Derivation of generalized quantum jump operators and comparison of the microscopic single photon detector models

The recent experiment of Parigi et al. Science 317, 1890 (2007)] shows, in agreement with theory, that subtraction of one photon can increase the expectation value of the number of photons in the thermal state. This observation agrees with the standard photon counting model in which the quantum jump superoperator (QJS) gives a count rate proportional to the number of photons. An alternate model for indirect photon counting has been introduced by Dodonov et al. Phys. Rev. A 72, 023816 (2005)]. In their model the count rate is proportional to the probability that there are photons in the cavity, and the cavity field is bidirectionally coupled with a two state quantum system which is unidirectionally coupled to a counting device. We give a consistent first principle derivation of the QJSs for the indirect photon counting scheme and establish the complete relations between the physical measurement setup and the QJSs. It is shown that the time-dependent probability for photoelectron emission event must include normalization of the conditional probability. This normalization was neglected in the previous derivation of the QJSs. We include the normalization and obtain the correct photoelectron emission rates and the correct QJSs and show in which coupling parameter regimes these QJSs are applicable. Our analytical results are compared with the exact numerical solution of the Lindblad equation of the system. The derived QJSs enable analysis of experimental photon count rates in a case where a one-to-one correspondence does not exist between the decay of photons and the detection events.