  We examine the emergence of periodicity in a non-interacting steady-state
quantum system without external drive inspired by quantum time crystals'
spontaneous time-translation symmetry breaking. Specifically, we consider a
lattice ring of non-interacting electrons undergoing weak local position
measurements. Our analysis uncovers time-periodic structures in steady-state
two-time correlation functions, with periodicity linked to the system's group
velocity. This study demonstrates a measurement-induced clock mechanism,
highlighting periodic behaviors in two-time correlators of a non-equilibrium
steady state, contributing to understanding time-periodic phenomena in
minimally interactive quantum systems.
