Fig. 3
Signatures of classical chaos in quantum FOTOCs. a Initial exponential growth of the FOTOC, \([1 - {\cal{F}}_X(t)]/(\delta \phi )^2\) and the initial state \(|\Psi _0^{\mathrm{c}}\rangle = |( - N/2)_x\rangle \otimes |0\rangle\) (see Supplementary Note 1 for examples of exponential growth in other states). We assume \(\delta \phi \ll 1/N\) such that we may equivalently use \({\mathrm{var}}(\hat X) \simeq [1 - {\cal{F}}_X(t)]/(\delta \phi )^2\) for the plotted data. The scrambling time t* is defined by the saturation of the FOTOC, which we extract from the first maximum and plot in the inset (blue data). We find \(t^ \ast \sim a_0 + {\mathrm{log}}(N)/\lambda _{\mathrm{Q}}\) with a0 a fit parameter (gray line). b Lyapunov exponent, λ, as a function of transverse field: Quantum λQ (red markers) and classical 2λL (solid lines). Superscript notation of the exponents denotes the initial polarization of the chosen coherent spin state. Top panel for \(|\Psi _0^{\mathrm{c}}\rangle\), the same state as (a), and bottom for \(|\Psi _0^y\rangle \equiv |( - N/2)_y\rangle \otimes |0\rangle\), here N = 104 particles. In both plots we observe \(\lambda _{\mathrm{Q}} \simeq 2\lambda _{\mathrm{L}}\). Error bars for λQ are a 95% confidence interval from an exponential fitted to the numerical data. Coupling g and detuning δ are same as Fig. 2. In (a) B/(2π) = 0.7 kHz (B/Bc = 0.2). Source data are provided as a Source Data file
