  If driven sufficiently strongly, superconducting microresonators exhibit
nonlinear behavior including response bifurcation. This behavior can arise from
a variety of physical mechanisms including heating effects, grain boundaries or
weak links, vortex penetration, or through the intrinsic nonlinearity of the
kinetic inductance. Although microresonators used for photon detection are
usually driven fairly hard in order to optimize their sensitivity, most
experiments to date have not explored detector performance beyond the onset of
bifurcation. Here we present measurements of a lumped-element superconducting
microresonator designed for use as a far-infrared detector and operated deep
into the nonlinear regime. The 1 GHz resonator was fabricated from a 22 nm
thick titanium nitride film with a critical temperature of 2 K and a
normal-state resistivity of $100\, \mu \Omega\,$cm. We measured the response of
the device when illuminated with 6.4 pW optical loading using microwave readout
powers that ranged from the low-power, linear regime to 18 dB beyond the onset
of bifurcation. Over this entire range, the nonlinear behavior is well
described by a nonlinear kinetic inductance. The best noise-equivalent power of
$2 \times 10^{-16}$ W/Hz$^{1/2}$ at 10 Hz was measured at the highest readout
power, and represents a $\sim$10 fold improvement compared with operating below
the onset of bifurcation.
