"Super-Heisenberg" and Heisenberg Scalings Achieved Simultaneously in the Estimation of a Rotating Field

The Heisenberg scaling, which scales as N^{-1} in terms of the number of particles or T^{-1} in terms of the evolution time, serves as a fundamental limit in quantum metrology. Better scalings, dubbed as "super-Heisenberg scaling," however, can also arise when the generator of the parameter involves many-body interactions or when it is time dependent. All these different scalings can actually be seen as manifestations of the Heisenberg uncertainty relations. While there is only one best scaling in the single-parameter quantum metrology, different scalings can coexist for the estimation of multiple parameters, which can be characterized by multiple Heisenberg uncertainty relations. We demonstrate the coexistence of two different scalings via the simultaneous estimation of the magnitude and frequency of a field where the best precisions, characterized by two Heisenberg uncertainty relations, scale as T^{-1} and T^{-2}, respectively (in terms of the standard deviation). We show that the simultaneous saturation of two Heisenberg uncertainty relations can be achieved by the optimal protocol, which prepares the optimal probe state, implements the optimal control, and performs the optimal measurement. The optimal protocol is experimentally implemented on an optical platform that demonstrates the saturation of the two Heisenberg uncertainty relations simultaneously, with up to five controls. As the first demonstration of simultaneously achieving two different Heisenberg scalings, our study deepens the understanding on the connection between the precision limit and the uncertainty relations, which has wide implications in practical applications of multiparameter quantum estimation.