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AIP conference proceedings

On the Purcell effect beyond the dipole approximation

Philip Trøst Kristensen, Jakob Egeberg Mortensen, Peter Lodahl, Søren Stobbe

Published Jan 1, 2012• 1 citation
Publisher

Abstract

We investigate spontaneous emission from excitons in quantum dots beyond the dipole approximation and show how the symmetry of the exciton wavefunction plays a crucial role. We show explicitly that for spherically symmetric excitons, the Purcell effect is independent of the exciton size and is governed by the local density of optical states at the center of the exciton only, which is identical to the result derived with the dipole approximation. This surprising result is a spontaneous emission counterpart to the shell theorem of classical mechanics and electrostatics and provides new insights to the physics of mesoscopic emitters as well as great simplifications in practical calculations.

DOI
10.1063/1.4750098

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We investigate spontaneous emission from excitons in quantum dots beyond the dipole approximation and show how the symmetry of the exciton wavefunction plays a crucial role. We show explicitly that for spherically symmetric excitons, the Purcell effect is independent of the exciton size and is governed by the local density of optical states at the center of the exciton only, which is identical to the result derived with the dipole approximation. This surprising result is a spontaneous emission counterpart to the shell theorem of classical mechanics and electrostatics and provides new insights to the physics of mesoscopic emitters as well as great simplifications in practical calculations.
We investigate spontaneous emission from excitons beyond the point source dipole approximation and show how the symmetry of the exciton wave function plays a crucial role. We find that for spherically symmetric wave functions, the Purcell effect is independent of the wave function size and therefore is given exactly by the dipole approximation theory. This surprising result is a spontaneous emission counterpart to the shell theorems of classical mechanics and electrostatics and provides insights into the physics of mesoscopic emitters as well as great simplifications in practical calculations.
We derive a rigorous theory of the interaction between photons and spatially extended excitons confined in quantum dots in inhomogeneous photonic materials. We show that beyond the dipole approximation, the radiative decay rate is proportional to a nonlocal interaction function, which describes the interaction between light and spatially extended excitons. In this regime, light and matter degrees of freedom cannot be separated and a complex interplay between the nanostructured optical environment and the exciton envelope function emerges. We illustrate this by specific examples and derive a series of important analytical relations, which are useful for applying the formalism to practical problems. In the dipole limit, the decay rate is proportional to the projected local density of optical states, and we obtain the strong and weak confinement regimes as special cases.
Abstract The spontaneous emission of a spherical semiconductor quantum dot is considered without using the dipole approximation. It is shown that the dipole-forbidden radiation of a shallow potential well with one electron level becomes more intense than the dipole-permitted radiation of a deeper potential well and that the potential deepening makes the dipole approximation closer. It is also shown that the cylindrically symmetric anisotropic radiation can be obtained from a spherical quantum dot if the direction of the quantization axis remains unchanged for the acts of photon emission.
Abstract Inhibition and enhancement of spontaneous emission rate, first proposed by Purcell, is discussed in terms of altered density-of-modes in various types of solid-state cavities and the Purcell factor is derived for two-level atomic system. The conditions for observing Purcell factor and experimental findings in micropillars embedded with Quantum Dots (QD)s are discussed. The strong interaction between photons and matter in the form of QD excitons, dispersion relations for ground and excited states separated by Rabi splitting are discussed. Experimental photoluminescence spectra showing anticrossing of exciton lines and cavity lines by temperature and other tuning methods are used to illustrate the strong light-matter interaction. The Jayness–Cummins model based on second quantization is presented to point out Rabi oscillation, Purcell effect, and other novel features. Microcavity structures for observing cavity quantum electrodynamics effects are mentioned. Microcavity lasers predicting thresholdless operation for unity spontaneous emission coupling factors are discussed.
In 1946, E. M. Purcell predicted that the radiative lifetime of an emitter is not an intrinsic property but can be modified by structuring the surrounding electromagnetic field [36]. By inserting a semiconductor quantum dot (QD) in an optical cavity, one can accelerate or inhibit its spontaneous emission. In the present article, we show that the QD spontaneous emission can be deterministically controlled to fabricate bright sources of quantum light. QDs in cavities: basics, motivation, first demonstrations Light-matter coupling We note f the ground state of the QD and e its excited state. For a cavity mode close to resonance with the QD optical transition, we consider only the states with 0 or 1 photon in the cavity mode. The states ∣ e, 0〉 and ∣ f, 1〉 are coupled through light–matter interaction, with a constant g , where h g = ∣〈 e, 0∣ Ed ∣ f, 1∣, with d the dipole of the optical transition e → f and E the electric field at the QD position. Each of the states ∣ e, 0 > and ∣ f, 1 > are also coupled to continua of states: continuum of the free-space optical mode, phonons of the semiconductor matrix, etc. [4]. Here, we consider only the coupling to the continuum of the free-space optical mode, related to the cavity losses, with a constant γ c . When g << γ c , the photon emitted by the recombination of an exciton efficiently escapes outside the cavity. The QD optical transition radiative recombination rate can be accelerated (Purcell effect) or inhibited.
Several methods have been developed to calculate the spontaneous emission rate for extended emitters in the vicinity of a nanophotonic structure.While taking the emitter's spatial extension, these methods have taken approximations regarding the full interaction, either by coupling to one single plasmonic mode or adding higher order rates individually thus neglecting interference terms.Here we apply a new framework [1] to extended emitters, by which, all these interactions can be considered, and we show both under, and over-estimation of the Purcell Factor when using a point-dipole.