undergo the same time delay $\Delta t_{10}$ when traveling along the long path (figure \[fig:time\_bins\_franson\]).

![\[fig:time\_bins\_franson\] Scheme to analyze energy-time entangled two-photon states proposed by Franson [@franson1989]. A photon pair is generated by SPDC within the coherence time $\tau_p^{coh}$ of a pump photon. Both photons are injected into two separated and unbalanced Mach-Zehnder interferometers with a time delay of $\Delta t_{10}$ imprinted on the photon in the longer arm. A qubit state is then measured by varying the phases $\phi_i$ and $\phi_s$ while performing coincidence measurements between output ports $A$ and $B$.](time_bins_franson.eps){width="100.00000%"}

For $\Delta t_{10}\ll\tau^{coh}_p$, where $\tau^{coh}_p$ denotes the coherence time of the pump photon, the state generated by the first pair of beam splitters is a coherent superposition $$\label{eq:psidisc_qubit}
\vert\psi\rangle^{(2)} = \sum_{j=0}^{1}\sum_{k=0}^{1} c_{jk}\vert j \rangle_i \vert k\rangle_s,$$ where we associate $\vert 0\rangle_{i,s}$ with the short and $\vert 1\rangle_{i,s}$ with the long path of the interferometer. Characteristic qubit interference fringes can be observed by a coincidence detection between the output ports $A$ and $B$ while varying the phases $\phi_i$ and $\phi_s$. The original experiment of Franson can be extended to higher dimensional qudits by endowing each interferometer with additional arms. Two four-arm interferometers were used in [@richart2012] to experimentally demonstrate 