Abstract
We present a formal treatment of the modification of spontaneous emission rate by a cavity (Purcell effect) in sub-wavelength semiconductor lasers. To explicitly express the assumptions upon which our formalism builds, we summarize the results of non-relativistic quantum electrodynamics (QED) and the emitter-field-reservoir model in the quantum theory of damping. Within this model, the emitter-field interaction is modified to the extent that the field mode is modified by its environment. We show that the Purcell factor expressions frequently encountered in the literature are recovered only in the hypothetical condition when the gain medium is replaced by a transparent medium. Further, we argue that to accurately evaluate the Purcell effect, both the passive cavity boundary and the collective effect of all emitters must be included as part of the mode environment.
Highlights
The fundamental system in cavity quantum electrodynamics is a two-level emitter interacting with the electromagnetic field in a cavity [1,2]
We present a formal treatment of the modification of spontaneous emission rate by a cavity (Purcell effect) in sub-wavelength semiconductor lasers
To explicitly express the assumptions upon which our formalism builds, we summarize the results of non-relativistic quantum electrodynamics (QED) and the emitter-field-reservoir model in the quantum theory of damping
Summary
The fundamental system in cavity quantum electrodynamics (cavity-QED) is a two-level emitter interacting with the electromagnetic field in a cavity [1,2]. In nano-scale lasers, enhanced emission together with a reduced number of cavity modes relative to large lasers can have significant effects, especially on sub-threshold behavior. We argue that the spontaneous emission probability of an electron-hole pair in a cavity is modified by the cavity itself, and, though indirectly, by the aggregate of other electron-hole pairs present This latter effect can be significant, yet is not readily included into existing models. We illustrate our result by evaluating the Purcell effect of a sub-wavelength semiconductor laser reported by Nezhad et al [8] For this example, we use an absorptive reservoir because only the cavity boundary is included in the environment. We discuss the importance of the aggregate emitter effect, and the difficulties it presents within the framework of the current model
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