Entanglement generation in a continuous-wave collective SU(4) bad-cavity laser

We theoretically investigate the collective dynamics of a dissipative SU(4) system consisting of two internal and two motional states. By unravelling the master equation into SU(4) quantum trajectory simulations, we show that the output light field exhibits properties similar to that of superradiant lasing. Entanglement between the two degrees of freedom allows the length of the collective atomic dipole to change with purely collective interactions, which is impossible in a system with only two states. We formulate an algorithm to trace out either degree of freedom, allowing us to calculate the macroscopic entanglement entropy that develops between the pseudospin (internal) and motional (external) degrees of freedom. We show this entanglement grows linearly with the number of atoms which is beyond what may be achieved with typical bipartite entanglement of two symmetric subsystems. Furthermore, we calculate the quantum Fisher information in steady-state in order to show how the multi-particle entanglement can be used to demonstrate beyond standard quantum limit metrology.