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Based on the fact that the quantum mechanical version of Hankel transform kernel (the Bessel function) is just the transform between | q , r 〉 and (s , r ′ | , two induced entangled state representations, and working with them, we derive fractional Fourier–Hankel transformation (FrFHT) caused by the operator e − i α (a 1 † a 1 + a 2 † a 2) e i π a 2 † a 2 , where e i π a 2 † a 2 is named the core operator and is essential to the fractional transformation. The fractional property (additive rule) of the FrFHT can be explicitly proved. [ABSTRACT FROM AUTHOR]

Abstract: By introducing the generalized Wigner operator for s-parameterized quasiprobability distribution and employing the technique of integration within ordered product (IWOP) of operators (normally ordered, Weyl ordered or antinormally ordered), we derive two new quantum-mechanical formulas for describing no counts registered on a photonic detector when a light field’s density operator is known, one involves ’s s-parameterized distribution function, and the other involves ’s coherent state mean value, when these information is known then using the new formulas to calculate no-photocount would be convenient. [Copyright &y& Elsevier]

By converting the triangular functions in the integration kernel of the fractional Fourier transformation to the hyperbolic function, i.e., tan α → tanh α, sin α → sinh α, we find the quantum mechanical fractional squeezing transformation (FrST) which satisfies additivity. By virtue of the integration technique within the ordered product of operators (IWOP) we derive the unitary operator responsible for the FrST, which is composite and is made of eiπa†a/2 and exp . The FrST may be implemented in combinations of quadratic nonlinear crystals with different phase mismatches. [ABSTRACT FROM AUTHOR]

By using the technique of integration within ordered product of operators, we put forward the combinatorial optical complex wavelet-fractional Fourier transform in the context of quantum optics. The unitary operator for this new transform is found and its normally ordered form is deduced. We apply this new transform to the two-mode vacuum state and the two-mode number state and explain that it can be used to analyze and identify various quantum optical states.

Enlightened by the special transformation in our preceding paper [J. Mod. Opt. 56, 1227 (2009)], we propose a new complex integration transformation corresponding to two mutually conjugate two-mode entangled states