Single-photon source over the terahertz regime

We present a proposal for a tunable source of single photons operating in the terahertz (THz) regime. This scheme transforms incident visible photons into quantum THz radiation by driving a single polar quantum emitter with an optical laser, with its permanent dipole enabling dressed THz transitions enhanced by the resonant coupling to a cavity. This mechanism offers optical tunability of properties such as the frequency of the emission or its quantum statistics (ranging from antibunching to entangled multi-photon states) by modifying the intensity and frequency of the drive. We show that the implementation of this proposal is feasible with state-of-the-art photonics technology.

Introduction-Terahertz (THz) radiation-lying at frequencies from 0.1 THz to 70 THz-has sparked a broad interest recently [1,2] due to its key relevance for addressing transition frequencies of vibrational and rotational levels in molecules [3], as well as single-particle and collective transitions in semiconductor materials [4].Such potential provides an avenue to harness light-matter interactions with relevant applications (primarily related to imaging and spectroscopy) in multiple areas, ranging from food sciences [5], medical diagnostics, and biology [6], to high-bandwidth communication [7] or security [8].
However, quantum THz technology is at a much more incipient stage than its visible, near-infrared or microwave counterparts [9,10].As already demonstrated in these spectral regimes, quantum light offers important technological advantages, such as metrological precision at the Heisenberg-limit [11], alternative quantum computing paradigms [12] or eavesdropping protection in remote communications [13].Through the development of THz quantum technology, these advances could be transferred and exploited in areas where THz radiation is of key relevance.This avenue would also mean an opportunity to reduce the experimental requirements inherent to current quantum optical implementations, since THz quantum platforms are expected to offer a compromise between the microwave regime, which demands cooling down to millikelvin temperatures and involve important scalability challenges, and the optical one, where materials are strongly absorptive and require nanometric-precision in fabrication.The common mechanism of deterministic single-photon emission enabled by optical dipole transitions in quantum emitters is drastically limited, if not absent, in the THz regime, because the electronic pure dephasing is orders of magnitude larger than the THz emission rate [14].There are, however, a few demonstrations of heralded quantum THz radiation sources based on spontaneous parametric down-conversion [15].
A promising route towards the emission of THz radiation is to exploit the dressing between electronic transitions and driving electric fields, i.e., the AC or dynam-ical Stark effect.This dressing splits the energy levels into doublets separated by the Rabi frequency Ω R [see Fig. 1(a)], which for certain values of the field intensity can lie in the THz regime.Crucially, in polar systems with broken inversion symmetry, radiative transitions among dressed states in the same Rabi doublet become dipole allowed and have been proposed as a possible channel of emission of THz radiation [16][17][18][19][20].However, to the best of our knowledge, only classical properties of the THz radiation generated-such as the emission spectrum-or semi-classical lasing limits have been considered in such systems.
In this work, we show the prospects of this mechanism with single polar emitters for the realization of quantum optics in the THz regime, demonstrating its ability for the transduction of classical visible light into THz radiation with diverse purely quantum properties, such as single photon emission, multi-photon emission and non-classical correlations between different frequencies of emission.We consider that the single polar emitter is dressed by an optical laser and that its resulting THz transitions -enabled among the two states of a Rabi doublet-couple to a THz nanophotonic cavity.The cavity provides a Purcell enhancement of the emission that is eventually radiated into free-space.This design exploits the tunability of the laser parameters and the THz nanocavity architecture to provide considerable brightness and a remarkable optical control of the quantum properties of the emission.
Model-We consider a single two-level system (TLS), consisting of a ground state |g⟩ and an excited state |e⟩, separated by the optical transition frequency ω 0 .The TLS is driven by a laser field E L of frequency ω L , which is also in the optical range.Furthermore, the TLS couples to a single cavity mode (annihilation operator â) with the THz frequency ω c and field E c (cf. Fig. 1(a) for a schematic representation).These features are described by the Hamiltonian (ℏ = 1): ∝ d ee describes the permanent dipole component, originating from asymmetries in the charge distribution of its ground state.For our purposes, the static dipole moment needs to be substantial, i.e., at least of the order of a few Debyes (D), as have been reported in systems such as colloidal quantum dots [21], excitonic systems [22], perovskites [23], simple polar molecules [24], macromolecules [25], non-polar molecules in matrices [26], color centers [27], 2D materials [28] and Rydberg atoms [29].
The coherent drive gives rise to two dressed eigenstates of the quantum emitter, split in energy by the Rabi frequency Ω R = √ ∆ 2 + Ω 2 , where ∆ = ω L − ω 0 and Ω = d ge • E L are the laser detuning and driving amplitude, respectively [19,30].These states are given by |+⟩ = s|e⟩ + c|g⟩ and |−⟩ = −c|e⟩ + s|g⟩, where we define s = sin θ, c = cos θ, with θ ≡ arctan(h) ∈ [0, π 4 ], and h is a dressing ratio defined as h ≡ Ω R −∆ Ω ∈ [0, 1] that identifies the limit of no dressing (h = 0) and the resonant limit of a fully-dressed emitter (h = 1).The σ±,z operators can be expressed straightforwardly in terms of the Pauli matrices of the dressed-state basis ζ±,z , i.e., σ± = cs ζz + s 2 ζ± − c 2 ζ∓ and σz = (s 2 − c 2 )ζ z − 2cs( ζ+ + ζ − ).By applying a rotating wave approximation to eliminate all terms oscillating at optical frequencies in Ĥ, and then moving to the dressed basis by writing the TLS operators in terms of ζ±,z , we obtain the following Hamiltonian [19] Here, χ = d ee •E c /2 is the coupling rate between the TLS and the THz cavity, which, importantly, depends on the permanent component of the dipole moment.This permanent dipole moment allows for cavity-emitter coupling terms of the form ∝ (â + â † )σ z in the original Hamiltonian, which, crucially, oscillate at THz frequencies, enabling resonant interactions between the THz cavity and the dressed emitter.Additionally, we take into account cavity photon loss with a rate κ and TLS excitation decay with the spontaneous emission rate in vacuum γ.The presence of counter-rotating terms in Eq. ( 1) requires a careful description of the interaction between the system and the bath to prevent unphysical processes such as the emission of photons at zero frequency.In particular, these terms induce a change in the time dependence of the field operator (â(t) ̸ = â(0)e −iωct ), which affects the typical secular approximation commonly made during the derivation of the master equation in the optical regime (similar to the situation found in the ultra-strong coupling regime [31,32]).As a result, the interaction between the cavity and the environment is described by the operator X+ = j,k>j ω kj /ω c ⟨j|(â + â † )|k⟩|j⟩⟨k| that encompasses all the positive-frequency transitions of (â + â † ) [33].Here, |k⟩ is the k-th eigenstate with energy ω k (sorted in ascending order) and ω kj = ω k − ω j .The scaling of X+ with ω kj is chosen to describe the coupling to an Ohmic bath [32].We also define [ X+ ] † = X− .The complete dynamics of the open quantum system is thus described by the Master equation [34] , where we have defined the Lindblad superoperator and where the usual decay term D(â) has been replaced by D( X+ ) [35].Similarly, the input-output relations are given by âout = âin + √ κ X+ [36], so that quantities such as the radiated photon flux will be given by κ⟨ X− X + ⟩.For the case of the emitter, the dressed operator for spontaneous emission remains identical to σ− .
In practice, we observe that the standard Lindblad description with D(â) gives qualitatively the same results as using D( X+ ), given that we are far from being in the strong-coupling limit (χ ≪ ω c ).On the other hand, the use of the proper input-output relations in terms of X± is crucial, since otherwise one would describe the unphysical emisson of photons with energies equal or close to zero.Even in cases in which these photons only make a minor contribution to the total photon flux emitted, they have a significant impact on the photon statistics, leading to important incorrect contributions to bunched photon statistics when Ω R < ω c .
To gain a better understanding of the dynamics, it is helpful to express the dissipative part in terms of the dressed qubit operators ζ±,z .After discarding offresonant terms based on the assumption that ω c ≫ γ, one obtains a combination of effective incoherent losses, pumping and dephasing, ρ , that all depend on the laser detuning, i.e., γ − = γs 4 , γ + = γc 4 , and γ z = γc 2 s 2 .It can be shown that this configuration drives the dressed-state population inversion (γ + > γ − ), if the laser is blue-detuned ∆ > 0 [19], which is the setting that we will choose for the rest of the paper.In order to achieve a high emission flux, a limit of interest is that of a saturated dressed emitter, reached when the pumping rate greatly exceeds its decay, γ + ≫ γ − .This situation takes place when the driving detuning is much larger than the Rabi doublet splitting (∆ ≫ Ω), corresponding to a small dressing ratio h ≪ 1.
Resonant mechanism of THz emission-By tuning the Rabi frequency Ω R in resonance with the cavity frequency, ω c , Jaynes-Cummings-like terms ∝ âζ + + â † ζ− in Eq. ( 1) become resonant and dominate the dynamics.In this regime, the system becomes efficient at absorbing optical radiation from the driving field and emitting THz photons, since the cavity couples to intra-doublet THz transitions and Purcell-enhances them.This regime of operation is sketched in Fig. 1(b) and demonstrated in Fig. 1(c), which shows the substantial increase in the cavity population when Ω R is tuned into this resonant regime.Around this point of operation, we can ignore off-resonant terms in Eq. (1) (provided ω c ≫ χ), and use the resulting effective Jaynes-Cummings Hamiltonian for the dressed states Ĥ Since under this approximation we have neglected counter-rotating terms, we can safely substitute X+ by â in the Lindblad term of the master equation and in the calculations of photon flux.This substitution enables us to obtain approximate analytical solutions, which provide valuable insights into the different emission regimes.
For this analytical calculation, we can assume that the cavity is nearly empty and treat it as a TLS (truncating the number of excitations at 1).Then, we obtain that the photon flux in the resonant condition (Ω R = ω c ) is given by: where we introduced the effective cooperativity C ≡ 16χ 2 κγ h 2 + h −2 −1 and an effective rate κ ≡ γ + + γ − + 4γ z +κ.The full expression as a function of ω c , which can be found in the Supp.Material, describes a Lorentzian centered around Ω R as shown in Fig. 1(c).
Notice that the introduced effective cooperativity C is closely connected to the standard expression of the cooperativity, C = 4χ 2 /κγ, but accounts for the effective coupling between the cavity and the dressed emitter, which depends on the detuning between emitter and drive via h so that C = 4C/(h 2 + h −2 ).In the strongly detuned case h ≪ 1, we have C ≈ 4Ch 2 .To provide an understanding of the relationship between these quantities, notice that a typical value of h for the parameters chosen in the text is h ≈ 0.2, meaning that C ≈ 0.16C.A natural limit to consider is when cavity losses represent the dominant decay channel, κ ≫ γ, which implies that κ ≈ κ.In that case, in the limit of small cooperativity, C ≪ 1, the photon flux acquires the simple form κ⟨â † â⟩ ≈ γ + C, meaning that the flux will increase as κ is decreased (so that C is increased).On the other hand, if the cooperativity is large C ≫ 1, we find that κ⟨â † â⟩ ≈ γ + .The flux reaches a maximum value when κ is decreased into the strong-coupling region κ = 4csχ, an exact value that we obtain by optimizing Eq. ( 2).This maximum flux is, again, simply given by κ⟨â † â⟩ ≈ γ + in the natural situation of γ ≪ χ and detuned driving, h ≪ 1.The fact that the maximum photon flux is given by γ + implies that the brightness of the THz source scales with the optical emission rate into free space.This relationship is noteworthy because the optical emission rate is significantly larger than its THz counterpart, since both scale with the emission frequency as ω 3 .A more detailed analytical study of the conditions of maximum flux, including a full expression valid for all regimes, are provided in the Supp.Material.If κ is further decreased to the point in which κ ̸ ≫ γ, the photon flux gets reduced below its maximum value of γ + since κ ̸ ≈ κ.We then conclude that the condition of operation that provides the highest possible photon flux of γ + is given by the conditions κ ≫ γ and C ≫ 1.
These analytical estimations are confirmed by exact, numerical results.Fig. 2(a) shows exact calculations of the output photon flux κ⟨ X− X+ ⟩ as a function of κ/γ and Ω R .On the other hand, Fig. 2(b) shows the flux at the resonance labeled I, Ω R = ω c , versus κ/γ .In both plots, Ω R is modified by fixing Ω and varying the detuning ∆.The orange line in Fig. 1(b) corresponds to the analytical formula in Eq. ( 2), confirming the validity of our analytical results.
Next, we consider the quantum statistics of the emission, measured through the zero-delay second-order correlation function g (2) We show numerical calculations of its steady-state value in Figs.2(c,d).Notably, we find that the resonance (I) coincides with a regime of strongly antibunched emission where g (2) (0) < 1, meaning that, in the regime in which the output flux is maximum, this platform operates as a single THz photon source.By truncating at 2 excitations, we can obtain the analytic expression g (2) (0) ≈ 2γ + /(γ + + 3κ), valid in the limit in which h ≪ 1 and C ≪ κ/γ (see Supp.Material for a general expression and further details).g (2) (0) is antibunched for κ > γ, but when κ is decreased beyond the strongcoupling regime, most of the antibunching will be lost [see kink in the curve in Fig. 2(e), after which g (2) (0) slowly trends towards 1].Note that in resonance (Ω R = ω c ), there is a small region of near-coherent states within the antibunched region, meaning that the antibunching can be made much stronger by setting the cavity slightly out of this resonance.This effect is more important the lower the κ, and more visible in the Ω-ramp in Fig. S1(b) in the Supp.Material.
Multi-photon resonances-Beyond the main resonant mechanism of THz photon emission at Ω R = ω c described so far, a sweep over the Rabi frequency as the one shown in Fig. 2(a,c,e) also unveils additional features in both the output flux and the emission statistics.In particular, small revivals in the output photon flux can be observed when the emitter frequency is exactly twice (II) or three times (III) the cavity frequency.These peaks are related to multi-photon processes enabled by the counter-rotating terms of the form ζ+ â † and ζz â † in Eq. ( 1), which we ignored in our analytical derivations presented above.Each peak corresponds to a n-th order process becoming resonant, as has been previously reported in other light-matter systems featuring interaction terms that do not conserve neither parity nor the total number of excitations [37][38][39].Indeed, at these points, the dynamics are governed by an effective n-th order Hamiltonian Ĥeff = λ n (â) n ζ+ + (â † ) n ζ− , where n = 2 or 3 for (II) and (III), respectively (further information with analytical expressions for λ n can be found in the Supp.Material).In the presence of dissipation, this gives rise to strongly correlated emission, which in our case corresponds to the simultaneous emission of multiple photons within a Rabi doublet, see Fig. 2(f).The activation of each of these resonances results in an extraordinary degree of optical tunability of the quantum statistics of the emission, as seen Fig. 2(e), where, by changing the Rabi frequency of the drive Ω R , g 2 (0) spans eight orders of magnitude from antibunching to superbunching.The tunability offered when the Rabi frequency Ω R is alternatively modified by optically tuning the laser power Ω instead of its detuning is very similar [cf.dashed orange line in Fig. 2(e)].However, the limits of c and s are inverted, which leads to bunching for low Ω and coherent states for large Ω.More results about the two tuning methods can be found in the Supp.Material.Overall, we find that modifying ∆ is a more versatile way to control the system, since the use of strong drivings to reach high values of Ω R can result in added pure dephasing (see Supp.Material).
Spectral Features-Beyond the demonstrated tunability of photon statistics, our proposal can also deliver broadband control over the emission frequency, oftentimes a limiting factor in sources of THz radiation.
To showcase this feature, we ramp Ω R and record the cavity emission spectrum S Γ (ω) = , where Γ is the bandwidth of the sensor.We focus on a particular case where κ = 0.158 THz, since that value exhibits both strong antibunching and a large output photon flux [see Figs.2(b) and (d)].The main frequency of emission is set by the dressed emitter and equal to Ω R .This feature can be clearly seen in Fig. 3(a), which shows S Γ (ω) as the Rabi frequency Ω R is varied.This indicates that the Jaynes-Cummings type of dynamics characteristic of the resonance (I) remains important even out of resonance.
A strong secondary signal in the spectrum is observed at the cavity frequency ω c , regardless of the value of Ω R .Finally, when Ω R > ω c , a third peak also emerges at a frequency ω 2 = Ω R − ω c , which is a signature of a two-photon processes in which the deexcitation of the dressed emitter within a Rabi doublet is accompanied by the emission of a photon at the cavity frequency ω c and a second photon of frequency ω 2 , matching the en- Γ (ω, ω) as a function of ω and ΩR for {χ, κ, γ, ωc}/2π = {0.05,0.158, 0.0005, 26} THz.We highlight some of the lines in the maps and denote the corresponding photon frequencies (black).(c) g (2) (0) (blue dashed), max g ergy conservation condition ω c + ω 2 = Ω R .This observation suggests non-trivial dynamics of emission of multimode correlated states, which should manifest as strong features the frequency-resolved second-order correlation function at zero delay, g Γ (ω 1 , ω 2 ) [40][41][42].To confirm this, we resort to the sensor method develop in Ref. [40] and compute this quantity through the correlations between two ancillary qubits, fixing the spectral resolution of these sensors equal to the cavity linewidth Γ = κ (see Supp.Material).We first compute the photon statistics for a given spectral frequency ω, i.e., g Γ (ω, ω), versus Ω R , as shown in Fig. 3(b).We observe that the main emission line ω = Ω R is strongly antibunched, as expected since emission at this frequency stems from firstorder processes originating from Jaynes-Cummings-like interaction terms.The other two lines that were clearly visible in the spectrum feature bunched statistics, evidencing their multi-photon character , and a new strongly bunched line at ω = Ω R /2, not visible in the spectrum, is also present.This line corresponds to two-photon pro-cesses in which both photons are emitted at the same frequency (instead of one of them being emitted at the cavity frequency).Since this process is not stimulated by the cavity, it is only visible in the statistics.These results suggest that frequency filtering can act as an extra control knob of the quantum statistics of the THz emission.Indeed, this is illustrated in Fig. 3(c), where we plot the minimum and maximum possible values of g Γ (ω, ω) for each Ω R , which ends up always being, respectively, lower or larger than the degree of coherence of unfiltered signal, g (2) (0).The large difference between these maximum and minimum values of g Γ (ω, ω) highlights the tunability offered by the method of frequency filtering in the THz regime.
Beyond the obvious potential of antibunched THz sources for quantum technologies, spectrally correlated emission like the type we are reporting also holds the potential of quantum applications exploiting nonclassical properties such as entanglement [43,44].To reveal potential non-classical correlations we inspect the cross-correlations between two different frequencies ω 1 and ω 2 .Correlations with non-classical character can be identified by the violation of the Cauchy-Schwarz inequality (CSI), reformulated as [45][46][47].Fig. 4 shows a typical map of R(ω 1 , ω 2 ) in frequencyfrequency space, where we chose a relatively large Rabi splitting Ω R /2π = 70 THz that allows for multiphoton processes to be observable.This map presents a plethora of features that evidences the richness and complexity of the different quantum processes of emission present in this THz source.Providing a complete catalogue of these features is outside of the scope of this text.However, we highlight that the dominant feature exhibiting a strong violation of the CSI is the anti-diagonal line described by the equation ω 1 + ω 2 = Ω R , corresponding to the joint emission of two photons by the deexcitation of the emitter within a Rabi doublet.For this line one would also find a violation of the Clauser-Horne-Shimony-Holt inequality [48] (result not shown).In summary, our results suggest that this source can emit entangled THz photon pairs via two-photon processes.Furthermore, we note that our observation of the two-photon resonant peak (II) in the output flux, corresponding to the case ω 1 = ω 2 = ω c , evidences that these processes can be Purcell-enhanced by a cavity in a mechanism of bundle emission [49].
Another relevant aspect is the ability to detect the single THz photons generated by our proposal.State-ofthe-art cooled THz detectors can achieve noise-equivalent power (NEP) of up to 10 −19 W Hz − 1 2 with responses below nanoseconds [58] that, together with the bandwidths here considered (∼ 0.158 THz), can provide a minimum detectable power close to P min = NEP × √ κ = 4 • 10 −14 W. Thus, even with a moderate output photon flux of 4 • 10 −4 THz (cf.Fig. 2(b)), and radiative decays κ rad of 50% of the total decay rate (κ = κ rad + κ abs , κ abs being the absorption rate in SiC), we can estimate an emitted power of P = κ rad ⟨ X− X+ ⟩ℏω c ≈ 6 • 10 −13 W. This estimation amounts to a signal-to-noise ratio of roughly ten, that together with future engineering of emitter interactions on nanostructures and further advances in material science, provide prospects for the creation of bright THz single-photon emitters.Furthermore, since we have shown that the brightness of our source is a function of the linewidth of the emitter, we expect that it could be further amplified via Purcell enhancement by adding a second cavity on resonance with the optical transition of the emitter.
Conclusions-We have shown that a single coherently driven emitter with a permanent dipole moment in a THz cavity can operate as a versatile source of quantum THz radiation, accessing a broad range of frequencies and photon statistics, and featuring a complex quantum correlations between different THz photons.The quantum sources that we propose call for exploring novel interfaces of optomechanical transductions of THz photons to optical ones [59][60][61], that in conjunction with optical singlephoton detectors, or via single electron transistors [62], can open the detection of nonclassical THz correlations necessary to harvest the field of THz quantum optics.Beyond the immediate applications of single THz sources for technologies such as imaging or quantum communications, our findings represent a step towards future quantum technologies in the THz, which may consist on more complex cavity setups [55] capable to enhance the multimode correlations that we identify here, and turn them into integrated bright sources of entangled light [30,49] and matter [63] at the THz.

Tunability via the laser amplitude
Here we provide further information and results on the implications of modifying the Rabi frequency Ω R by tuning the laser amplitude, rather than the laser frequency.Fig. S1 is the analogue of Fig. 2 in the main text, except that now ∆ instead of Ω is kept constant, thus showing an alternative way to tune the quantum statistics with the laser amplitude.The results are similar, the main differences being an overall lower flux and a lower value of g (2) (0).Notice that, in this situation, h is about three times larger here than in the main text.For instance, in the resonance (I), we obtain h ≈ 0.67, in contrast to the value h ≈ 0.2 corresponding to the results presented in the main text.

Validity of the Effective Hamiltonian
To check the validity of our assumption that at the specific points the (II) and (III) the Hamiltonian is indeed dominated by terms proportional to λ n [(â) n ζ+ + (â † ) n ζ− ] we compare the (∆, Ω)-dependence of the effective coupling strengths λ n obtained via perturbation theory with the n-th Glauber correlation function ⟨( X− ) n ( X+ ) n ⟩, which gives the probability of at least encountering n photons.We have and As can be seen in Fig. S2, there is a good agreement between the effective two-and three-photon transition rates and the correlation functions at second and third order, respectively, validating our interpretation of the results at (II) and (III).

Difference between the Standard and the Dressed Master Equation
Fig. S3 shows that the choice of the Master equation does not have any discernable impact on the results.Fig. S4 shows the major difference that the change in the input-output relation makes.Note how in the resonances (especially Ω R = ω c ) the deviation is negligible as long as κ is not too large .The largest discrepancy for g (2) (0) is found for Ω R < ω c .where ω TO /2π = 23.61THz and ω LO /2π = 28.91THz are the transverse and longitudinal optical phonon frequencies, Γ SiC /2π = 0.084 THz is the absorption damping, and ϵ ∞ = 7 is the static permittivity.These values are taken from the experimental fitting in [53], neglecting anisotropic effects in the SiC response.The spectral density in Fig. S5(a) presents a number of peaks, originating from the surface phonon polariton resonances sustained by the cavity.This is defined in terms of the electromagnetic Dyadic Green's function and the static dipole moment as J(ω) = ω 2 πϵ0ℏc 2 d ee Im{G(r, r, ω)}d ee .It can be shown [51], that in the quasi-static limit it can be expressed as a sum of Lorentzian terms of the form where χ n is the electromagnetic coupling strength for mode n, ω n its natural frequency, and Γ n its damping rate (including both radiative and absorption channels).
Here, we will only focus on the two lowest-frequency modes (n = 1, 2), which have strong dipolar and quadrupolar characters, respectively.The inset of Fig. S5(a) demonstrates that only these two modes contribute significantly to the THz emission from the cavity (weighted by the radiative contribution to the total spectral density [52]).The surface phononic resonances at higher-frequencies, and particularly the pseudomode at 28.25 THz, are dark, and remain effectively decoupled from the far-field of the cavity.Through a Lorentzian fitting of the numerical J(ω), we can extract the parameters for these two modes: Note that we have splitted the mode damping rate, κ n , into its radiative, κ rad n , and absorption components, and that the latter is given by the loss in the SiC permittivity, κ abs n = Γ SiC , i.e., κ n = κ rad n + Γ SiC .Fig. S5(b) and (c) show maps of the electric field amplitude parallel to the emitter orientation (dimer axis) for the n = 1 and n = 2 surface

Figure S5 .
FigureS5.Plot of the spectral density (a), J(ω)[51] at the center of the 50 nm gap between two 1 µm diameter SiC spheres.The emitter orientation is parallel to the dimer axis and we have taken |dee|=50 D for its static dipole moment (also in accordance with experiments[21]).J(ω) was obtained by means of full electrodynamic simulations using the Finite Element Solver of Maxwell's Equations implemented in Comsol Multiphysics.The inset of (a) shows the radiative spectral density for the cavity.Panels (b) and (c) show amplitude maps for the electric field component along the dimer axis and for the n = 1 and n = 2 surface phononic modes.Colors render the field amplitude in linear scale from black (minimum) to yellow (maximum).