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A Measure of Monopole Inertia in the Quantum Spin Ice Yb$_2$Ti$_2$O$_7$

An important and continuing theme of modern solid state physics is the realization of exotic excitations in materials (e.g. quasiparticles) that have no analogy (or have not yet been observed) in the actual physical vacuum of free space. Although they are not fundamental particles, such quasiparticles do constitute the most basic description of the excited states of the "vacuum" in which they reside. In this regard the magnetic textures of the excited states of spin ices, magnetic pyrochlore oxides with dominant Ising interactions, are proposed to be modeled as effective magnetic charge monopoles. Recent inelastic neutron scattering experiments have established the pyrochlore material Yb$_2$Ti$_2$O$_7$ (YbTO) as a quantum spin ice, where in addition to the Ising interactions there are substantial transverse terms that may induce quantum dynamics and - in principle - coherent monopole motion. Here we report a combined time domain terahertz spectroscopy (TDTS) and microwave cavity study of YbTO to probe its complex dynamic magnetic susceptibility. We find that the form of the susceptibility is consistent with monopole motion and a magnetic monopole conductivity can be defined and measured. Using the unique phase sensitive capabilities of these techniques, we observe a sign change in the reactive part of the magnetic response. In generic models of monopole motion this is only possible through introducing inertial effects, e.g. a mass dependent term, to the equations of motion. Analogous to conventional electric charge systems, measurement of the conductivity's spectral weight allows us to derive a value for the magnetic monopole mass, which we find to be approximately 1800 electron masses. Our results establish the magnetic monopoles of quantum spin ice as true coherently propagating quasiparticles of this system.