Quasiperiodic trajectories drawn by the Bloch vector of the thermal multiphoton Jaynes-Cummings model

We study the time evolution of the Bloch vector of the thermal multiphoton Jaynes-Cummings model (JCM). If the multiphoton JCM incorporates thermal fluctuations, its corresponding Bloch vector evolves unpredictably, traces a disordered trajectory, and exhibits quasiperiodicity. However, if we plot the trajectory as a discrete-time sequence with a constant time interval, it reveals unexpected regularities. First, we show that this plot is invariant under a scale transformation of a finite but non-zero time interval. Second, we numerically evaluate the times at which the absolute value of the z-component of the Bloch vector is nearly equal to zero. At those times, the density matrix of the two-level system approximates a classical ensemble of the ground and excited states. We demonstrate that some time values can be derived from the denominators of the fractions of certain approximations for irrational numbers. The reason underlying these findings is that the components of the Bloch vector for the thermal multiphoton JCM are described with a finite number of trigonometric functions whose dimensionless angular frequencies are irrational numbers in the low-temperature limit.