Nonlinear Heisenberg Limit via Uncertainty Principle in Quantum Metrology

The Heisenberg limit is acknowledged as the ultimate precision limit in quantum metrology, traditionally implying that root mean square errors of parameter estimation decrease linearly with the time T of evolution and the number N of quantum gates or probes. However, this conventional perspective fails to interpret recent studies of "super-Heisenberg" scaling, where precision improves faster than linearly with T and N. In this work, we revisit the Heisenberg scaling by leveraging the position-momentum uncertainty relation in parameter space and characterizing precision in terms of the corresponding canonical momentum. This reformulation not only accounts for time and energy resources, but also incorporates underlying resources arising from noncommutativity and quantum superposition. By introducing a generating process with indefinite time direction, which involves noncommutative quantum operations and superposition of time directions, we obtain a quadratic increment in the canonical momentum, thereby achieving a nonlinear-scaling precision limit with respect to T and N. Then we experimentally demonstrate in quantum optical systems that this nonlinear-scaling enhancement can be achieved with a fixed probe energy. Our results provide a deeper insight into the Heisenberg limit in quantum metrology, and shed new light on enhancing precision in practical quantum metrological and sensing tasks.