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## Integral-form solution of the Caldeira-Leggett density operator equation obtained by virtue of thermo entangled state representation

Ye Qian, Chen Qian-Fan, Fan Hong-Yi
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• #### 摘要

开放量子系统, 即系统-热库模型, 可以用一个关于密度算符的主方程来描述, 比如, 用来描述固态物理中耗散现象的Caldeira-Leggett 主方程. 虽然已经有人为了求解此主方程的约化密度矩阵的精确表达式而做过一些努力, 但迄今还未见有解答. 本文使用了一种全新的方法来求解Caldeira-Leggett 方程, 用这个新方法可以得到积分形式的显式表达. 该方法的要点在于利用有序算符内积分技术把关于密度算符的微分方程首先转化成关于密度态矢量的微分方程, 再将密度态矢量投影到热纠缠态表象中, Caldeira-Leggett 方程就转变成了关于波函数的微分方程, 而波函数是函数. 这样就可以使用数学中求解微分方程的方法来求解出波函数. 再次利用有序算符内积分技术, 再将波函数转化为态矢量和算符, 就得到了Caldeira-Leggett 方程的积分形势解.

#### Abstract

Open quantum system, namely system-reservoir model, is described by a master equation of density operator. For example, the Caldeira-Leggett eqaution describes dissipative phenomenon of solid physics. Although some efforts have been made to derive the exact expression of this master equation, so far as we know, it has not been reported in the literature. The purpose of this paper is to provide a new approach to solving the Caldeira-Leggett equation, via this approach the explicit integral-form expression of ρ(t) can be obtained. The main point of this approach is to convert equation of density operator into an equation of density state vector, and then project density state vector into thermo entangled state representation and convert it into wave function by using the technique of integration within an ordered product of operators. Thus the master equation for Caldeira-Leggett model is converted into an differential equation of wave function. Wave function is also a function. The wave function can be obtained via the approach to solving the differential equation in mathematics. It can be converted into a density state vector and density operator. Using the technique of integration within an ordered product of operators again, the integra-form solution of the Caldeira-Leggett equation is obtained.

#### 参考文献

 [1] Feynman R P, Vernon F L 1963 Ann. Phys. 24 118 [2] Caldeira A O, Leggett A J 1983 Physics 121A 587 [3] Breuer H P, Petruccione F 2002 The Theory of Open Quantum Systems (New York: Oxford University Press Inc.) [4] Fan H Y 1997 Representation and Transformation Theory in Quantum Mechanics Progress of Dirac's Symbolic Method (Shanghai: Shanghai Scientific and Technical Publishers) [5] Gradshteyn I S, Ryzhik I M 1980 Tables of Integrals, Series, and Products (London: Fourth Edition Academic Press) [6] Fan H Y 2005 From Quantum Mechanics to Quantum Optics-Development of the Mathematical Physics (Shanghai: Shanghai Jiao Tong University Press)

#### 施引文献

•  [1] Feynman R P, Vernon F L 1963 Ann. Phys. 24 118 [2] Caldeira A O, Leggett A J 1983 Physics 121A 587 [3] Breuer H P, Petruccione F 2002 The Theory of Open Quantum Systems (New York: Oxford University Press Inc.) [4] Fan H Y 1997 Representation and Transformation Theory in Quantum Mechanics Progress of Dirac's Symbolic Method (Shanghai: Shanghai Scientific and Technical Publishers) [5] Gradshteyn I S, Ryzhik I M 1980 Tables of Integrals, Series, and Products (London: Fourth Edition Academic Press) [6] Fan H Y 2005 From Quantum Mechanics to Quantum Optics-Development of the Mathematical Physics (Shanghai: Shanghai Jiao Tong University Press)
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##### 出版历程
• 收稿日期:  2012-04-25
• 修回日期:  2012-05-28
• 刊出日期:  2012-11-05

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