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The time evolution of a closed quantum system is connected to its Hamiltonianthrough Schroedinger's equation. The ability to estimate the Hamiltonian iscritical to our understanding of quantum systems, and allows optimization ofcontrol. Though spectroscopic methods allow time-independent Hamiltonians to berecovered, for time-dependent Hamiltonians this task is more challenging. Here,using a single trapped ion, we experimentally demonstrate a method forestimating a time-dependent Hamiltonian of a single qubit. The method involvesmeasuring the time evolution of the qubit in a fixed basis as a function of atime-independent offset term added to the Hamiltonian. In our system theinitially unknown Hamiltonian arises from transporting an ion through a static,near-resonant laser beam. Hamiltonian estimation allows us to estimate thespatial dependence of the laser beam intensity and the ion's velocity as afunction of time. This work is of direct value in optimizing transportoperations and transport-based gates in scalable trapped ion quantuminformation processing, while the estimation technique is general enough thatit can be applied to other quantum systems, aiding the pursuit of highoperational fidelities in quantum control.

Time-dependent Hamiltonian estimation for Doppler velocimetry of trapped ions