量子相空间分布函数在几种量子通道中的演化
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摘要
相空间一词最早出现在经典物理的哈密顿动力学中,为了形象地说明哈密顿正则方程,将质点的坐标q与动量p组成的坐标架张成了一个相空间。相空间中相体积的概念被用来描写统计力学中的状态数、系综和热力学几率,相体积在演化中的不变性称为刘维定理。但是,在量子论中,根据海森堡的不确定性原理,人们不能同时精确地测量微观粒子的位置q和动量p,即不能确定到一个相点。于是自然就会想到在相空间中定义准分布函数来研究微观粒子的量子态及其运动。鉴于此,魏格纳(Wigner)引入了对应密度算符ρ的准经典分布函数W(q,p),它的边缘分布分别对应于坐标空间和动量空间测量到粒子的几率,赋予了相空间以新的意义。但是魏格纳函数本身并不总是正定的,故不能作为一个概率分布函数(只能称之为准概率分布函数)。在魏格纳函数定义的基础上Husimi引入一个新的分布函数——Husimi函数,由于它总是正定的故可以作为一个新的概率分布函数。本文将基于热纠缠态表象和用有序算符内的积分技术(IWOP技术)来研究魏格纳函数和Husimi函数如何在各种量子通道中的时间演化规律。主要内容如下：
一、1)简要介绍了量子力学表象积分的新技术,讨论了正规序算符内的积分技术(IWOP技术)和外尔(Weyl)编序算符内的积分技术(IWWOP技术)。2)阐述了开放系统量子理论的基本思想和研究方法,对密度矩阵的Kraus算符和表示作一扼要地说明。3)并回顾了应用热纠缠态表象方法如何求解多种常见量子退相干模型下的密度算符主方程。
二、作为IWWOP技术在量子相空间理论中的应用,我们探讨什么量子化方案能将经典相空间中一条射线(以δ(x-λq-σp)表示)保持算符δ-函数(射线)形式,即量子化为δ(x-λQ-νP).我们发现它是Weyl量子化。
三、介观电路的量子化是研究量子计算机和超导量子电路的一个重要课题。本文采用一个全新的观点和方法,将介观RLc电路的量子耗散代之以相应的密度算符在振幅衰减通道的演化来研究,得出了该回路密度矩阵量子耗散的解析形式。
四、我们发现无论何种退相干模式,魏格纳函数随时间的演化在物理上总可以归结为魏格纳算符随时间演化的问题。在此基础上我们利用密度矩阵的算符和(Kraus算符和)表示推导出振幅衰减通道、激光过程、位相扩散(或阻尼)模型以及扩散极限情况下的噪声通道等若干量子通道中的魏格纳算符随时间的演化规律。
五、我们将Husimi函数随时间的演化归结为相应的粗粒高斯平滑魏格纳算符随时间演化的问题,基于此我们成功地推导出振幅衰减通道中的Husimi函数随时间的演化情况。
六、对于一个场强与外源相关的谐振子我们建立了非线性数-相压缩态,在这个态我们发现当粒子数起伏增加时,相位起伏作相应地减小。我们推导出其数-相不确定关系。
Phase space is a word first appeared in Hamiltonian dynamics of classical physics. Coordinate frame constituted by coordinate q and momentum p spans a phase space, which vividly illustrates the Hamiltonian canonical equations. The concept of phase volume in phase space is used to describe the number of state, ensemble and thermodynamic probability in statistical mechanics, phase volume invariance in evolution is called Liouville's theorem. However, in quantum theory, people can't accurately measure the position and momentum of microscopic particles simultaneously according to the Heisenberg uncertainty principle, that is to say, one can't determine a phase point. So it is naturally to think that one can define a quasi distribution function in phase space to study the quantum states of microscopic particles and their motion. In view of this, Wigner introduced the density operator p corresponding to the classical distribution function W(q,p), its marginal distributions are corresponding to the observed probabilities of the particle in coordinate space and momentum space respectively, which gives new meaning to the phase space. But the Wigner function itself is not always positive definite, so it can't be as a probability distribution function (can only be called quasi probability distribution function). On the basis of the definition of the Wigner function Husimi introduced a new distribution function——Husimi function. Because it is always positive definite, thus it can be used as a new probability distribution function. This article will study how the Wigner function and Husimi function in various kinds of quantum channel evolute over time by using the thermo entangled state representation method and based on IWOP technique. The main contents are as follows:
1.1)The new technique of quantum representation integration is briefly introduced. We discussed the technique of integration within an ordered product (IWOP) of operators and the technique of integration within the Weyl ordered product (IWWOP) of operators.2) The basic ideas and research methods of quantum open system are expounded and the Kraus operator sum reprensentation of density matrix is briefly illustrated.3) We review how to solve the master equations of density operators under a variety of quantum decoherence models by using the method of thermo entangled state representation.
2. As the application of IWWOP technique in the quantum phase space theory, we discussed what kind of quantization scheme can make a classical phase space ray (expressed with δ(x-λq-σp)) keep the form of δ-function, namely δ(x-λQ-vP). We found that it is Weyl quantization scheme.
3. The quantization of a mesoscopic circuit is an important subject of studying quantum computer and superconducting quantum circuit. Using a new viewpoint and a new method, we replaced the problem of the quantum dissipation of mesoscopic RLC circuit by studying the evolution of the corresponding density operator in the amplitude dissipative channel and obtained the analytic form of the density matrix of the circuit was.
4. We found that no matter what kind of decoherence modes time evolution of their Wigner functions can be always ascribed to time evolution of their Wigner operators. Based on which, we used the Kraus operator sum representation of density operator to derive the time evolution law of the Wigner operator in several quantum channels such as amplitude dissipative channel, laser process, phase diffusion (or damping) model and noise channel in diffusion limit.
5. Time evolution of Husimi functions can be always ascribed to time evolution of their coarse-Guassin-smoothing-Wigner-operator. Based on which, we derive the time evolution law of Husimi function in amplitude dissipative channel.
6. For an harmonic oscillator with a field intensity related external source we establish the nonlinear number-phase squeezed state, in this state we find that while the number fluctuation increases, the phase fluctuation decreases correspondingly. The number-phase uncertainty relationship is exactly derived.
一、1)简要介绍了量子力学表象积分的新技术,讨论了正规序算符内的积分技术(IWOP技术)和外尔(Weyl)编序算符内的积分技术(IWWOP技术)。2)阐述了开放系统量子理论的基本思想和研究方法,对密度矩阵的Kraus算符和表示作一扼要地说明。3)并回顾了应用热纠缠态表象方法如何求解多种常见量子退相干模型下的密度算符主方程。
二、作为IWWOP技术在量子相空间理论中的应用,我们探讨什么量子化方案能将经典相空间中一条射线(以δ(x-λq-σp)表示)保持算符δ-函数(射线)形式,即量子化为δ(x-λQ-νP).我们发现它是Weyl量子化。
三、介观电路的量子化是研究量子计算机和超导量子电路的一个重要课题。本文采用一个全新的观点和方法,将介观RLc电路的量子耗散代之以相应的密度算符在振幅衰减通道的演化来研究,得出了该回路密度矩阵量子耗散的解析形式。
四、我们发现无论何种退相干模式,魏格纳函数随时间的演化在物理上总可以归结为魏格纳算符随时间演化的问题。在此基础上我们利用密度矩阵的算符和(Kraus算符和)表示推导出振幅衰减通道、激光过程、位相扩散(或阻尼)模型以及扩散极限情况下的噪声通道等若干量子通道中的魏格纳算符随时间的演化规律。
五、我们将Husimi函数随时间的演化归结为相应的粗粒高斯平滑魏格纳算符随时间演化的问题,基于此我们成功地推导出振幅衰减通道中的Husimi函数随时间的演化情况。
六、对于一个场强与外源相关的谐振子我们建立了非线性数-相压缩态,在这个态我们发现当粒子数起伏增加时,相位起伏作相应地减小。我们推导出其数-相不确定关系。
Phase space is a word first appeared in Hamiltonian dynamics of classical physics. Coordinate frame constituted by coordinate q and momentum p spans a phase space, which vividly illustrates the Hamiltonian canonical equations. The concept of phase volume in phase space is used to describe the number of state, ensemble and thermodynamic probability in statistical mechanics, phase volume invariance in evolution is called Liouville's theorem. However, in quantum theory, people can't accurately measure the position and momentum of microscopic particles simultaneously according to the Heisenberg uncertainty principle, that is to say, one can't determine a phase point. So it is naturally to think that one can define a quasi distribution function in phase space to study the quantum states of microscopic particles and their motion. In view of this, Wigner introduced the density operator p corresponding to the classical distribution function W(q,p), its marginal distributions are corresponding to the observed probabilities of the particle in coordinate space and momentum space respectively, which gives new meaning to the phase space. But the Wigner function itself is not always positive definite, so it can't be as a probability distribution function (can only be called quasi probability distribution function). On the basis of the definition of the Wigner function Husimi introduced a new distribution function——Husimi function. Because it is always positive definite, thus it can be used as a new probability distribution function. This article will study how the Wigner function and Husimi function in various kinds of quantum channel evolute over time by using the thermo entangled state representation method and based on IWOP technique. The main contents are as follows:
1.1)The new technique of quantum representation integration is briefly introduced. We discussed the technique of integration within an ordered product (IWOP) of operators and the technique of integration within the Weyl ordered product (IWWOP) of operators.2) The basic ideas and research methods of quantum open system are expounded and the Kraus operator sum reprensentation of density matrix is briefly illustrated.3) We review how to solve the master equations of density operators under a variety of quantum decoherence models by using the method of thermo entangled state representation.
2. As the application of IWWOP technique in the quantum phase space theory, we discussed what kind of quantization scheme can make a classical phase space ray (expressed with δ(x-λq-σp)) keep the form of δ-function, namely δ(x-λQ-vP). We found that it is Weyl quantization scheme.
3. The quantization of a mesoscopic circuit is an important subject of studying quantum computer and superconducting quantum circuit. Using a new viewpoint and a new method, we replaced the problem of the quantum dissipation of mesoscopic RLC circuit by studying the evolution of the corresponding density operator in the amplitude dissipative channel and obtained the analytic form of the density matrix of the circuit was.
4. We found that no matter what kind of decoherence modes time evolution of their Wigner functions can be always ascribed to time evolution of their Wigner operators. Based on which, we used the Kraus operator sum representation of density operator to derive the time evolution law of the Wigner operator in several quantum channels such as amplitude dissipative channel, laser process, phase diffusion (or damping) model and noise channel in diffusion limit.
5. Time evolution of Husimi functions can be always ascribed to time evolution of their coarse-Guassin-smoothing-Wigner-operator. Based on which, we derive the time evolution law of Husimi function in amplitude dissipative channel.
6. For an harmonic oscillator with a field intensity related external source we establish the nonlinear number-phase squeezed state, in this state we find that while the number fluctuation increases, the phase fluctuation decreases correspondingly. The number-phase uncertainty relationship is exactly derived.
引文
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