Semiclassical description of regular-to-chaotic tunneling in cavities
Final Report Abstract
The design of optical cavities with directional lasing emission based on the knowledge of classical phase-space structures has lead to a large variety of experiments. The quality factors of long-lived lasing modes in such cavities is governed by dynamical tunneling by which low-loss modes can access phase-space regions with enhanced decay due to refractive escape. This effect is particularly enhanced in the presence of periodic ray-trajectories and their associated non-linear resonances in systems with mixed regular-chaotic ray dynamics via the mechanism of resonance-assisted tunneling. While an excellent theoretical description of dynamical tunneling in simple model systems was available before the start of this project, a detailed understanding of experimentally feasible systems, like optical cavities with a generic mixed phase space, was an open question. In particular, an intuitive picture of dynamical tunneling in optical microcavities and consequently of a ray-based description of lasing emission was missing. This project focused on advancing the theoretical understanding of dynamical tunneling and its resonance-assisted enhancement in optical microcavities. To this end we firstly developed a quantum mechanical description of resonance-assisted tunneling in simple model systems with a classical phase space in which regular and chaotic motion coexist. This allowed for going beyond the regime that was accessible by existing perturbative methods and built the foundation for an intuitive semiclassical description of resonance-assisted tunneling. Such a semiclassical picture in terms of complex paths connecting regions of regular and chaotic motion constitutes the first main result of this project. It provides an explanation of the strength of tunneling by a few easily accessible properties of the underlying classical dynamics and establishes an analytical approximation capturing the universal aspects of resonance-assisted tunneling. Based on this we extended the semiclassical description to optical microcavities focusing again on generic systems whose classical ray dynamics exhibits a mixed phase space. By mapping the classical ray description to a situation similar to the one treated in the first part of the project we obtained a semiclassical description for optical systems. The theory provides a relation of the quality factor of long-lived optical modes to complex classical ray trajectories connecting regular phase-space regions confined by total internal reflection with chaotic regions which exhibit refractive escape. The strength of this tunneling effect and hence the quality factors are explained using a few easily accessible properties of the underlying classical ray dynamics. Moreover, in order to benchmark semiclassical calculations we improved state of the art numerical methods for computing states of optical microcavities to unprecedented efficiency and accuracy. Besides being an important part of the project this opens the door to investigate various questions arising in optical microcavities in the short wave-length regime and connect to experimental results.
Publications
- Perturbation-free prediction of resonance-assisted tunneling in mixed regular-chaotic systems, Phys. Rev. E, 94, 062220 (2016)
N. Mertig, J. Kullig, C. Löbner, A. Bäcker, and R. Ketzmerick
(See online at https://doi.org/10.1103/physreve.94.062220) - Complex-path prediction of resonanceassisted tunneling in mixed systems, Phys. Rev. E, 95, 020202(R) (2017)
F. Fritzsch, A. Bäcker, R. Ketzmerick, and N. Mertig
(See online at https://doi.org/10.1103/physreve.95.020202) - Resonance-assisted tunneling in deformed optical microdisks with a mixed phase space, Phys. Rev. E, 100, 042219 (2019)
F. Fritzsch, R. Ketzmerick, and A. Bäcker
(See online at https://doi.org/10.1103/physreve.100.042219) - Resonance-assisted tunneling in four-dimensional normal-form Hamiltonians, Phys. Rev. E, 99, 042213 (2019)
M. Firmbach, F. Fritzsch, R. Ketzmerick, and A. Bäcker
(See online at https://doi.org/10.1103/physreve.99.042213)
