1205.4308 (Hiroo Azuma et al.)

Hidden structures in time evolution of Bloch vector under thermal
Jaynes-Cummings model    [PDF]

Hiroo Azuma, Masashi Ban

We reveal hidden structures of 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, the Bloch vector seems to be in complete disorder and confusion. Because of the thermal photon distribution, both its norm and direction change hard at random, so that the Bloch vector shows a quasichaotic behaviour. However, if we take a different viewpoint compared with ones that we have been used to, we can find some novel structures in the Bloch vector's trajectories plotted at constant time intervals. In this paper, at first, we try to give an explanation of emergence of the quasichaotic behaviour by drawing an analogy between the dynamics of the Bloch vector and that of a compressible fluid. Next, we discuss the following two facts: (1) If we adjust the time interval $\Delta t$ properly, figures consisting of plotted dots 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).

View original: http://arxiv.org/abs/1205.4308