Continuous monitoring of driven-dissipative quantum optical systems is a crucial element in the implementation of quantum metrology, providing essential strategies for achieving highly precise measurements beyond the classical limit. In this context, the relevant figure of merit is the quantum Fisher information of the radiation field emitted by the driven-dissipative sensor. Saturation of the corresponding precision limit as defined by the quantum Cramer-Rao bound is typically not achieved by conventional, temporally local continuous measurement schemes such as counting or homodyning. To address the outstanding open challenge of efficient retrieval of the quantum Fisher information of the emission field, we design a novel continuous measurement strategy featuring temporally quasilocal measurement bases as captured by matrix product states. Such measurement can be implemented effectively by injecting the emission field of the sensor into an auxiliary open system, a `quantum decoder' module, which `decodes' specific input matrix product states into simple product states as its output field, and performing conventional continuous measurement at the output. We devise a universal recipe for the construction of the decoder by exploiting time reversal transformation of quantum optical input-output channels, thereby establishing a universal method to achieve the quantum Cramer-Rao precision limit for generic sensors based on continuous measurement. As a by-product, we establish an effective formula for the evaluation of the quantum Fisher information of the emission field of generic driven-dissipative open sensors. We illustrate the power of our scheme with paramagnetic open sensor designs including linear force sensors, fibre-interfaced nonlinear emitters, and driven-dissipative many-body sensors, and demonstrate that it can be robustly implemented under realistic experimental imperfections.

We propose a model describing the formation of both dual (quantum) and classical Shapiro steps in small Josephson junctions. According to this model, the dual Shapiro steps are formed at relatively low frequency of the microwave signal and low microwave power, while the classical steps are formed in the opposite limit of high frequency and power. The crossover between the two regimes is controlled by a single parameter - the effective relaxation time of the environment. The model accounts for the effect of a large inductor in the bias circuit, which has been used in recent experiments to protect the junction from the high frequency noise of the environment. We predict the possibility of observing both types of steps in the same sample. Our model describes the I-V curves observed in the experiments with reasonable accuracy, thus opening up an opportunity for quantitative fitting of the data.

We consider the Rabi Hamiltonian which exhibits a quantum phase transition (QPT) despite consisting only of a single-mode cavity field and a two-level atom. We prove QPT by deriving an exact solution in the limit where the atomic transition frequency in unit of the cavity frequency tends to infinity. The effect of a finite transition frequency is studied by analytically calculating finite-frequency scaling exponents as well as performing a numerically exact diagonalization. Going beyond this equilibrium QPT setting, we prove that the dynamics under slow quenches in the vicinity of the critical point is universal, that is, the dynamics is completely characterized by critical exponents. Our analysis demonstrates that the Kibble-Zurek mechanism can precisely predict the universal scaling of residual energy for a model without spatial degrees of freedom. Moreover, we find that the onset of the universal dynamics can be observed even with a finite transition frequency.

The Rabi model, a two-level atom coupled to a harmonic oscillator, can undergo a second-order quantum phase transition (QPT) [M. -J. Hwang et al, Phys. Rev. Lett. 115, 180404 (2015)]. Here we show that the Rabi QPT accompanies critical behavior in the higher energy excited states, i.e., the excited-state QPT (ESQPT). We derive analytic expressions for the semiclassical density of states, which shows a logarithmic divergence at a critical energy eigenvalue in the broken symmetry (superradiant) phase. Moreover, we find that the logarithmic singularities in the density of states leads to singularities in the relevant observables in the system such as photon number and atomic polarization. We corroborate our analytical semiclassical prediction of the ESQPT in the Rabi model with its numerically exact quantum mechanical solution.