Quantum metrology in a driven-dissipation down-conversion system beyond the parametric approximation

We investigate quantum metrology in a degenerate down-conversion system composed of a pump mode and two degenerate signal modes. In the conventional parametric approximation, the pump mode is assumed to be constant, not a quantum operator. We obtain the measurement precision of the coupling strength beyond the parametric approximation. Without a dissipation, the super-Heisenberg limit can be obtained when the initial state is the direct product of classical state and quantum state. When the pump mode suffers from a single-photon dissipation, the measurement uncertainty of the coupling strength is close to 0 as the coupling strength approaches 0 with a coherent driving. The direct photon detection is proved to be the optimal measurement. This result has not been changed when the signal modes suffer from the two-photon dissipation. When the signal modes also suffer from the single-mode dissipation, the information of the coupling strength can still be obtained in the steady state. In addition, the measurement uncertainty of the coupling strength can also be close to 0 and become independent of noise temperature as a critical point approaches. Finally, we show that a driven-dissipation down-conversion system can be used as a precise quantum sensor to measure the driving strength.