Introduction

Strongly confined electromagnetic fields at the nanoscale allow to enhance radiation-matter interaction in the optical or near-infrared domain to such a level that scattering of single radiation quanta may become important. On the one hand, this is a key target of modern nanophotonics research, e.g., for the realization of all-optical sensing, switching, and routing at the lowest possible control power, up to the level of a single energy quantum1. On the other hand, it constitutes a potential playground to explore fundamental quantum manybody physics related to strongly correlated states in open systems2. Within this context, single-photon nonlinear optics has been predicted in a variety of nanophotonic systems3,4,5,6,7,8,9,10, and demonstrated through the single-photon blockade effect, i.e., inhibition of the absorption of a single quantum of energy due to the presence of another one, both in atomic11 and solid-state12,13,14 cavity-quantum-electrodynamics (CQED). In these realizations, a single quantum emitter — an atom or a quantum dot — strongly interacts with a single confined mode of a suitably engineered electromagnetic resonator.

Among the different routes to the electromagnetic confinement, plasmonic systems allow for enhancement of radiation-matter interaction to scales well below the diffraction limit, hardly reachable with conventional optical means. In particular, surface plasmons, i.e., mixed radiation-matter excitations that arise at metal-dielectric interfaces due to the coupled oscillation of the electromagnetic field and the electron charge density15, are currently attracting considerable interest to develop quantum plasmonic devices16,17. One of the ultimate goals is to explore the use of surface plasmon excitations to enhance nonlinear interactions with single quantum emitters18,19,20. From the point of view of classical electromagnetism, optical bistability in metal nanoantennas involving a third-order nonlinear medium has been extensively analyzed21. On the other hand, in the limit of extreme localization of the plasmonic field, a truly quantum regime can be reached, characterized by the nonlinear interaction among single quanta of energy. Only a few pioneering works have proposed to reach such regime of single-plasmon blockade by using non-resonant material nonlinearities. In particular, early attempts of measuring a plasmonic analog of the Coulomb blockade effect have been reported in ref. 22 by detecting quantized steps in the low-power transmission through sub-wavelength metallic pinholes. More recently, theoretical proposals were focusing on confined surface plasmons in monolayer graphene nanocavities23,24, to realize a plasmon blockade effect in the mid-infrared range. As continuous-wave excitation is generally assumed, the actual observability of such an effect in a realistic experimental set-up may be strongly hindered by the extremely short plasmonic lifetimes, especially when turning to conventional noble metals at optical frequencies. In general, there has been little consideration on the possibility of overcoming these limitations with pulsed excitation techniques.

At variance from the conventional CQED systems, which typically rely on long lifetimes, thus limiting the maximum single-photon emission rates, here we propose a system where the intrinsically small material nonlinearity25 can be enhanced due to electromagnetic field confinement, so that a regime of quantum nonlinearity can be reached even without the use of long-lifetime quantum emitters. This potentially provides an extremely fast source of quantum states of radiation with order-of-magnitude improvements over full-dielectric systems, to be eventually employed in quantum information. Specifically, in the present work we analyze the quantized surface plasmonic excitations of a metal nanodimer, a nanostructure largely within reach of state-of-the-art technology in terms of size and gaps between its constitutive elements26,27,28,29. Localized surface plasmons are characterized by a finite linewidth due to both radiation losses and dissipation inside the metal. We show that the single-plasmon blockade can be reached quite straightforwardly under an external coherent excitation, e.g., an external laser input, by infiltrating the interstitial region of the dimer with a sufficiently nonlinear optical material (e.g., molecular dye solutions), as sketched in Fig. 1(a). Even if a similar system was proposed in ref. 22, the actual probing of plasmon blockade in terms of quantum correlation measurements, which we theoretically address in the present work, was not considered before.

Figure 1
figure 1

(a) Sketch of the system under consideration in this work. A nanodimer made up of two metallic nanoparticles (MNPs) is embedded in a nonlinear medium with a third-order susceptibility χ(3) and it is coherently pumped at frequency ωp from the exterior. The dimer sustains a localized surface plasmon with a decay rate γ0 = γrad + γnr, due to radiation losses and dissipation. (b) Scheme of the energy levels of the system, highlighting the single-plasmon-blockade mechanism.

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Although it is easy to compute the response of the metallic nanoparticle with different kinds of external excitations, such as plane waves or oscillating dipoles, establishing a quantum-mechanical theory of the plasmonic excitations is a far more complex task, due to the intrinsically lossy (radiative and non-radiative) character of surface plasmons20,24,30,31. In the following, we show that, by relying on the formalism of quasinormal modes, it is indeed possible to rigorously demonstrate that the system can be modeled with a quantum master equation involving a single bosonic operator, similarly to non-absorbing photonic systems. Within this theoretical picture, the physical basis of the effect can be grasped by looking at the first excitation levels of the system, which are schematically reported in Fig. 1(b). We suppose to coherently pump the nanodimer at a frequency ωp, in resonance with the bare surface plasmon energy ħω0. The simultaneous presence of two highly-confined plasmons in the spatial region occupied by the nonlinear material is associated with a large nonlinear interaction, U, which produces a shift of the double-excitation energy. The magnitude of U essentially depends on the degree of localization of the electromagnetic field, which can be quantified through the effective volume of the plasmonic mode, Veff. As we discuss in the following, although the formal expression for Veff is the same as for dielectric systems, it is essential to ensure the correct normalization of the plasmon eigenfield in agreement with the theory of quasinormal modes. Moreover, due to their intrinsically lossy character, plasmonic levels present a finite linewidth, γ0. When the effective volume is sufficiently small (for the system under consideration,  μm3), the magnitude of U could result in a shift of the two-plasmon excitation frequency (Δω = 2U/ħ, see the scheme in Fig. 1) that is larger than the natural linewidth of the mode, calculated to be around 100 meV for the typical nanodimers under consideration. As a result, the system cannot absorb a second plasmon but upon re-emission of the first one, thus effectively becoming a source of single plasmonic excitations.

Finally, we theoretically show that single-plasmon blockade could be experimentally measured, despite the fs-scale plasmon lifetime, by using a pulsed excitation source. Besides opening another route to the experimental study of quantum plasmonic effects, truly meant as the mutual interaction of single plasmon excitations at the nanoscale, these results could enable an ultra-high-rate source of single radiation quanta at visible wavelengths.

Results and Discussion

We consider localized surface plasmons of gold nanodimers in the context of the boundary element method (BEM) formalism developed in ref. 32, which can be applied to any system of locally homogeneous regions of space separated by abrupt interfaces. In Fig. 2(a) we plot the semiclassical decay rate of a dipole emitter with momentum p located at the center of the 10-nm gap between two gold nanodisks and directed along the inter-particle axis (see inset), showing a clear peak around ω = 1.7 eV due to coupling with a (longitudinal) surface plasmon of the nanoantenna. In this work, we use the experimental dielectric function of evaporated gold33 and we assume a background refractive index nB = 1.5, to account in an averaged way for the surrounding dielectrics. The rate computed with the MNPBEM toolbox (blue solid curve), a publicly available implementation of the BEM34, is compared with a completely independent calculation (dots) within the discrete dipole approximation (DDA)35,36, showing very good agreement. Both quantities are normalized to the dipole free-space decay rate Γfree = ω3nBp2/(3πε0ħc3).

Figure 2
figure 2

(a) Normalized decay rate of a dipolar emitter located at the center of the 10-nm interstitial region between two 60-nm-diameter, 15-nm-high gold nanodisks (see inset), vs the emitter energy. The calculation is performed with the BEM, DDA, and Fermi golden rule (FGR) applied to Eq. (4). (b) Cross section of the intensity of the QNM electric field, , along the dimer middle plane. The units for the QNM field are chosen in agreement with Eq. (4).

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It is recognized that the definition of localized plasmons in metal nanostructures, and, more generally, of the natural oscillation modes of leaky optical systems, can be made rigorous in the framework of quasi normal modes (QNMs)37,38,39, i.e., the solutions of a non-Hermitian differential equation with complex eigenfrequencies . We have computed QNMs with the following procedure. In the BEM formalism, the electric field in each region is decomposed into an incident and a scattered part, E(r, ω) = Einc(r, ω) + Esc(r, ω), the latter being identified with the field generated by a (fictitious) charge and current distribution on the enclosing surface. After discretizing the surface into a collection of N representative points, the charge and current distribution is calculated from the solution of a linear problem of the form Σx = a, where Σ(ω) is a N × N matrix and a is a vector constructed from the incident field. For brevity, we omit the full equations of the method and we refer to ref. 32 for the definition of the involved quantities (see Supplementary Information). For our purposes, it suffices to notice that the QNMs of the system are obtained by the condition , which corresponds to the nonlinear eigenvalue problem

for the generalized eigenvector . More importantly, the quasi normal field (QNF), , can be computed as the electric field generated by the surface charge and current distribution obtained from the eigenvector (see Supplementary Information). We have solved Eq. (1) working within the MNPBEM toolbox and using the two-sided Rayleigh functional iterative algorithm40. For instance, for the plasmon associated to the peak in Fig. 2(a), we obtain eV, and the QNF intensity plotted in Fig. 2(b).

At variance from the well-known relation , which holds for normal modes in non-dispersive media, QNMs satisfy a much more complex normalization condition37,38,39. Instead of explicitly calculating the normalization integral, we follow an implicit approach to normalize the field, similar to that proposed in ref. 41. Indicating with r0 the position of the dipole emitter at the center of the dimer gap, we normalize the QNM by imposing the following relation38,41 for :

(the factor 2ε0 is for dimensional convenience). We further improved the procedure by factoring out the singular term and imposing the relation only to the residues, with great advantages in terms of numerical accuracy and computational efficiency (see Supplementary Information).

The usual starting point for the quantization of plasmonic systems is the system-bath approach30, where the electric field is expanded on a continuum of frequency-dependent bosonic operators. Following ref. 30 and introducing the collective modes , Eq. (2) implies that for ω ≈ ω0 and γ0r/c ≪ 1 the field operator can be approximated as

with and the QNF introduced previously (see Supplementary Information). This model corresponds to a structured bosonic bath with a Lorentzian spectral density, whose central frequency and linewidth are directly related to the QNM eigenfrequency. According to a well-known theoretical result30,42, the dynamics of such a model is equivalent to that of a single bosonic mode, , which can thus be interpreted as the plasmon destruction operator, coupled to a flat reservoir with dissipation rate γ0. Thus, the electric field operator can be equivalently written

with the system dynamics being described by a density-matrix master equation in the Markov approximation:

This represents a commonly employed approach in the recent field of quantum plasmonics20,24,30,31.

To check its validity, in Fig. 2 we also show (red dashed curve) the emitter decay rate calculated in the weak coupling regime by applying the Fermi golden rule to Eq. (4), i.e., and normalized to Γfree(ω). The very good agreement with the semi-classical result near the resonance frequency confirms the validity of our model and justifies the single-mode approximation of Eq. (4).

We assume the plasmonic dimer to be embedded in a nonlinear medium, e.g., with a strong third-order nonlinear susceptibility χ(3), and we write the Hamiltonian in the form

where the first term accounts for the energy of the plasmon mode, including the contribution of metal dispersion (as implied by the QNM normalization condition). The second term is a four-operator product deriving from the third-order susceptibility of the nonlinear medium around the dimer and accounting for two-plasmon scattering. In a sense, this term is the analog of Hubbard interaction in electron systems. If we assume the χ(3) to be nonzero only in a region outside the nanodimers and dispersionless, we can use the same expression for the nonlinear interaction energy already introduced for dielectric systems8:

The master equation approach would also allow to include nonlinear loss terms, such as the nonlinear absorption introduced by an imaginary part of the χ(3) response, which we neglect due to the small pumping rates considered here (see, e.g. ref. 8). Moreover, we are considering a scalar nonlinear response (i.e., no tensorial coupling between the different components), since we are mainly interested in a proof-of-principle theoretical demonstration of the quantum nonlinear behavior in a standard experimental setup as a function of the possible values of the χ(3) elements. In this respect, we can simplify the nonlinear coupling energy to , where the effective volume of the confined plasmon is defined as

the integration being limited to the domain Ω represented by the bounded physical volume filled with the nonlinear medium. Veff essentially reflects the degree of localization of the QNF in region Ω.

As a final approximation, we are neglecting any nonlocal effects of the χ(3) response, which are known to arise for gap sizes certainly smaller than 1 nm43,44. We also stress that recent experimental investigations at the single or few molecules level within the sub-nm gap size between a metallic microsphere and a planar surface have been thoroughly explained by purely local theoretical modeling45.

By solving the eigenproblem (1) within the MNPBEM toolbox, we have characterized the lowest-frequency longitudinal plasmon modes of two kinds of nanodimers: nanodisk dimers and bowtie dimers made of two mirroring triangular nanoparticles, for various sizes and inter-particle distances. The results for the effective length and the linewidth ħγ0 are summarized in Fig. 3(a,b), respectively. The effective length depends mostly on the gap size, reflecting the plasmonic field enhancement in the interstitial region, whereas the linewidth is mainly affected by the particle size (due to radiative decay) and plasmon frequency (due to metal dissipation). Single plasmon blockade is based on a delicate interplay between Veff and γ0, thus both quantities must be taken into account when choosing the best geometry. It is already evident, however, that bowtie dimers are favored with respect to nanodisks. In any case, we notice that even the smallest gap size considered in our simulations, i.e. 5 nm, is much larger than what is currently achieved28,29. The integral in Eq. (8) has been performed in the region external to the metal and bounded by a box spanning the same height of the dimer along z and extending 10 nm beyond the particle extremities in the xy plane. This choice is compatible, for instance, with a film of nonlinear medium coated on top of a substrate. In general, we observed little variation of the values of the effective length when varying the bounding box, as long as the interstitial region between the metal nanostructures is entirely included in the integration volume. We also notice that the structures under consideration are well described within the regime of classical electromagnetic theory, and do not require to include quantum effects due to the microscopic nature of the metallic surfaces46,47.

Figure 3
figure 3

(a) Effective length, , and (b) decay rate, ħγ0, for the lowest-frequency longitudinal plasmon in dimers of different gold nanoparticles (see insets), versus the interstitial gap size. The curves refer to the case of disks with diameter of 40, 60, and 80 nm and bowtie dimers obtained from equilater triangles with edge of 40 and 60 nm (curvature radius = 5 nm). The height of all particles is 15 nm.

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Following our previous considerations, the key quantity for quantifying the occurrence of blockade effects in the system is the ratio between the double-excitation energy shift and the plasmon linewidth, or, equivalently, the ratio U/(ħγ0). In Fig. 4 we plot this ratio as a function of the nonlinear susceptibility of the infiltrated medium for a few cases of potential interest, selected from the results presented in Fig. 3. Three possible nanoplasmonic dimers are considered, either with disk or bowtie geometry. We expect that plasmon blockade starts to play a significant role in the quantum dynamics of the system when the energy shift becomes comparable with the linewidth, i.e., when . As it can be seen from Fig. 4, this condition corresponds to values of the nonlinear susceptibility .

Figure 4
figure 4

Ratio between the nonlinear coupling energy and the plasmon linewidth, U/(ħγ0), as a function of the third-order nonlinear susceptibility of the infiltrated nonlinear medium, for some structures of Fig. 3. Gap sizes are: 15 nm (D80), 10 nm (B60), and 5 nm (B40). The horizontal dotted line corresponds to the value .

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Such predictions are confirmed by detailed numerical solutions of the quantum master equation (5) based on the modeling above. In order to take into account the realistic implementation of a quantum nanoplasmonic device, we assume an experimental configuration allowing for a coherent driving of the localized plasmon, which can be described by the Hamiltonian

where F(t)/ħ represents the effective rate of plasmon excitation at the fixed external frequency ωp (e.g., imposed by a pumping laser). As a figure of merit for quantum nonlinear behavior, we focus on the degree of antibunching in the second-order plasmon correlations48, . In Fig. 5(a) we plot the normalized function at zero time delay (τ = t′ − t = 0), defined as , under continuous wave excitation (F/ħ = 0.01γ0) and as a function of the nonlinear susceptibility of the infiltrated nonlinear material, which is supposed to have the same refractive index nB = 1.5 of the background (see Supplementary Information). In all cases, the condition g(2)(0) < 0.5 can be considered as a threshold for single-plasmon blockade, in analogy to photon counting statistics in CQED experiments11,12,14. Ideally, when g(2)(0) → 0 the probability of detecting two photons at the output of the device is negligible soon after having detected one. This corresponds to the single-plasmon-blockade regime, which we have schematically shown in Fig. 1(b). We assume that the external driving field is in resonance with the bare plasmonic mode. Indeed, this corresponds to the condition of maximum plasmon antibunching8. In general, increasing the detuning between the pump and the plasmonic resonance is detrimental for the observation of the blockade effect, especially for the case of positive detuning (ωp > ω0), which could result in the direct excitation of the two-plasmon state blue-shifted by the nonlinear interaction.

Figure 5
figure 5

(a) Second-order correlation function at zero time delay as a function of the third-order nonlinear susceptibility magnitude, for the same structures as Fig. 4. A schematic representation of the proposed experiment is given in the inset. (b) Second-order correlation under pulsed excitation for the B60 structure and χ(3) = 10−15 m2/V2 (corresponding to the circled result in the previous panel). The pulse sequence is represented in the top of the figure.

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In agreement with the more qualitative analysis of Fig. 4, values of the order of χ(3) ~ 10−16 m2/V2 are predicted from the results of Fig. 5(a) to be sufficient to reach the single-plasmon blockade regime in the structure with the tightest electromagnetic field confinement, i.e. the bowtie structure with a 5-nm gap between the triangle tips. Such nonlinear values can be achieved, e.g., in glasses doped with metallic nanoparticles49,50. In such cases, care must be taken in the choice of metallic doping, since the plasmon excitations of the nanoparticles embedded in the glass matrix might interact with the target surface plasmon excitation of the nanodimer, thus making the whole analysis more complicated. In any case, stronger nonlinearities are necessary for structures with inter-particle gaps larger than 10 nm. On the other hand, such high values of the third-order nonlinearity might be within reach of available materials. For instance, values on the order of χ(3) ~ 10−15–10−13 m2/V2 have been measured in organic dye molecules in the visible range51,52. Therefore, the use of organic molecules appears a promising route for the experimental observation of plasmon blockade. In order to avoid complications due to strong interaction between surface plasmons and optical excitations of the nonlinear medium, as well as avoiding nonlocality of the nonlinear response, it is important to select a frequency range characterized by a significant third-order nonlinearity together with an essentially flat first-order response.

These results suggest that the single-plasmon blockade regime might be within reach. However, experimentally detecting this effect might still be challenging due to the extremely short plasmon lifetime. For a decay rate ħγ0 ~ 100 meV, one expects τp = 1/γ0 ~ 6.6 fs: this is beyond any possible resolution of single-counting detection. However, such an experiment could still be performed under pulsed excitation, where sufficiently short pulses would overcome the issue of short lifetime under cw excitation. In Fig. 5(b) we show that this is indeed possible: a train of gaussian pulses with 10 fs duration is sent on the structure B60 assuming χ(3) ~ 10−15 m2/V2 [see Fig. 5(a)]. The suppression of the zero-time delay peak in the unnormalized function G(2)(τ = t′ − t), calculated with the master equation after a two-time super-operator evolution5,9 (see Supplementary Information for details), is the signature of single-plasmon blockade in this system (in this simulation, t = 12 fs). In particular, this device demonstrates a single-plasmon source on demand, since each pulse triggers the re-emission of a single-plasmon from the system before a second one can be excited. Given the short plasmon lifetimes, an ultra-high emission rate can be achieved in principle (about 5 THz in this case), beyond state-of-art demonstrations based on plasmon enhancement of single emitters spontaneous emission rate53. As such, this system can be turned into an ultra-fast single-plasmon source conditioned to the availability on a seemingly fast laser source. For instance, ultra-high repetition rate femtosecond lasers have been shown up to 200 GHz range54. We notice that the usefulness of such an ultra-fast source could be currently hindered by the lack of sufficiently fast single-photon detectors, although a ps-time resolution for photon pair correlation measurements has been shown by use of a streak camera-based technique in the visible/near-infrared range55. On the other hand, such a fast system response might be employed, e.g., in engineering novel quantum devices, such as single-photon switches or ultra-fast single photon detectors, which we leave for further exploration.

A possible experimental configuration is schematically described in the inset of Fig. 5(a): a waveguide below the nanostructured dimer excites the localized plasmon mode, and scattered radiation is collected from a near-field tip (e.g., a scanning near-field optical microscope probe), where the output is sent to a Hanbury Brown-Twiss set-up for correlation measurements. Coincidences counts from the outgoing pulses would reveal the plasmon blockade regime, similarly to experiments performed with CQED systems12,14.

In summary, we have theoretically shown the potential interest of metallic nanodimers interacting with ordinary nonlinear media to achieve the quantum plasmonic regime, as defined by the true interaction between two plasmonic quanta confined in the same spatial region. The main contribution given in this work is twofold: on the one hand, we have provided an effective method to estimate the single-plasmonic confinement in the hot-spot of the dimer nanostructures. On the other hand, we have shown how the single-plasmon blockade can be evidenced by time-resolved second-order correlation measurements under pulsed excitation, thus overcoming the intrinsically short plasmon lifetimes, which we believe might stimulate further experimental research in quantum plasmonics, turning short lifetime into a resource.

Additional Information

How to cite this article: Alpeggiani, F. et al. Visible quantum plasmonics from metallic nanodimers. Sci. Rep. 6, 34772; doi: 10.1038/srep34772 (2016).