Experimental demonstration of super-Heisenberg quantum metrology with indefinite gate order

The Heisenberg precision limit, a 1/N scaling for ensemble measurement with N independent elements, is widely believed to represent the ultimate precision limit of quantum metrology. Several proposals have challenged this belief in the past, for example using non-linear interactions among the probes. Nevertheless, the Heisenberg limit is found to stand firm and remain observed by these proposals with respect to relevant resources, such as the total energy of the probes. Here in this work, we demonstrate a quantum metrology protocol surpassing the Heisenberg limit by probing two groups of independent processes in a superposition of distinct alternative orders. With each process creating a phase space displacement, our setup achieves the super-Heisenberg limit 1/N^2 in the estimation of a geometric phase associated with the two sets of N displacements. In contrast to previous studies, our results only require a single photon probe whose initial energy is independent of N, and are shown to outperform every reported setup where the displacements are probed in a definite order. Our experiment demonstrates indefinite causal order interferometry in a continuous-variable system and opens up experimental investigations of quantum metrology setups boosted by indefinite causal order.