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Acta Sinica Quantum Optica

Normally Ordering and Anti-normally Ordering Some Useful Operator Formulas in Quantum Optics Theory by Virture of the IWOP Technique

Xuefen Xu

Published Jan 1, 2007• 0 citations
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Abstract

By using the technique of integration within an ordered product(IWOP) of operators,which is proposed in Ref.[1],the paper derives the useful light-field operator formulas(in ACTA SINICA Quantum Optica,Vol.8(2002) p8-12 written by Chen Chang-yuan) in a convenient and easily be remembered way.Some other operator formulas about two-variable Hermite polynomials are also derived.

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By using the technique of integration within an ordered product(IWOP) of operators,which is proposed in Ref.[1],the paper derives the useful light-field operator formulas(in ACTA SINICA Quantum Optica,Vol.8(2002) p8-12 written by Chen Chang-yuan) in a convenient and easily be remembered way.Some other operator formulas about two-variable Hermite polynomials are also derived.
By virtue of the operator Hermite polynomial method [Fan H Y and Zhan D H 2014 Chin. Phys. B 23 060301] we find a new special function which is useful in quantum optics theory, whose expansion involves both power-series and Hermite polynomials, i.e., By virtue of the operator Hermite polynomial method and the technique of integration within ordered product of operators (IWOP) we derive its generating function. The circumstance in which this new special function appears and is applicable is considered.
By virtue of the entangled state representation we concisely derive some new operator identities with regard to the two-variable Hermite polynomial (TVHP). By them and the technique of integration within an ordered product (IWOP) of operators we further derive new generating function formulas of the TVHP. They are useful in quantum optical theoretical calculations. It is seen from this work that by combining the IWOP technique and quantum mechanical representations one can derive some new integration formulas even without really performing the integration.
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.
For directly normalizing the photon non-Gaussian states (e.g., photon added and subtracted squeezed states), we use the method of integration within an ordered product (IWOP) of operators to derive some new bosonic operator-ordering identities. We also derive some new integration transformation formulas about one- and two-variable Hermite polynomials in complex function space. These operator identities and associative integration formulas provide much convenience for constructing non-Gaussian states in quantum engineering.