Energy transfer and coherence in coupled oscillators with delayed coupling: a classical picture of two-level systems

The Frimmer-Novotny model to simulate two-level systems by coupled oscillators is extended by incorporating a constant time delay in the coupling. The effects of the introduced delay on system dynamics and two-level modeling are then investigated and found substantial. Mathematically, introducing a delay converts the dynamical system from a finite one into an infinite-dimensional system. The resulted system of delay differential equations is solved using the Krylov method with Chebyshev interpolation and post-processing refinement. The calculations and analyses reveal the critical role that a delay can play. It has oscillatory effects as the main dynamical eigenmodes move around a circle with a radius proportional to the coupling strength and an angle linear with the delay. This alteration governs the energy transfer dynamics and coherence. Accordingly, both, the delay and the coupling strength dictate the stability of the system. The delay is the main related parameter as for certain intervals of it, the system remains stable regardless of the coupling. A significant effect occurs when one of the main modes crosses the imaginary axis, where it becomes pure imaginary and dampingless. Thus, the two states energies can live and be exchanged for an extremely long time. Furthermore, it is found that the delay alters both the splitting and the linewidth in a way further influencing the energy transfer and coherence. It is found also that the delay should not be large to have significant effect. For example, for an optical system with 500 nm wavelength, the critical delay can be in tens of attoseconds.