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Journal of Optics B Quantum and Semiclassical Optics

Operator ordering in quantum optics theory and the development of Dirac s symbolic method

Hong-Yi Fan

Published Jun 13, 2003• 326 citations
Publisher

Abstract

We present a general unified approach for arranging quantum operators of optical fields into ordered products (normal ordering, antinormal ordering, Weyl ordering (or symmetric ordering)) by fashioning Dirac's symbolic method and representation theory. We propose the technique of integration within an ordered product (IWOP) of operators to realize our goal. The IWOP makes Dirac's representation theory and the symbolic method more transparent and consequently more easily understood. The beauty of Dirac's symbolic method is further revealed. Various applications of the IWOP technique, such as in developing the entangled state representation theory, nonlinear coherent state theory, Wigner function theory, etc, are presented.

DOI
10.1088/1464-4266/5/4/201

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We present a general unified approach for arranging quantum operators of optical fields into ordered products (normal ordering, antinormal ordering, Weyl ordering (or symmetric ordering)) by fashioning Dirac's symbolic method and representation theory. We propose the technique of integration within an ordered product (IWOP) of operators to realize our goal. The IWOP makes Dirac's representation theory and the symbolic method more transparent and consequently more easily understood. The beauty of Dirac's symbolic method is further revealed. Various applications of the IWOP technique, such as in developing the entangled state representation theory, nonlinear coherent state theory, Wigner function theory, etc, are presented.
Via the route of developing Dirac's symbolic method by virtue of the technique of integration within an ordered product (IWOP) of operators we derive various entangled state representations, squeezed states, and operator ordering formulae. We show how the entangled states can be applied to many aspects of quantum optics theory. The applications of the entangled states in the interdisciplinary study of quantum optics and Fourier optics are also introduced. All these discussions exhibit the power of Dirac's symbolic method with the aid of the IWOP technique.
In this paper by virtue of the technique of integration within an ordered product (IWOP) of operators and the intermediate coordinate-momentum representation in quantum optics, we derive the normal ordering and antinormal ordering products of the operator ( fQ + gP ) n when n is an arbitrary integer. These products are very useful in calculating their matrix elements and expectation values and obtaining some useful mathematical formulae. Finally, the applications of some new identities are given.
By the newly developed technique of integration within an ordered product (IWOP) of 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 can not 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. 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 to 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 IWOP technique are fully used. All these confirms 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".
Usually complicated unitary operators in exponential form are hard to handle and the transformation stated by them are not physically clear until these operators are disentangled. By virtue of the technique of integration within an ordered product (IWOP) of operators, we can disentangle some complicated unitary operators and then reveal their physical role. Many new unitary operators can be found after new quantum mechanical representations are constructed by virtue of the IWOP technique. The unitary operators for permutation, Hilbert transform, Householder transform, and Hardmad transform can also be introduced. The IWOP technique thus bridges this mathematical gap between classical mechanics and quantum mechanics and provides a new route connecting classical transformations to quantum mechanical unitary operators. In this way, the symbolic method exhibits further beauty and elegance and can be increasingly used for developing various fields of quantum physics.
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”.