Nanopolaritons: Vacuum Rabi splitting with a single quantum dot in the center of a dimer nanoantenna
S. Savasta, R. Saja, A. Ridolfo, O. Di Stefano, P. Denti, F. Borghese
aa r X i v : . [ qu a n t - ph ] M a r Nanopolaritons: Vacuum Rabi splitting with a single quantumdot in the center of a dimer nanoantenna
S. Savasta, R. Saja, A. Ridolfo, O. Di Stefano, P. Denti, F. Borghese
Dipartimento di Fisica della Materia e Ingegneria Elettronica,Universit`a di Messina Salita Sperone 31, I-98166 Messina, Italy (Dated: October 1, 2018)
Abstract
The demonstration of enhanced spontaneous emission of nanoscaled optical emitters near metal-lic nanoparticles and the recent realization of a nanolaser based on surface plasmon amplificationby stimulated emission of radiation (spaser) encourage the search for strong coupling regime at thenanoscale. Here we propose the concept of nanopolaritons. We demonstrate with accurate scat-tering calculations that the strong coupling regime of a single quantum emitter (a semiconductorquantum dot) placed in the gap between two metallic nanoparticles can be achieved. The largestdimension of the investigated system is only 36 nm. Nanopolaritons will advance our fundamentalunderstanding of surface plasmon enhanced optical interactions and could be used as ultra-compactelements in quantum-information technology. /γ SP ∼ −
100 fs which can be very helpful for ultrafast signal processing but lowers theirperformances as effective resonators, so discouraging the search for strong coupling betweenlocalized SPs and quantum emitters at the nanoscale. An outstanding demonstration of thiscavity-like behavior of metallic nanoparticles is the recent realization of a nanolaser based onsurface plasmon amplification by stimulated emission of radiation (spaser) [17]. Despite the2arge losses exhibited by SPs, the Rabi splitting between extended SP waves and organicexcitons was demonstrated in Refs. [18–20]. In these cases the strongly coupled systeminvolves many quantum emitters and extends over many optical wavelengths. Vacuum Rabisplitting has also been observed in hybrid exciton-plasmonic crystals which consist of goldnanovoids (diameter d = 600 nm) covered with an organic film [21]. Recently the efficientcoupling between an individual optical emitter and propagating SPs confined to a conduct-ing nanowire has been studied both theoretically [22] and experimentally [23]. Although thissystem does not display vacuum Rabi splitting, light escaping from the emitter is efficientlytransferred to the nanowire. The potential of this system as an ultracompact single-photontransistor has been theoretically demonstrated [24].Here we investigate the requirements to be satisfied by SP-nanosystems coupled to one oremore quantum emitters in order to display the vacuum Rabi splitting. We present accuratecalculations demonstrating that a silver dimer nanoantenna coupled to a single quantum dotdisplays the Rabi doublet. We start considering a metal nanoparticle or nanostructure sur-rounded by the resonant medium that spatially overlaps with the SP eigenmode and whoseemission line at energy ¯ hω spectrally overlaps with the SP eigenmode. In the following wewill address two different kinds of active media: i) a dielectric matrix doped with N quan-tum emitters, e.g. dye molecules (Fig. 1a); ii) a single semiconductor quantum dot (Fig.2a). In a simplified picture of coupled harmonic oscillators [25], where the SP resonance isdescribed as a single mode with a frequency independent decay rate γ SP [26], the resonant( ω SP = ω ) interaction of the SP mode with one optically active electronic transition givesrise to modes with energyΩ ± = ω − i ( γ SP + γ ) / ± p g − ( γ SP − γ ) / , (1)where γ is the full width at half maximum (FWHM) of the quantum emitters emissionline and g is the active medium-SP coupling rate. It is proportional to the dipole momentassociated to the transition and to the SP mode density at the transition energy. If theSP-mode interacts with N quantum emitters, assuming that all of them experience thesame SP field intensity, and that the coupling for one emitter is g , the resulting totalcoupling increases according to g = √ N g . The vacuum Rabi splitting between the twoenergies Re(Ω + ) and Re(Ω − ) appears when g > ( γ SP − γ ) /
16. In addition, in order tosee an evident splitting or complete oscillations before decay in the time domain the further3ondition g > ( γ SP + γ ) / γ of theelectronic transitions in many cases of interest is much smaller than γ SP ≥
50 meV. Thecriterion for strong coupling can therefore be approximated by g > γ SP /
4. Despite thelinewidth of SP modes is very large, this criterion can be satisfied thanks to the substantialincrease in the coupling strength g due to the concentration of photons in nanosized volumesenabled by SPs. Silver nanoparticles perform much better than their gold counterparts sincethe imaginary part of the Ag dielectric function drops to much lower values. Moreover thecoating of such particles with a dielectric medium can be exploited to shift the dipole plasmonresonance at longer wavelengths, where the imaginary part of the dielectric function is lowerand where it is more likely to find resonant quantum emitters. At resonance ( ω SP = ω ),the interaction with a resonant field-mode increases the spontaneous emission rate of theemitter according to the relation [27]Γ = Γ + 4 g /γ SP , (2)where Γ is the spontaneous decay rate in the absence of the metallic nanoparticle. Thespontaneous emission rate Γ of a quantum emitter nearby a metallic nanoparticle can alsobe expressed as [14, 28] Γ = Γ ρ ( r , ω ) , (3)where ρ ( r , ω ) = D ( r , ω ) /D ( ω ) is the enhancement of the field mode density at theposition and at the transition energy of the emitter with respect to the free space value.From Eqs (2) and (3) one can easily evaluate g from the knowledge of Γ , γ SP and ρ : g = Γ γ SP ( ρ ( r , ω ) − / . (4)For example typical dye molecules have free-space radiative decay times τ SE ∼ = ¯ h/τ SE = 2 . µ eV. A silver nanosphere of radius a = 7 nm displays amode density of the order ρ ( r = 10 nm) ∼ R is the distance from the centerof the sphere. The FWHM of the Ag SP dipole resonance coated by a dielectric shell withrefractive index n ∼ . γ SP ∼
60 meV. From such data it results g ∼ . N ≥
50 molecules in order to achieve strong coupling. Thisanalysis also can show that strong coupling regime with a single quantum emitter could alsobe achieved, although more demanding. Semiconductor quantum dots can display radiative4
50 400 450 500 550 600 E x t i n c t i on c r o ss s e c t i on s
380 400 420 440 460 480 500 E x t i n c t i on c r o ss s e c t i on s Wavelength (nm) κ = Inputphotons activemoleculeScatteredphotonsSP waves (a) (b)(d)(c) (e) pea k w a v e l eng t h ( n m ) Peak cross section
X 10 - (nm )
350 400 450 500 550 600
Wavelength (nm) refractive index of the host shell
FIG. 1: Vacuum Rabi splititng with a silver core interacting with a dielectric shell doped withactive organic molecules. a Schematic of the strong interaction between surface plasmons and theactive organic molecules in the surrounding shell. The input light excites either the SP waves orthe organic molecules. In each case the excitation is coherently exchanged between them beforethe excitation is scattered out. b Calculated extinction cross sections as function of the wavelengthof the input field obtained for different extinction coefficients κ of the doped shell. A significantRabi splitting appears for κ ≥ × − . c Extinction cross sections spectra obtained for differentrefractive indexes n b of the dielectric host shell. A clear anti-crossing is observed owing to strongcoupling between the organic molecules and the localized surface plasmon mode. d Dependence ofthe two Rabi peak-wavelengths on the refractive index of the host shell n b . e Dependence of thetwo Rabi peak extinction cross sections on the refractive index of the host shell n b . decay rates of the order of τ = 400 ps. In this case a mode density of the order of ρ > g ∆ > γ SP γ /
4, where 0 < ∆ < γ ≪ γ SP .The above analysis provides an estimate of the requirements for achieving the vacuumRabi splitting at the nanoscale but suffers from a number of approximations and oversim-plifications: i) broad SP resonances cannot be fully modeled as ideal single mode resonanceswith constant damping; ii) the interaction with quantum emitters can switch on multipolarcontributions which alters the SP density of states; iii) The strong gradients displayed bythe localized SP fields as well as polarization effects prevents the possibility for the quantumemitters to experience the same field intensity. In the following we present detailed scat-tering calculations for two different systems: a) nanoparticles with a silver spherical corecovered by a dielectric shell doped with active organic molecules; b) a single quantum dotin between a pair of silver nanospheres. The optical properties of these coupled systems canbe exactly calculated through the formalism of the multipole expansion of the fields [30].This formalism based on generalizations of the Mie theory [31] is indeed able to take intoaccount all the multiple scattering processes that occur among the involved scatterers. Inparticular the optical properties of core/shell spheres are calculated using the extension ofthe Mie theory to radially non-homogeneous spheres by Wyatt [32]. We consider as incidentfield a monochromatic linearly polarized plane wave. The scattering cross-section and ab-sorption cross-sections are defined via Poyntings theorem [31]. The scattering cross-section σ scat is defined as the total integrated power contained in the scattered field normalized bythe irradiance of the incident field and the absorption cross-section σ abs is defined by thenet flux through a surface surrounding the scattering system normalized by the incidentfield irradiance, and is thus a measure of how much energy is absorbed by the system. Inthe following we calculate extinction cross sections σ ext = σ scat + σ abs as a function of theincident-field wavelength, being extinction spectroscopy a widely adopted technique for thecharacterization of nano- and micro-particles. We employ a silver core of radius r Ag = 7 nm,whose frequency-dependent dielectric permittivity is taken from Ref. [33], surrounded by adoped dielectric shell giving rise to a whole structure of radius r = 22 nm. The dielectricshell is described in its simplest way as a medium with a single resonance at the energy E
6y the following permittivity, ǫ = ǫ b + AE − E − iEγ (5)where ǫ b is the background dielectric constant mainly due to the host matrix medium, E isthe dispersionless exciton energy, γ is the total homogeneous plus inhomogeneous broaden-ing of the excitonic resonance, and the constant A is proportional to the oscillator strengthof the transition. It depends on the dipole moment of the single active molecule and onthe density of doping molecules. We use parameters corresponding to the organic moleculestetra-(2,6-t-butyl)phenol-porphyrin zinc (4TBPPZn) exploited for the first demonstration ofthe strong exciton-photon coupling in an organic semiconductor microcavity: E = 2 .
88 eVcorresponding to a wavelength λ ∼
430 nm and γ = 57 meV. The concentration used inRef[34] corresponds to A ∼ . × − (eV) , giving a peak extinction coefficient κ = 7 × − .Fig. 1b displays extinction cross section spectra as a function of the wavelength of the inputfield for different extinction coefficients κ displayed in the figure. Increasing the extinctioncoefficient (e.g by increasing the molecules density), the spectra evolve from a single peak atthe resonance wavelength of the SP-mode to a doublet characteristic of the strong couplingregime. We adopted a matrix with background dielectric constant ǫ b = n = 3 .
01 able tored-shift the SP dipole mode resonance of the silver nanoparticle at resonance with E . Thisshift also determines an useful reduction of the SP mode linewidth thanks to the reductionof κ Ag at larger wavelengths. In order to investigate the coupling between the SP modeand the active molecules, the energies of the two subsystems must to be tuned through eachother. For tuning through resonance, different methods could be employed. Here we simplystudy structures with different values of n b corresponding to dielectric matrices with differentrefractive indexes. Figure 1c shows spectra calculated with different background refractiveindexes ranging from 1.35 to 2.1. Over the entire range of refractive indexes the energiesof the two contributions to the spectrum are well separated and avoid crossing each other.This anticrossing behavior which is more evident in Fig. 1d is characteristic of true strongcoupling, the regime of reversible exchange of energy back and forth between the quantumemitters and the localized SP-mode that is, vacuum Rabi oscillations. It is interesting toobserve that at large detuning (e.g. for n b = 2 .
1) the extinction cross section of excitons at λ = 430 nm is smaller by more than one order of magnitude than that of the SP-mode. Thisis consequence of the very strong absorption and scattering cross sections of SP resonances7s compared to that of the shell, although doped with resonant molecules. Only at resonancethanks to the strong coupling the two peaks display the same intensity both correspondingto mixed SP-exciton modes as shown in Fig. 1(e). Calculations here presented focus on thescattering process, however we expect as well striking modifications (beyond perturbativenanoantennas currently investigated [15]) of the active molecules fluorescence induced bythe strong coupling effect. In particular we expect that the strong coupling will transfer tothe light emitted by the active molecules (or dots) the broad spectral and giant scatteringcross sections of metallic nanoparticles. In addition these systems can allow the realizationof the spaser in the strong coupling regime [35].For a number of quantum information tasks involving quantum operations on singlequbits the vacuum Rabi splitting with a single quantum emitter is highly desirable. Semi-conductor quantum dots, have dipole moments 50-100 times larger than those of atoms andmolecules, while at the same time behaving very close to ideal two-level quantum emitters[36]. Nevertheless our analysis based on Eq. (4) showed that placing a single quantum dota few nanometers close to a metallic nanoparticle is not sufficient to produce vacuum Rabisplitting. However it is well known that the electromagnetic field in the gap region of apair of strongly coupled nanoparticles can be drastically amplified, resulting in an extraor-dinary enhancement factor large enough for single-molecule detection by surface enhancedRaman scattering (SERS) [29, 37–39]. We exploit this so-called hot spot phenomenon inorder to demonstrate that the vacuum Rabi splitting with a single quantum emitter withina subwavelength nanosystem can be achieved. We employ a pair of silver spheres of radius r Ag = 7 nm separated by a gap d = 8 nm embedded in a dielectric medium with permettiv-ity ǫ r = 3. Individual nanoparticle plasmons hybridize to give two new splitted modes : abonding and an antibonding combination [40]. The net dipole moment of the antibondingconfiguration is zero, this mode is not easily excited by light (dark mode). In contrast, thebonding configuration corresponds to two dipole moments moving in phase. It is easily ex-cited by input light and produces an extraordinary enhancement of the field mode density inthe gap between the particles. We consider a spherical quantum dot with radius r rmQD = 2nm, whose lowest energy exciton is resonant with the dimer bonding mode. Because of itssymmetry, a spherical quantum dot has three bright excitons with optical dipoles parallel tothe three direction x, y, and z respectively. The resulting frequency dependent permettivityis described by Eq. (5) with ǫ b = 3 and A = µ ¯ h/ ( ǫ V ), where µ is the dipole moment, and8
00 420 440 460 480 500 µ /e = 0.7 nm µ /e = 0.5 nm µ /e = 0.3 nm µ /e = 0.1 nm E x t i n c t i on c r o ss s e c t i on Wavelength (nm)
440 460 480 E x t i n c t i on c r o ss s e c t i on Wavelength (nm) C
430 440 450 460 470 480420440460480 µ / e = 0.5 nm P ea k w a v e l eng t h ( n m ) Exciton wavelength (nm)
K E r A g r QD d e b (b)(a)(c) (d) FIG. 2: Vacuum Rabi splitting with a single quantum dot in the center of a dimer nanoantenna. a Sketch of the system and of the excitation. b Calculated extinction cross sections as functionof the wavelength of the input field obtained for different dipole moments of the quantum dot. c Extinction cross sections spectra obtained for different resonant energies E of the quantum dotexciton ( µ/e = 0 . d Dependence of the two Rabi-peaks(extinction cross sections) wavelengths on the exciton transition wavelength λ = h c/E ( µ/e = 0 . V is the dot volume. Fig. 2a display a sketch of the system and of the input field polarizedalong the trimer axis in order to provide the largest field enhancement at the dot position.Fig. 2b shows the extinction cross section spectra calculated for different dipole moments µ = er , being e the electron charge. For r = 0 . r = 0 . r = 0 . λ = h c/E ( µ/e = 0 . µ/e ∼ . [1] Vahala, K. J. Optical microcavities. Nature , 839846 (2003).
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