SU(1,1)李代数玻色算符正规和反正规乘积的简单实现

通过指数算符的分解,给出了SU(1,1)李代数玻色指数算符的正规和反正规乘积.     

量子变换理论及其应用概述

概括地介绍了量子变换理论的基本思想,并就这一基本理论在指数上二次型的正规与反正规乘积表示、二次型哈密顿量的配分函数和波函数,以及量子信息中连续变量系统的纠缠度和保真度的计算做了简要的说明。     

利用特殊函数和类比法有序化排列正负指数幂算符

研究算符函数的有序化排列是一项重要的数理任务.本文利用特殊函数和正规乘积排序与反正规乘积排序间的互换法则法导出了幂算符(αα^+)^±n和(α^+α)^±n的正规与反正规乘积排序.进一步,利用类比法得到了算符(XP)^±n和(PX)^±n的坐标-动量排序与动量-坐标排序式.最后,对新得到的这些算符结果的应用进行一些讨论.     

EPR型连续变量纠缠态的正规乘积方法求解

提出了一种求解EPR型连续变量纠缠态在Fock表象中的具体形式的方法.该方法利用量子场论中的正规乘积的性质,通过对正规乘积形式的玻色子算符函数的运算,导出了Fock表象中两粒子EPR型连续变量纠缠态(即两粒子算符X1-X2和P1 P2的共同本征态)的具体形式.该方法可以进一步推广至多粒子EPR型纠缠态相应具体形式的求解.因而,这是一种求解此类纠缠态在Fock表象中具体形式的普遍技术.     

级联光束分离器的纠缠特性及其应用

光束分离器是一个具有广泛应用的线性光学器件,它在非经典量子态特别是纠缠态的制备中具有重要作用.基于单个光束分离器的表象表示,本文进一步考察多个级联光束分离器的纠缠特性,特别是结合有序算符内的积分技术推导了级联光束分离器的正规乘积、紧指数表示及级联算符的表象表示.作为应用,本文利用两个级联光束分离器获得了量子力学表象及其Schmidt分解,并结合量子条件测量制备了qubit态的叠加态.本文的研究方法已被直接推广至多个光束分离器级联情况,相关研究内容为多模纠缠态、多模qubit态的制备提供了一种有效的途径,且为由光束分离器组成的线性器件系统总作用的算符正规乘积及其紧指数表示提供了一般方法.     

Schwinger玻色子表示中转动算符的对易性质

本文发展Schwinger玻色子表示中的角动量理论, 其中包括用正规乘积内积分法导出转动算符对易子[e^iaJx,e^iβJy]的正规乘积表达式及不同次序的转动对相干态波函数的影响. 还给出了e^σJ-e^λJ+等算符的正规乘积表达式及其对原子相干态的应用.     

扩散过程中弱相干光场的退相干

研究了扩散过程中弱相干光场量子特性的演化.利用正规乘积、反正规乘积和Weyl编序算符内的积分技术,采用热纠缠态表象求解密度矩阵主方程,利用Kraus算符给出扩散过程中密度算符解的表达式,导出初态为弱相干态的量子态密度算符演化规律.讨论了扩散对光场压缩效应和反聚束效应的影响.结果表明:随着扩散过程的进行,弱相干场压缩深度和压缩范围均在减小;扩散初期光场呈反聚束效应,扩散时间大于一定值后反聚束效应消失.     

在量子光学框架中研究魏格纳-维利分布

我们在量子光学框架中研究光信号的魏格纳-维利分布,指出利用魏格纳算符和纠缠魏格纳算符的显示正规乘积形式以及压缩算符的纠缠态表象,这方面的研究就可做到数学上简明和物理上有吸引力.     

Wigner算符的正规乘积形式和相干态形式的应用

本文导出了Wigner算符的正规乘积形式和相干态形式及其若干应用, 其中包括若干新量子算符公式的导出, Moyal定理的相干态推广, 计算以前文献未曾得到的若干与经典函数对应的量子Weyl算符以及若干与量子算符对应的Weyl经典函数.     

算符二次量子化形式的一种导出方式

首先通过单体算符在粒子数表象下的形式得到单粒子跃迁算符(定义为粒子产生湮没算符的乘积,即a+iaj)的具体展开式,同时给出它与产生湮没算符的对易关系,然后以此为基础,在不区分全同粒子系统的统计学性质的情况下,简洁地得到二体算符和任意n体算符的二次量子化形式.     

Extended two-parameter squeezed states

In a recent paper, a new λ-dependent squeezed states was proposed [J. Beckers, N. Debergh and F.H. Szafraniec, Phys. Lett. A 243 (1998) 256–260] in which the usual Fock-space creation operator has been modified by introducing a simple translational term. In this paper, we generalize this approach by adding a new term proportional to the creation operator using exponential quadratic operators.     

Free Meixner States

Free Meixner states are a class of functionals on non-commutative polynomials introduced in [Ans06]. They are characterized by a resolvent-type form for the generating function of their orthogonal polynomials, by a recursion relation for those polynomials, or by a second-order non-commutative differential equation satisfied by their free cumulant functional. In this paper, we construct an operator model for free Meixner states. By combinatorial methods, we also derive an operator model for their free cumulant functionals. This, in turn, allows us to construct a number of examples. Some of these examples are shown to be trivial, in the sense of being free products of functionals which depend on only a single variable, or rotations of such free products. On the other hand, the multinomial distribution is a free Meixner state and is not a product. Neither is a large class of tracial free Meixner states which are analogous to the simple quadratic exponential families in statistics. This work was supported in part by NSF grant DMS-0613195.     

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.     

Analytic Expression of Arbitrary Matrix Elements for Boson Exponential Quadratic Polynomial Operators

Making use of the transformation relation among usual, normal, and antinormal ordering for the multimode boson exponential quadratic polynomial operators （BEQPO＇s）I we present the analytic expression of arbitrary matrix elements for BEQPO＇s. As a preliminary application, we obtain the exact expressions of partition function about the boson quadratic polynomial system, matrix elements in particle-number, coordinate, and momentum representation, and P representation for the BEQPO＇s.     

A Pair of Resonance Stripe Solitons and Lump Solutions to a Reduced(3+1)-Dimensional Nonlinear Evolution Equation

With symbolic computation, some lump solutions are presented to a(3+1)-dimensional nonlinear evolution equation by searching the positive quadratic function from the Hirota bilinear form of equation. The quadratic function contains six free parameters, four of which satisfy two determinant conditions guaranteeing analyticity and rational localization of the solutions, while the others are free. Then, by combining positive quadratic function with exponential function, the interaction solutions between lump solutions and the stripe solitons are presented on the basis of some conditions. Furthermore, we extend this method to obtain more general solutions by combining of positive quadratic function and hyperbolic cosine function. Thus the interaction solutions between lump solutions and a pair of resonance stripe solitons are derived and asymptotic property of the interaction solutions are analyzed under some specific conditions. Finally, the dynamic properties of these solutions are shown in figures by choosing the values of the parameters.     

Scaling Laws for the Multidimensional Burgers Equation with Quadratic External Potential

The reordering of the multidimensional exponential quadratic operator in coordinate-momentum space (see X. Wang, C.H. Oh and L.C. Kwek (1998). J. Phys. A.: Math. Gen. 31:4329–4336) is applied to derive an explicit formulation of the solution to the multidimensional heat equation with quadratic external potential and random initial conditions. The solution to the multidimensional Burgers equation with quadratic external potential under Gaussian strongly dependent scenarios is also obtained via the Hopf-Cole transformation. The limiting distributions of scaling solutions to the multidimensional heat and Burgers equations with quadratic external potential are then obtained under such scenarios. AMS Subject Classifications: 60G60, 60G15, 62M15, 60H15     

How to decompose arbitrary continuous-variable quantum operations

We present a general, systematic, and efficient method for decomposing any given exponential operator of bosonic mode operators, describing an arbitrary multimode Hamiltonian evolution, into a set of universal unitary gates. Although our approach is mainly oriented towards continuous-variable quantum computation, it may be used more generally whenever quantum states are to be transformed deterministically, e.g., in quantum control, discrete-variable quantum computation, or Hamiltonian simulation. We illustrate our scheme by presenting decompositions for various nonlinear Hamiltonians including quartic Kerr interactions. Finally, we conclude with two potential experiments utilizing offline-prepared optical cubic states and homodyne detections, in which quantum information is processed optically or in an atomic memory using quadratic light-atom interactions.     

s-Ordered Exponential of Quadratic Forms Gained via IWSOP Technique

Using the generalized -ordered Wigner operator, in which is a vector over the field of complex numbers, the technique of integration within an s-ordered product of operators (IWSOP) has been extended to multimode case. We derive the -ordered form of the widely applicable multimode exponential of quadratic form , each mode being in some particular order s _i , applying this method.     

Newton-Leibniz integration for ket-bra operators (II)—application in deriving density operator and generalized partition function formula

Via the route of applying Newton-Leibniz integration rule to Dirac’s symbolic operators, we show that the density operator e^−βH, where H is multi-mode quadratic interacting boson operators, is a mapping of symplectic transformation in the coherent state representation appearing in the form of non-symmetric ket-bra operator integration. By virtue of the technique of integration within an ordered product (IWOP) of operators, we deduce its normally ordered form which directly leads to the generalized partition function formula and the Wigner function. Some new representations, such as displacement-squeezing correlated squeezed coherent states, constructed by the IWOP technique, also bring convenience in deriving partition functions.