Shell theorem for spontaneous emission

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.

Figure 1

A shell theorem for spontaneous emission. (a) An extended emitter in inhomogeneous surroundings (exemplified by a QD above a dielectric mirror) coherently excites and scatters light (indicated by dashed lines) throughout the finite volume of the emitter itself. This may lead to nontrivial interference effects which significantly complicates Purcell effect calculations. (b) We derive a surprising theorem stating that for spherically symmetric exciton wave functions ${\chi }_{⊚}\left(r\right)$, the Purcell effect is determined only by the photonic response in the center of the emitter. (c) This nontrivial result is valid also for shell-type emitters in which the exciton wave function vanishes in the center.

Figure 2

Purcell factors in a PC cavity. (a) Normalized mode profile $|\tilde{\mathbf{f}}{(x,y)|}^2$ of the PC cavity mode. (b) Purcell factor for $y$-polarized QDs as a function of QD position $x_0$ along the line $y_0=0$. For elliptically shaped QDs oriented along the $x$ axis (${\beta }_x=66$nm and ${\beta }_y={\beta }_x/10$, dashed red) or the $y$ axis (${\beta }_y=66\mathrm{nm}$ and ${\beta }_x={\beta }_y/10$, dashed-dotted blue) deviations from the DA (solid gray) are observable, but for circular QDs (${\beta }_{x/y}=22$ nm, circles; and ${\beta }_{x/y}=66$nm, squares) the agreement with the DA is perfect, in accordance with Eq. (5). (c) Purcell factor at $x_0=y_0=0$ for $y$-polarized QDs of different shapes (see inset) and sizes; ${\beta }_x+{\beta }_y=22$ nm (solid red) and ${\beta }_x+{\beta }_y=66$nm (dashed blue).

Figure 3

Purcell factor for $x$- or $y$-polarized QDs of different shapes in GaAs as a function of distance $z_0$ to a silver mirror, as illustrated in the inset. For oblate ellipsoidal QDs (${\beta }_z=5$ nm and ${\beta }_{x/y}=75$nm, solid red) as well as for prolate ellipsoidal QDs (${\beta }_{x/y}=5$ nm and ${\beta }_z=75$nm, dashed blue) deviations from the DA result (solid gray) are evident, but for spherical dots (${\beta }_{x/y/z}=50$nm, circles) the agreement with the DA is perfect.