energy-time entangled ququarts. Instead of using interferometers, we discretize the time domain of the idler and signal photons into time bins (figure \[fig:time\_bins\]) $$\label{eq:time_bins}
f_{j}^{i,s}(t) = 
\cases{
  \sqrt{\frac{2\pi}{\Delta t_j}} & for $\vert t-t_{j}\vert<\Delta t_j/2$ \cr
  0 & otherwise \cr
}$$ analogous to (\[eq:freq\_bins\]) in the frequency domain with the help of the SLM.

![\[fig:time\_bins\] Franson’s original scheme adapted to time bins implemented by a SLM. Under specific conditions for $\Delta t_{10}$ and $\Delta t_j$ (see text) projections onto a superposition of $\vert 0 \rangle_i \vert 0\rangle_s$ and $\vert 1 \rangle_i \vert 1\rangle_s$ can be measured.](time_bins.eps){width="100.00000%"}

The coefficients in (\[eq:psidisc\_qubit\]) are now related to the joint temporal amplitude $\Upsilon(t_i,t_s)$ of the SPDC photons through $$c_{jk}=\int_{t_j-\frac{\Delta t_j}{2}}^{t_j+\frac{\Delta t_j}{2}}\int_{t_k-\frac{\Delta t_k}{2}}^{t_k+\frac{\Delta t_k}{2}}dt_{i}dt_{s}\Upsilon\left(t_{i},t_{s}\right),$$ where $\Upsilon\left(t_{i},t_{s}\right)$ is the Fourier transform of $\Gamma(\omega_i,\omega_s)$. Since the SLM manipulates the frequency spectrum of the entangled photons, the time bins of (\[eq:time\_bins\]) are Fourier transformed to obtain $$\label{eq:time_bins_freq}
f_{j}^{i,s}(\omega) = \sqrt{\frac{\Delta t_j}{2\pi}}e^{-i\omega t_j}\mathrm{sinc}\left(\frac{\omega \Delta t_j}{2}\right),$$ where orthonormality holds for any $j$ and $k$ provided that $|t_j-t_k|>(\Delta t_j+\Delta t_k)/2$. In order to attain entangled qubits, the time delay $\Delta t_{10}=t_1-t_0$ has to exceed the coherence time $\tau^{coh}_{i,s}$ of the idler and signal photon to avoid single photon interference. A coincidence window 