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.

The Dicke model is a paradigmatic quantum-optical model describing the interaction of a collection of two-level systems with a single bosonic mode. Effective implementations of this model made it possible to observe the emergence of superradiance, i.e., cooperative phenomena arising from the collective nature of light-matter interactions. Via reservoir engineering and analogue quantum simulation techniques, current experimental platforms allow us not only to implement the Dicke model but also to design more exotic interactions, such as the two-photon Dicke model. In the Hamiltonian case, this model presents an interesting phase diagram characterized by two quantum criticalities: a superradiant phase transition and a spectral collapse, that is, the coalescence of discrete energy levels into a continuous band. Here, we investigate the effects of both qubit and photon dissipation on the phase transition and on the instability induced by the spectral collapse. Using a mean-field decoupling approximation, we analytically obtain the steady-state expectation values of the observables signaling a symmetry breaking, identifying a first-order phase transition from the normal to the superradiant phase. Our stability analysis unveils a very rich phase diagram, which features stable, bistable, and unstable phases depending on the dissipation rate. The Dicke model 1 describes the interaction of a collection of two-level systems with a single bosonic mode. In the thermodynamic limit, this model exhibits a superradiant phase transition at zero temperature 1-5. Namely, the ground-state number of photons changes non analytically from zero to finite values as the light-matter coupling strength is increased across a critical value. The relevance of the Dicke model to capture the physics of light-matter coupling near the critical point is the object of an ongoing debate. In particular, an obstacle to the observation of a superradiant phase transition arises due to the presence of the so-called diamagnetic term. In this regard, resolutions of gauge ambiguities have been recently proposed, such as employing modified unitary transformations or going beyond the two-level system description 6-9. It is possible, however, to circumvent this controversy entirely by using driven systems and bath engineering to simulate effective Hamiltonians. Thanks to this approach, it was possible to observe the superradiant phase transition in several platforms, in particular atomic systems in cavity 10-12 , and trapped ions 13,14. Other proposals have been put forward, using NV centers array 15 , or superconducting circuits 16. In general, the implementation of analogue quantum simulations 17,18 provides an ideal playground to test driven-dissipative physics in a controlled setting. Their experimental feasibility has motivated increasing research efforts devoted to the study of driven-dissipative quantum optical models, as it is known that noise and dissipation can drastically change the properties of the steady-state phase diagrams and the emergence of p...

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In the context of phase estimation with Gaussian states, we introduce a quantifiable definition of metrological advantage that takes into account thermal noise in the preparation procedure. For a broad set of states, isotropic non-pure Gaussian states, we show that squeezing is not only necessary, but sufficient, to achieve metrological advantage. We interpret our results in the framework of resource theory, and discuss possible sources of advantage other than squeezing. Our work is a step towards using phase estimation with pure and mixed state to define and quantify nonclassicality. This work is complementary with studies that defines nonclassicality using quadrature displacement estimation.arXiv:1807.08045v3 [quant-ph]

Phase transitions represent a compelling tool for classical and quantum sensing applications. It has been demonstrated that quantum sensors can in principle saturate the Heisenberg scaling, the ultimate precision bound allowed by quantum mechanics, in the limit of large probe number and long measurement time. Due to the critical slowing down, the protocol duration time is of utmost relevance in critical quantum metrology. However, how the long-time limit is reached remains in general an open question. So far, only two dichotomic approaches have been considered, based on either static or dynamical properties of critical quantum systems. Here, we provide a comprehensive analysis of the scaling of the quantum Fisher information for different families of protocols that create a continuous connection between static and dynamical approaches. In particular, we consider fully-connected models, a broad class of quantum critical systems of high experimental relevance. Our analysis unveils the existence of universal precision-scaling regimes. These regimes remain valid even for finite-time protocols and finite-size systems. We also frame these results in a general theoretical perspective, by deriving a precision bound for arbitrary time-dependent quadratic Hamiltonians.

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