## 2 Experimental Details  
Figure 1 schematically shows the experimental setup of the QML Er:Y\({}_{2}\)O\({}_{3}\) ceramic laser. A home-developed Er:Y\({}_{2}\)O\({}_{3}\) ceramic sample with the doping concentration of 7 at.% was cut into the size of 2 mm \(\times\) 3 mm \(\times\) 12.5 mm and polished in Brewster angle to reduce the Fresnel reflection. Single-pass absorption of the sample for the 976 nm pump light was \(\sim\)90%. A 10 W single-mode Yb-fiber laser at 976 nm (linewidth \(\sim\)0.1 nm) is utilized to provide high intensity pump. The fiber laser output has a collimated beam of \(\sim\)750 \(\upmu\)m diameter with a beam quality factor (M\({}^{2}\)) of \(\sim\)1.05 and a polarization extinction ratio of \(>\)99.9:1. An isolator was built in the collimating laser head to avoid the feedback of the pump light. The fiber output beam was first expanded with a 1:4 telescope system, comprising two plane-convex lens of 50 mm (F1) and 200 mm (F2) focal length, to \(\sim\)3 mm diameter. It was then focused into the Er:Y\({}_{2}\)O\({}_{3}\) ceramic by a 75 mm focal length lens. It was estimated that the pump laser beam diameter at the center of the gain medium was \(\sim\)105 \(\upmu\)m in the sagittal and \(\sim\)120 \(\upmu\)m in the tangential plane. The polarization direction of the pump light was adjusted using a half-wave plate (HWP) so as to match the Brewster-angle-cut surface of the gain medium. For efficient heat removal, the Er:Y\({}_{2}\)O\({}_{3}\) ceramic wrapped with indium foil was mounted inside water-cooled copper with a maintained temperature of \(\sim\)10 \({}^{\circ}\)C. The radius of curvature of plane-concave mirrors R\({}_{1}\), R\({}_{2}\), and R\({}_{3}\) were 100 mm, 100 mm, and 50 mm, respectively. The plane-concave mirrors were all coated with the reflectivity of \(>\)99.8% at 2650-2950 nm and the transmission of \(>\)97% at 965-980 nm for filtering out unabsorbed pump light and avoiding adverse effect on the semiconductor saturable absorber (SESAM). The transmittance of the output coupler was 1% in the 2650-2950 nm wavelength range. The physical length of the z type resonator was \(\sim\)1.1 m in total, corresponding to a round-trip repetition rate of \(\sim\)130 MHz. In the gain medium, the cavity mode diameter was estimated to be \(\sim\)140 and \(\sim\)90 \(\upmu\)m in the tangential and sagittal plane, and \(\sim\)50 \(\upmu\)m on the SESAM surface, respectively. The SESAM (BATOP, SAM-2800-3-10 ps, Jena, Germany) was designed for operating in the 2700-2900 nm wavelength range and used as a high reflective end-mirror. Its modulation depth and saturation fluence were \(\sim\)2% and \(\sim\)90 \(\upmu\)J/cm\({}^{2}\), respectively.  
The QML pulse trains were monitored by an infrared detector (VIGO System S.A., PVM-10.6, Ozarow Mazowiecki, Poland) with a response time of 1.5 ns and a 1 GHz digital oscilloscope (Keysight,DSO-S 104A, Santa Rosa, CA, USA). An optical spectrum analyzer (Thorlabs, OSA 205, Newton, NJ, USA) with the resolution of \(\sim\)130 pm at 3 \(\upmu\)m was used to recorded the output spectra. A 3 GHz spectrum analyzer (Agilent, N9320B, Santa Clara, CA, USA) with a resolution bandwidth (RBW) of 10 Hz was used to analyzed the radio frequency (RF) spectrum.