  We show that Ramsey spectroscopy of fermionic alkaline-earth atoms in a
square-well trap provides an efficient and accurate estimate for the
eigenspectrum of a density matrix whose $n$ copies are stored in the nuclear
spins of $n$ such atoms. This spectrum estimation is enabled by the high
symmetry of the interaction Hamiltonian, dictated, in turn, by the decoupling
of the nuclear spin from the electrons and by the shape of the square-well
trap. Practical performance of this procedure and its potential applications to
quantum computing, quantum simulation, and time-keeping with alkaline-earth
atoms are discussed.
