Fig. 4: Analysis of two-level dynamics at small N_B.

a The dressed ground level time crystal frequency in the simulation (thin magenta line) follows that extracted from the experiment (thick black line). The experimental line is obtained by tracing the maximum in the Fourier spectrum shown in Fig. 3 (oscillations are filtered out by the long time window). The ground state frequency initially corresponds to ω_B, and after the avoided crossing at t = 3.3 s it corresponds to ω_S. Simultaneously, the excited level frequency (simulation - thin cyan line, experiment - dotted thick dark red line) switches from ω_S to ω_B. Oscillations of the frequencies after the crossing arise owing to the population oscillations. b We can hence use the simulation to extract the undressed frequencies ω_B (thick red line) and ω_S (thin blue line), which cross at t = 3.3 s (dotted vertical line). The crossing rate d∣ω_S − ω_B∣/dt ≈ dω_B/dt is accelerated intrinsically by the feedback ω_B(N_B), as illustrated by the different slopes of the dash-dotted and dashed red lines. The Landau-Zener population transfer magnitude is determined by this speed-up. Frequencies are plotted in the rotating frame Δω = ω − ω₀. The corresponding populations are shown in the panels below: c The measured signal amplitude (ground level - thick black line, excited state - dotted thick dark red line) agrees with the simulated dressed populations (ground level - thin magenta line, excited state-thin cyan line). Both the total population from simulation and the measured signal are normalised to one at the crossing, and the filling factors applied for comparison in Fig. 3 (see Methods) are not used here. d The undressed populations N_B (thick red line) and N_S (thin blue line) are extracted from the simulation. The total population is normalised to one at the crossing. The black line shows the bulk population with the exponential decay compensated, \({N}_{{{{{{{{\rm{B}}}}}}}}}^{0}={N}_{{{{{{{{\rm{B}}}}}}}}}\exp (t/{\tau }_{{{{{{{{\rm{B}}}}}}}}})/{N}_{{{{{{{{\rm{B}}}}}}}}}(t=0)\). After the crossing, the compensated bulk population averages at \({N}_{{{{{{{{\rm{B}}}}}}}}}^{0}=0.61\) (horizontal dash line), corresponding to the fraction of population transferred to the excited state by the Landau-Zener mechanism.