the continuous temporal tensor product decomposition of the Fock space $\mathfrak{F}=\mathfrak{F}_{t]} \otimes \mathfrak{F}_{(t}$ into past and future components, which is of basic importance in what follows. Write $$\mathbb{E}_{11} [ X \otimes F ] = \langle \eta 1_\xi \vert  (X \otimes F) \vert \eta
1_\xi \rangle = \langle \eta  \vert X \vert \eta \rangle \langle 1_\xi \vert  F  \vert 1_\xi \rangle$$ for the expectation with respect to the product state $\vert \eta 1_\xi \rangle$, where the field is in the single photon state. Here and in what follows $X$ is a bounded system operator acting on $\mathfrak{H}_S$, and $F$ is a field operator acting on the Fock space $\mathfrak{F}$. Similarly, we may define the expectation when the field is in the vacuum state, $$\mathbb{E}_{00} [ X \otimes F ] = \langle  \eta 0 \vert  (X \otimes F) \vert \eta
0  \rangle = \langle \eta \vert X \vert \eta \rangle \langle 0 \vert  F \vert 0  \rangle .$$ We will also have need for the cross-expectations $$\begin{aligned}
\mathbb{E}_{10}[ X \otimes F ] = \langle \eta 1_\xi \vert  (X \otimes F)  \vert \eta 0
\rangle, \ \text{and} \ \mathbb{E}_{01}[ X \otimes F ] = \langle \eta 0  \vert
(X \otimes F) \vert  \eta 