  Quantum states cohere and interfere. Quantum systems composed of many atoms
arranged imperfectly rarely display these properties. Here we demonstrate an
exception in a disordered quantum magnet that divides itself into nearly
isolated subsystems. We probe these coherent clusters of spins by driving the
system beyond its linear response regime at a single frequency and measuring
the resulting "hole" in the overall linear spectral response. The Fano shape of
the hole encodes the incoherent lifetime as well as coherent mixing of the
localized excitations. For the disordered Ising magnet,
$\mathrm{LiHo_{0.045}Y_{0.955}F_4}$, the quality factor $Q$ for spectral holes
can be as high as 100,000. We tune the dynamics of the quantum degrees of
freedom by sweeping the Fano mixing parameter $q$ through zero via the
amplitude of the ac pump as well as a static external transverse field. The
zero-crossing of $q$ is associated with a dissipationless response at the drive
frequency, implying that the off-diagonal matrix element for the two-level
system also undergoes a zero-crossing. The identification of localized
two-level systems in a dense and disordered dipolar-coupled spin system
represents a solid state implementation of many-body localization, pushing the
search forward for qubits emerging from strongly-interacting, disordered,
many-body systems.

Disordered Systems and Neural Networks