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Physical Review A

Anomalous commutation relation and modified spontaneous emission inside a microcavity

Masahito Ueda, Nobuyuki Imoto

Published Jul 1, 1994• 26 citations
Publisher

Abstract

Usual quantum-optical operator relations for a beam splitter are shown to lead to an anomalous commutation relation inside a microcavity. The physical origin of this anomaly is identified as self-interference of the mode whose coherence length is longer than the round-trip length of the cavity. Altered spontaneous emission of an excited atom is found to be a direct manifestation of this anomalous commutation relation. The anomalous Heisenberg uncertainty relations, which are derived from the commutation relation according to the Schwartz inequality, cannot be detected by probing the internal field with a beam splitter. The anomalous commutation relation, however, can be related to the change in the effecitve reflectivity of the beam splitter. The similarity and difference between an excited atom and a probe beam splitter are discussed.

DOI
10.1103/physreva.50.89

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Usual quantum-optical operator relations for a beam splitter are shown to lead to an anomalous commutation relation inside a microcavity. The physical origin of this anomaly is identified as self-interference of the mode whose coherence length is longer than the round-trip length of the cavity. Altered spontaneous emission of an excited atom is found to be a direct manifestation of this anomalous commutation relation. The anomalous Heisenberg uncertainty relations, which are derived from the commutation relation according to the Schwartz inequality, cannot be detected by probing the internal field with a beam splitter. The anomalous commutation relation, however, can be related to the change in the effecitve reflectivity of the beam splitter. The similarity and difference between an excited atom and a probe beam splitter are discussed.
It has been previously shown [Phys. Rev. A50, 89 (1994)PLRAAN1050-294710.1103/PhysRevA.50.89] that cavity-based electromagnetic confinement leads to an “anomalous” fields operators commutation relation that is undetectable by probing the cavity with a beam splitter. However, using this commutator in the case of parametric fluorescence (spontaneous parametric down conversion) when it occurs inside an open cavity implies a strong intensification of this process. This prediction can validate, or not, this commutation relation. The ab initio approach used is based entirely on vacuum field fluctuations and does not resort to the concept of density of states. Finally, through a generalization of creation and annihilation operators in the presence of noise, this approach raises fundamental questions about quantum modes. We expect this work to stimulate new theoretical developments and related experiments, which might lead to new applications in quantum nonlinear optics.
This chapter presents simple theoretical models which take into account the effect of dephasing of the vacuum fields on the spontaneous emission in a microcavity. This would be necessary to evaluate the limit of the capability of microcavity-based solid-state optical devices. The dephasing of the quantized fields, i.e., the vacuum fields, which is our main concern, results from the energy exchange between fields and medium polarizations interacting with phonon reservoirs, that is, from the dispersion. A symmetric structure of the system allows us to use the one-way time-evolution of field b, in order to obtain a motion equation of the field, which is in general derived from a round-trip time-evolution of the field. The alteration of the spontaneous emission in a microcavity can be viewed as a manifestation of the self-interference of the quantized fields, which are vacuum fields.
The transverse distributions of intensity inside and outside an optical cavity produced by spontaneous emission from atoms inside the cavity are reviewed. The mode properties inside a closed passive one-dimensional cavity are summarised and the spatial distributions of emitted light for a passive cavity with partially transmitting mirrors are discussed. A qualitative theory describes the effects of an amplifying medium inside the cavity, covering the regimes both below and above the lasing threshold. Expressions are derived for the angular spread of the output light in the presence of gain. A theory based on rigorous general relations between gain and noise provides more complete results for the spontaneous emission noise from an amplifying slab within an optical cavity.
In an active ``microcavity'' the condition of maximum ``enhancement'' of spontaneous emission corresponds to resonant coupling of atoms with a single field mode. We demonstrate that this exceptional condition results in the effective cancellation of spontaneous emission, in the onset of stimulated emission at exceedingly low excitation levels, and in an anomalously high stimulated-emission gain. In the context of phase-transition theory the active microcavity behaves as a statistical ensemble undergoing an order-disorder transition at an extremely high value of the critical temperature.