Experimental super-Heisenberg quantum metrology with indefinite gate order

The Heisenberg limit, corresponding to a root mean square error vanishing as 1/N with the number N of independent processes probed in an experiment, is widely believed to be an ultimate limit to the precision of quantum metrology.In this work, we experimentally demonstrate a quantum metrology protocol surpassing Heisenberg limit by implementing indefinite (a superposition of) orders of two groups of independent processes. Each process creates a phase space displacement, and the precision to estimate the geometric phase introduced by a total number of 2N processes approaches the super-Heisenberg limit of 1/N². In our setup, the polarization of a single photon coherently controls the order of displacements on the transverse modes of the radiation field, resulting into indefinite order and allowing us to outperform every setup where the displacements are probed in a definite order. Our experiment features a realization of coherent control over the order in a continuous-variable system, and can be applied to the measurement of various important parameters.