Measurements and Filtering Theory

We consider the following problem: a field interacts with a system S and after this interaction no direct measurement is made on S, but one extracts information on S by measuring the field before (input) and after (output) the interaction. For example, in the case of an atom which decays emitting radiation, if one knows its initial state and detects the radiated photons, then one can deduce some information on its new state. The idea is to deduce information on the system emitting the radiation from the measured radiation. The emitted radiation is a typical example of an output process. Other typical examples of output fields are the field operators (quadratures) evolved at time t. More generally, one is interested in the statistics of the output field with respect to a given initial state. Often, e.g. in quantum optics, by signal one means the mean value of the output process and by noise its variance. Typical choices of input fields are (Math) or linear combinations thereof. The corresponding output fields are where U _t is an evolution operator involving the interaction between the system and the field.