## 2 Materials and Methods  
### Polarization in Quantum Interference  
Figure 1 shows the quantum interference pattern between entangled photons in a Mach Zehnder interferometer obtained by scanning the reference arm relative to the sample arm while measuring the coincidence rate [17]. In a dispersionless sample, the wavenumber of the entangled photons remains unchanged, and the interference profile resembles the standard Hong-Ou-Mandel dip, displayed in Figure 1, scan of a blank microscope slide. For a dispersive medium, we express the wavenumber \(k(\omega)\) in the Taylor series about the center wavelength of the entangled photons \(\omega_{0}\) and the down-converted shifted frequency \(\omega\):  
\[k(\omega)=k_{0}+\alpha(\omega-\omega_{0})+\beta(\omega-\omega_{0})^{2}+\dots \tag{1}\]  
and approximating the group velocity with the first two terms  
\[v_{g}\simeq 1/\alpha-2\beta(\omega-\omega_{0})/\alpha^{2} \tag{2}\]  
The annihilation operators corresponding to the two output ports of a 50:50 fiber beam splitter are then described as [14]  
\[ \begin{split} a_{1}(\omega_{1})&=|s\rangle\frac{i} {\sqrt{2}}\sqrt{R}a_{s}(\omega_{1})e^{ik(\omega_{1})d}+|r\rangle\frac{1}{ \sqrt{2}}\sqrt{T}a_{i}(\omega_{1})e^{i\omega_{1}\delta l/c}\\ a_{2}(\omega_{2})&=|s\rangle\frac{1}{\sqrt{2}} \sqrt{T}a_{s}(\omega_{2})e^{ik(\omega_{2})d}+|r\rangle\frac{i}{\sqrt{2}} \sqrt{R}a_{i}(\omega_{2})e^{i\omega_{2}\delta l/c}\end{split}  \tag{3}\]  
where \(a_{i}\) and \(a_{s}\) are the reference arm and signal arm operators, \(d\) is the sample thickness, \(\delta l\) the optical path length translation along the direction of propagation, and \(|s\rangle\) is the [PERSON] vector that describes the polarization profile of the sample photons after the birefringent medium and \(|r\rangle\) is the [PERSON] vector of the reference beam. In an isotropic medium \(|s\rangle\equiv|r\rangle\), all sample photons have the same polarization as the reference photons. \(T\) and \(R\) are the transmission and reflection coefficients where \(T+R=1\)[2]. The probability of detecting a coincidence between the two detectors is:  
Figure 1: Normalized coincidences, HOM interference pattern. The sub-graph shows single counts. \[P_{c}\propto\int d\omega_{1}\int d\omega_{2}\langle\Psi|a_{1}^{\dagger}(\omega_{1}) a_{2}^{\dagger}(\omega_{2})a_{1}(\omega_{1})a_{2}(\omega_{2})|\Psi\rangle \tag{4}\]  
Substitution of Equations (1) and (3) into Equation (4) and considering the canonical commutation relation of the operators  
\[ \begin{gathered}\langle\Psi|a_{1}^{\dagger}(\omega_{1})a_{2}^{ \dagger}(\omega_{2})a_{1}(\omega_{1})a_{2}(\omega_{2})|\Psi\rangle=\\ \frac{1}{4}R^{2}\langle r,s|r,s\rangle a_{i}^{\dagger}(\omega_{1 })a_{s}^{\dagger}(\omega_{2})a_{i}(\omega_{1})a_{s}(\omega_{2})+\\ \frac{1}{4}T^{2}\langle s,r|s,r\rangle a_{s}^{\dagger}(\omega_{1 })a_{i}^{\dagger}(\omega_{2})a_{s}(\omega_{1})a_{s}(\omega_{2})\\ +\frac{1}{4}RT\langle r,s|s,r\rangle a_{i}^{\dagger}(\omega_{1}) a_{s}^{\dagger}(\omega_{2})a_{s}(\omega_{1})a_{s}(\omega_{2})e^{id(k(\omega_{1})-k( \omega_{2}))+id|\delta l/c(\omega_{2}-\omega_{1})}\\ +\frac{1}{4}TR\langle s,r|r,s\rangle a_{s}^{\dagger}(\omega_{1}) a_{i}^{\dagger}(\omega_{2})a_{i}(\omega_{1})a_{s}(\omega_{2})e^{id(k(\omega_{2})-k( \omega_{1}))+id|\delta l/c(\omega_{1}-\omega_{2})}\end{gathered}  \tag{5}\]  
re-writing \(\omega^{\prime}=\omega_{1}-\omega_{0}=\omega_{0}-\omega_{2}\), also the coincidence window is much larger than the bandwidth of the entangled photons the time integral yields an effective \(\delta\) function [14]:  
\[ \begin{gathered}=\frac{1}{4}|\delta(\omega_{p}-\omega_{1}-\omega _{2})|^{2}|f(\omega^{\prime})|^{2}[R^{2}+T^{2}\\ -2 RT\langle s,r|r,s\rangle cos(2\omega^{\prime}(\delta l/c-\alpha d ))]\end{gathered}  \tag{6}\]  
The function \(f\) describes the spectral profile of the down-converted light and is proportional to the coherence of the entangled photons emitted from the crystal.