  Nonlinear damping, a force of friction that depends on the amplitude of
motion, plays an important role in many electrical, mechanical and even
biological oscillators. In novel technologies such as carbon nanotubes,
graphene membranes or superconducting resonators, the origin of nonlinear
damping is sometimes unclear. This presents a problem, as the damping rate is a
key figure of merit in the application of these systems to extremely precise
sensors or quantum computers. Through measurements of a superconducting
circuit, we show that nonlinear damping can emerge as a direct consequence of
quantum fluctuations and the conservative nonlinearity of a Josephson junction.
The phenomenon can be understood and visualized through the flow of
quasi-probability in phase space, and accurately describes our experimental
observations. Crucially, the effect is not restricted to superconducting
circuits: we expect that quantum fluctuations or other sources of noise give
rise to nonlinear damping in other systems with a similar conservative
nonlinearity, such as nano-mechanical oscillators or even macroscopic systems.
