Jamil Daboul and Salomon S Mizrahi 2005 J. Phys. A: Math. Gen. 38 427 doi:10.1088/0305-4470/38/2/010
O(N) symmetries, sum rules for generalized Hermite polynomials and squeezed states
Jamil Daboul1 and Salomon S Mizrahi2
Show affiliationsQuantum optics has been dealing with coherent states, squeezed states and many other non-classical states. The associated mathematical framework makes use of special functions as Hermite polynomials, Laguerre polynomials and others. In this connection we here present some formal results that follow directly from the group O(N) of complex transformations. Motivated by the squeezed states structure, we introduce the generalized Hermite polynomials 
, which include as particular cases, the Hermite polynomials as well as the heat polynomials. Using generalized raising operators, we derive new sum rules for the 
, which are covariant under O(N) transformations. The 
and the associated sum rules become useful for evaluating Wigner functions in a straightforward manner. As a byproduct, we use one of these sum rules, on the operator level, to obtain raising and lowering operators for the Laguerre polynomials and show that they generate an sl(2, R) 
su(1, 1) algebra.
- PACS
- MSC
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81R30 Coherent states (See also 22E45); squeezed states (See also 81V80)
- Subjects
- Dates
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Issue 2 (14 January 2005)
Received 12 July 2004, in final form 4 November 2004
Published 15 December 2004
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O(N) symmetries, sum rules for generalized Hermite polynomials and squeezed states
Jamil Daboul and Salomon S Mizrahi 2005 J. Phys. A: Math. Gen. 38 427
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