Fig. 1: LZSM interference mechanisms in bosonic systems.

For cases studied in this article, we show the level structure of the undriven system (left); the photon number n of the driven, but not-modulated, system as a function of the pump-to-cavity detuning Δ (center); and the LZSM pattern emerging when the cavity eigenfrequency is periodically modulated with strength ζ and frequency Ω (right). a In the qubit regime of infinite nonlinearity, the system consists only of the ground and excited states. This level structure gives rise to a single excitation peak (\(\left\vert 0\right\rangle \to \left\vert 1\right\rangle\)) at detuning Δ = 0. Thus, the standard LZSM interference pattern emerges. b In the Kerr regime, where the anharmonicity is larger than the loss (∣χ∣ > κ), the system consists of many uneven-spaced states with different numbers of excitations. When Δ = nχ, multiphoton transitions \(\left\vert 0\right\rangle \to \left\vert n\right\rangle\) occur for a large enough drive. This multi-photonic transition structure is periodically repeated around each standard LZSM peak. c In the Duffing regime, where the anharmonicity is smaller than the loss (∣χ∣ < κ), the uneven-spaced states are broadened by dissipation and cannot be distinguished. The drive excites multiple levels, resulting in a deviation from a Lorentzian shape, and the Kerr nonlinearity competes with detuning, giving rise to bistability. Such a deviation and the presence of bistability are imprinted in each LZSM peak. d In the linear regime (χ = 0), all levels are equispaced. When driven, only a Lorentzian peak appears at Δ = 0, similar to the qubit regime. Upon modulation of the resonator frequency, the LZSM interference is also indistinguishable from that in the qubit regime.