Schrödinger's Swing: A Child's Swing and a Cat Qubit in One Normal Form

The paper Schrödinger’s Swing by Doug Doucette shows that a child pumping a playground swing and a superconducting Kerr-cat qubit are governed, after averaging, by the same one-parameter Hamiltonian normal form, despite one being a classical object of tens of kilograms and the other a microwave circuit holding a few photons at millikelvin temperatures. The identification is established in three steps: an averaging theorem that reduces the parametrically pumped pendulum to a slow autonomous Hamiltonian, an exact identity that equates the pendulum Hamiltonian with that of a junction whose Josephson energy is modulated, and an exact computation of the classical symbol of the resulting Kerr parametric oscillator. A transfer theorem then converts scaled quantities of one system into those of the other through an explicit dictionary. Three blind numerical tests confirm the correspondence: the locked amplitude measured on the swing predicts the mean photon number of the cat state, the slow breathing frequency of the swing predicts the protective energy gap of the qubit, and the phase-space area of one lobe of the swing’s separatrix predicts the number of protected level pairs, all without adjustable parameters. The paper marks with equal care what does not transfer—quantum interference, zero-point shifts, and dissipation. Finally, the thinness ratio is computed for the exact swing and reveals a sharp passage from thin to thick chaos as the pump depth increases, linking the classical skeleton of the two wells to the broader thin-and-thick-chaos framework.