Quantum corrections to the entropy in a driven quantum Brownian motion model

The quantum Brownian motion model is a typical model in the study of nonequilibrium quantum thermodynamics. Entropy is one of the most fundamental physical concepts in thermodynamics.In this work, by solving the quantum Langevin equation, we study the von Neumann entropy of a particle undergoing quantum Brownian motion. We obtain the analytical expression of the time evolution of the Wigner function in terms of the initial Wigner function. The result is applied to the thermodynamic equilibrium initial state, which reproduces its classical counterpart in the high temperature limit. Based on these results, for those initial states having well-defined classical counterparts, we obtain the explicit expression of the quantum corrections to the entropy in the weak coupling limit. Moreover, we find that for the thermodynamic equilibrium initial state, all terms odd in h are exactly zero. Our results bring important insights to the understanding of entropy in open quantum systems.     

Wigner's function and other distribution functions in mock phase spaces

This review deals with the methods of associating functions with quantum mechanical operators in such a manner that these functions should furnish conveniently semiclassical approximations. We present a unified treatment of methods and results which usually appear under expressions such as Wigner's function, Weyl's association, Kirkwood's expansion, Glauber's coherent state representation, etc.; we also construct some new associations.Section 1 gives the motivation by discussing the Thomas-Fermi theory of an atom with this end in view.Section 2 introduces new operators which resemble Dirac delta functions with operator arguments, the operators being the momenta and coordinates. Reasons are given as to why this should be useful. Next we introduce the notion of an operator basis, and discuss the possibility and usefulness of writing an operator as a linear combination of the basis operators. The coefficients in the linear combination are c-numbers and the c-numbers are associated with the operator (in that particular basis). The delta function type operators introduced before can be used as a basis for the dynamical operators, and the c-numbers obtained in this manner turn out to be the c-number functions used by Wigner, Weyl, Kirkwood, Glauber, etc. New bases and associations can now be invented at will. One such new basis is presented and discussed. The reasons and motivations for choosing different bases is then explained.The copious and seemingly random mathematical relations between these functions are then nothing else but the relations between the expansion coefficients engendered by the relations between the different bases. These are shown and discussed in this light. A brief discussion is then given to possible transformation of the p, q labels.Section 3 gives examples of how the semiclassical expansions are generated for these functions and exhibits their equivalence.The mathematical paraphernalia are collected in the appendices.     

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.     

CALCULATION OF THE KINETIC-ENERGY DISTRIBUTIONS OF FISSION-FRAGMENTS FROM MONTE-CARLO SIMULATION TO THE FOUR-DIMENSIONAL LANGEVIN EQUATION

The four-dimensional Langevin equation for two collective coordinates (the distance bet ween the centers of mass of the dascent fragments and the neck parameter) and their conjugate momenta is used as a dynamical equation to describe the descent of Brownian particles from the saddle-point to the scission points Monte-carlo method is used to slove the Langevin equation.The variances of the kinetic-energy distributions of nuclear fission-fragments in the range 322/A<40 have been calculated.The results of calculation are in good agreement with the experimental data.     

Power-law tails from multiplicative noise

We show that the well-known linear Langevin equation, modeling the Brownian motion and leading to a Gaussian stationary distribution of the corresponding Fokker-Planck equation, is changed by the smallest multiplicative noise. This leads to a power-law tail of the distribution for sufficiently large momenta. At finite ratio of the correlation strength for the multiplicative and the additive noises the stationary energy distribution becomes exactly the Tsallis distribution.     

Towards Nonlinear Quantum Fokker-Planck Equations

It is demonstrated how the equilibrium semiclassical approach of Coffey et al. can be improved to describe more correctly the evolution. As a result a new semiclassical Klein-Kramers equation for the Wigner function is derived, which remains quantum for a free quantum Brownian particle as well. It is transformed to a semiclassical Smoluchowski equation, which leads to our semiclassical generalization of the classical Einstein law of Brownian motion derived before. A possibility is discussed how to extend these semiclassical equations to nonlinear quantum Fokker-Planck equations based on the Fisher information.     

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.     

Quenched master fields

We derive an exact algebraic (master) equation for the euclidean master field of any large-N matrix theory, including quantum chromodynamics. The master equation is the quenched Langevin equation. The master field, a translationally covariant function of (uniform) random momenta and (gaussian) random noise, is easily constructed in perturbation theory.     

Correspondence between quantum-optical transform and classical-optical transform explored by developing Dirac’s symbolic method

By virtue of the new technique of performing integration over Dirac’s ket–bra operators, we explore quantum optical version of classical optical transformations such as optical Fresnel transform, Hankel transform, fractional Fourier transform, Wigner transform, wavelet transform and Fresnel–Hadmard combinatorial transform etc. In this way one may gain benefit for developing classical optics theory from the research in quantum optics, or vice-versa. We cannot only find some new quantum mechanical unitary operators which correspond to the known optical transformations, deriving a new theorem for calculating quantum tomogram of density operators, but also can reveal some new classical optical transformations. For examples, we find the generalized Fresnel operator (GFO) to correspond to the generalized Fresnel transform (GFT) in classical optics. We derive GFO’s normal product form and its canonical coherent state representation and find that GFO is the loyal representation of symplectic group multiplication rule. We show that GFT is just the transformation matrix element of GFO in the coordinate representation such that two successive GFTs is still a GFT. The ABCD rule of the Gaussian beam propagation is directly demonstrated in the context of quantum optics. Especially, the introduction of quantum mechanical entangled state representations opens up a new area in finding new classical optical transformations. The complex wavelet transform and the condition of mother wavelet are studied in the context of quantum optics too. Throughout our discussions, the coherent state, the entangled state representation of the two-mode squeezing operators and the technique of integration within an ordered product (IWOP) of operators are fully used. All these have confirmed Dirac’s assertion: “...for a quantum dynamic system that has a classical analogue, unitary transformation in the quantum theory is the analogue of contact transformation in the classical theory”.     

Dynamical properties of colloidal systems: 1. Derivation of stochastic transport equations

To describe dynamical properties of a system of interacting Brownian particles stochastic transport equations are derived for the positions of the particles and their concentration fluctuations. This is achieved by an expansion of the Langevin equation for the momenta in terms of the reciprocal of the friction coefficient. As a by-product this procedure gives a new derivation of the generalized Smoluchowski equation. Using a local equilibrium approximation for the configurational distribution function a mode-mode coupling equation is derived for the local concentration, which still depends on the random forces of the solvent. For the interaction free case the relation to the ordinary diffusion approach is established.     

Entanglement Rule of Wigner Operators at a Beamsplitter

It is known that beamsplitter can be used to produce quantum entanglement, in this paper we examine this topic from the point of view of Wigner operators. Using Weyl-ordering of the Wigner operator and the Weyl ordering invariance of Weyl ordered operators under similarity transformation we derive the entanglement rule of Wigner operators at a beamsplitter.     

Universal invariants in quantum mechanics and physics of optical and particle beams

We give a brief review of the theory of quantum universal invariants and their counterparts in the physics of light and particle beams. The invariants concerned are certain combinations of the second- and higher-order moments (variances) of quantum-mechanical operators, or the transverse phase-space coordinates of the paraxial beams of light or particles. They are conserved in time (or along the beam axis) independently of the concrete form of the coefficients of the Schrödinger-like equations governing the evolution of the systems, provided that the effective Hamiltonian is either a generic quadratic form of the generalized coordinate-momenta operators or a linear combination of generators of some finite-dimensional algebra (in particular, any semisimple Lie algebra). Using the phase space representation of quantum mechanics (paraxial optics) in terms of the Wigner function, we elucidate the relation of the quantum (optical) invariants to the classical universal integral invariants of Poincaré and Cartan. The specific features of Gaussian beams are discussed as examples. The concept of the universal quantum integrals of motion is introduced, and examples of the “universal invariant solutions” to the Schrödinger equation, i.e., self-consistent eigenstates of the universal integrals of motion, are given.     

On the quantum langevin equation

The quantum Langevin equation is the Heisenberg equation of motion for the (operator) coordinate of a Brownian particle coupled to a heat bath. We give an elementary derivation of this equation for a simple coupled-oscillator model of the heat bath.Deceased.     

Symmetric quadratic Hamiltonians with pseudo-Hermitian matrix representation

We prove that any symmetric Hamiltonian that is a quadratic function of the coordinates and momenta has a pseudo-Hermitian adjoint or regular matrix representation. The eigenvalues of the latter matrix are the natural frequencies of the Hamiltonian operator. When all the eigenvalues of the matrix are real, then the spectrum of the symmetric Hamiltonian is real and the operator is Hermitian. As illustrative examples we choose the quadratic Hamiltonians that model a pair of coupled resonators with balanced gain and loss, the electromagnetic self-force on an oscillating charged particle and an active LRC circuit.     

Preserving Coherence of Coherent State Studied via Weyl-Wigner Correspondence

By means of the Weyl correspondence and the explicit normally orderedexpression of the Wigner operator we convert the time evolution equation ofcoherent states, governed by some Hamiltonian operators, into seeking forconsistent solution of a set of evolution equtions of classical variableswhich can meet the requirment that an initial coherent state remainscoherent all the time.     

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.     

Asymptotics of High Order Noise Corrections

We consider an evolution operator for a discrete Langevin equation with a strongly hyperbolic classical dynamics and noise with finite moments. Using a perturbative expansion of the evolution operator we calculate high order corrections to its trace in the case of a quartic map and Gaussian noise. The asymptotic behaviour is investigated and is found to be independent up to a multiplicative constant of the distribution of noise.