Quantum optics of traveling-wave attenuators and amplifiers

We use a continuous-mode quantization scheme to derive relations between the output- and input-field operators for traveling-wave propagation along attenuating and amplifying optical fibers. These relations provide complete information on the temporal and longitudinal spatial developments of the signal field. They are used here to obtain the effects of propagation on the first and second moments of the photocount in direct detection and of the signal field measured in balanced homodyne detection. Some of the results are similar to those obtained for attenuation or amplification of standing waves in cavities, and, for example, the survival of any input squeezing still limits the maximum gain to twofold. There are, however, additional propagation effects for the traveling-wave system. Thus, in direct detection, it is necessary to take account of the changes in gain profile with propagation distance, and in homodyne detection there are fundamental quantum-mechanical restrictions on the minimum field uncertainties that can be achieved in measurements at separated space-time points. These uncertainty properties are derived in detail and illustrated by the example of a squeezed input signal.