Many-body dynamics for the Dicke model based on the quantum measurement

The Dicke model (DM) is a well-suited system for exploring enhanced metrology and quantum information processing. An entangled spin-boson cat-state in the DM is obtained when the spin and boson degrees of freedom interact in the strong coupling limit. This entangled spin-boson cat-state is a metrological resource to the spin cat-state in terms of the spin degrees of freedom. However, it is difficult to experimentally explore features of coherence of the entangled spin-boson cat state, and it is hard to control and measure the boson and spin degrees of freedom simultaneously. Treating the spin subsystem as the main system of interest, we consider the bosonic degrees of freedom as an environment that can be controlled and probed to some extent. We investigate a measurement process on the bosonic system to extract partial information about the spin system.
We also study the physics of how the measurement affects the quantum phase transition of the DM for various limits.