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arXiv:quant-ph/0402029

Spontaneous emission of homogeneous broadening molecules on a micro-droplet's surface and local-field correction

Yun‐Jie Xia, Guang-Can Guo, P. T. Leung

Published Feb 4, 2004• 0 citations
arXivHTMLPDF

Abstract

We consider the spontaneous emission of a broadening molecule on the surface of a micro-sphere in this paper. The density of states for the micro-cavity is derived from quasi-normal models(QNM's) expansion of the correlation functions of electromagnetic fields. Through detailed analysis we show that only weak coupling between a broadening atom(molecule) and the electromagnetic fields exists in a dielectric sphere cavity whether the sphere is small or big. From these results we find the explicit expression of the spontaneous emission decay rate for a surfactant broadening molecule on the surface of a micro-droplet with radius $a$, in which only $1/a$ and $1/a^{2}$ components exhibit. Then we apply this expression to a real experiment and obtain a consistent result with the experiment. We also show that the real-cavity model of local field correction is accurate, and reveal that the local-field correction factor can be measured precisely and easily by fluorescence experiments of surfactant molecules. Moreover, the spontaneous decay of a surfactant molecular on droplet's surface is sensitive to the atomic broadening, so that the fluorescence experiment in a micro-sphere cavity can be used to estimate the radiative broadening.

DOI
10.48550/arxiv.quant-ph/0402029

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We consider the spontaneous emission of a broadening molecule on the surface of a micro-sphere in this paper. The density of states for the micro-cavity is derived from quasi-normal models(QNM's) expansion of the correlation functions of electromagnetic fields. Through detailed analysis we show that only weak coupling between a broadening atom(molecule) and the electromagnetic fields exists in a dielectric sphere cavity whether the sphere is small or big. From these results we find the explicit expression of the spontaneous emission decay rate for a surfactant broadening molecule on the surface of a micro-droplet with radius aaa, in which only 1/a1/a1/a and 1/a21/a^{2}1/a2 components exhibit. Then we apply this expression to a real experiment and obtain a consistent result with the experiment. We also show that the real-cavity model of local field correction is accurate, and reveal that the local-field correction factor can be measured precisely and easily by fluorescence experiments of surfactant molecules. Moreover, the spontaneous decay of a surfactant molecular on droplet's surface is sensitive to the atomic broadening, so that the fluorescence experiment in a micro-sphere cavity can be used to estimate the radiative broadening.
The spontaneous emission of a broadened molecule on the surface of a microsphere is studied. Through detailed analysis, we show that only weak coupling between a broadened atom (molecule) and the electromagnetic fields exists in a spherical dielectric cavity whether the sphere is small or large. From these results, we find an explicit expression for the spontaneous emission decay rate for a broadened surfactant molecule on the surface of a microdroplet with radius a , in which only 1/ a and 1/ a 2 components are exhibited. Then we apply this expression to a real experiment and obtain a consistent result with the experiment.
In this chapter, the authors develop a consistent theory of spontaneous emission in an optical micro cavity from a broadened atom. They derive the atomic broadening from first principles by assuming the presence of reservoirs coupled to the atoms. The simplest model of the atom is the so-called two-level atom, which has only two energy levels, the excited and the ground level. However, realistic atoms are surrounded by media that affect the atoms, giving them homogeneous or inhomogeneous broadening. The important parameter of coupling ratio of spontaneous emission to a desired quasi mode cannot be exactly treated by the present mathematical model of the cavity, although it is easily seen qualitatively that the ratio will decrease with increasing atomic broadening. The authors assume that each atom emits independently and ignored the possible interaction among atoms during spontaneous emission. Solving the master equation for the density matrix elements, the readers have derived formulas for spontaneous emission rate and emission spectrum.
The lifetime of an excited atom near an absorbing dielectric surface is calculated from an exact solution of a microscopic Hamiltonian model, which includes the effects of dispersion, local-field correction, and near-field Coulomb interaction. Results for the total decay rate are shown to be in excellent agreement with those based on classical electromagnetic theory and to yield the well-known result for the rate of nonradiative energy transfer in the limit of very small distance from the surface. \textcopyright{} 1996 The American Physical Society.
A significant limitation in the detection and spectroscopy of individual dye molecules in liquid room temperature samples is the loss of fluorescence signal through chromophore photobleaching. Although often considered a fundamental restriction on sensitivity, it has been shown that the probability per excitation cycle of photobleaching may be reduced by enhancing the spontaneous emission rate of the chromophore located inside a microdroplet, which acts as a optical cavity. Studies of glycerol microdroplets containing R6G dye molecules, and levitated in a three-dimensional electrodynamic trap, have demonstrated a significant enhancement of both the fluorescence decay rate [1] and the total fluorescence yield [2] of the chromophore. Modification of the emission rate is greatest for molecules located near the surface of small (< 10 μm) microdroplets, where the influence of the droplet cavity modes of the droplet is most significant. To exploit this surface sensitivity, surfactant forms of rhodamine dyes were recently studied [3]. Fluorescence emission of the dye molecules, constrained to the droplet surface and with fixed transition moments perpendicular to the surface normal, were shown to be significantly enhanced at small droplet sizes. Additionally, at larger droplet sizes corresponding to higher droplet Q’s, preferential emission into the cavity modes was observed, however the detection of fluorescence photons was delayed by the storage time of the cavity, which may be on the same order as the fluorescence lifetime. These measurements, performed on relatively concentrated dye solutions, suggest that microcavity effects may be exploited to greatly enhance single molecule detection sensitivity in glycerol microdroplets. Furthermore, the large cavity storage time of the high-Q droplets opens the possibility for re-excitation of the molecule following emission into the droplet resonance, and subsequent stimulated emission, that is, single-molecule lasing.
Starting from a quantum Maxwell equation for the vector potential operator, we study the spontaneous emission properties of a two-level atom interacting with the electromagnetic field in a lossless and inhomogeneous dielectric structure and express the spontaneous emission rate, the external quantum efficiency, and the spontaneous emission factor in terms of the classical Green function of the dielectric structure. Comparing these results with the corresponding classical results, we show that the above quantities can be calculated from the radiation field of a classical dipole. Based on this correspondence, a numerical algorithm is developed to calculate the modification of spontaneous emission in microcavities. The line shape of the emission spectrum of the light source and the decay of the cavity resonant modes are automatically taken into account in this algorithm. The approximate expressions for the spontaneous emission rate and the spontaneous emission factor are discussed.
Spontaneous decay of an excited atom near a dispersing and absorbing microsphere of given complex permittivity that satisfies the Kramers-Kronig relations is studied, with special emphasis on a Drude-Lorentz permittivity. Both the whispering gallery field resonances below the band gap (for a dielectric sphere) and the surface-guided field resonances inside the gap (for a dielectric or a metallic sphere) are considered. Since the decay rate mimics the spectral density of the sphere-assisted ground-state fluctuation of the radiation field, the strengths and widths of the field resonances essentially determine the feasible enhancement of spontaneous decay. In particular, strong enhancement can be observed for transition frequencies within the interval in which the surface-guided field resonances strongly overlap. When material absorption becomes significant, then the highly structured emission pattern that can be observed when radiative losses dominate, reduces to that of a strongly absorbing mirror. Accordingly, nonradiative decay becomes dominant. In particular, if the distance between the atom and the surface of the microsphere is small enough, the decay becomes purely nonradiative.