polariton emission. The red dashed line indicates the energy of the investigated condensate state. (c) Normalized visibility of the interference fringes as a function of the path difference.[]{data-label="fig:EkSpace_Interference"}](./fig_Coherence_time_neu_v04a.pdf){width="\linewidth"}

The RR is mounted on a motorized linear stage, that allows us to vary the path difference $\Delta s$ between the emission collected from both interferometer arms. Fig. \[fig:EkSpace\_Interference\](a)-(b) show two selected interferograms of the $I(E,k)$ emission pattern for large (Fig. \[fig:EkSpace\_Interference\](a)) and short (Fig. \[fig:EkSpace\_Interference\](b)) path differences $\Delta s$, respectively. To investigate the temporal coherence properties of the polariton condensate we analyzed the normalized visibility of the interference fringes, $$\begin{aligned}
    \label{eq:Inorm}
    V_{{\mathrm{norm}}} & = \frac{I_ {\text{interf}} - I_{RR} - I_{PM}}{2\sqrt{I_{RR}  I_{PM}}} = g^1(\Delta t) \cos(\phi_{12}),\end{aligned}$$ as a function of the temporal delay $\Delta t = \frac{\Delta s}{c}$, where $c$ is the speed of light (cf. Fig. \[fig:EkSpace\_Interference\](c)). Here, $I_{{\mathrm{interf}}}$ is the intensity of the interference pattern, and $I_{{\mathrm{RR}}}$, $I_{{\mathrm{PM}}}$ are the intensities of the RR and PM arm, respectively, $g^1$ is the first-order coherence function and $\phi_{12}$ is the phase difference between the emission from the individual interferometer arms. By assuming a Gaussian decay of $g^1(\Delta t) = g^1(0) \exp[
- (\pi / 2) (\Delta t ^2 / \tau_{{\mathrm{coh}}}^2)]$[@Saleh.2008] we determined 