Non-Markovian Dynamics of Metamaterial Waveguides Coupled to Two-Level Systems

Introduction.—In recent years, metamaterials have attracted much attention. Because of their precise geometric structure and size, metamaterials have peculiar physical properties. It has been extensively studied in many directions [1–8]. Like hyperbolic metamaterial waveguide [9–12], or, the emergence of metamaterials with artificially engineered subwavelength composites offers a new perspective on lightmanipulation and exhibits intriguing optical phenomena [13–17] and so on. Metamaterials have the advantage of being man-made materials that can be engineered to produce strange properties. The NZI waveguide in metamaterials is a flexible design platform for dispersion engineering [18, 19] because of its unique and complex dispersion relation, which contains rich physics. The research of NZI waveguide has been very extensive, and its application also has broad prospects [20, 21]. The NZI waveguide can produce rich physical results because of its refractive index close to zero frequency . The study of metamaterial waveguide coupled with two-level system shows that it has a wide application in quantum device design [22, 23]. The coupling system between metamaterial waveguides and quantum radiators has been extensively studied under Markovian approximation [2, 4, 24]. However, the weak coupling condition of Markovian approximation is broken under the system due to the existence of singularities, which does not reflect the interesting and rich physics of the system due to the special dielectric constant. This has led to extensive interest in studying Non-Markovian dynamics, and there have been many excellent results in recent years, such as Non-Markovian dynamics in open systems[25–30], Non-Markovian processes in cavity quantum electrodynamics[31–37], and some experimental verifications[38, 39]. We go beyond Markovian approximation to study Non-Markovian dynamics of systems. This shows us the rich physics of the system, which is different from under the Markovian approximation or other simplified assumptions.

In this paper, we investigate the non-Markovian dynamics of a metamaterial waveguide coupled to a two-level system. We focus on the effects of the nonlocality and singularities in the dielectric constant of the waveguide on the dynamics of the system. We use a time-convolutionless master equation approach to obtain the reduced density matrix of the system, which we then use to calculate the dynamics of the system. Our results show that the nonlocality and singularities in the dielectric constant have a significant impact on the dynamics of the system. We find that the nonlocality leads to a slower decay of the coherence of the two-level system, while the singularities lead to a faster decay. Our study provides insights into the rich physics of metamaterial waveguides coupled to quantum systems and highlights the importance of considering non-Markovian dynamics in such systems.