  One of the central challenges in condensed matter physics is to comprehend
systems that have strong disorder and strong interactions. In the strongly
localized regime, their subtle competition leads to glassy electron dynamics
which ceases to exist well before the insulator-to-metal transition is
approached as a function of doping. Here, we report on the discovery of glassy
electron dynamics deep inside the good metal regime of an electron-doped
quantum paraelectric system: KTaO$_3$. We reveal that upon excitation of
electrons from defect states to the conduction band, the excess injected
carriers in the conduction band relax in a stretched exponential manner with a
large relaxation time, and the system evinces simple aging phenomena - a
telltale sign of glassy dynamics. Most significantly, we observe a critical
slowing down of carrier dynamics below 35 K, concomitant with the onset of
quantum paraelectricity in the undoped KTaO$_3$. Our combined investigation
using second harmonic generation technique, density functional theory and
phenomenological modeling demonstrates quantum fluctuation-stabilized soft
polar modes as the impetus for the glassy behavior. This study addresses one of
the most fundamental questions regarding the potential promotion of glassiness
by quantum fluctuations and opens a route for exploring glassy dynamics of
electrons in a well-delocalized regime.
