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Journal of the Optical Society of America B

Detection of “anomalies” inside microcavities through parametric fluorescence: a formalism based on modulated commutation relations and consequences on the concept of density of states

Serge Gauvin, Joseph Zyss, Cory Walker

Published Mar 28, 2019• 3 citations
Publisher

Abstract

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.

DOI
10.1364/josab.36.000c62

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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.
The interest toward strong electromagnetic confinement structures through the use of microcavities is of great interest, from both the fundamental understanding and the technological application sides. Here, we focus our attention to second order nonlinear processes inside such structures. From a basic point of view, electromagnetic confinement leads, physically, to the manipulation of vacuum field fluctuations and, formally, to an anomalous commutation relation [Ueda, M. and Imoto, N., "Anomalous commutation relation and modified spontanenous emission inside a microcavity," Phys. Rev. A, 50(1), 89-92 (1994)]. However, an attempt to probe the resulting anomaly with a beam splitter located inside a cavity is theoretically proved inadequate for this task. For this reason, parametric fluorescence (PF, parametric down conversion) and second harmonic generation (SHG) are considered as potential tools to probe vacuum field fluctuations in confinement structures. Indeed, it was recently showed that PF can be strongly intensified, by two orders of magnitude or more, when it occurs inside a high confinement (open) cavity. Therefore, PF could efficiently be used to probe the expected anomaly. Additionally, in consideration of the simplest possible experimental scheme to probe the anomaly and because it is generally admitted that SHG is the time reversal of degenerate parametric fluorescence, here our attention is also directed toward SHG. Formally, we show that the equations describing SHG and PF are not symmetrical regarding quantum noise. It turns out that, conversely to the case of PF, the intensification of vacuum field fluctuations tends to inhibit SHG. We interpret this result as due to the reduction of waves coherence induced by stronger quantum noise. In conclusion, albeit both processes could be used as probes to prove or disprove the realm of quantum anomalies, PF appears as a more convenient tool.
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 long been established that inside microcavities the resulting electromagnetic (optical) confinement might lead to strong enhancement (Purcell effect) or inhibition (Kleppner effect) of spontaneous emission. In both cases, the formulation is based on the concept of density of optical modes. However, and without resorting to this concept, there is an alternative point of view entirely based on the modulation of the vacuum field fluctuations. Consequently, a different interpretation of lasing, which does not involve the concept of density of modes, is conceivable.
Ultracompact nonlinear optical devices utilizing two-dimensional (2D) materials and nanostructures are emerging as important elements of photonic circuits. Integration of the nonlinear material into a subwavelength cavity or waveguide leads to a strong Purcell enhancement of the nonlinear processes and compensates for a small interaction volume. The generic feature of such devices which makes them especially challenging for analysis is strong dissipation of both the nonlinear polarization and highly confined modes of a subwavelength cavity. Here we solve a quantum-electrodynamic problem of the spontaneous and stimulated parametric down-conversion in a nonlinear quasi-2D waveguide or cavity. We develop a rigorous Heisenberg-Langevin approach which includes dissipation and fluctuations in the electron ensemble and in the electromagnetic field of a cavity on equal footing. Within a relatively simple model, we take into account the nonlinear coupling of the quantized cavity modes, their interaction with a dissipative reservoir and the outside world, amplification of thermal noise and zero-point fluctuations of the electromagnetic field, and other relevant effects. We derive closed-form analytic results for relevant quantities such as the spontaneous parametric signal power and the threshold for parametric instability. We find a strong reduction in the parametric instability threshold for 2D nonlinear materials in a subwavelength cavity and provide a comparison with conventional nonlinear photonic devices.