Experimental demonstration of nonlinear quantum metrology with optimal quantum state

Nonlinear quantum metrology may exhibit better precision scalings. For example, the uncertainty of an estimated phase may scale as Δϕ∝1/N² 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 Δϕ∝1/(N²-1) 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.