ISSN 1063-7761, Journal of Experimental and Theoretical Physics, 2014, Vol. 119, No. 2, pp. 227–241. © Pleiades Publishing, Inc., 2014. Original Russian Text © I.E. Protsenko, A.V. Uskov, V.M. Rudoi, 2014, published in Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki, 2014, Vol. 146, No. 2, pp. 265–280. 227 1. INTRODUCTION The problem of spontaneous emission of several atoms has been solved in superradiance theory (SRT) [1–3]. The approach used in [1–3] is a generalization of the theory [4, 5] of spontaneous emission in the free space, in which the equations for the probability amplitudes of states of the atoms + field system are solved. The field is treated as a thermostat: a photon emitted by an atom never returns to it from an infi- nitely large number of photon states; the excitation of each of these states is a small perturbation. The first goal of this article is a generalization of the approach developed in [1–3] to the case when not another atom, like in [1–3], but a metal nanoparticle with frequen- cies of localized plasmon resonances (LPRs) of har- monic electron density oscillations close to the fre- quency of a radiative transition in the emitter is located near the atom or a quantum resonant emitter (molecule or quantum dot). Radiation of quantum emitters near metal nanoparticles attracts consider- able attention, in particular, in connection with sur- face enhanced Raman scattering (SERS), surface enhanced fluorescence (SEF), and other effects [6, 7]. Resonant radiation from emitters near metal nanopar- ticles has been studied by many researchers (see, e.g., [8, 9] and reviews [6, 10, 11]), but without using SRT approaches. The latter may provide additional infor- mation, e.g., describe the nonexponential decay of excited states of emitters and generation of plasmons, including in the case of detuning of the resonance fre- quency of the emitter from the LPR frequency. It will be shown below that the SRT-based approach is con- venient and can easily be generalized to the case of many particles; its results can be represented in the form of analytical formulas or the solutions to systems of linear differential equations that can easily be inte- grated numerically. A quantum emitter and a nanopar- ticle exchange a photon; in this case, quantum effects may be significant and can naturally be taken into account using the SRT approach. Another goal of this study is application of SRT to solve the specific problem of calculating the rate of the emitter transition from the excited to the ground state near an ellipsoidal metal nanoparticle, which has res- onant and nonresonant LPR modes with respect to the emitter. The difference between the rate of spontane- ous emission of the emitter near nanobodies, in a res- onator, etc., from the rate of its spontaneous emission in the free space is characterized by the Purcell factor [12]. A modification of the Purcell factor to the case of inexact resonance for spontaneous emission in a Fabry–Perot resonator was proposed in [5]. In our study, the SRT approach is generalized to calculate the Purcell factor of emitters near nanoparticles for detuning from resonance and in the presence of non- radiative losses. It will be shown that in cases when the classical approach [13] or the quantum-mechanical approach [14] to calculating the Purcell factor is appli- cable, the results obtained using these approaches and the SRT methods in our work coincide. At the generation threshold of conventional lasers, population inversion of the states of their active Relaxation of Excited States of an Emitter near a Metal Nanoparticle: An Analysis Based on Superradiance Theory I. E. Protsenko a, c , A. V. Uskov a, c , and V. M. Rudoi b, c a Lebedev Physical Institute, Russian Academy of Sciences, Moscow, 119991 Russia b Frumkin Institute of Physical Chemistry and Electrochemistry, Russian Academy of Sciences, Moscow, 119071 Russia c New Energy Technologies Ltd., Skolkovo, Moscow, 143025 Russia e-mail: protsenk@gmail.com Received December 31, 2013 Abstract—Methods of superradiance theory are employed for determining the relaxation rate of the excited state of a resonant emitter (atom, molecule, or quantum dot) near a metal nanoparticle under resonant exci- tation of plasmons in it, viz., modes of spatially uniform (dipole) harmonic oscillations of the electron den- sity. Detuning from resonance and nonradiative loss suppressing radiation from the emitter near the nanopar- ticle surface are taken into account. The results are used to estimate the threshold conditions for generating a plasmon (“dipole”) nanolaser. It is shown that the threshold conditions of induced (laser) generation of plasmons for the emitter at a distance of 5–40 nm from an ellipsoidal nanoparticle are satisfied for relatively low emitter pumping rates (on the order of the rate of spontaneous emission of the emitter into the free space). DOI: 10.1134/S1063776114080147 ATOMS, MOLECULES, OPTICS