  We investigate the transient dynamics of a lumped-element oscillator based on
a dc superconducting quantum interference device (SQUID). The SQUID is shunted
with a capacitor forming a nonlinear oscillator with resonance frequency in the
range of several GHz. The resonance frequency is varied by tuning the Josephson
inductance of the SQUID with on-chip flux lines. We report measurements of
decaying oscillations in the time domain following a brief excitation with a
microwave pulse. The nonlinearity of the SQUID oscillator is probed by
observing the ringdown response for different excitation amplitudes while the
SQUID potential is varied by adjusting the flux bias. Simulations are performed
on a model circuit by numerically solving the corresponding Langevin equations
incorporating the SQUID potential at the experimental temperature and using
parameters obtained from separate measurements characterizing the SQUID
oscillator. Simulations are in good agreement with the experimental
observations of the ringdowns as a function of applied magnetic flux and pulse
amplitude. We observe a crossover between the occurrence of ringdowns close to
resonance and adiabatic following at larger detuning from the resonance. We
also discuss the occurrence of phase jumps at large amplitude drive. Finally,
we briefly outline prospects for a readout scheme for superconducting flux
qubits based on the discrimination between ringdown signals for different
levels of magnetic flux coupled to the SQUID.
