Abstract
The Purcell effect is defined as a modification of the spontaneous emission rate of a quantum emitter at the presence of a resonant cavity. However, a change of the emission rate of an emitter caused by an environment has a classical counterpart. Any small antenna tuned to a resonance can be described as an oscillator with radiative losses, and the effect of the environment on its radiation can be modeled and measured in terms of the antenna radiation resistance, similar to a quantum emitter. We exploit this analogue behavior to develop a general approach for calculating the Purcell factors of different systems and various frequency ranges including both electric and magnetic Purcell factors. Our approach is illustrated by a general equivalent scheme, and it allows resenting the Purcell factor through the continuous radiation of a small antenna at the presence of an electromagnetic environment.
Highlights
In the weak-coupling regime when χ γ, γdis, the hybridization of the quantum emitter and resonator eigenstates is weak
D1 2 /(12πε[0] c3) = ω03 d 2 /(3πε[0] c3)[20], q = ω/c is wavenumber in free space, Es(rd) is the scattering part of electric field evaluated at the quantum emitter position rd, and the quantum emitter has a dipole moment d1 oscillating at the frequency ω0
According to Eq (1), the magnitude of the Purcell factor does not depend on the magnitude of the transition dipole moment d, because the scattered field value is directly proportional to the dipole moment, the numerator and denominator are free of d
Summary
We notice that a quantum emitter and a nanoantenna (which is a classical resonant scatterer), both of these objects, in the absence of tunneling effects interact purely electromagnetically, and their coupling is governed by Maxwell’s equations Both of them can be described in terms of resonant RLC-circuits. We introduce an alternative equivalent circuit for radiating systems comprising an optical emitter and a nanoantenna This circuit model illustrates a simple algorithm for calculating the additional term R12 is incorporated into the expression of the input resistance Rin in presence of object “2”. The current that is induced in a short dipole antenna of effective length l reads as I = El/Z, where Z = Rrad + Rdis + jX is the total impedance of the particle, see Fig. 2(a). The Purcell factor in accordance with Eq (16) takes the form 1+
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