appear in the single photon master equations (\[eq:rho-dyn-a-11\])-(\[eq:rho-dyn-a-00\]), in contrast to the vacuum case (which corresponds to (\[eq:rho-dyn-a-00\])).

Embedding {#sec:the-matrix}
---------

In this section we construct a suitable embedding for the system and single photon field, and show how the system of master equations from Section \[sec:master\] can be compactly represented as a single equation for a larger system. This embedding will be used in subsequent sections to derive the quantum filter. We should emphasize, however, that our embedding is not the same as that used in [@HPB04], [@GJN11a], [@GJNC11]. The embedding is illustrated in Figure \[fig:extend-1\].

![System embedded in the extended system. While the analysis does not employ any coupling between the system and ancilla two-level system, the ancilla, system and field are assumed to be initialized in a superposition state $\vert \Sigma \rangle$ defined in equation (\[eq:super\]).[]{data-label="fig:extend-1"}](images/extend-2)

Recall that the system and field are defined on a Hilbert space $\mathfrak{H}
= \mathfrak{H}_S \otimes \mathfrak{F}$. We define an extended space $$\tilde {\mathfrak{H}}= \mathbb{C}^2 \otimes \mathfrak{H} = \mathfrak{H}
\oplus \mathfrak{H}$$ which includes the system, field and an ancilla two-level system. Let $\vert e_0 \rangle$ and $\vert e_1 \rangle$ be an orthonormal basis for $\mathbb{C}^2$, $$\vert e_0  \rangle = \left[
\begin{array}{c}
