用纠缠态表象和有序算符内的积分方法发展量子力学相空间理论
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摘要
本论文利用有序算符内的积分方法(IWOP)和量子纠缠态表象发展量子力学相空间理论。量子相空间分布函数允许人们用尽可能多的经典语言来描述系统的量子特性并作为量子力学算符的表象工具来使用,最近被作为用来研究量子信息和量子计算机的有用工具。量子相空间的Wigner函数理论可以广泛地应用于处理量子光学、量子化学中的各种问题,尤其是用Wigner分布函数研究光场密度矩阵以及光的量子相干性以及激光理论等,而Husimi分布函数对于研究量子-经典对应、量子混沌也有其特殊的应用。本课题发现从量子纠缠的新概念和用IWOP技术的新方法可以较大地丰富量子相空间理论。本文主要内容包括:
探索了从Wigner函数求P表示的途径,提出了由已知Wigner函数导出P表示的公式,并通过实例说明了该公式的用法,该项工作丰富了量子光场的经典表述理论。在纠缠Wigner算符的基础上提出纠缠Husimi算符的新概念,发现纠缠Husimi算符就是一个双模压缩相干态纯态密度矩阵,它为我们提供了一种简洁精炼的算符版式来计算双模量子态的Husimi分布函数。在算符的Weyl编序乘积积分技术的基础上导出两个Weyl编序算符的乘积公式,并进一步用纠缠态表象将该公式推广到纠缠形式,从而使Weyl-Wigner对应理论得到丰富和发展。发现了Weyl对应在研究Husimi算符中的新应用,提出了一种简便地寻找Husimi算符的方法,即把粗粒函数看作是Husimi算符的Weyl经典对应函数。由正规序Wigner算符的拉登变换引入了两个互为共轭的中介坐标-动量表象,在此基础上我们建立了相应的量子相空间理论,其中包括引入适合该空间的新的Wigner算符;并在该表象的基础上,建立了广义Fredholm算符方程,求出了它的解,并运用该方程导出有关厄米多项式的算符公式;揭示广义Wigner算符与统计学中的随机变量的二维正态分布形式上的相似,这对于研究量子态的tomogram(是英文Tomography的派生词)有用。作为纠缠Husimi算符理论的应用,我们计算并作图研究了单双模组合压缩态的Wigner函数和Husimi函数及其特性;计算并作图研究了激发压缩真空态的Husimi函数及其特性。利用双粒子纠缠态表象求解了带运动耦合的两个相互作用粒子的密度矩阵,带运动耦合的相互作用出现在分子物理、两个互感耦合电路量子化的问题中。充分运用有序算符内的积分技术和纠缠态表象,我们首次引入了均匀磁场(UMF)中电子态的Husimi算符,且把它表示为,即Husimi算符实际上是一纯压缩相干态γ,εκ投影子,它为我们提供了一种简便的算符版式来研究不同电子态的Husimi分布特性,论证了Husimi (边缘)分布是Wigner (边缘)分布的高斯扩展型。
Based on the technique of integration within an ordered product (IWOP) of operators and the quantum entangled state representation we develop the quantum phase space theory. The phase-space distribution function allows one to describe the quantum aspects of a system with as much classical language allowed and have been used as representation tools for quantum-mechanical operators. They provide the ideal link to explore and understand the transition to classical mechanics and to display in phase-space quantum effects. They have also recently proposed as a useful tool for studies related to quantum information and computation. The Wigner function theory can be applied to dealing with all kinds of problems in quantum optics and quantum chemistry, especially be used to study the density matrix of the light field, the quantum interference and the laser theory. The Husimi functions are known to be good tools to study the quantum-classical correspondence and quantum chaos. All these work develop the quantum phase space theory further. So our study is significant. The main content includes:
We explore how to solve the P represenrtation from the Wigner function and present a formula to obtain the P represenrtation from the known Wigner function. We also show how to use the formula through several examples. In doing so, the Weyl-Wigner corresponding theory can be enriched and developed. Based on the entangled Wigner operator we bring forward the new concept of the entangled Husimi operator, find that the entangled Husimi operator is a pure two-mode squeezed coherent state density matrix, which provides us with a neat and concise operator version to calculate the Husimi distribution function of the two-mode quantum state.
Based on the explicit Weyl-ordered form of Wigner erator and the technique of integration within Weyl ordered product of operators we derive the Weyl-ordered operator product formula. The formula is then generalized to the entangled form with the help of entangled state representations. In so doing the Weyl-Wigner correspondence theory gets enriched and developed. We find a new application of the Weyl correspondence in studying the Husimi operator and present a simple way to find the Husimi operator, i.e. regarded exp as the classical Weyl correspondence function connecting the Husimi operator.
Through the Radon transformation of the normally ordered Wigner operator we introduce two mutually conjugate intermediate coordinate- momentum representations. Based on them we construct the appropriate quantum phase space theory which includes the new Wigner operator adapting to this space and construct the appropriate generalized Fredholm operator equation and then find its solution. We then deriving the Hermite polynomials operator identities by applying the Fredholm equation. We also reveal the connection between the generalized Wigner operator and the 2-dimension normal distribution in statistics, which is useful to study the quantum tomogram.
As the application of the entanglement Husimi operator theory we calculate the Wigner function and the Husimi function of the one- and two-mode combination squeezed state ( OTCSS ) , study their characters through drawing the three-dimensional graphics. we also study the Husimi distribution of the excited squeezed vacuum state ( ESVS ).The theoretical calculation of Husimi function can help experimentalists to judge the quality of the experiment. We solve the problem of the density matrix for two interacting particles with kinetic coupling by using the bipartite entangled state representation. Such kind of interaction often appear as describing internal potential in molecular physics theory, the mutual magnetic inductance between two quantized coupling circuits, etc.
For the first time we introduce the operator for studying Husimi distribution function in phase space (γ,ε) for electron's states in uniform magnetic field by virtue of the IWOP technique and the entanglement state representation. Using the Wigner operator in the entangled state <λ| representation we find that is just a pure squeezed coherent state density operator, which brings much convenience for studying the Husimi distribution of various electron's states . We also demonstrate that the marginal distributions of the Husimi function are Gaussian-broadened version of the Wigner marginal distributions.
探索了从Wigner函数求P表示的途径,提出了由已知Wigner函数导出P表示的公式,并通过实例说明了该公式的用法,该项工作丰富了量子光场的经典表述理论。在纠缠Wigner算符的基础上提出纠缠Husimi算符的新概念,发现纠缠Husimi算符就是一个双模压缩相干态纯态密度矩阵,它为我们提供了一种简洁精炼的算符版式来计算双模量子态的Husimi分布函数。在算符的Weyl编序乘积积分技术的基础上导出两个Weyl编序算符的乘积公式,并进一步用纠缠态表象将该公式推广到纠缠形式,从而使Weyl-Wigner对应理论得到丰富和发展。发现了Weyl对应在研究Husimi算符中的新应用,提出了一种简便地寻找Husimi算符的方法,即把粗粒函数看作是Husimi算符的Weyl经典对应函数。由正规序Wigner算符的拉登变换引入了两个互为共轭的中介坐标-动量表象,在此基础上我们建立了相应的量子相空间理论,其中包括引入适合该空间的新的Wigner算符;并在该表象的基础上,建立了广义Fredholm算符方程,求出了它的解,并运用该方程导出有关厄米多项式的算符公式;揭示广义Wigner算符与统计学中的随机变量的二维正态分布形式上的相似,这对于研究量子态的tomogram(是英文Tomography的派生词)有用。作为纠缠Husimi算符理论的应用,我们计算并作图研究了单双模组合压缩态的Wigner函数和Husimi函数及其特性;计算并作图研究了激发压缩真空态的Husimi函数及其特性。利用双粒子纠缠态表象求解了带运动耦合的两个相互作用粒子的密度矩阵,带运动耦合的相互作用出现在分子物理、两个互感耦合电路量子化的问题中。充分运用有序算符内的积分技术和纠缠态表象,我们首次引入了均匀磁场(UMF)中电子态的Husimi算符,且把它表示为,即Husimi算符实际上是一纯压缩相干态γ,εκ投影子,它为我们提供了一种简便的算符版式来研究不同电子态的Husimi分布特性,论证了Husimi (边缘)分布是Wigner (边缘)分布的高斯扩展型。
Based on the technique of integration within an ordered product (IWOP) of operators and the quantum entangled state representation we develop the quantum phase space theory. The phase-space distribution function allows one to describe the quantum aspects of a system with as much classical language allowed and have been used as representation tools for quantum-mechanical operators. They provide the ideal link to explore and understand the transition to classical mechanics and to display in phase-space quantum effects. They have also recently proposed as a useful tool for studies related to quantum information and computation. The Wigner function theory can be applied to dealing with all kinds of problems in quantum optics and quantum chemistry, especially be used to study the density matrix of the light field, the quantum interference and the laser theory. The Husimi functions are known to be good tools to study the quantum-classical correspondence and quantum chaos. All these work develop the quantum phase space theory further. So our study is significant. The main content includes:
We explore how to solve the P represenrtation from the Wigner function and present a formula to obtain the P represenrtation from the known Wigner function. We also show how to use the formula through several examples. In doing so, the Weyl-Wigner corresponding theory can be enriched and developed. Based on the entangled Wigner operator we bring forward the new concept of the entangled Husimi operator, find that the entangled Husimi operator is a pure two-mode squeezed coherent state density matrix, which provides us with a neat and concise operator version to calculate the Husimi distribution function of the two-mode quantum state.
Based on the explicit Weyl-ordered form of Wigner erator and the technique of integration within Weyl ordered product of operators we derive the Weyl-ordered operator product formula. The formula is then generalized to the entangled form with the help of entangled state representations. In so doing the Weyl-Wigner correspondence theory gets enriched and developed. We find a new application of the Weyl correspondence in studying the Husimi operator and present a simple way to find the Husimi operator, i.e. regarded exp as the classical Weyl correspondence function connecting the Husimi operator.
Through the Radon transformation of the normally ordered Wigner operator we introduce two mutually conjugate intermediate coordinate- momentum representations. Based on them we construct the appropriate quantum phase space theory which includes the new Wigner operator adapting to this space and construct the appropriate generalized Fredholm operator equation and then find its solution. We then deriving the Hermite polynomials operator identities by applying the Fredholm equation. We also reveal the connection between the generalized Wigner operator and the 2-dimension normal distribution in statistics, which is useful to study the quantum tomogram.
As the application of the entanglement Husimi operator theory we calculate the Wigner function and the Husimi function of the one- and two-mode combination squeezed state ( OTCSS ) , study their characters through drawing the three-dimensional graphics. we also study the Husimi distribution of the excited squeezed vacuum state ( ESVS ).The theoretical calculation of Husimi function can help experimentalists to judge the quality of the experiment. We solve the problem of the density matrix for two interacting particles with kinetic coupling by using the bipartite entangled state representation. Such kind of interaction often appear as describing internal potential in molecular physics theory, the mutual magnetic inductance between two quantized coupling circuits, etc.
For the first time we introduce the operator for studying Husimi distribution function in phase space (γ,ε) for electron's states in uniform magnetic field by virtue of the IWOP technique and the entanglement state representation. Using the Wigner operator in the entangled state <λ| representation we find that is just a pure squeezed coherent state density operator, which brings much convenience for studying the Husimi distribution of various electron's states . We also demonstrate that the marginal distributions of the Husimi function are Gaussian-broadened version of the Wigner marginal distributions.
引文
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