Visible quantum plasmonics from metallic nanodimers

We report theoretical evidence that bulk nonlinear materials weakly interacting with highly localized plasmonic modes in ultra-sub-wavelength metallic nanostructures can lead to nonlinear effects at the single plasmon level in the visible range. In particular, the two-plasmon interaction energy in such systems is numerically estimated to be comparable with the typical plasmon linewidths. Localized surface plasmons are thus predicted to exhibit a purely nonclassical behavior, which can be clearly identified by a sub-Poissonian second-order correlation in the signal scattered from the quantized plasmonic field under coherent electromagnetic excitation. We explicitly show that systems sensitive to single-plasmon scattering can be experimentally realized by combining electromagnetic confinement in the interstitial region of gold nanodimers with local infiltration or deposition of ordinary nonlinear materials. We also propose configurations that could allow to realistically detect such an effect with state-of-the-art technology, overcoming the limitations imposed by the short plasmonic lifetime.

Scientific RepoRts | 6:34772 | DOI: 10.1038/srep34772 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 nonlinearity 25 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 elements [26][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.
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 plasmons 20,24,30,31 . 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 ω p , in resonance with the bare surface plasmon energy ħω 0 . 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, V eff . As we discuss in the following, although the formal expression for V eff 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, γ 0 . When the effective volume is sufficiently small (for the system under consideration, , the magnitude of U could result in a shift of the two-plasmon excitation frequency (Δ ω = 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.  (3) and it is coherently pumped at frequency ω p from the exterior. The dimer sustains a localized surface plasmon with a decay rate γ 0 = γ rad + γ nr , due to radiation losses and dissipation. (b) Scheme of the energy levels of the system, highlighting the single-plasmon-blockade mechanism.

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 gold 33 and we assume a background refractive index n B = 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 BEM 34 , 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 Γ free = ω 3 n B p 2 /(3πε 0 ħc 3 ).
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 ω . 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) = E inc (r, ω) + E sc (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 Σ x = 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 Σ ω =  det ( ) 0, which corresponds to the nonlinear eigenvalue problem for the generalized eigenvector  x. More importantly, the quasi normal field (QNF), (r), 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 algorithm 40 . For instance, for the plasmon associated to the peak in Fig. 2(a), we obtain ω . − . i 1 68 0 07 eV, and the QNF intensity plotted in Fig. 2 At variance from the well-known relation ∫ ε | | = r r r d ( ) ( ) 1 2  , which holds for normal modes in non-dispersive media, QNMs satisfy a much more complex normalization condition [37][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 r 0 the position of the dipole emitter at the center of the dimer gap, we normalize the QNM by imposing the following relation 38,41 for ω ω → : sc 0 0   (the factor 2ε 0 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 approach 30 , where the electric field is expanded on a continuum of frequency-dependent bosonic operators. Following ref. 30 and introducing the collective modes ω p ( ), Eq. (2) implies that for ω ≈ ω 0 and γ 0 r/c ≪ 1 the field operator can be approximated as and r ( )  the QNF introduced previously (see Supplementary Information). This model corresponds to a structured bosonic bath with a Lorentzian spectral density, whose central frequency are directly related to the QNM eigenfrequency. According to a well-known theoretical result 30,42 , the dynamics of such a model is equivalent to that of a single bosonic mode, p, which can thus be interpreted as the plasmon destruction operator, coupled to a flat reservoir with dissipation rate γ 0 . Thus, the electric field operator can be equivalently written i t i t 0 0 0 0 with the system dynamics being described by a density-matrix master equation in the Markov approximation: 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 0 0  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 systems 8 : 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 eff 1 4 the integration being limited to the domain Ω represented by the bounded physical volume filled with the nonlinear medium. V eff 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 nm 43,44 . 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 modeling 45 .
Scientific RepoRts | 6:34772 | DOI: 10.1038/srep34772 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 V eff 1/3 and the linewidth ħγ 0 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 V eff and γ 0 , 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 achieved 28,29 . 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 surfaces 46,47 .
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/(ħγ 0 ). 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 γ > .  (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 ω p (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 correlations 48 , . In Fig. 5(a) we plot the normalized function at zero time delay (τ = t′ − t = 0), defined as 2 , under continuous wave excitation (F/ħ = 0.01γ 0 ) and as a function of the nonlinear susceptibility of the infiltrated nonlinear material, which is supposed to have the same refractive index n B = 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 experiments 11,12,14 . 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 antibunching 8 . 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 (ω p > ω 0 ), which could result in the direct excitation of the two-plasmon state blue-shifted by the nonlinear interaction. In agreement with the more qualitative analysis of Fig. 4, values of the order of χ (3) ~ 10 −16 m 2 /V 2 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 nanoparticles 49,50 . 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 m 2 /V 2 have been measured in organic dye molecules in the visible range 51,52 . 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 ħγ 0 ~ 100 meV, one expects τ p = 1/γ 0 ~ 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 m 2 /V 2 [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 evolution 5,9 (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 rate 53 . 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 range 54 . 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 range 55 . 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 systems 12,14 .
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.