## 2 Experimental Setup  
Figure 1 shows a schematic drawing of the experimental setup with the EDFA on the bottom right side as the GS continuous-wave (CW) light source. After collimation of its fiber output, the unpolarized light beam is subsequently linear polarized by a Glan-Thompson prism resulting in a linear horizontal polarization state. Through a beam splitter, a reference beam and an object beam are generated, respectively. The object beam light passes a gas sample cell and the total transmitted light is coupled into a single-mode fiber (SMF) serving as the spectral bucket detection branch, i.e., without any spectral resolution. In the reference beam spectral resolution can either be achieved by various,  
in wavelength variable tunable interference bandpass filters with a full width at half maximum (FWHM) of approximately 10 nm or by an Echelette grating if even higher resolution of up to 0.5 nm is requested. Also the reference beam is then fiber-coupled and both beams are superimposed by a fiber combiner forming an overall Mach-Zehnder-like interferometer configuration. Finally, the light is focused onto the semiconductor photocathode of a photomultiplier (PMT). The photomultiplier in use incorporates a GaAsP photocathode (\(E_{\mathrm{g}}\approx 2.04\) eV) which has been selected regarding the source wavelengths in order to guarantee pure TPA detection. The here-employed TPA interferometry technique for the determination of \(g^{(2)}(\tau)\) goes back to the work of [PERSON] et al. [34] who demonstrated for the first time the photon bunching effect for black body sources with multiple THz wide optical spectra. The nonlinear TPA process requires two photons to be absorbed within a time frame given by the Heisenberg uncertainty. Hence ultra-fast intensity-intensity correlation detection \(\langle I(t)I(t)\rangle\) is enabled. By implementing a time delay \(\tau\) in the object arm via a variable optical delay line intensity auto-correlation measurements \(\langle I_{\mathrm{ref}}(t,\lambda_{\mathrm{ref}})I_{\mathrm{obj}}(t+\tau)\rangle\) are performed. By reading out the photon counts from the detector output while varying the optical path of one interferometer arm using a high precision motorized linear translation stage, a TPA interferogram \(I_{TPA}(\tau)\) is recorded. According to theory [47], \(I_{TPA}(\tau)\) comprises four terms: a constant one, the non-normalized second-order auto-correlation function \(G^{(2)}(\tau)=\langle I(t)I(t+\tau)\rangle\) and two fast oscillating terms \(F_{1}(\tau)\) and \(F_{2}(\tau)\) following the center angular frequency \(\omega_{0}=2\pi\upsilon_{0}\) and the frequency duplication \(2\omega_{0}\) of the emitted light, respectively. Low-pass filtering of \(I_{\mathrm{TFA}}\) subsequently to an FFT allows determining \(G^{(2)}(\tau)\) according to  
\[\frac{I_{TPA}^{low-pass}(\tau)}{I_{\mathrm{ref}}+I_{\mathrm{obj}}}=1+2\cdot G ^{(2)}(\tau) \tag{1}\]  
Finally, the key value for GM \(g^{(2)}(\tau)\), being GI or GS, respectively, is obtained according to \(g^{(2)}(\tau)=G^{(2)}(\tau)/G^{(2)}(\tau\gg\tau_{c})\) where \(\tau_{c}\) is the first order coherence time. In order to proof the concept of correlated photon spectroscopy by exploiting amplified spontaneous emission from the EDFA source [33; 36; 37], we selected acetylene (C\({}_{2}\)H\({}_{2}\)), also called ethyne as sample gas due to the match between the emission spectrum of the EDFA and the absorption features of acetylene having an absorption band from approx. 1510 to 1540 nm based on a simultaneous excitation of symmetric and asymmetric stretch bands within the CH-groups of the molecule [48] complemented by differently strong rotational excitations leading to a total of approx. 50 strong absorption lines. Under standard pressure conditions they exhibit a FWHM smaller than 0.5 nm. By increasing the gas pressure, broadening can be achieved either by self broadening or by cross broadening, e.g., by nitrogen.