Ultra-subharmonic bifurcation in a driven nonlinear oscillator - Part 1/2: classical and quantum manifestations

Ultra-subharmonic (USH) bifurcation refers to a broad class of dynamical transitions wherein a driven nonlinear oscillator is resonantly excited to one of q equiprobable stable states by some p/q-th USH of a drive, for naturals p and q. Historically, this bifurcation class has attracted mostly theoretical interest due to the intricate topology of phase space trajectories and its connections to chaos. Recently, particular instances of USH processes have been revisited experimentally in Josephson circuits, specifically in the novel low-dissipation quantum regime, for the purpose of processing quantum information. However, the general physical structure of USH processes — their perturbative order and time-averaged dynamics, remains unexamined, particularly for bridging the classical and quantum regimes. We show that the time-averaged dissipative dynamics governing any USH process in the low-dissipation regime, whether classical or quantum, can always be represented as motion on an effective static Hamiltonian surface endowed with discrete q-fold symmetry. Moreover, semiclassically, this so-called metapotential unifies two distinct quantum manifestations of USH bifurcation: multiphoton nonlinear resonances and protected multistability.