Phases Transitions In A Cross-cavity Quantum Rabi Model Possessing PT Symmetric Structurem

We study a system in which two cavities interact with a single two-level atom.One cavity has gain and the other cavity has dissipation.These two cavities sat-isfy parity and time inversion symmetry(PT symmetry)and form a cross cavity.In this system,the dissipative Rabi phase transition and the spontaneous PT symmetry breaking exist simultaneously.The entire system can be described by a master equation.Through the semi-classical approximation,ignoring quantum oscillations,the standard Heisenberg Langevin equation can be obtained from the master equation.When the system is in the PT asymmetric state,there will be a steady-state solution for the semi-classical equation we obtained.By obtaining the steady solution we will find a critical pointg_c,which corresponds to the Rabi phase transition.when the system is in the PT-symmetric state,there will be a set of oscillating solutions,and there is also a critical point.Using the unitary transformation in the regions of g<g_cand g>g_c,an effective master equation is obtained,and a corresponding phase diagram is obtained by solving it.After analysis,it is found that the normal phase and superradiative phase appearing in the previous dissipative quantum rabies model still exist in our system.When the system changes from the normal phase to the superradiative phase,the ef-fective coupling changes from being in direct proportion to g~2to being inversely proportional to g~2.At the same time,the PT asymmetric and symmetric phases also affect the dissipative quantum Rabi model.For the PT asymmetric phase,as time approaches infinity,the entire system will be in a stable state,because energy will eventually be dissipated into the surrounding environment.When the PT asymmetric phase is changed to a symmetric phase,the gain and loss of the system can be rebalanced.The original dissipative quantum Rabi model behaves like the decay-free Rabi model.This means that in addition to the normal and superradiative phases that existed in the previous dissipative quantum Rabi mod-el,there will also be an oscillating phase in the current system,which is entirely due to the PT symmetrical structure.Through a self-consistent equation,the os-cillation frequency can be determined,and the corresponding phase diagram can be made.After analyzing the oscillation frequency,it can be found that the oscil-lation frequency performance in the oscillation phase I(OP I)and the oscillation phase II(OP II)are completely different.When we fix the attenuation rate?,as the coupling strength g increases,the oscillation frequency also increases with OP I and decreases with OP II.The oscillation frequency is almost independent of the attenuation rate.It can also be seen that another important feature of the oscillating frequency is the discontinuity at the edge transitions from the oscillat-ing phase to the normal and superradiative phases.This results in a discontinuity in the order parameters of the previous equation,corresponding to a first-order quantum phase transition.