Stabilizing a discrete time crystal against dissipation

Eigenstate phases such as the discrete time crystal exhibit an inherent instability upon the coupling to an environment, which restores equipartition of energy and therefore acts against the protecting nonergodicity. Here, we demonstrate that a discrete time crystal can be stabilized against dissipation using coherent feedback. For a kicked random Ising chain subject to a radiative decay, we show that the time crystalline signal can survive through a mechanism of constructive interference upon reflecting the emitted photons by a mirror. We introduce a matrix product operator algorithm to solve the resulting non-Markovian dynamics. We find that the stabilization mechanism is robust against weak imperfections.