Fundamental limitations in spontaneous emission rate of single-photon sources

Rate of single-photon generation by quantum emitters (QEs) can be enhanced by placing a QE inside a resonant structure. This structure can represent an all-dielectric micro-resonator or waveguide and thus be characterized by ultra-low loss and dimensions on the order of wavelength. Or it can be a metal nanostructure supporting localized or propagating surface plasmon-polariton modes that are of subwavelength dimensions, but suffer from strong absorption. In this work, we develop a physically transparent analytical model of single-photon emission in resonant structures and show unambiguously that, notwithstanding the inherently high loss, the external emission rate can be enhanced with plasmonic nanostructures by two orders of magnitude compared to all-dielectric structures. Our analysis provides guidelines for developments of new plasmonic configurations and materials to be exploited in quantum plasmonics.

 and free-space 0 1    SE rates, the Purcell factor F , is given by [4] 3 2 0 6 , 2 where n is the medium refractive index inside the cavity.
It is clear from Eq. (1) that the SE rate can be enhanced by using either an optical cavity having a small volume and high finesse or, preferably, both. In recent years, following intensive investigations (see a recent review [2]) two classes of nanostructures have emerged as the candidates for use in SE control and enhancement. The first class is all-dielectric microcavities [ Fig. 1(a)], including those formed by photonic crystals, in which extremely high quality factors ( 4 10 Q  ) can be achieved, while the volume remains relatively large, on the order of 3 ( 2 ) n  [5,6]. The second class includes "plasmonic nano-cavities" incorporating metals [ Fig. 1(b)], in which volumes are much smaller than 3 ( 2 ) n  [7], but the Q-factor is typically small ( 100 Q  ) due to large loss in metals [8]. Despite large volume of work, it is still not clear which route (alldielectric or plasmonic) can lead to the highest SE rate enhancement. Nor is it apparent whether fundamental limits of SE modification can be found in these configurations. The goal of this Letter is to provide the answers to these questions.
To do so we consider theoretically QE coupling to localized surface plasmons (LSPs) and dielectric micro-cavities from the viewpoint of assessing fundamental factors limiting the achievable SE rates in these configurations. We then compare the QE coupling to propagating surface plasmon-polariton (SPP) modes and to dielectric waveguide modes, arguing that the usage of plasmonic configurations is advantageous in both cases.
For dielectric, i.e., diffraction limited and lossless, cavities one can obtain [Eq. (1)] the following upper limit for the Purcell factor: The fundamental issue with this configuration is related to the fact that an increase in Q is strictly connected with the corresponding decrease in the cavity emission rate: cav Q    , where  is the cavity resonant frequency. This decrease will inevitably limit an increase in the SE rate out of the cavity, since it cannot exceed the cavity emission rate. Intuitively, the optimum coupling is achieved when the two rates are equal, S E cav   , i.e., each photon emitted into the cavity leaves it before the next one appears and no "bottleneck" is formed. This condition also happens to define the boundary when the QEcavity coupling enters the strong-coupling regime with energy oscillating coherently between the QE and the cavity in the process of Rabi oscillations [2].
Here, we introduced the frequency / 2     in order to facilitate example calculations. The above relation is practically the same as that derived within the weak-coupling approximation [Eq. (3)], and we conclude that the time required for a photon to leave the optimum cavity is 2 T   , and thus the maximum rate of photons is The above relation represents the fundamental diffraction-determined limit for the photon rate out of dielectric cavities. Considering, for example, a QE with the lifetime of 10 ns, i.e., with , and the SE being centered at the wavelength of 1 m, i.e., 14 1 3 10 s     , one obtains from Eq. (6) that the optimum quality factor (for the diffraction-limited cavity) should be ~ 5100, which would ensure the maximum rate of single photons of ~ 54 GHz [Eq. (7)]. Note that, for larger cavities, both these values should be proportionally modified: the cavity (optimum) quality factor should be larger resulting consequently in a lower (out-of-cavity) emission rate [see Eq. (3)]. From the viewpoint of Rabi oscillations, larger cavities imply weaker vacuum fields and thus smaller Rabi frequencies, which in turn require smaller optimum cavity emission rates and larger quality factors [see Eq. (6)]. At any rate, this level of cavity quality factors has already been realized and even exceeded bringing QE-cavity systems in the strong-coupling regime [5,6].
Let us now consider the QE coupling to a generic LSP sustained by a plasmonic nanostructure [7]. The fundamental issue with this configuration is related to the fact that the LSP quality factor is relatively low and principally limited (in the electrostatic approximation [8]) by the electron collision frequency 14 -1 10 s m  in metals, when adopting the Drude model for describing the metal dielectric function [9]. One should also take into account the radiation channel of the LSP dissipation (characterized by the emission rate rad  ). When a QE interacts efficiently with an LSP field, i.e., when the QE is sufficiently close to the corresponding plasmonic nanostructure, photons are emitted primarily via the LSP radiation [10]. The SE rate of the QE-LSP system can therefore be written in the weak-coupling approximation as follows: Here, the LSP volume LSP V should be understood as an effective volume occupied by the LSP field, whose calculation is, in general, a complicated issue due to energy dissipation [11], but whose value (for strongly confined modes) is typically of the same order of magnitude as the nanostructure volume itself. Also the Purcell factor should be used with care when considering plasmonic nanostructures [12]. The LSP emission rate can be estimated by the considering the LSP being due to an electrical dipole resonance [13], with free electrons in metal oscillating (without dissipation) and generating the corresponding dipole moment (see Supplementary Material). Introducing the effective nanostructure volume: we can link the dipole magnitude and the LSP mode energy and find the emission rate using the classical formula for the radiating dipole: single-photon SE was demonstrated with quantum dots coupled to gap-plasmon based nanocavities [16], and large SE enhancements in metal nanostructures (found using the antenna RLC-circuit approach) were suggested for improving the performance of light-emitting diodes [17]. It should further be noted that the difference in the limits obtained for these two classes would, for a given metal, increase for QEs with shorter lifetimes radiating at longer wavelengths, since Eqs. (7) and (12)]. Finally, the estimated SE rate is seen just at the limit of the weak-coupling approximation, indicating that the strong-coupling regime ( 0 SE    ) is within the reach for strongly confined QE-LSP configurations as indeed was very recently demonstrated [18]. Let us now turn our attention to the SE enhancement for QEs located in waveguides, starting with the dielectric case [ Fig.  2(a)]. If a waveguide mode is strongly confined as, e.g., in high dielectric contrast ridge, nanowire and photonic crystal waveguides, the SE occurs mainly into the propagating waveguide modes with the rate enhancement that can be described by the Purcell factor for waveguides [19]: where g n is the mode group index and wm A is the modes size, i.e., the mode cross sectional area whose definition is a somewhat complicated issue [19]. Under the condition of diffraction-limited performance one obtains the upper limit for the Purcell factor: Fig. 2. Schematic configurations of (a) dielectric waveguide with a QE inside, and (b) tapered gap SPP waveguide configuration providing a strong mode confinement at the place where a QE is located. Possible realizations of the corresponding twodimensional waveguides are illustrated to the right, depicting cross sections perpendicular to the propagation direction and the locations of the supported mode fields.
The only possibility to significantly increase the SE rate into diffraction-limited waveguide modes is therefore to make use of slow-down effects that can conveniently be realized with photonic crystal waveguides (ensuring also tight mode confinement) near the band edge [20]. The fundamental issue with this configuration is related to the fact that the slow-down effect is of a very narrow bandwidth causing also a drastic increase in the propagation losses, so that even an optimistic estimate would be 100 g n  [20,21]. Consequently, this implies that the Purcell factor is at best limited by 30 with the maximum rate of photons estimated (for the same QE) to be < 3 GHz. We now turn our attention to the plasmonic waveguides supporting propagation of surface plasmon-polariton (SPP) modes, laterally confined far beyond the diffraction limit [19,22]. The fundamental issue with this configuration is related to the fact that the propagation loss in SPP-based waveguides increases drastically for strongly confined modes. This problem, known since the very inception of research in plasmonics, can be mitigated by coupling a strongly confined SPP waveguide to a low-loss (dielectric) waveguide before the SPP energy is dissipated in the metal [23]. Typically, one would first adiabatically taper out a very narrow lossy SPP waveguide to a relatively wider and lower loss SPP waveguide as has been demonstrated in [25] an subsequently couple to a dielectric, e.g., Si-based, waveguide [ Fig. 2(b)] as has already been successfully and efficiently realized in gap SPP waveguides [24]. Taking into account the propagation loss incurred in the narrowest part of a plasmonic waveguide while neglecting the power loss elsewhere (i.e., to absorption and radiation out of the waveguide as well as during propagation in adiabatic tapers and coupling to lossless photonic waveguides), the SE rate can be written by modifying Eq. (13) as follows: where L is the length of the most narrow part of a plasmonic waveguide [ Fig. 2(b)] with the SPP mode being characterized by the wavelength SPP  and the propagation length SPP L .
Let us consider a tapered configuration supporting gap SPP (GSP) modes [25], leaving its coupling to a wider GSP waveguide [ Fig. 2(b)] and further to a dielectric waveguide out of analysis. Another similar configuration is a V-groove, or indeed any trench waveguide, with the width w being in this case an averaged trench width. Using the limiting case of very small gap width ( w   ) for approximating the GSP wavelength [22] one can estimate the corresponding Purcell factor with the help of Eq. (13) as follows: . 2 Here | m  is the real part of the metal permittivity,  16)] is similar to 3 1 R scaling found for metal nanowires [23], signifying the fact that very large Purcell factors can be achieved with plasmonic waveguides. Considering the same system parameters as in the above case of QE-LSP coupling and the GSP configuration with a challenging but reasonable gap of 4 nm (e.g., a 0.9-nm-wide gap was realized in the recent experiments [18] Plasmonics offers unique possibilities for the manipulation of light at the nanoscale resulting in extreme light concentration and giant local field enhancements, phenomena that can advantageously be exploited in many fundamental and applied disciplines, including quantum optics. The field of quantum plasmonics is still relatively new [7], and its case is yet to be presented and tried, given the inevitable dissipation found in any plasmonic configuration [13]. In this Letter, we attempted to analyze "pros" and "contras" for a particular problem in quantum optics, viz., the realization of efficient and bright single-photon sources that would enable the generation of single photons with high repetition rates. We have considered QE coupling to dielectric cavities (waveguides) and localized (propagating) SPPs assessing fundamental factors that limit the achievable SE rates in these configurations. It has been found that the latter allows one to obtain the SE rate larger by almost two orders of magnitude than the former one. It is worthwhile to note (see Supplementary Material) that the optimized metal structure with today's lossy metals offer SE rate enhancements that are just a few times below the theoretical maximum attainable in the hypothetical [14] limit of lossless plasmonic structures. It is our view that QE enhancement, where the rate rather than overall external efficiency (as in the case of LED) of emission is the ultimate measure of performance, is one of the few application niches where plasmonics can shine despite the inherent metal loss. We believe that the present analysis will also be of great help when looking for new plasmonic configurations and materials to be exploited in quantum plasmonics.

SUPPLEMENTARY
This document provides supplementary information to "Fundamental limitations in spontaneous emission rate of single-photon sources," Optica volume, first page (year). It details the derivations of the fundamental limits for the spontaneous emission rate out of dielectric cavities within both weak and strong coupling approximations as well as that of the emission rate for localized surface plasmons due to resonant electrical dipoles sustained by plasmonic nanostructures under the condition of of the nanostructure volume being sufficiently small. Furthermore, the hypothetical case of lossless plasmonic structures is analyzed from the viewpoint of SE rate enhancement limitations.
For a non-absorbing cavity, which is characterized by the quality factor Q and volume V and contains a properly located and oriented quantum emitter (QE), and assuming that the cavity is in resonance with the QE radiative transition at the wavelength  , the Purcell factor is given by [ Taking into account that 1   results immediately in Eq. (3) given in the main text.
The above condition was obtained by making use of the Purcell factor, i.e., within the weak-coupling approximation. We demonstrate below that a similar relation can be found using a more rigorous approach that considers vacuum Rabi oscillations. Introducing the cavity emission into coupled equations describing vacuum Rabi oscillations one obtains: