  A pendulum prepared perfectly inverted and motionless is a prototype of
unstable equilibria and corresponds to an unstable hyperbolic fixed point in
the dynamical phase space. Unstable fixed points are central to understanding
Hamiltonian chaos in classical systems. In many-body quantum systems,
mean-field approximations fail in the vicinity of unstable fixed points and
lead to dynamics driven by quantum fluctuations. Here, we measure the
non-equilibrium dynamics of a many-body quantum pendulum initialized to a
hyperbolic fixed point of the phase space. The experiment uses a spin-1 Bose
condensate, which exhibits Josephson dynamics in the spin populations that
correspond in the mean-field limit to motion of a non-rigid mechanical
pendulum. The condensate is initialized to a minimum uncertainty spin state,
and quantum fluctuations lead to non-linear spin evolution along a separatrix
and non-Gaussian probability distributions that are measured to be in good
agreement with exact quantum calculations up to 0.25 s. At longer times, atomic
loss due to the finite lifetime of the condensate leads to larger spin
oscillation amplitudes compared to no loss case as orbits depart from the
separatrix. This demonstrates how decoherence of a many-body system can result
in more apparent coherent behaviour. This experiment provides new avenues for
studying macroscopic spin systems in the quantum limit and for investigations
of important topics in non-equilibrium quantum dynamics.
