Fig. 9: Analysis of dissipative quantum chaos using the SSQT criterion detailed in ref. ⁴⁵ and generalized to Floquet states here.

a Theoretical indicator of chaos \(\langle \cos \theta \rangle\) introduced in ref. ⁸¹ computed on the full Floquet-Liouvillian spectrum (red line and circles) and on the eigenvalues selected by the SSQT criterion (green line and squares). While the spectral analysis on the full Liouvillian indicates the presence of chaos independently of the drive amplitude for the parameters considered in the plot, the SSQT criterion identifies the broadening of the Duffing peaks in Fig. 7a–c (gray rectangle) with a dissipative quantum chaotic phase for the Floquet steady state \({\hat{\rho }}_{{\rm{ss}}}^{F}\). When the number of selected eigenvalues is smaller than 100 a statistically significant analysis can not be carried out, and we set \(\langle \cos \theta \rangle =0\). b Purity \(\,{\text{Tr}}\,({[{\hat{\rho }}_{{\rm{ss}}}^{{\rm{F}}}]}^{2})\) of the Floquet steady state \({\hat{\rho }}_{{\rm{ss}}}^{{\rm{F}}}\). The onset of steady-state quantum chaos in (a) coincides with the drop of the purity of the steady state below 0.1. We use the parameters of Fig. 7a, the cutoff in the Hilbert space is fixed to 90, and \({c}_{\min }\) is selected as discussed in the text.