Two Pendulums and a Condensate

Two weakly coupled identical pendulums and a Bose–Einstein condensate held in a double-well potential belong to the same dynamical class. Both feature two modes of equal signature locked near a one-to-one resonance, an exchange symmetry, and a conserved total action that fixes the overall level of the motion. On that level the systems reduce to a single Hamiltonian whose behavior is controlled by two invariants: the ratio of nonlinear frequency shift to coupling strength, and the strength of the second harmonic of the relative phase. Self-trapping, in which the imbalance between the two sides never reverses sign, occurs precisely when the nonlinear-to-coupling ratio exceeds a threshold set by those invariants and by the initial imbalance and phase. When all the energy begins on one side, the threshold takes one of two simple values according to whether any detuning favors or opposes that side. The out-of-phase stationary state loses stability and bifurcates at half that threshold, while the period of energy exchange diverges logarithmically as the threshold is approached. These laws are confirmed by direct integration of pendulum pairs, ensembles of random quartic oscillators, and the one-dimensional Gross–Pitaevskii equation; a pendulum pair whose coupling is tuned to match the condensate’s invariants reproduces the entire self-trapping boundary of the atomic system. When the conserved action decays slowly through damping or atom loss, the release from self-trapping follows an adiabatic area rule obeyed by both systems. The corresponding quantum dimer, however, tunnels on a time that grows exponentially with atom number, so the classical laws hold only up to that quantum departure.