Dissipative phase transitions in the driven-dissipative Kerr parametric oscillator offer a promising route for realizing criticality-enhanced quantum sensing based on continuous measurements. However, achieving such enhancement through realistic measurement schemes remains an outstanding challenge. Here, we extend the backaction-evasion strategy introduced in our earlier work for the Gaussian linear case [arXiv:2511.22248 (2025)] to analyze how the quantum and classical Fisher information scale with the Kerr nonlinearity at dissipative critical points. Our results show that backaction-evading homodyne monitoring achieves enhanced photon-number scaling that surpasses the standard quantum limit, and significantly outperforms alternative protocols such as continuous photon counting. As an additional methodological contribution, we also implement and benchmark time-discrete approximation schemes with improved statistical convergence properties. We use these methods to compute the classical Fisher information for continuous homodyne detection, and demonstrate that they provide efficient access to this quantity near dissipative critical points, thereby extending the reach of existing methods.
Dissipative phase transitions in the driven-dissipative Kerr parametric oscillator offer a promising route for realizing criticality-enhanced quantum sensing based on continuous measurements. However, achieving such enhancement through realistic measurement schemes remains an outstanding challenge. Here, we extend the backaction-evasion strategy introduced in our earlier work for the Gaussian linear case [arXiv:2511.22248 (2025)] to analyze how the quantum and classical Fisher information scale with the Kerr nonlinearity at dissipative critical points. Our results show that backaction-evading homodyne monitoring achieves enhanced photon-number scaling that surpasses the standard quantum limit, and significantly outperforms alternative protocols such as continuous photon counting. As an additional methodological contribution, we also implement and benchmark time-discrete approximation schemes with improved statistical convergence properties. We use these methods to compute the classical Fisher information for continuous homodyne detection, and demonstrate that they provide efficient access to this quantity near dissipative critical points, thereby extending the reach of existing methods.
Continuous monitoring of open quantum-optical systems offers a promising route towards quantum-enhanced sensing. In such sensing schemes, the ultimate precision limit is dictated by the quantum Cramér-Rao bound. However, reaching this fundamental bound with established continuous measurement techniques has remained an outstanding challenge. Here, we present an approach tailored to open quantum-optical sensors operating near dissipative critical points. The key insight of our approach is to exploit a synergy between the intrinsic critical dynamics and the unavoidable quantum measurement backaction of continuous homodyne detection. By carefully aligning the homodyne phase to the squeezing angle of the sensor state, quantum backaction can be evaded in the critical dynamics, thereby significantly boosting the attainable precision. We demonstrate the broad applicability of this approach across several sensor models and for both single- and multi-parameter estimation, showing that it consistently narrows the gap toward the ultimate precision bound.
We propose a high-precision phase estimation scheme in a hybrid interferometer by synergistically combining a Kerr nonlinear phase shifter and multi-photon subtraction operations. Using a coherent state and a vacuum state as input resources, we systematically evaluate the phase sensitivity via homodyne detection and analyze the quantum Fisher information as well as the quantum Cramér-Rao bound under both ideal and lossy conditions. Our results show that the joint integration of Kerr nonlinearity and multi-photon subtraction yields remarkable advantages over either technique used alone. The proposed scheme enables the phase sensitivity to surpass the standard quantum limit, exceed the conventional Heisenberg scaling (1/N), and approach the super-Heisenberg scaling (1/N2)-a direct consequence of Kerr nonlinearity. More precisely, the super-Heisenberg scaling $\propto $ 1/N2 is the ultimate precision limit permitted by the k=2 Kerr nonlinearity and does not violate the fundamental Heisenberg limit for linear phase accumulation. Even under moderate internal photon loss, the system maintains high precision and exhibits enhanced robustness to decoherence. The Kerr nonlinearity introduces an intensity-dependent phase shift proportional to the squared photon number, while multi-photon subtraction tailors non-Gaussian states to strengthen phase information extraction. Compared with existing schemes based on hybrid interferometers or SU(1,1) interferometers, our architecture achieves superior precision and stronger loss resilience. All components are experimentally accessible with current quantum optical technologies. This work provides a promising route for practical high-precision quantum metrology and quantum sensing.
Quantum mechanics imposes a limit on the precision of a continuous position measurement of a harmonic oscillator, due to backaction arising from quantum fluctuations in the measurement field. This standard quantum limit can be surpassed by monitoring only one of the two non-commuting quadratures of the motion, known as backaction-evading measurement. This technique has not been implemented using optical interferometers to date. Here we demonstrate, in a cavity optomechanical system operating in the optical domain, a continuous two-tone backaction-evading measurement of a localized gigahertz-frequency mechanical mode of a photonic-crystal nanobeam cryogenically and optomechanically cooled close to the ground state. Employing quantum-limited optical heterodyne detection, we explicitly show the transition from conventional to backaction-evading measurement. We observe up to 0.67 dB (14%) reduction of total measurement noise, thereby demonstrating the viability of backaction-evading measurements in nanomechanical resonators for optical ultrasensitive measurements of motion and force.
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.
Quantum metrology, a cornerstone of quantum technologies, exploits entanglement and superposition to achieve higher precision than classical protocols in parameter-estimation tasks. When combined with critical phenomena such as phase transitions, the divergence of quantum fluctuations is predicted to enhance the performance of quantum sensors. Here, we implement a critical quantum sensor using a superconducting parametric (i.e., two-photon driven) Kerr resonator. The sensor, a linear resonator terminated by a superconducting quantum interference device, operates near the critical point of a finite-component second-order dissipative phase transition obtained by scaling the system parameters. We analyze the performance of a frequency-estimation protocol and show that quadratic precision scaling with respect to the system size can be achieved with finite values of the Kerr nonlinearity. Since each photon emitted from the cavity carries more information about the parameter to be estimated compared to its classical counterpart, our protocol opens up perspectives for faster or more precise metrological protocols. Our results demonstrate that quantum advantage in a sensing protocol can be achieved by exploiting a phase transition.
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 Cramér-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 a 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 the time-reversal transformation of quantum-optical input-output channels, thereby establishing a universal method to achieve the quantum Cramér-Rao precision limit for generic sensor designs 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 drivendissipative open sensors.We illustrate the power of our scheme with paramagnetic open sensor designs including linear force sensors, fiber-interfaced nonlinear emitters, and driven-dissipative many-body sensors, and demonstrate that it can be robustly implemented under realistic experimental imperfections.