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arXiv:1612.01231 · Journal of Physics B Atomic Molecular and Optical Physics

On correctness of the two-level model for description of active medium in quantum plasmonics

Eugeny D. Chubchev, E S Andrianov, A A Pukhov, A P Vinogradov, A A Lisyansky

Published Aug 2, 2017• 6 citations
arXivHTMLPDF

Abstract

Abstract In order to simplify the theoretical description of spasers, a gain medium is commonly represented by a two-level system (TLS). A realistic model, however, should have four levels. By using the Lindblad equations we develop a description of such a system and show that depending on ratios of the Rabi frequency and the rate of relaxation of the polarization, a four-level system (FLS) may be reduced to one of two effective TLSs that reproduce the key properties of a FLS.

DOI
10.1088/1361-6455/aa8375

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Abstract In order to simplify the theoretical description of spasers, a gain medium is commonly represented by a two-level system (TLS). A realistic model, however, should have four levels. By using the Lindblad equations we develop a description of such a system and show that depending on ratios of the Rabi frequency and the rate of relaxation of the polarization, a four-level system (FLS) may be reduced to one of two effective TLSs that reproduce the key properties of a FLS.
Optical-intensity-dependent gain and mixing susceptibilities of amplifying semiconductor materials have been calculated using the density-matrix approach. The model, which takes intraband relaxation and carrier-injection rate into account, allows for numerical computation of these susceptibilities, whatever the medium saturation is. As a simplification, an equivalent two-level system is derived which represents the gain and nearly degenerate four-wave mixing susceptibilities. Such an equivalence, which leads to the same field-induced polarization as the conventional rate equation model, also connects the two-level (or rate equation) parameters to the real characteristics of the semiconductor material.>
Many papers on semiconductor gain media have presented models for two-level homogeneously and inhomogeneously broadened media. In this paper, we show why and where these models become inadequate. The problems originate from the fact that the two-band semiconductor gain medium is an inhomogeneously broadened four-level medium with rapid cross relaxation due to carrier-carrier scattering, and all four levels have appreciable probability in gain media. The four states are given by |n e n h k s>, where n e (= 0 or 1) and n h (= 0 or 1) are the number of electrons or holes, respectively, for momentum k and spin s. The two-level system is a one-electron system with no states corresponding to the partially excited semiconductor states |0 e 1 h k> and |l e 0 h k>, which involve two electrons. The Pauli exclusion principle never gets a chance to enter the picture. In the absence of many-body effects, semiconductor gain can be fitted in an ad hoc fashion by a two-level model, but problems that involve the level probabilities individually, such as the laser photon statistics, elude a two-level description.
The intracavity absorption (or gain) coefficient of two-level atoms embedded inside a Fabry–Perot cavity has been demonstrated to be well represented by an equivalent two-level system. The parameters of this new two-level system are simply related to that of the medium and to the cavity characteristics. When associated with well-known results of fluorescence spectra and nearly degenerate four-wave mixing in two-level media, this equivalence provides a universal description of intracavity resonance fluorescence and relaxation oscillations in any nonlinear absorptive, amplifying, or dispersive Fabry–Perot cavities.
Abstract In this article, we theoretically investigate a spaser (surface plasmon amplification by stimulated emission of radiation) comprising a spherical silver nanoparticle surrounded by a four-level gain medium of quantum dots. The spaser system is pumped coherently and incoherently with the same excitation rate, and the characteristics of the resultant coherent localized surface plasmon (LSP) mode are compared for the two pumping scenarios. We provide a detailed analytical expression for the steady state and demonstrate that the incoherent pump is more suitable for the continuous spaser mode. The reason is better understood by studying the temporal evolution of the number of LSPs N n , where the LSP oscillation starts earlier for an incoherent drive and relaxes to a steady state with a large value of N n . At a large pump rate, the spaser curve shows saturation. In addition, we have found that the resonance peak of the spaser field is independent of coherent and incoherent pumping, whereas the peak amplitude of the field depends on the pump rate.
In Chapter 1, we introduced the standard rate equations (SRE) for a laser containing two-level atoms between which stimulated emission is possible. But real lasers exhibit much more complicated energy level schemes. Throughout this chapter, we consider several models in which the light–matter interaction is described by population equations for all the levels involved in the laser operation. Although they are mathematically more complicated, we shall investigate these equations in the same way as in Chapter 1, i.e. by formulating dimensionless equations and by analyzing the stability properties of the steady states. The basic ingredients of the SRE model used up to here, namely pumping and relaxation processes, play key roles in the efficiency of lasers. But already during the pioneering days of the laser, it appeared important to investigate three- and four-level models in order to obtain more reliable information on quantities such as the power conversion efficiency or the response time. The common extensions of the two-level SRE typically consider an open two-level system, or three- or four-level systems, depending on the nature of the active medium. Restriction to as few as four levels is again a crude simplification of the complex population dynamics occurring in most lasers. But it is rather surprising to see how good these simple kinetic models are. As we demonstrate by studying specific examples, the solution of the three- or four-level rate equations differs only slightly from the solution of the SRE.