  Superconducting circuits with Josephson junctions are promising candidates
for developing future quantum technologies. Of particular interest is to use
these circuits to study effects that typically occur in complex
condensed-matter systems. Here, we employ a superconducting quantum bit
(qubit),a transmon, to carry out an analog simulation of motional averaging, a
phenomenon initially observed in nuclear magnetic resonance (NMR) spectroscopy.
To realize this effect, the flux bias of the transmon is modulated by a
controllable pseudo-random telegraph noise, resulting in stochastic jumping of
the energy separation between two discrete values. When the jumping is faster
than a dynamical threshold set by the frequency displacement of the levels, the
two separated spectral lines merge into a single narrow-width,
motional-averaged line. With sinusoidal modulation a complex pattern of
additional sidebands is observed. We demonstrate experimentally that the
modulated system remains quantum coherent, with modified transition
frequencies, Rabi couplings, and dephasing rates. These results represent the
first steps towards more advanced quantum simulations using artificial atoms.
