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arXiv:2204.05577 · Results in Physics

Quantum estimation of Kerr nonlinearity in driven-dissipative systems

Dong Xie, Chunling Xu, An Min Wang

Published Sep 2, 2022• 7 citations
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Abstract

We mainly investigate the quantum measurement of Kerr nonlinearity in the driven-dissipative system. Without the dissipation, the measurement precision of the nonlinearity parameter χ scales as “super-Heisenberg scaling” 1/N2 with N being the total average number of particles (photons) due to the nonlinear generator. Here, we find that “super-Heisenberg scaling” 1/N3/2 can also be obtained by choosing a proper interrogation time. In the steady state, the “super-Heisenberg scaling” 1/N3/2 can only be achieved when the nonlinearity parameter is close to 0 in the case of the single-photon loss and the one-photon driving or the two-photon driving. The “super-Heisenberg scaling” disappears with the increase of the strength of the nonlinearity. When the system suffers from the two-photon loss in addition to the single-photon loss, the optimal measurement precision will not appear at the nonlinearity χ=0 in the case of the one-photon driving. Counterintuitively, in the case of the two-photon driving we find that it is not the case that the higher the two-photon loss, the lower the measurement precision. It means that the measurement precision of χ can be improved to some extent by increasing the two-photon loss.

DOI
10.1016/j.rinp.2022.105957

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We mainly investigate the quantum measurement of Kerr nonlinearity in the driven-dissipative system. Without the dissipation, the measurement precision of the nonlinearity parameter χ scales as “super-Heisenberg scaling” 1/N2 with N being the total average number of particles (photons) due to the nonlinear generator. Here, we find that “super-Heisenberg scaling” 1/N3/2 can also be obtained by choosing a proper interrogation time. In the steady state, the “super-Heisenberg scaling” 1/N3/2 can only be achieved when the nonlinearity parameter is close to 0 in the case of the single-photon loss and the one-photon driving or the two-photon driving. The “super-Heisenberg scaling” disappears with the increase of the strength of the nonlinearity. When the system suffers from the two-photon loss in addition to the single-photon loss, the optimal measurement precision will not appear at the nonlinearity χ=0 in the case of the one-photon driving. Counterintuitively, in the case of the two-photon driving we find that it is not the case that the higher the two-photon loss, the lower the measurement precision. It means that the measurement precision of χ can be improved to some extent by increasing the two-photon loss.
Super-Heisenberg scaling, which scales as N − β with β > 1 in terms of the number of particles N or T − β in terms of the evolution time T , is better than Heisenberg scaling in quantum metrology. It has been proven that super-Heisenberg scaling can be achieved when the Hamiltonian of the system involves many-body interactions or time-dependent terms. We demonstrate that nonlinear quantum scrambling facilitates the achievement of super-Heisenberg scaling T − β when the generator of the parameter is time-independent. More importantly, in dissipative systems, we can still obtain super-Heisenberg scaling. In the optical cavity system, an exponential improvement in measurement precision over time can be achieved by combining injected external squeezing and intracavity squeezing. Our work provides an optimal method for leveraging nonlinear resources to enhance the measurement precision of the driving field.
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 between the pump mode and two degenerate signal modes 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. This does not require the use of entanglement resources which are not easy to prepare. 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 the critical point between the normal and superradiance phase approaches. Finally, we show that a driven-dissipation down-conversion system can be used as a precise quantum sensor to measure the driving strength.
We address multiparameter quantum estimation for coherently driven nonlinear Kerr resonators in the presence of loss. In particular, we consider the realistic situation in which the parameters of interest are the loss rate and the nonlinear coupling, whereas the amplitude of the coherent driving is known and externally tunable. Our results show that this driven-dissipative model is asymptotically classical, i.e., the Uhlmann curvature vanishes, and the two parameters may be jointly estimated without any additional noise of quantum origin. We also find that the ultimate bound to precision, as quantified by the quantum Fisher information (QFI), increases with the interaction time and the driving amplitude for both parameters. Finally, we investigate the performance of quadrature detection, and show that for both parameters the Fisher information oscillates in time, repeatedly approaching the corresponding QFI.
Driven-dissipative systems have recently attracted great attention due to the existence of novel physical phenomena with no analog in the equilibrium case. The Keldysh path-integral theory is a powerful tool to investigate these systems. However, it has still been a challenge to study strong nonlinear effects implemented by recent experiments, since in this case the photon number is few and quantum fluctuations play a crucial role in the dynamics of the system. Here we develop an approach for deriving exact steady states of driven-dissipative systems by introducing the Keldysh partition function in the Fock-state basis and then mapping the standard saddle-point equations into Keldysh-Heisenberg equations. We take the strong Kerr nonlinear resonators with and without the nonlinear driving as two examples to illustrate our method. It is found that, in the absence of the nonlinear driving, the exact steady state obtained does not exhibit bistability and agrees well with the complex PPP-representation solution. While in the presence of the nonlinear driving, the multiphoton resonance effects are revealed and are consistent with the qualitative analysis. Our method provides an intuitive way to explore a variety of driven-dissipative systems especially with strong correlations.
We investigate quantum-enhanced metrology in a triple point criticality and discover that quantum criticality does not always enhance measurement precision. We have developed suitable adiabatic evolution protocols to effectively restrain excitations, which could accelerate the adiabatic evolutions and lead to an exponential super-Heisenberg scaling. This scaling behavior is quite valuable in practical parameter estimating experiments with limited coherence time. Dissipation can strengthen the super-Heisenberg scaling until decoherence increases to dominate in the dissipative dynamics. Additionally, measurement precisions beyond Heisenberg scaling can be experimentally achieved in the trapped ion system. Our findings strongly indicate that criticality-enhanced metrology can indeed significantly enhance measurement precisions to a super-Heisenberg scaling when combining a triple point and beneficial parameter modulations, which will be conducive to the exploration of other super-Heisenberg scaling and their applications.
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/N1/N1/N), and approach the super-Heisenberg scaling (1/N21/N^{2}1/N2)-a direct consequence of Kerr nonlinearity. More precisely, the super-Heisenberg scaling $\propto $ 1/N21/N^{2}1/N2 is the ultimate precision limit permitted by the k=2k=2k=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.