Means and covariances of photon numbers in multimode Gaussian states

In the field of continuous variables and particularly of Gaussian states an important role is played by the statistics of photon numbers. This issue has been considered in pioneering works for the one- and two-mode cases, but a systematic approach seems not to be available. In this paper, the most general multimode mixed Gaussian state, which is generated starting from a multimode thermal state and processed by the most general Gaussian unitary, is considered. In the $N$-mode the photon numbers are represented by $N$ random variables, one for each mode, and the target is the evaluation of the means and of the covariances. The means describe statistically the amount of photon numbers in each mode, while the covariances give the correlations between the photon numbers in the different modes. For both means and covariances a closed-form result is obtained, expressed in simple and compact formulas. The main tools are provided by the representation of Gaussian unitaries (given by a cascade of an $N$-mode squeeze operator, an $N$-mode rotation operator, and a parallel set of $N$ single-mode displacement operators) and by the corresponding Bogoliubov transformations.

Figure 1

Mean and variance of photon numbers in a single-mode pure Gaussian state vs $r$. On the left for $\theta =\pi /4$ and three values of $\alpha$. On the right for $\alpha =2+i3$ and three values of $\theta$.

Figure 2

Mean and variance in a single-mode noisy Gaussian state vs $r$ for four values of $\mathcal{N}$ and fixed values of $\alpha =1$. For the plot on the right we also used $\theta =0.5$.

Figure 3

Normalized covariance ${\varepsilon }_{12}$ vs $r$ of the photon numbers in a CS squeezed state for $\alpha =1$ and five values of $\theta$