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

Abstract

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 ∆t properly, figures consisting of plotted dots at the constant time interval acquire scale invariance under replacement of ∆t by s∆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).

Equivalence of a compressible inviscid flow and the Bloch vector under the thermal Jaynes–Cummings model

“…As explained in Ref. [17], for the quasiperiodic system, the trajectory of the motion never intersects itself during a finite time interval in the phase space because of incommensurate angular frequencies.…”

“…Here, we evaluate numerical errors caused by the above treatment [17]. For example, we can estimate the upper bound of numerical errors for L 1 (t) as…”

“…On the one hand, in Ref. [17], we consider quasiperiodicity to be the most important factor in understanding confusion of the Bloch vector's trajectories under the thermal JCM. On the other hand, in this paper, we try to find physical meanings of it by drawing analogy between the Bloch vector and a compressible fluid.…”

“…In Ref. [17], Azuma and Ban show that the time evolution of the Bloch vector under the thermal JCM has a quasiperiodic structure. Because of the quasiperiodicity, the dynamics of the Bloch vector has countably infinite incommensurate angular frequencies, ω n = √ n for n = 1, 2, 3, 5, ..., where the variable n comes from the number of photons of a cavity mode and the series n = 1, 2, 3, 5, ... lets {ω n } be incommensurate.…”

“…We introduce the compressible fluid for explaining it to be equivalent to the evolution of the Bloch vector in Eq. (17). As a result of this procedure, we obtain these quantities, as the density of the fluid, the pressure and the external force per unit mass.…”