counts (net counts) with 1$\sigma$ standard deviations using Poisson statistics. The solid curves are fits to the data points by means of (\[eq:fitd2\]), (\[eq:fitd3\]) and (\[eq:fitd4\]).](frequency_bin_qudits.eps){width="100.00000%"}

Equations (\[eq:fitd2\]), (\[eq:fitd3\]) and (\[eq:fitd4\]) are used to fit the data points. The corresponding visibilities (table \[tab:freq\_bin\]) are calculated with the aid of the fitting parameter $\lambda_d$ and the relation (\[eq:cglmp\_lamb\]). We find $V_d>V_d^c$ for all $d$ which demonstrates the existence of frequency-bin entanglement.

[@lllll]{} $d$ & $V_d^c$& $V_d$&\
2&0.707&0.903$\pm$0.018\
3&0.775&0.860$\pm$0.019\
4&0.817&0.959$\pm$0.008\

Time-bin basis
--------------

According to (\[eq:time\_bin\_dt0\]), a qubit is encoded in time bins via the applied transfer function $$\label{eq:M_time_bins_SLM}
M^{i,s}(\omega)=\vert u_0^{i,s}\vert+\vert u_1^{i,s}\vert\e^{i(\omega t_1+\phi_{i,s})}.$$ Here, the time bin $f_0^{i,s}(t)$ is fixed at $t_0=0$ fs and bin $f_1^{i,s}(t)$ is delayed by $t_1$. To obtain maximally entangled states and fulfill condition (\[eq:M\_condition\]), we choose $\vert u_0^{i,s}\vert=\vert u_1^{i,s}\vert=1/2$. This is equivalent to use beam splitters with $T=R=1/2$ in the Franson experiment (figure \[fig:time\_bins\_franson\]). Although we set $\Delta t_0=\Delta t_1=0$ fs to make the individual bins as narrow as possible, the time bins, however, are always of finite width since the transfer function of (\[eq:M\_time\_bins\_SLM\]) is limited in $\omega$ due to the finite aperture of the SLM. Note, that in the present experiment the coherence time of 