The 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.
© 2016 Optical Society of America
Efficient and bright single-photon sources, which enable the generation of single photons with high repetition rates, are crucial components for quantum communication and computation systems [1,2]. The common approach to the realization of single-photon sources is to make use of spontaneous emission (SE) from two-level systems emitting one photon at a time—so-called quantum emitters (QEs) that can be selected from various atomic or molecular structures, such as dye molecules, quantum dots, and color centers in crystals.
The intrinsic radiative lifetime of a QE placed within an unconstrained dielectric is of the order of 10 ns in the visible or near-IR spectral range, which is certainly too long to ensure high repetition rates of single-photon emission. The SE rate can, however, be increased by placing a QE in a suitable photonic environment with an increased electromagnetic local density of states . Thus, for a non-absorbing cavity characterized by the quality factor and volume and containing a properly located and oriented QE and being in resonance with the QE radiative transition at the wavelength , the ratio between the modified and free-space SE rates, the Purcell factor , is given by 
It is clear from Eq. (1) that the SE rate can be enhanced by using an optical cavity having either a small volume or a high finesse or, preferably, both. In recent years, following intensive investigations (see a recent review ) two classes of nanostructures have emerged as the candidates for use in SE control and enhancement. The first class is all-dielectric micro-cavities [Fig. 1(a)], including those formed by photonic crystals, in which extremely high quality factors () can be achieved, while the volume remains relatively large, on the order of [5,6]. The second class includes “plasmonic nanocavities” incorporating metals [Fig. 1(b)], in which volumes are much smaller than , but the Q-factor is typically small () due to large loss in metals . Despite the large volume of work, it is still not clear which route (all-dielectric 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:2]. Using the above condition, one can evaluate the optimum quality factor ensuring the highest out-of-cavity SE rate and, consequently, the fundamental limit for the SE emission rate enhanced by a dielectric cavity (see Supplement 1):
The above condition was obtained by considering the SE modification as described with the Purcell factor [Eq. (1)], which is valid only in the weak-coupling approximation, when . It is instructive to show 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 can arrive at the following expression for the out-of-cavity emission rate (see Supplement 1):3), and
Considering the relations obtained, one realizes that the most important parameter determining the emission temporal behavior is the ratio [Fig. 1(c)]. Well-developed Rabi oscillations are observed for . In this case, the out-of-cavity emission rate oscillates accordingly, and the emission process stretches over long time periods. In the opposite limit, , there is a long non-oscillatory response, and it is found that the emission rate is reaching maximum values and the emission takes the shortest time, when (see Supplement 1)—roughly the critical coupling condition, with the optimum cavity quality factor given by3)], and we conclude that the time required for a photon to leave the optimum cavity is , and thus the maximum rate of photons is
Considering, for example, a QE with the lifetime of 10 ns, i.e., with , and the SE being centered at the wavelength of 1 μm, i.e., , one obtains from Eq. (6) that the optimum quality factor (for the diffraction-limited cavity) should be , which would ensure the maximum rate of single photons of [Eq. (7)]. Note that, for larger cavities, both of 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 . 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 ) by the electron collision frequency in metals, when adopting the Drude model for describing the metal dielectric function . One should also take into account the radiation channel of the LSP dissipation (characterized by the emission rate ). 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 . The SE rate of the QE–LSP system can therefore be written in the weak-coupling approximation as follows: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 .
The LSP emission rate can be estimated by considering the LSP being due to an electrical dipole resonance , with free electrons in metal oscillating (without dissipation) and generating the corresponding dipole moment (see Supplement 1). Introducing the effective nanostructure volume,
Relating the LSP mode volume associated with the Purcell factor [Eq. (8)] and the effective nanostructure volume , which for a spherical nanoparticle is simply equal to the particle volume, is a challenging issue that can hardly be dealt with in a simple and general way. Since these volumes are of the same order of magnitude for highly confined modes that we are interested in, we assume hereafter that . Combining Eqs. (8) and (10), we obtain a key result—the fundamental loss-determined limit for LSP-enhanced photon rates:
Considering the same QE as above, and a silver (gold)-based LSP nanostructure characterized by and , one obtains from Eq. (12) the maximum SE rate , setting thus the maximum rate of single photons at , which is two orders of magnitude larger than that obtained above for dielectric cavities. One can also use Eq. (10) to deduce that this SE rate requires the LSP volume corresponding to a -radius spherical nanoparticle. Recently, ultrafast () single-photon SE was demonstrated with quantum dots coupled to gap-plasmon-based nanocavities , and large SE enhancements in metal nanostructures (found using the antenna RLC-circuit approach) were suggested for improving the performance of light-emitting diodes . 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 [cf. 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 () is within reach for strongly confined QE–LSP configurations as indeed was very recently demonstrated .
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, e.g., in high dielectric contrast ridge, nanowire, and photonic crystal waveguides, the SE occurs mainly into the propagating waveguide modes with rate enhancement that can be described by the Purcell factor for waveguides :18]. Under the condition of diffraction-limited performance, one obtains the upper limit for the Purcell factor: 19]. The fundamental issue with this configuration is related to the fact that the slow-down effect is of a very narrow bandwidth, also causing a drastic increase in the propagation losses, so that even an optimistic estimate would be [19,20]. 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 .
We now turn our attention to the plasmonic waveguides supporting the propagation of SPP modes, laterally confined far beyond the diffraction limit [18,21]. 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 quantum 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 . Typically, one would first adiabatically taper out a very narrow lossy SPP waveguide to a relatively wider and lower loss SPP waveguide, as discussed in  in relation to nanofocusing and subsequently couple to a dielectric, e.g., Si-based, waveguide [Fig. 2(b)] as has already been successfully and efficiently realized in gap SPP (GSP) waveguides . 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:2(b)] with the SPP mode being characterized by the wavelength and the propagation length .
Let us consider a tapered configuration supporting GSP modes , 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 -groove, or indeed any trench waveguide, with the width being in this case an averaged trench width. Using the limiting case of very small gap width () for approximating the GSP wavelength  one can estimate the corresponding Purcell factor with the help of Eq. (13) as follows:16)] is similar to scaling found for metal nanowires , 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 ), one obtains and, consequently, . The latter values are significantly larger than the best estimate for photonic crystal waveguides, and the scaling indicates that even much larger values of the Purcell factor are within the reach. The presence of the exponential loss factor in the expression for the enhanced SE rate [Eq. (15)] emphasizes the importance of a proper choice of the narrow gap length . It seems reasonable to suggest that this length should be close to the mode wavelength, (this length is sufficient to transform the QE radiated field into the mode field ), so that the role of the loss factor can be neglected (for silver and gold, ), at least in the present estimations.
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 be advantageously exploited in many fundamental and applied disciplines, including quantum optics. The field of quantum plasmonics is still relatively new , and its case is yet to be presented and tried, given the inevitable dissipation found in any plasmonic configuration . In this Letter, we attempted to analyze the “pros” and “cons” 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 Supplement 1) that the optimized metal structure with today’s lossy metals offers SE rate enhancements that are just a few times below the theoretical maximum attainable in the hypothetical  limit of lossless plasmonic structures. It is our view that QE enhancement, where the rate rather than the 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.
European Research Council (ERC) (341054); Army Research Office (ARO) (W911NF-15-1-0629).
See Supplement 1 for supporting content.
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