frequency bins through amplitude modulation (figure \[fig:freq\_bins\]).

![\[fig:freq\_bins\]Measured SPDC spectrum overlaid with a schematic frequency-bin pattern. The transmitted amplitude $\vert u_j^{i,s}\vert$ (white bars) of each bin can be adjusted through amplitude modulation by means of the SLM.](frequency_bins.eps){width="50.00000%"}

The corresponding $f_{j}^{i,s}(\omega)$ are defined according to $$\label{eq:freq_bins}
f_{j}^{i,s}(\omega) = 
\cases{
  1/\sqrt{\Delta\omega_j} & for $\vert\omega-\omega_{j}\vert<\Delta\omega_j/2$ \cr
  0 & otherwise, \cr
}$$ where we impose $|\omega_j-\omega_k|>(\Delta\omega_j+\Delta\omega_k)/2$ for all $j,k$ to guarantee that adjacent bins do not overlap. If we further assume a continuous wave pump, we restrict (\[eq:psidisc\]) to its diagonal form $$\label{eq:psidiscdiag}
\vert\psi\rangle^{(d)} = \sum_{j=0}^{d-1} c_{j}\vert j \rangle_i \vert j\rangle_s.$$

### Time-bin basis

Time-bin entangled photons are typically manipulated in the temporal domain by interferometers with variable optical path lengths [@marcikic2002; @thew2004; @richart2012]. Up to now, time bins have been preferentially used to encode quantum information and to analyze the temporal properties of the down-converted photons, a concept that was first proposed by Franson [@franson1989]. A Franson interferometer was imitated by shaping the spectrum of entangled photons in the telecom wavelength regime using a wave shaper in combination with conventional electronic coincidence counting in [@lukens2013]. In Franson’s interferometric scheme, each photon of an entangled pair enters an unbalanced Mach-Zehnder interferometer where both photons 