Fig. 2. Signatures of higher-order photon processes.

a, b Simulated Wigner quasiprobability distribution representing the initial time evolution of the coherent state $∣\alpha ⟩$ with α = 1.7 for small and large quantum phase fluctuations. This value of α was chosen to match η/2 for η = 3.4 (see text). As quantum phase fluctuations increase, the evolution remarkably transitions from diffusive to nondiffusive. c–f Simulated transition frequency shifts δ_n = ω_n − ω₀, where ω_n is the transition frequency between energy levels n and n + 1, represented versus photon number at the starred external flux value (c, d), and versus external flux (e, f). We observe the transition from an ordered to an alternating arrangement that asymptotically approaches a Bessel function (dashed lines). Simulations correspond to numerical diagonalization and time propagation of Eq. (1) with parameters $\mathrm{\Omega }/2\pi =2.86\text{GHz},{\mathcal{E}}_J/h=0.795\text{GHz}$ and η = 0.34 for (a), (c), (e) and η = 3.4 for (b), (d), (f) as indicated by the top axis.