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 $P\left(\alpha \right)$, the photon number distribution $P\left[n\right]$, and the experimentally relevant autocorrelation functions $g^{\left(m\right)}$. 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 ($g^{\left(m\right)}\to 1$) is derived, revealing that they can be generated from a large class of initial states.

Figure 1

(a) Numerical results for the autocorrelation functions $g^{\left(m\right)}$ (measured on the left $y$ axis) for $m=2$ (blue, solid), $m=3$ (blue, dashed), and $m=4$ (blue, dot-dashed) and mean $〈n〉$ [red, big dots, (measured on the right $y$ axis)] of an initially thermal $\ell$-PSS. (b) Photon number distributions of the same states for $\ell \in [0\cdots 12]$ photon subtractions.

Figure 2

Photon number distribution (blue solid curve) of the $\ell$-PSS after $\ell \in [10,100,10000]$ subtractions in comparison to the Poisson distribution with the same mean photon number ${〈n〉}_0$ (black dashed curve). The photon number distribution of the initial thermal state with ${〈n〉}_0=1$ is depicted on the left by the orange solid curve. The boxes show the number of photon subtractions $\ell$ and the corresponding autocorrelations $g^{\left(m\right)}$ of order $m=2,3,4$. Note that the $x$ and $y$ axes have different scaling for each $\ell .$