%% APPENDIX %%

\section*{Appendix: light power imbalance in the interferometers}
\label{sec:a}

In the derivation of the second order correlation function Eq.~(\ref{eqn:g2}) in
Sec.~\ref{sec:idea} we assumed a balanced optical power in the two arms of each
interferometer (Eq.~(\ref{eqn:first-splitter-output})). Practical power
imbalances, e.g., due to different losses in delay fibres, affect the resulting
bunching signature. The expression for the resulting second-order correlation
function
\begin{equation}
  g^{(2)}(\tau)=\frac{\left\langle E^{*}(t)E^{*}(t+\tau)E(t)E(t+\tau) \right\rangle}{\left\langle E^{*}(t) E(t) \right\rangle}
\end{equation}
would need to use a modified electric field compared to Eq.~(\ref{eq:field}). 
We exemplify this with a simple example. 
Consider a single interferometric loop ($n=1$) comprising two beam paths. If
we normalize $\left\langle E^{*}(t) E(t) \right\rangle =1$, and assign 
relative powers $\eta$ and $1-\eta$ in the two paths (corresponding to fields $E_A=\sqrt{\eta}$
and 
$E_B=\sqrt{1-\eta}$, respectively), such that the interesting power ratio $\eta$ ranges
from 0 to 0.5. Then, the second-order correlation function $g^{(2)}(\tau)$ becomes:
\begin{align}
g^{(2)}(\tau) &= \left\langle E^{*}(t)E^{*}(t+\tau)E(t)E(t+\tau) \right\rangle \notag =\\
&= \sum_{j,k,l,m=A}^{B} \left\langle E^{*}_{j}(t)E^{*}_{k}(t+\tau)E_{l}(t)E_{m}(t+\tau) \right\rangle\,,
\end{align}
summing over the contributions of the two arms $A,B$.
The remaining non-zero terms comprise:
\begin{align}
  &\hspace*{-1cm}\sum\limits_{A}^{B} \left\langle
      E^{*}_{j}(t)E^{*}_{j}(t+\tau)E_{j}(t)E_{j}(t+\tau) \right\rangle =
  \notag \\
  &= \sqrt{\eta}^4 + \sqrt{1-\eta}^4 \notag \\
  &= \eta^2 + (1-\eta)^2\,,\\
%\end{align}
%\begin{align}
&\hspace*{-1cm}\sum_{j,k=A}^{B} \left\langle
                E^{*}_{j}(t)E^{*}_{k}(t+\tau)E_{j}(t)E_{k}(t+\tau)
                \right\rangle = \notag \\
&= \sqrt{\eta}^2 \sqrt{1-\eta}^2 + \sqrt{1-\eta}^2 \sqrt{\eta}^2 \notag \\
&= 2 \eta (1-\eta)\,,
\end{align}
and
\begin{align}
&\hspace*{-5mm}\sum_{j,k=A}^{B} \left\langle
                E^{*}_{j}(t)E^{*}_{k}(t+\tau)E_{k}(t)E_{j}(t+\tau)
                \right\rangle = \notag \\
&= \big( \sqrt{\eta}^2 \sqrt{1-\eta}^2 + \sqrt{1-\eta}^2 \sqrt{\eta}^2 \big) |g^{(1)}(\tau)|^{2} \notag \\
&= 2 \eta (1-\eta) |g^{(1)}(\tau)|^{2}\,.
\end{align}
Summing up the non-zero contributions results in the description of the photon bunching dependence on the power ratio $\eta$ in combining two phase-independent light beams:
\begin{align}
g^{(2)}(\tau) &= \eta^{2} + (1-\eta)^{2} + 2\eta(1-\eta) + 2\eta(1-\eta)|g^{(1)}(\tau)|^{2} \notag \\
&= 1 + 2\eta(1-\eta)|g^{(1)}(\tau)|^{2}
\end{align}

\begin{figure}
	\includegraphics[width=\columnwidth]{figures/data_plot/g2_vs_alpha.eps}
	\caption{\label{fig:ratio}
(a) Second-order timing correlation $g^{(2)}(\tau)$ extracted from the output
of a single-loop asymmetric Mach-Zehnder interferometer, with different light
power ratios $\eta$ incident at the second beamsplitter: $\eta=0.5:0.5=1$ (green), $\eta=0.15:0.85=0.117$ (purple), and $\eta=0:1=0$ (orange). 
%The solid black curves shows the fit of the curves of $g^{(2)}(\tau)$ for $\eta$ = 0.50 and 0.15 to Equations 5.6 and 5.5. 
(b) The photon bunching amplitude, $g^{(2)}(0)-1$, scales with different power contributions by the non-delayed light field $E(t)$ over the total output.
The red datapoints show the bunching peaks $g^{(2)}(0)-1$ fitted from the measurements of $g^{(2)}(\tau)$ at varying power ratios.
The black solid curve shows the ideal scaling relationship between
$g^{(2)}(0)-1$ against the light power ratio $\eta$.
The blue dashed curve shows the dependency with a visibility of 0.9 adjustment.
%$\eta$ using Equation 5.5 if the laser fully emits coherent light.
%while the the blue dashed curve shows the same relationship accounting for the fraction of coherent light emitted by the laser $\rho$ = 0.940.
}
\end{figure}

A graphical representation for different power ratios $\eta$ is shown in Fig.~\ref{fig:ratio}(a).
The dynamics of temporal photon bunching in excess of the random
Poissonian floor can be described simply by $2\eta(1-\eta)$.

Additional reductions of the photon bunching signature can be expected from
other contributions to interferometer amplitude imbalance, e.g., a
limited mode matching or a polarization mismatch. Fig.~\ref{fig:ratio}(b)
shows measured photon bunching amplitudes, increasing with a better power
matching in the two arms. However, the interferometric visibility
$|g^{(1)}(\tau)|$ of the source laser and other instrumentation imperfections
might account for the deviation of experimental data points from the theoretical curve in Fig.~\ref{fig:ratio}(b).
%The blue dashed curve in Fig.\,\ref{fig:ratio} shows the dependency with a visibility of 0.9 adjustment. 

A more general adaptation of different power imbalances in multiple interferometers
($n>1$) can be treated in a similar way as exemplified here to generalize the electric field expression Eq.~(\ref{eqn:first-splitter-output}).