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arXiv:2410.06379

Effective models for quantum optics with multilayer open cavities

Astghik Saharyan, Juan-Rafael Álvarez, Axel Kuhn, S. Guérin

Published Oct 8, 2024• 0 citations
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Abstract

Effective models to describe the dynamics of an open cavity have been extensively discussed in the literature. In many of these models the cavity leakage to the outside is treated as a loss introduced phenomenologically. In contrast to these, we focus here on characterizing the outgoing photon using a novel approach where the outside is treated as part of the system. In such a global system, in order to separately characterize the photon inside and outside cavity, we demonstrate a first-principle derivation of a coherent cavity-reservoir coupling function for cavities with mirrors consisting of a stack of dielectric layers. In particular, we show that due to the effects induced by the multilayer nature of the cavity mirror, even in the standardly defined high-finesse cavity regime, the cavity-reservoir system description might differ from the one where the structure of the mirror is neglected. Based on this, we define a generalized cavity response function and a cavity-reservoir coupling function, which account for the longitudinal geometric structure of the cavity mirror. This allows us to define an effective reflectivity for the cavity with a multilayer mirror as if it was sitting in a well-defined effective mirror plane. We estimate the error of such a definition by considering cavities of different lengths and mirror structures. Finally, we apply this model to characterize the dynamics of a single photon produced in such a cavity and propagating outside.

DOI
10.48550/arxiv.2410.06379

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Effective models to describe the dynamics of an open cavity have been extensively discussed in the literature. In many of these models the cavity leakage to the outside is treated as a loss introduced phenomenologically. In contrast to these, we focus here on characterizing the outgoing photon using a novel approach where the outside is treated as part of the system. In such a global system, in order to separately characterize the photon inside and outside cavity, we demonstrate a first-principle derivation of a coherent cavity-reservoir coupling function for cavities with mirrors consisting of a stack of dielectric layers. In particular, we show that due to the effects induced by the multilayer nature of the cavity mirror, even in the standardly defined high-finesse cavity regime, the cavity-reservoir system description might differ from the one where the structure of the mirror is neglected. Based on this, we define a generalized cavity response function and a cavity-reservoir coupling function, which account for the longitudinal geometric structure of the cavity mirror. This allows us to define an effective reflectivity for the cavity with a multilayer mirror as if it was sitting in a well-defined effective mirror plane. We estimate the error of such a definition by considering cavities of different lengths and mirror structures. Finally, we apply this model to characterize the dynamics of a single photon produced in such a cavity and propagating outside.
Over the last decades, quantum optics has evolved from high-quality-factor cavities in the early experiments toward new cavity designs involving leaky modes. Despite very reliable models, in the concepts of cavity quantum electrodynamics, photon leakage is most of the time treated phenomenologically. Here, we take a different approach, and starting from first principles, we define an inside-outside representation which is derived from the original true-mode representation, in which one can determine the effective Hamiltonian and Poynting vector. Unlike the phenomenological model, they allow a full description of a leaking single photon produced in the cavity and propagating in free space. This is applied for a laser-driven atom-cavity system. In addition, we propose an atom-cavity nonresonant scheme for single-photon generation, and we rigorously analyze the outgoing single photon in time and frequency domains for different coupling regimes. Finally, we introduce a particular coupling regime ensuring adiabatic elimination for which the pulse shape of the outgoing single photon is tailored using a specifically designed driving field envelope.
We evaluate the exact dipole coupling strength between a single emitter and the radiation field within an optical cavity, taking into account the effects of multilayer dielectric mirrors. Our model allows one to freely vary the resonance frequency of the cavity, the frequency of light or atomic transition addressing it, and the design wavelength of the dielectric mirror. The coupling strength is derived for an open system with unbound frequency modes. For very short cavities, the effective length used to determine their mode volume and the lengths defining their resonances are different, and also found to diverge appreciably from their geometric length, with the radiation field being strongest within the dielectric mirror itself. Only for cavities much longer than their resonant wavelength does the mode volume asymptotically approach that normally assumed from their geometric length.
We present a rigorous first principles derivation of the heuristic approach behind most of cavity QED: the quantization of the field in a high-Q cavity as if the mirrors were perfect with losses modelled by a coupling with the outside. We derive the Hamiltonian describing this coupling, thereby allowing us to avoid the coarse-graining approximation implicit in the usual Langevin or master equation treatments of cavity damping. We obtain an explicit expression for the coupling constant that describes the non-Markovian nature of the reservoir in the regime beyond the validity of the coarse-graining approximation.
In this chapter, a one-dimensional optical cavity with output coupling is con-sidered. The optical cavity has transmission loss at one or both of the end surfaces. The classical, natural cavity mode is defined, and decaying or growing mode functions are derived using the cavity boundary conditions. A series of resonant modes appears. But these modes are not orthogonal to each other and are not suitable for quantum-mechanical analysis of the optical field inside or outside of the cavity. Hypothetical boundaries are added at infinity in order to obtain orthogonal wave mode functions that satisfy the cavity and infinity boundary conditions. These new mode functions are suitable for field quantization, where each mode is quantized separately and the electric field of an optical wave is made up of contributions from each mode. Some results of quantization are described in the next chapter. Chapter 3 deals with the usual quasimode model: a perfect cavity with distributed internal loss or with a fictitious loss reservoir.
Interfacing solid-state emitters with photonic structures is a key strategy for developing highly efficient photonic quantum technologies. Such structures are often organized into two distinct categories: nanocavities and waveguides. However, any realistic nanocavity structure simultaneously has characteristics of both a cavity and waveguide, which is particularly pronounced when the cavity is constructed using low-reflectivity mirrors in a waveguide structure with good transverse light confinement. In this regime, standard cavity quantum optics theory breaks down, as the waveguide character of the underlying dielectric is only weakly suppressed by the cavity mirrors. By consistently treating the photonic density of states of the structure, we provide a microscopic description of an emitter including the effects of phonon scattering over the full transition range from waveguide to cavity. This generalized theory lets us identify an optimal regime of operation for single-photon sources in optical nanostructures, where cavity and waveguide effects are concurrently exploited.