From fractional Fourier transformation to quantum mechanical fractional squeezing transformation

By converting the triangular functions in the integration kernel of the fractional Fourier transformation to the hyperbolic function, i.e., tanα→tanhα,sinα→sinhα, 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 e^iπa*a/2 and expia/2(a²+a^*2). The FrST may be implemented in combinations of quadratic nonlinear crystals with different phase mismatches.