## 1 Introduction
In the THz range, two different techniques have been adopted to demonstrate the high level of coherence achieved by QCL-FC sources: Shifted wave interference Fourier transform spectroscopy (SWIFTS) [29] and, more recently, Fourier analysis of comb emission (FACE) [30; 31]. Thanks to these properties, QCL-FCs can be employed in spectroscopic setups, such as dual-comb spectroscopy (DCS) setups, as proposed in 2016 with an etalon signal [32] (i.e., simulating a real, molecular absorption), with spectra of ammonia gas [33] and water vapor [34; 35], and even exploiting frequency referencing in a hybrid DCS spectrometer [36]. The second huge limitation of THz QCLs is the need of cryogenic cooling, ultimately hindering the miniaturization of QCL-based setups, which can now be overcome with an alternative approach based on difference frequency generation (DFG) in mid-infrared QCLs [37], referred as THz DFG-QCLs [38; 39; 40]. Mid-infrared QCLs are engineered to provide mid-infrared gain for pumps and to possess giant second-order nonlinearity \(\chi^{(2)}\) for THz DFG in the QCL active region [38; 40; 41]. Since nonlinear processes, such as DFG, do not require any population inversion, THz DFG-QCLs are able to operate at room temperature, similar to other mid-infrared QCLs. As a consequence, these are currently the only electrically pumped, monolithic semiconductor THz sources operable at room temperature in the 0.6 to 6 THz frequency range [42; 43]. This key operation mode has strongly motivated, in recent years, further research on DFG-based QCL lasers, although the achievable emitted power is lower than in directly THz emitting lasers. By adopting a strongly coupled upper-state (dual-upper-state: DAU) active region design approach [41; 44], which does not need two stacked laser active regions for dual wavelength mid-IR pumps, continuous-wave (CW) performance of THz DFG-QCLs has considerably improved in the past few years [45; 46]. In the initial research and development of THz DFG-QCLs, narrow-linewidth, widely tunable devices have been extensively developed. The potential use of single mode devices as a metrological grade reference for heterodyne detection has also been investigated [47]. This technology has been recently migrated to DFG QCL devices operating in multimode regime, in which broadband THz emission is generated via nonlinear mixing between a single mid-IR pump frequency selected by a largely detuned distributed feedback (DFB) grating, and FP modes of the second mid-IR pump selected by the laser cavity [48]. The potential operation of THz combs has been assessed for multimode THz DFG-QCL devices, initially at 78 K [40], and subsequently at room temperature [49]. However, these were evaluated against the spectral coherence of mid-IR emission spectra, and by retrieval of a single and narrow intermodal beatnote (IBN). However, this is a condition that is necessary, but not sufficient, to prove comb operation, as the difference between multimode operation and comb emission ultimately relies in the equal mode spacing, and a phase relation (binding the modes one another) being constant over time. In this work, we demonstrate the detection of the modes emitted by a DFG QCL-FC with a multi-heterodyne technique, which eventually returns a resolution much higher than Fourier transform based spectrometers (in the order of few GHz). This enables a direct comparison of the mode spacing with the electrically detected IBN frequency, which results in a very good agreement. Unfortunately, the signal-to-noise ratio provided by the experimental setup, limited by the THz power-per-mode emitted by the DFG QCL device, does not allow for a characterization of the device phase relation, that will be attempted with next generation devices.