Quasiperiodicity in time evolution of the Bloch vector under the thermal Jaynes-Cummings model

We study a quasiperiodic structure in the time evolution of the Bloch vector, whose dynamics is governed by the thermal Jaynes-Cummings model (JCM). Putting the two-level atom into a certain pure state and the cavity field into a mixed state in thermal equilibrium at initial time, we let the whole system evolve according to the JCM Hamiltonian. During this time evolution, motion of the Bloch vector seems to be in disorder. Because of the thermal photon distribution, both a norm and a direction of the Bloch vector change hard at random. In this paper, taking a different viewpoint compared with ones that we have been used to, we investigate quasiperiodicity of the Bloch vector's trajectories. Introducing the concept of the quasiperiodic motion, we can explain the confused behaviour of the system as an intermediate state between periodic and chaotic motions. More specifically, we discuss the following two facts: (1) If we adjust the time interval $\Delta t$ properly, figures consisting of plotted dots at the constant time interval acquire scale invariance under replacement of $\Delta t$ by $s\Delta t$, where $s(>1)$ is an arbitrary real but not transcendental number. (2) We can compute values of the time variable $t$, which let $|S_{z}(t)|$ (the absolute value of the $z$-component of the Bloch vector) be very small, with the Diophantine approximation (a rational approximation of an irrational number).