factors containing trigonometric expressions according to[^3]:

$$\begin{aligned}
\exp[-i \Omega A_1(B_1+B_2)\sin(\zeta+\phi_1)] =
\sum_{k=-\infty}^{\infty} \exp[i k (\zeta+\phi_1)] J_k\left[-\Omega
A_1(B_1+B_2)\right] \cr && \label{expan1}\end{aligned}$$

and

$$\begin{aligned}
&&\exp\left[-i \Omega A_2 B_2 \sin\left(\zeta + B_1 p + A_1 B_1
\sin(\zeta+\phi_1)+\phi_2\right)\right] = \cr &&
\sum_{m=-\infty}^{\infty} \exp\left[i m \left(\zeta + B_1 p + A_1
B_1 \sin(\zeta+\phi_1)+\phi_2\right)\right] J_m\left[-\Omega A_2
B_2\right] ~, \label{expan2}\end{aligned}$$

where one can still expand

$$\begin{aligned}
\exp[i m A_1 B_1 \sin(\zeta+\phi_1)] = \sum_{l=-\infty}^{\infty}
\exp[i l (\zeta+\phi_1)] J_l\left[ m A_1 B_1\right] ~.\label{expan3}\end{aligned}$$

Assuming, for the moment, a dependence of the laser phases $\phi_1$ and $\phi_2$ on $\zeta$, and collecting terms that have a dependence on $\zeta$ we can define

$$\begin{aligned}
\bar{f}_{\zeta}(k+l+m-\Omega)=\int_{-\infty}^{\infty} d\zeta
f_{0\zeta}(\zeta) \exp[i (k+l+m-\Omega)\zeta] \exp[i (k+l) \phi_1 +
i m \phi_2]\cr && \label{fbarz}\end{aligned}$$

and obtain from Eq. (\[ffft3\]):

$$\begin{aligned}
\bar{\rho}(\Omega) && = \frac{1}{\sqrt{2\pi}} \sum_{m,k,l}
\bar{f}_{\zeta}(k+l+m-\Omega) J_k\left[-\Omega A_1(B_1+B_2)\right]
J_m\left[-\Omega A_2 B_2\right] J_l\left[ m A_1 B_1\right]\cr &&
\times \int_{-\infty}^{\infty} dp \exp\left[-\frac{p^2}{2}\right]
\exp[-i \Omega (B_1+B_2)p+im B_1p]~. \label{ffft4}\end{aligned}$$

The integration over $p$ can be carried out using

$$\begin{aligned}
\frac{1}{N_0} \int_{-\infty}^{\infty} dp \exp[-i \Omega p (B_1+B_2)
+ i m p B_1] f_{0p}(p) = \exp[(\Omega  (B_1+B_2)-m  B_1)^2/2] \cr
\label{uno}\end{aligned}$$

which yields

$$\begin{aligned}
\bar{\rho}(\Omega) && =  \sum_{m,k,l} \bar{f}_{\zeta}(k+l+m-\Omega)
J_k\left[-\Omega A_1(B_1+B_2)\right] J_m\left[-\Omega A_2 B_2\right]
J_l\left[ m A_1 B_1\right]\cr && \times \exp[(\Omega  (B_1+B_2)-m
B_1)^2/2] ~.  \label{ffft5}\end{aligned}$$

