$\langle \psi_1 \vert
\psi_2 \rangle$. The expected value of an operator $X$ when the system is in state $\vert \psi \rangle$ is denoted $\mathbb{E}_\psi[ X ] = \langle \psi \vert X \vert \psi \rangle$. For operators $A$ and $B$ we write $
\langle A, B \rangle = \mathrm{tr}[ A^\ast B ].
$

Problem Formulation {#sec:problem}
===================

We consider a quantum system $S$ coupled to a quantum field $B$, as shown in Figure \[fig:filter-one-1\]. The field $B$ has two components, the input field $B_{in}$ and the output field, $B_{out}$. In this paper we consider two non-classical cases for the state $\vert \Psi \rangle$ of the input field (i) a single photon state $\vert \Psi \rangle = \vert 1_\xi \rangle$, where $\xi$ is a complex valued function such that $\int_0^\infty \vert
\xi(s) \vert^2 ds =1$ (representing the wave packet shape), or (ii) a superposition of coherent states $\vert \Psi \rangle  = \sum_j \alpha_j \vert f_j \rangle$, where $\vert f_j \rangle$ are coherent states and the complex numbers $\alpha_j$ ($j=1,\ldots,n$) are normalized weights.

As illustrated in Figure \[fig:filter-one-1\], the field interacts with the quantum system $S$, and the results of this interaction provide information about the system that may be obtained through continuous 