Noise properties of quantum amplifiers with frequency-dependent gain

The power spectrum is calculated of the output of an optical amplifier with frequency-dependent gain for the case in which the input consists of one excited mode that is in an arbitrary quantum state and an infinite set of other modes that are in the vacuum state. The power spectrum is found to be independent of the particular quantum state of the excited input mode and to depend only on the shot-noise level of the amplified output and on the number of noise photons generated by the amplifier at the upper and lower frequency sidemodes of the excited input mode. For the case of a quantum-noise-limited amplifier and a coherent-state input, the results agree with the single-mode treatment of the amplifier only for the case in which the gain experienced by the sidemodes is equal to that experienced by the excited input mode.