Dissipative phase transition in systems with two-photon driving and dissipation near the critical point

Driven-dissipative phase transition in an open quantum systems nowdays is among the most rapidly developing fields of quantum optics. An example of such a system is depicted in Fig. 1 (a) , which shows a superconducting microwave cavity undergoing two-photon driving at g rate as well as two-photon dissipation at γ rate [1] . In this system, the presence of the two-photon dissipation does not destroy, but stabilizes the quantum state, also known as Schrödinger cat state, and it is possible to prepare a dynamically protected qubit on its basis [2] . Numerical methods are commonly applied to study such quantum systems. However, their use typically does not allow to uncover physical phenomena underlying the evolution of the system. In the limit of a large number of photons, a qualitative picture of the occurring phenomena can be obtained using the so-called semiclassical approximation. Nevertheless, this approach completely neglects the quantum fluctuations, which can play a significant role, even in the case of large average number of photons in the system.