Correspondence between quantum-optical transform and classical-optical transform explored by developing Dirac’s symbolic method
Correspondence between quantum-optical transform and classical-optical transform explored by developing Dirac’s symbolic method
Hong-yi Fan1,3 , Li-yun Hu2 ( )
1. Department of Physics, Shanghai Jiao Tong University, Shanghai 200030, China; 2. College of Physics & Communication Electronics, Jiangxi Normal University, Nanchang 330022, China; 3. Department of Material Science and Engineering, University of Science and Technology of China, Hefei 230026, China
Abstract :By virtue of the new technique of performing integration over Dirac’s ket–bra operators, we explore quantum optical version of classical optical transformations such as optical Fresnel transform, Hankel transform, fractional Fourier transform, Wigner transform, wavelet transform and Fresnel–Hadmard combinatorial transform etc. In this way one may gain benefit for developing classical optics theory from the research in quantum optics, or vice-versa. We cannot only find some new quantum mechanical unitary operators which correspond to the known optical transformations, deriving a new theorem for calculating quantum tomogram of density operators, but also can reveal some new classical optical transformations. For examples, we find the generalized Fresnel operator (GFO) to correspond to the generalized Fresnel transform (GFT) in classical optics. We derive GFO’s normal product form and its canonical coherent state representation and find that GFO is the loyal representation of symplectic group multiplication rule. We show that GFT is just the transformation matrix element of GFO in the coordinate representation such that two successive GFTs is still a GFT. The ABCD rule of the Gaussian beam propagation is directly demonstrated in the context of quantum optics. Especially, the introduction of quantum mechanical entangled state representations opens up a new area in finding new classical optical transformations. The complex wavelet transform and the condition of mother wavelet are studied in the context of quantum optics too. Throughout our discussions, the coherent state, the entangled state representation of the two-mode squeezing operators and the technique of integration within an ordered product (IWOP) of operators are fully used. All these have confirmed Dirac’s assertion: “...for a quantum dynamic system that has a classical analogue, unitary transformation in the quantum theory is the analogue of contact transformation in the classical theory”.
Key words :
Dirac’s symbolic method
IWOP technique
entangled state of continuum variables
entangled Fresnel transform
Collins formula
Generalized Fresnel operator
complex wavelet transform
complex Wigner transform
complex fractional Fourier transform
symplectic wavelet transform
entangled symplectic wavelet transform
Symplectic-dilation mixed wavelet transform
fractional Radon transform
new eigenmodes of fractional Fourier transform
收稿日期: 2011-07-28
出版日期: 2012-06-01
Corresponding Author(s):
Hu Li-yun,Email:hlyun2008@126.com
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