A. Design of the cavity platform

To achieve the highest passive mechanical stability, we have designed a monolithic cavity nanopositioning stage [Figs. 2(a), 2(b), and 2(c)] based on flexure-mechanical elements and piezo actuators for fine tuning that can cover an (x, y, z)-volume of (70 × 70 × 10) µm3 at room temperature that reduces to ∼ (10 × 10 × 1.5) µm3 at 10 K [Fig. 2(d)]. We calibrate the lateral scanning range by cavity transmission images of a planar mirror where a grid pattern with a pitch of 10 μm has been machined by CO2-laser shots. During the cooldown, repeated scanning cavity images reveal a shift of the fiber position to the right due to differential thermal contraction of the piezo actuators and the frame. In addition, the scanning range decreases due to the capacity reduction of the piezo stacks. For coarse tuning over a volume of (2 × 2 × 2) mm3, the stage incorporates electromotors and gear work. The central mechanical element is a flexural lever arm to which the cavity fiber is fixed, which itself is glued into a steel needle to increase mechanical stability. The assembly is preloaded by three piezo-electric actuators that allow bending motions to achieve three-dimensional nanopositioning of the fiber tip. There are screws to control the prestress on the x- and y-piezo actuators, which are also used to bend the fiber needle and thus allow for angular alignment of the fiber tip. The lever arm design aims for low weight and small dimensions in combination with a high stiffness. This ensures high mechanical resonance frequencies of several kHz in order to avoid coupling to acoustic noise. Furthermore, the noise sources are reduced by clamping the fiber and all cables connected to the top frame. The positioning along the cavity axis (z) requires particular attention, since it should simultaneously allow one to tune across several longitudinal cavity resonances, i.e., a positioning range of a few micrometers, and to controllably maintain resonance conditions with sub-picometer resolution. The required overall dynamic range of 107 goes beyond the capability of available voltage sources to drive a single piezo. We thus physically separate the ultra-fine tuning with a thin piezo plate with ∼100 nm expansion range (fine piezo) from the fine tuning with a piezo stack with a few micrometer tuning range (coarse piezo) under cryogenic conditions. Both piezos are separately driven by suitably filtered voltage sources and amplifiers. To obtain an even wider range of cavity length tunability over a few millimeters, a DC motor, which was slightly modified to work under cryogenic conditions, is used to move the piezo actuators in the longitudinal direction. We clean the motors and gear works from grease in warm soap water and acetone, shorten the motor axis, and increase the bearing diameter to avoid the motors getting stuck at low temperatures. The wide tuning range is necessary to compensate for thermal contraction upon the cooldown of the platform. We minimize the effects of thermal contraction by using titanium for the main frame as well as the lever arm, which has a low thermal expansion coefficient (8.6 × 10−6 K−1). The planar mirror is mounted inside a copper mirror holder, which is thermally isolated from the main frame by glass spheres. It is pressed against the main frame by a spring to ensure mechanical stiffness while keeping the flexibility to move the mirror laterally with two additional DC motors over several millimeters.