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Physical Review B

Shell theorem for spontaneous emission

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

Published Nov 20, 2013• 23 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.88.205308

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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 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 present a general, semimicroscopic, self-consistent treatment of spontaneous emission for a two-level atom in a dielectric microcavity with arbitrary shape. Both lossless and absorbing media are considered. Our approach is based on linear-response theory and the coupled-dipole method. We illustrate our method on spherical and cubic cavities. We discuss the Purcell effect, the frequency shift, the local-field effect, and the dipole-dipole coupling experienced by a source placed in a microcavity.
Spontaneous emission in dipole approximation is studied theoretically using both source-field theory and a Schrodinger picture approach. Using source-field theory we obtain formal equations for the Poynting vector and energy density without making the rotating wave approximation (RWA) and Weisskopf-Wigner approximation (WWA). The initial condition at t=0 is one in which the atom is in an excited state and the field in the vacuum state. The source-field expressions are evaluated within the the RWA and WWA and are found to satisfy Poynting's theorem. To explore the consequences of not making the RWA and WWA, the Poynting vector and energy density are calculated using perturbation theory. We use a Schrodinger picture approach and essentially reproduce and complement the results of Compagno, Passante, and Persico [J. Mod. Optics 37:8, 1377 (2007)] and those of Power and Thirunamachandran [Phys. Rev. A 45, 54 (1992)] obtained using a Heisenberg picture approach. The theory involves a sum over field mode frequencies and both finite cutoffs and convergence factors are used to carry out the sums. It is shown that the perturbation theory calculation leads to unphysical values for atomic state populations for all times when a sum over all field frequencies is taken, even if a convergence factor is used. It is also proved that the fields calculated using source-field theory always satisfy Poynting's theorem for ct not equal to R, where R is the distance from the atom.
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.
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.
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.