Nonlinear quantum metrology may exhibit better precision scalings. For example, the uncertainty of an estimated phase may scale as $\mathrm{\Delta }\varphi \propto 1/N^2$ under quadratic phase accumulation, which is $1/N$ times smaller than the linear counterpart, where N is probe number. Here, we experimentally demonstrate the nonlinear quantum metrology by using a spin-I ($I>1/2$) nuclear magnetic resonance (NMR) ensemble that can be mapped into a system of $N=2I$ spin-1/2 particles and the quadratic interaction can be utilized for the quadratic phase accumulation. Our experimental results show that the phase uncertainty can scale as $\mathrm{\Delta }\varphi \propto 1/(N^2-1)$ by optimizing the input states, when N is an odd number. In addition, the interferometric measurement with quadratic interaction provides a new way for estimating the quadrupolar coupling strength in an NMR system. Our system may be further extended to exotic nonlinear quantum metrology with higher order many-body interactions.

非线性量子计量学可能表现出更好的精度缩放。例如，估计相位的不确定性可以缩放为$\mathrm{△}\phi \propto \mathrm{1个}/{否}^{\mathrm{2个}}$在二次相位累积下，这是$\mathrm{1个}/否$比线性对应物小一倍，其中N是探针数。在这里，我们通过使用自旋I ($我>\mathrm{1个}/\mathrm{2个}$) 核磁共振 (NMR) 系综，可以映射到一个系统中$否=\mathrm{2个}我$自旋 1/2 粒子和二次相互作用可用于二次相位累积。我们的实验结果表明，相位不确定性可以缩放为$\mathrm{△}\phi \propto \mathrm{1个}/({否}^{\mathrm{2个}}-\mathrm{1个})$当N为奇数时，通过优化输入状态。此外，具有二次相互作用的干涉测量为估算核磁共振系统中的四极耦合强度提供了一种新方法。我们的系统可以进一步扩展到具有更高阶多体相互作用的奇异非线性量子计量学。