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Alexandria Engineering Journal

Directional spontaneous emission in a mono-mode waveguide with a circular cross section

S. Al‐Awfi

Published Apr 6, 2025• 2 citations
Publisher

Abstract

Exploring the spontaneous emission control of an excited two-level atom interacting with the vacuum within a circular waveguide surrounded by a perfect electric conductor is the main objective of this work. This type of emission is dependent on the intrinsic properties of the emitter, such as the emitter’s transition frequency and dipole moment direction, as well as the surrounding electromagnetic environment. Due to being statistical in nature, the fields do not have a well-defined phase, and thus the emission is considered incoherent or in an uncontrollable direction. We propose an approach to direct the emission from an atom embedded in a circular waveguide . To achieve this, multimode waveguides should be excluded because the total emission is simply presented in three orthogonal directions. In contrast, with mono-mode waveguides, the single photon transports in a single-mode waveguide coupled to a cavity embedded with an atom. Thus, the electric dipole moment vector is forced to take the direction of the only mode in which it is allowed to propagate. Besides giving rise to a single value for the emission, this also provides a way to control direction. Such waveguides have already been produced by recent advances in microfabrication technology.

DOI
10.1016/j.aej.2025.03.124

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arXiv:2109.11940•Sep 24, 2021•0 citations
Exploring the spontaneous emission control of an excited two-level atom interacting with the vacuum within a circular waveguide surrounded by a perfect electric conductor is the main objective of this work. This type of emission is dependent on the intrinsic properties of the emitter, such as the emitter’s transition frequency and dipole moment direction, as well as the surrounding electromagnetic environment. Due to being statistical in nature, the fields do not have a well-defined phase, and thus the emission is considered incoherent or in an uncontrollable direction. We propose an approach to direct the emission from an atom embedded in a circular waveguide . To achieve this, multimode waveguides should be excluded because the total emission is simply presented in three orthogonal directions. In contrast, with mono-mode waveguides, the single photon transports in a single-mode waveguide coupled to a cavity embedded with an atom. Thus, the electric dipole moment vector is forced to take the direction of the only mode in which it is allowed to propagate. Besides giving rise to a single value for the emission, this also provides a way to control direction. Such waveguides have already been produced by recent advances in microfabrication technology.
This thesis is concerned with the theory of spontaneous emission in the context of cavity QED and the guiding of atoms by laser light in confined geometries. We focus on three different geometrical shapes of waveguide, namely the parallel plate guide and two kinds of cylindrical waveguide: one with a rectangular cross section and another with a circular cross section. These waveguides are assumed to be characterised by two distinct properties. Firstly, all three types of guide have perfect conductor boundary conditions, which means that the guide walls are infinitely conducting and, consequently, lossless. Secondly, the guides are assumed to have subwavelength dimensions (less than the atom electric dipole transition wavelength #lambda#). The first consequence of this is that the spontaneous emission process is possible only via a few cavity modes. For each type of structure, we examine the confinement of electromagnetic fields within it and proceed to quantise these fields by following the standard procedure, incorporating the boundary conditions at the guide walls. This allows the position-dependent spontaneous emission rate to be evaluated for an electric dipole located at an arbitrary point within the guide. Asymptotic limits of the spontaneous emission rate are then derived which serve as useful checks of the calculations. For the rectangular guide, we are able to recover the results appropriate for the parallel-plate case when side a of the rectangular cross section becomes large, while side b is kept fixed. For the circular guide, we are able to recover the results appropriate for the free space case when the radius of the guide becomes large. The second part of the thesis deals with the theory of the motion of atoms in the spatially varying light fields inside the three different shapes of waveguide. In each structure we examine the forces that act on the atom and solve the equation of motion leading to the atom dynamics when an electric dipole within the guide is subject to an excited p-polarised cavity mode. The characteristics of the atomic motion in the guide in the presence of a cavity mode are then explored. We show that under these conditions the atom becomes subject to a transverse dipole potential and a dissipative force, both of which vary across the guide and are also functions of the atom velocity along the mode propagation direction. The dipole potential is responsible for the transverse trapping (and hence the channelling) of atoms at specific regions of the cross section, while the dissipative force controls the longitudinal motion of the channelled atoms. The conditions facilitating atom guiding are explored using typical parameters for sodium atoms in guides of subwavelength dimensions. The roles of the van der Waals and Casimir-Polder potentials on the atomic motion in waveguides is clarified theoretically in the context of the parallel-plate guide and we show that such potentials are effective only at relatively short distances from the guide walls. Finally, we show that atomic motion in a hollow circular guide is subject to a quantised torque arising from the orbital angular momentum effects associated with any excited waveguide mode of order l > 0. We find that an appropriately detuned atom immersed in such a mode may be trapped radially in a vibrational state and be made to exhibit novel rotational effects associated with the light torque, including a rotational frequency shift. (author)
Quantum mechanical treatment of light inside dielectric media is important to understand the behavior of an optical system. In this paper, a two-level atom embedded in a rectangular waveguide surrounded by a perfect electric conductor is considered. Spontaneous emission, propagation, and detection of a photon are described by the second quantization formalism. The quantized modes for light are divided into two types: photonic propagating modes and localized modes with exponential decay along the direction of waveguide. Though spontaneous emission depends on all possible modes including the localized modes, detection far from the source only depends on the propagating modes. This discrepancy of dynamical behaviors gives two different decay rates along space and time in the correlation function of the photon detection.
Spontaneous emission into micro- or nano-scale structures can be enhanced or inhibited by engineering the local density of optical states using closed quantum cavities. When electromagnetic fields are confined within small mode volumes of a high-Q symmetrical closed quantum cavity, the Purcell effect significantly accelerates the natural spontaneous emission. Due to the statistical nature of spontaneous emission, the emitted fields do not have a well-defined phase; consequently, the emission is incoherent and lacks directional control. The situation becomes more complicated in multi-mode cavities, as selecting a specific mode in a particular direction requires precise detuning between the emitter transition frequency and the cavity mode frequency to ensure strong coupling. In single-mode cavities, only the dominant mode can resonate, and only in a specific direction; thus, an appropriate transition frequency range can be easily determined. Such cavities allow for greater manipulation of the local electromagnetic density of states, facilitating effective control of the direction of spontaneous emission. This property is essential for creating efficient single-photon sources, photonic circuits, and on-chip quantum information.
In this work we study the spontaneous emission of a two-level atom placed inside a hollow cylindrical waveguide of infinite length and with perfectly electric conducting walls. The atom, whose electric dipole moment is assumed to be oriented along the symmetry axis of the waveguide, couples to a photonic band structure with a non-linear dispersion relation. We address two different regimes of coupling between the atom and the photonic band structure. In particular, we refer to the regime where the internal frequency of the a.tom is close to a band edge as the strong coupling regime and to the regime where the internal frequency is far from a band edge as the weak coupling regime. In the weak coupling regime, we find that the a.tom decays exponentially with a modified decayed rate due to the presence of the waveguide, and that the line-shape of the emitted photons within every band takes the form of a Lorentzian. In the strong coupling regime, we find a non-exponential decay of the atom, which is due to non-Markovian dynamics of the system. Moreover, the population of the atom presents oscillations in time and it converges to a non-zero positive value, indicating that the a.tom never fully decays. In this regime the spectrum of the emitted photons becomes more involved.
Unidirectional (chiral) emission of light from a circular dipole emitter into a waveguide is only possible at points of perfect circular polarization (C points), with elliptical polarizations yielding a lower directional contrast. However, there is no need to restrict engineered systems to circular dipoles, and with an appropriate choice of dipole unidirectional emission is possible for any elliptical polarization. Using elliptical dipoles, rather than circular, typically increases the size of the area suitable for chiral interactions (in an exemplary mode by a factor ∼30), while simultaneously increasing coupling efficiencies. We propose illustrative schemes to engineer the necessary elliptical transitions in both atomic systems and quantum dots.
Unidirectional (chiral) emission of light from a circular dipole emitter into a waveguide is only possible at points of perfect circular polarisation (C points), with elliptical polarisations yielding a lower directional contrast. However, there is no need to restrict engineered systems to circular dipoles and with an appropriate choice of dipole unidirectional emission is possible for any elliptical polarization. Using elliptical dipoles, rather than circular, typically increases the size of the area suitable for chiral interactions (in an exemplary mode by a factor ∼30\sim 30∼30), while simultaneously increasing coupling efficiencies. We propose illustrative schemes to engineer the necessary elliptical transitions in both atomic systems and quantum dots.