Optimal signal-to-quantum noise ratio in squeezed displaced number states

For an arbitary quantum state of radiation with frequency omega, the optimum signal-to-quantum noise ratio rho for fixed energy (or power per unit frequency) (h) over bar omegaN(s) has the value 4N(s)(N-s + 1), attainable by the squeezed coherent vacuum. The question arises as to what rho is attainable when the vacuum is imperfect, and there are n photons present. We show that in such a situation the attainable signal-to-quantum noise ratio rho is reduced by a factor (2n + 1)(2).