We show that applying feedback and weak measurements to a quantum system induces phase transitions beyond the dissipative ones. Feedback enables controlling essentially quantum properties of the transition, i.e., its critical exponent, as it is driven by the fundamental quantum fluctuations due to measurement. Feedback provides the non-Markovianity and nonlinearity to the hybrid quantumclassical system, and enables simulating effects similar to spin-bath problems and Floquet time crystals with tunable long-range (long-memory) interactions.

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We propose and numerically analyze a new method of cooling for generic polarizable particles. It combines the ideas of cavity self-organization and feedback cooling. In particular, we propose to control the periodic potential for the atoms using the Bragg back-reflection of the probe light. The feedback loop is designed to maximize the reflection, which corresponds to tight localization of the atoms in the potential wells. The model, with realistic parameters, has been numerically simulated and found to demonstrate pronounced cooling and spatial localization effects. On the basis of the same numerical simulations, the time scale of the cooling process was shown to be comparable to the trapping time in conventional dipole traps.