  The time evolution of a closed quantum system is connected to its Hamiltonian
through Schroedinger's equation. The ability to estimate the Hamiltonian is
critical to our understanding of quantum systems, and allows optimization of
control. Though spectroscopic methods allow time-independent Hamiltonians to be
recovered, for time-dependent Hamiltonians this task is more challenging. Here,
using a single trapped ion, we experimentally demonstrate a method for
estimating a time-dependent Hamiltonian of a single qubit. The method involves
measuring the time evolution of the qubit in a fixed basis as a function of a
time-independent offset term added to the Hamiltonian. In our system the
initially unknown Hamiltonian arises from transporting an ion through a static,
near-resonant laser beam. Hamiltonian estimation allows us to estimate the
spatial dependence of the laser beam intensity and the ion's velocity as a
function of time. This work is of direct value in optimizing transport
operations and transport-based gates in scalable trapped ion quantum
information processing, while the estimation technique is general enough that
it can be applied to other quantum systems, aiding the pursuit of high
operational fidelities in quantum control.
