Correspondence between dissipative phase transitions of light and time crystals

Understand

We predict the emergence of a time crystal generated by an incoherently driven and dissipative nonlinear optical oscillator, where the nonlinearity also comes from dissipation.

• We show that a second-order dissipative phase transition of light occurs in the frame rotating at the cavity frequency, while a boundary (dissipative) time crystal emerges in the laboratory frame.
• We relate these two phenomena by using the Liouvillian superoperator associated with the Lindblad master equation and its symmetries.
• These results connect the emergence of a second-order dissipative phase transition and a dissipative time crystal in the thermodynamic limit, allowing to interpret them as the same phenomenon in terms of the Liouvillian spectrum, but just in different frames.