标题： 量子范德波尔振子的精确稳态：临界现象与增强计量学 显示英文标题

标题： Exact steady state of quantum van der Pol oscillator: critical phenomena and enhanced metrology

摘要： 开放多体系统的量子临界性因其涌现现象和普适性而引起了广泛关注。 这里我们使用复数$P$-表示法，给出了量子范德波尔振子的精确稳态。 我们展示了阈值对应于一种耗散相变，其稳态性质出现突变，并增强了计量性能。 通过解析稳态研究了临界行为和有限尺寸效应。 此外，我们在热力学极限下，在临界点和时间晶体相中都获得了发散的量子费舍尔信息（QFI），但只有临界点的QFI接近海森堡极限。 我们进一步证明，稳态光子数是具有最大信噪比的优化估计可观测量。 我们表明，海森堡极限的计量性能源于对易率的增强大于光子数的标准差。 我们的工作揭示了时间晶体、耗散相变和增强计量之间的内在关系。 显示英文摘要

摘要： Quantum criticality of open many-body systems has attracted lots of interest for emergent phenomena and universality. Here we present the exact steady state of the quantum van der Pol oscillator using the complex $P$-representation. We show the threshold corresponds to a dissipative phase transition with abrupt changes of steady-state properties and enhanced metrology. The critical behaviors and finite-size effects are investigated through the analytical steady state. Moreover, we obtain divergent quantum Fisher information (QFI) in the thermodynamic limit both at the critical point and in the time crystal phase, but only the QFI at the critical point approaches the Heisenberg limit. We further prove that the steady-state photon number is the optimized estimated observable with the largest signal-to-noise ratio. We show that the Heisenberg-limited metrology originates from the larger enhancement of the susceptibility than the standard deviation of the photon number. Our work reveals the underlying relation between the time crystal, dissipative phase transition, and enhanced metrology.