Physical systems close to a quantum phase transition exhibit a divergent susceptibility, suggesting that an arbitrarily-high precision may be achieved by exploiting quantum critical systems as probes to estimate a physical parameter. However, such an improvement in sensitivity is counterbalanced by the closing of the energy gap, which implies a critical slowing down and an inevitable growth of the protocol duration. Here, we design different metrological protocols that make use of the superradiant phase transition of the quantum Rabi model, a finite-component system composed of a single two-level atom interacting with a single bosonic mode. We show that, in spite of the critical slowing down, critical quantum optical systems can lead to a quantum-enhanced time-scaling of the quantum Fisher information, and so of the measurement sensitivity.In a system close to a critical point, small variations of physical parameters may lead to dramatic changes in the equilibrium state properties. The possibility of exploiting this sensitivity for metrological purposes is well known, and it has already been applied in classical devices, e.g. in superconducting transition-edge sensor [1]. Besides, the development of quantum metrology has extensively shown that quantum states can outperform their classical counterparts for sensing tasks [2]. Therefore, a question naturally arises: what sensitivity can be achieved using interacting systems close to a quantum-critical point? In the last few years, this question has attracted growing interest and it has been addressed by different methods [3][4][5][6][7][8][9]. These studies may be roughly divided in two classes.The first approach, which we will call the "dynamical" paradigm [5,7], focus on the time evolution induced by a Hamiltonian close to a critical point. In this approach, one prepares a probe system in a suitably chosen state, lets it evolve according to the critical Hamiltonian, and finally measures it. This bear close similarity to the standard interferometric paradigm of quantum metrology [2]. On the other hand, the "static" approach [3, 6] is based on the equilibrium properties of the system. It consists in preparing and measuring the system ground state in the unitary case, or the system steady-state when open quantum systems are considered. In proximity of the phase transition the susceptibility of the equilibrium state diverges, and so it does the achievable measurement precision. Unfortunately, the time required to prepare the equilibrium state diverges as well, both in the unitary [10] and in the driven-dissipative case [11,12], a behavior called critical slowing down. Only very re-cently, it has been demonstrated that for a large class of spin models these two approaches are formally equivalent [9], and that they both make it possible to achieve the optimal scaling limit of precision with respect to system size and to measurement time. These results were obtained considering spin systems that undergo quantum phase transitions in the thermodynamic limit, where the numb...

The controllability of current quantum technologies allows to implement spin-boson models where two-photon couplings are the dominating terms of light-matter interaction. In this case, when the coupling strength becomes comparable with the characteristic frequencies, a spectral collapse can take place, i.e. the discrete system spectrum can collapse into a continuous band. Here, we analyze the thermodynamic limit of the two-photon Dicke model, which describes the interaction of an ensemble of qubits with a single bosonic mode. We find that there exists a parameter regime where two-photon interactions induce a superradiant phase transition, before the spectral collapse occurs. Furthermore, we extend the mean-field analysis by considering second-order quantum fluctuations terms, in order to analyze the low-energy spectrum and compare the critical behavior with the one-photon case.