: it is needed, in this formalism, in order to obtain, e.g., the vacuum polarization tensor. Since we are also interested in the plasma modes, the fluctuations of one-particle quantities—and their spectrum—(in particular, of the four current) are calculated in view of their use in the fluctuation-dissipation theorem. We also outline a microscopic proof of this theorem, on the basis of a BBGKY hierarchy for the covariant Wigner functions, and point out the existence of an effective plasma frequency.     

等时量子输运理论

在普遍量子情形下,建立了协变Ｗｉｇｎｅｒ函数的能量矩的等时动力论链和链的截断方案,得到了与低阶矩相关的物理密度量的输运方程和约束方程,并且证明高阶矩都能通过低阶矩表示．     

量子扩散通道中Wigner算符的演化规律

众所周知,量子态的演化可用与其相应的Wigner函数演化来代替.因为量子态的Wigner函数和量子态的密度矩阵一样,都包含了概率分布和相位等信息,因此对量子态的Wigner函数进行研究,可以更加快速有效地获取量子态在演化过程的重要信息.本文从经典扩散方程出发,利用密度算符的P表示,导出了量子态密度算符的扩散方程.进一步通过引入量子算符的Weyl编序记号,给出了其对应的Weyl量子化方案.另外,借助于密度算符的另一相空间表示-Wigner函数,建立了Wigner算符在扩散通道中演化方程,并给出了其Wigner算符解的形式.本文推导出了Wigner算符在量子扩散通道中的演化规律,即演化过程中任意时刻Wigner算符的形式.在此结论的基础上,讨论了相干态经过量子扩散通道的演化情况.     

Covariant Relativistic Statistical Mechanics of Many Particles

In this paper the quantum covariant relativistic dynamics of many bodies is reconsidered. It is emphasized that this is an event dynamics. The events are quantum statistically correlated by the global parameter τ. The derivation of an event Boltzmann equation emphasizes this. It is shown that this Boltzmann equation may be viewed as exact in a dilute event limit ignoring three event correlations. A quantum entropy principle is obtained for the marginal Wigner distribution function. By means of event linking (concatenations) particle properties such as the equation of state may be obtained. We further reconsider the generalized quantum equilibrium ensemble theory and the free event case of the Fermi-Dirac and Bose-Einstein distributions, and some consequences. The ultra-relativistic limit differs from the non-covariant theory and is a test of this point of view.     

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.     

Entanglement Involved in Pair Coherent State Studied via Wigner Function Formalism

We calculate Wigner function, tomogram of the pair coherent state byusing its Schmidt decomposition in the coherent state representation. It turns out that the Wigner function can be seen as the quantum entanglement (QE) between two two-variable Hermite polynomials (TVHP) and the tomogram is further simplified as QE of two single-variable Hermite polynomials. The Husimi function of pair coherent state is also calculated.     

The Wigner molecule in a 2D quantum dot

The charge density and pair correlation function of three interacting electrons confined within a two-dimensional disc-like hard-wall quantum dot are calculated by full numerical diagonalization of the Hamiltonian. The formation of a Wigner molecule in the form of equilateral triangular configuration for electrons is observed as the size of the dot is increased.     

Wigner function and tomogram of the Hermite polynomial state

Using the coherent state representation of Wigner operator and the technique of integration within an ordered product (IWOP) of operators, this paper derives the Wigner function for the Hermite polynomial state (HPS). The tomogram of the HPS is calculated with the aid of newly introduced 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.     

从离散Wigner函数的角度探讨量子相干性度量

量子相干性是量子信息处理的基本要素,在量子计算中扮演着重要的角色.为了便于讨论量子相干性在量子计算中的作用,本文从离散Wigner函数角度对量子相干性进行了探讨.首先对奇素数维量子系统的离散Wigner函数进行了分析,分离出表征相干性的部分,提出了一种可能的基于离散Wigner函数的量子相干性度量方法,并对其进行了量子相干性度量规范的分析;同时也比较了该度量与l_1范数相干性度量之间的关系.重要的是,这种度量方法能够明确给出量子相干性程度与衡量量子态量子计算加速能力的负性和之间不等式关系,由此可以解析地解释量子相干性仅是量子计算加速的必要条件.     

Poincaré covariance and κ-Minkowski spacetime

A fully Poincaré covariant model is constructed as an extension of the κ-Minkowski spacetime. Covariance is implemented by a unitary representation of the Poincaré group, and thus complies with the original Wigner approach to quantum symmetries. This provides yet another example (besides the DFR model), where Poincaré covariance is realised à la Wigner in the presence of two characteristic dimensionful parameters: the light speed and the Planck length. In other words, a Doubly Special Relativity (DSR) framework may well be realised without deforming the meaning of “Poincaré covariance”.     

Quantum corrections for Boltzmann equation

We present the lowest order quantum correction to the semiclassical Boltzmann distribution function, and the equation satisfied by this correction is given. Our equation for the quantum correction is obtained from the conventional quantum Boltzmann equation by explicitly expressing the Planck constant in the gradient approximation, and the quantum Wigner distribution function is expanded in powers of Planck constant, too. The negative quantum correlation in the Wigner distribution function which is just the quantum correction terms is naturally singled out, thus obviating the need for the Husimi’s coarse grain averaging that is usually done to remove the negative quantum part of the Wigner distribution function. We also discuss the classical limit of quantum thermodynamic entropy in the above framework. Supported by the National Natural Science Foundation of China (Grant No. 10404037) and the Scientific Research Fund of GUCAS (Grant No. 055101BM03)     

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.     

复合函数算符的微商法则及其在量子物理中的应用

给出了在量子物理学、量子统计学、算符排序理论、矩阵论以及控制理论中有着重要用途的复合函数算符的一般微分法则,利用这一法则研究了Wigner算符和Weyl对应规则中的积分问题,证明了两类典型的算符恒等公式.给出了Wigner算符的有序算符内的微分形式,并得到了一些重要函数的新的微分式.最后,引入了一个参数型的Wigner算符来统一正规序、Weyl编序以及反正规序三种算符排序.     

Evolution of quantum states via Weyl expansion in dissipative channel

Based on the Weyl expansion representation of Wigner operator and its invariant property under similar transformation, we derived the relationship between input state and output state after a unitary transformation including Wigner function and density operator. It is shown that they can be related by a transformation matrix corresponding to the unitary evolution. In addition, for any density operator going through a dissipative channel, the evolution formula of the Wigner function is also derived. As applications, we considered further the two-mode squeezed vacuum as inputs, and obtained the resulted Wigner function and density operator within normal ordering form. Our method is clear and concise, and can be easily extended to deal with other problems involved in quantum metrology, steering, and quantum information with continuous variable.     

Quantum Kinetic Theory of a Massless Scalar Model in the Presence of a Schwarzschild Black Hole

We employ quantum kinetic theory to investigate local quantum physics in the background of spherically symmetric and neutral black holes formed through the gravitational collapse. For this purpose in mind, we derive and study the covariant Wigner distribution function near to and far away from the black‐hole horizon. We find that the local density of the particle number is negative in the near‐horizon region, while the entropy density is imaginary. These pose a question whether kinetic theory is applicable in the near‐horizon region. We elaborate on that and propose a possible interpretation of how this result might nevertheless be self‐consistently understood.     

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.     

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.     

Magnetic phase diagram in a quasi-two-dimensional electron gas

Using spin density functional theory within the framework of the local spin density approximation with Perdew-Zunger type exchange-correlation energy, ferromagnetism in a quasi-two-dimensional electron gas (Q-2DEG) is studied. The electronic and magnetic structures of a thin film are calculated as a function of film thickness and electron density. Ferromagnetism in the Q-2DEG is found to appear at a higher electron density than in the three-dimensional electron gas. Unless a film is very thin, with decreasing electron density, a magnetic phase transition occurs from a spin-unpolarized fluid to a Wigner film with surface magnetism, in which the spin polarization localizes only in the neighborhood of surfaces. Further decreasing density induces another transition to a fully spin-polarized ferromagnetic Wigner film.