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A pendulum prepared perfectly inverted and motionless is a prototype ofunstable equilibria and corresponds to an unstable hyperbolic fixed point inthe dynamical phase space. Unstable fixed points are central to understandingHamiltonian chaos in classical systems. In many-body quantum systems,mean-field approximations fail in the vicinity of unstable fixed points andlead to dynamics driven by quantum fluctuations. Here, we measure thenon-equilibrium dynamics of a many-body quantum pendulum initialized to ahyperbolic fixed point of the phase space. The experiment uses a spin-1 Bosecondensate, which exhibits Josephson dynamics in the spin populations thatcorrespond in the mean-field limit to motion of a non-rigid mechanicalpendulum. The condensate is initialized to a minimum uncertainty spin state,and quantum fluctuations lead to non-linear spin evolution along a separatrixand non-Gaussian probability distributions that are measured to be in goodagreement with exact quantum calculations up to 0.25 s. At longer times, atomicloss due to the finite lifetime of the condensate leads to larger spinoscillation amplitudes compared to no loss case as orbits depart from theseparatrix. This demonstrates how decoherence of a many-body system can resultin more apparent coherent behaviour. This experiment provides new avenues forstudying macroscopic spin systems in the quantum limit and for investigationsof important topics in non-equilibrium quantum dynamics.

Non-equilibrium dynamics of an unstable quantum pendulum