A QUANTUM WAVELET FOR QUANTUM OPTICS

A quantum, or operator-valued, wavelet is defined for a general densit y operator rho, in a basis generated by a general observable THETA by defining an operator-valued dilation. The scale changing part of the d ilation is shown to correspond to the Yuen squeeze operator. The wavel et gives a family of operator-valued coefficients which represent a gi ven density operator in the eigenbasis of THETA, possibly a complete s et of commuting observables. The wavelet is given in both the Heisenbe rg and Schrodinger pictures. Then an inverse problem is formulated whi ch allows an unknown density operator to be calculated in terms of the family of all wavelet operators. It is interesting that a limiting pr ocess is required to obtain a unique inverse, when one exists. Then th e Heisenberg-picture dilation is applied to two known examples: the un itary process of phase sensitive amplification and the irreversible pr ocess of number amplification.