Differences and similarities between lasing and multiple-photon subtracted states

We examine the effect that the subtraction of multiple photons has on the statistical characteristics of a light field. In particular, we are interested in the question whether an initial state transforms into a lasing state, i.e.,~a (phase diffused) coherent state, after infinitely many photon subtractions. This question is discussed in terms of the Glauber P-representation $P(\alpha)$, the photon number distribution $P[n]$, and the experimentally relevant autocorrelation functions $g^{(m)}$. We show that a thermal state does not converge to a lasing state, although all of its autocorrelation functions at zero delay time converge to one. This contradiction is resolved by the analysis of the involved limits, and a general criterion for an initial state to reach at least such a pseudo-lasing state ($g^{(m)}\to 1$) is derived, revealing that they can be generated from a large class of initial states.