光子学报 ›› 2020, Vol. 49 ›› Issue (10): 1027001-1027001.doi: 10.3788/gzxb20204910.1027001

• 量子光学 • 上一篇    下一篇

光学分数傅里叶变换的量子理论

张科1(), 李兰兰1, 余海军1, 杜建明1, 范洪义2()   

  1. 1.淮南师范学院 电子工程学院,安徽 淮南 232038
    2.中国科学技术大学 材料科学与工程系,合肥 230026
  • 收稿日期:2020-05-15 接受日期:2020-06-01 出版日期:2020-10-25 发布日期:2020-10-13
  • 通讯作者: 范洪义 E-mail:zk626217@163.com;fhym@ustc.edu.cn
  • 作者简介:张科(1986-),男,实验师,博士,主要研究方向为量子光学和量子信息学.Email:zk626217@163.com
  • 基金资助:
    国家自然科学基金(1775208);安徽省教育厅自然科学重点项目(KJ2019A0688);淮南师范学院重点研究项目(2019XJZD04)

Quantum Theory of Optical Fractional Fourier Transform

Ke ZHANG1(), Lan-lan LI1, Hai-jun YU1, Jian-ming DU1, Hong-yi FAN2()   

  1. 1.School of Electronic Engineering,Huainan Normal University,Huainan,Anhui 232038,China
    2.Department of Material Science and Engineering,University of Science and Technology of China,Hefei 230026,China
  • Received:2020-05-15 Accepted:2020-06-01 Online:2020-10-25 Published:2020-10-13
  • Contact: Hong-yi FAN E-mail:zk626217@163.com;fhym@ustc.edu.cn
  • Supported by:
    National Natural Science Foundation of China(1775208);Natural Science Foundation of Anhui Higher Education Institutions of China(KJ2019A0688);Key Projects of Huainan Normal University(2019XJZD04)

摘要:

采用以算符为宗量的厄密多项式理论找出生成分数傅里叶变换的算符,即将分数傅里叶变换纳入量子理论.分析坐标—动量互换算符在分数傅里叶变换加法律中的作用.在推导过程中,充分利用了算符厄密多项式的广义母函数公式以及编好序的积分理论.算符厄密多项式理论的核心是HnQ=:2Qn:,即把复杂的特殊函数结构算符用正规排序的幂级数来代替,极大地简化了运算过程.

关键词: 分数傅里叶变换, 坐标—动量互换算符, 算符厄密多项式理论, 编好序的积分理论

Abstract:

The aim of this paper is to find out the operator for generating fractional Fourier transform in Hermitian polynomial theory with the operator as the argument, and to incorporate fractional Fourier transform into quantum theory. The role of coordinate-momentum exchanging operator is explored in playing FFrT's addition rule. In the whole derivation the generalized generating function formula of operator Hermitian polynomials and the integration method within ordered product of operators are used. The core of operator Hermite polynomial theory is the operator identity HnQ=:2Qn:, which turns the operator of complex special function into power series in normal ordering, as a result,this greatly simplifies calculations.

Key words: Fractional Fourier transformation, Coordinate-momentum exchanging operator, Operator Hermite polynomial theory, Integration method within ordered product of operators

中图分类号: