Controlling Quantum Chaos with Stochastic Feedback in the Quantum Cat Map

In this work, we study a stochastic control protocol applied to the quantum cat map, a canonical model of quantum chaos, to uncover universal features of measurement and feedback-induced phase transitions. The natural instability of the map tends to spread quantum states, while the control introduces probabilistic weak resets that counteract the map's action; this competition between the two effects drives a transition between chaotic and controlled (stabilized) regimes. Near the unstable fixed point of the map, the dynamics can be effectively described by an inverted harmonic oscillator, capturing the universal features of the control transition. In the compact phase space of the cat map, however, the sharp transition of the unbounded model becomes a smooth quantum crossover. Using numerical simulations, we track how the quantum state evolves as the control strength and the rate of control (weak measurement) are varied. We identify clear signatures of phase transitions in the wavefunction's steady-state properties. These results capture the universal aspects of measurement and feedback-driven phase transitions in quantum systems.