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January 06, 2012

C 2012 American Chemical Society

Plasmon Blockade in NanostructuredGrapheneAlejandro Manjavacas,†,* Peter Nordlander,‡ and F. Javier Garcıa de Abajo†,§,*

†IQFR-CSIC, Serrano 119, 28006 Madrid, Spain and ‡Department of Physics and Astronomy, M.S. 61, Rice University, Houston, Texas 77005-1892, United States.§Present address: Currently on sabbatical at Optoelectronics Research Centre, University of Southampton, Southampton SO17 1BJ, U.K.

Controlling lightmatter interactionat the nanoscale is one of the para-mount goals of nanophotonics,13

which has stimulated a large deal of work inrecent years intended to understand andcharacterizemetallic nanostructures that sup-port surface plasmons.48 These excitationsare capable of confining light fields in regionswell below the diffraction limit, while theysimultaneously display large enhancementof the electromagnetic field.9,10 Because ofthese properties, surface plasmons have beenused in different applications ranging fromultrasensitive biosensing11,12 to nano-opticalcircuits.13 Additionally, the large fields asso-ciated with surface plasmons can producestrong interactions with other photonic ele-ments, such as quantum emitters, thus giv-ing rise to newhybrid plasmon-emitter systemsexhibiting unique optical response.1424

The possibility of reaching the quantumregime using plasmonic systems faces diffi-cult challenges but offers great opportuni-ties.2527 For instance, the nonlinear behav-ior arising from the quantum nature of both afermionic emitter and light has been exten-sively investigated on theoretical and experi-mental grounds over the past decade.28,29 Aninteresting example is the photon blockadeeffect,30 which has been experimentally ob-served inatomscoupled tocavities,31 aswell asin superconducting circuits.32 The essentialingredient of these experiments is the achieve-ment of strong lightmatter coupling, char-acterized by an interaction energy exceedingthe damping introduced by cavity losses.Nonetheless, reaching the strong-coupling re-gime is a delicate task that has so far beenreserved to a handful of cavity quantum elec-trodynamics (QED) experiments.The extension of this regime to quantum

emitters interacting with plasmons sup-ported by metallic nanoparticles has beenrecently pursued2024 in order to realizequantum behavior in more robust and com-pact systems than the elaborate QED setups.

However, this goal seems to be very difficultto achieve using standard plasmonic materi-als such as noble metals due to the relativelylarge losses inherent in thesematerials, whichlimit the lifetime of the plasmons to∼10 opti-cal cycles. A promising alternative approachconsists of using plasmons supported bydoped graphene nanodisks, which have re-cently been shown to be capable of producingstrong coupling33 as a consequence of theirlonger lifetimes (∼100 optical cycles34). A richvariety of graphene plasmons have beeninvestigated3336 that should allow themanip-ulation of plasmon propagation and trapping.

* Address correspondence toa.manjavacas@csic.es,j.g.deabajo@csic.es.

Received for review December 3, 2011and accepted January 6, 2012.

Published online10.1021/nn204701w

ABSTRACT

Among the many extraordinary properties of graphene, its optical response allows one to

easily tune its interaction with nearby molecules via electrostatic doping. The large

confinement displayed by plasmons in graphene nanodisks makes it possible to reach the

strong-coupling regime with a nearby quantum emitter, such as a quantum dot or a molecule.

In this limit, the quantum emitter can introduce a significant plasmonplasmon interaction,

which gives rise to a plasmon blockade effect. This produces, in turn, strongly nonlinear

absorption cross sections and modified statistics of the bosonic plasmon mode. We characterize

these phenomena by studying the equal-time second-order correlation function g(2)(0), which

plunges below a value of 1, thus revealing the existence of nonclassical plasmon states. The

plasmon-emitter coupling, and therefore the plasmon blockade, can be efficiently controlled

by tuning the doping level of the graphene nanodisks. The proposed system emerges as a new

promising platform to realize quantum plasmonic devices capable of commuting optical

signals at the single-photon/plasmon level.

KEYWORDS: quantum plasmonics . graphene plasmons . plasmon blockade .nonclassical plasmons . nanophotonics . strong coupling . graphene nanodisk

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The very small plasmon damping in graphene iscaused by the extraordinary electrical properties ofsingle-layer carbon,3739 which together with the pos-sibility ofmodifying theplasmonic spectrumof graphenenanostructures via doping40 opens new paths for devel-oping quantumplasmonic devices capable of controllinglightmatter interaction at the quantum level.In this article, we show that quantum effects can

introduce a strong nonlinear optical response in asystem formed by a doped graphene nanodisk and anearby quantum emitter. We predict a plasmon block-ade effect under feasible conditions, which allowsactively switching the optical properties of the result-ing device in a manner analogous to the photonblockade effect that has been extensively studied incooled atoms coupled to photonic cavities. We showthat, in contrast to photon blockade, thewavelength atwhich plasmon blockade takes place can be straight-forwardly tuned simply by changing the charge in thegraphene via electrostatic gating.41 The plasmon block-ade effect produces nonclassical distributions whenmul-tiple plasmon states are generated. Our study empha-sizes nanostructured graphene coupled to quantumemitters as a viable platform to implement quantuminformation devices in robust, tunable structures.The system under study is depicted in Figure 1. We

consider a graphene nanodisk of diameter D doped toa Fermi energy EF relative to the Fermi energy for theneutral disk. This graphene nanostructure supportslocalized surface plasmons33 of energy pωp. Specifi-cally, we focus on a dipolar plasmon with polarizationparallel to the disk and m = 1 azimuthal symmetry. Thequantum emitter, which is placed 10 nm above the

nanodisk, is described as a two-level systemwith excitedand ground states |væ and |Væ separated by an energy pω0.The entire system is illuminated by an external laserof frequency ωL. In the absence of any losses, thedynamical evolution of the interacting system can bedescribed by a Hamiltonian that consists of three terms:H = Hfree þ Hint þ Hext. The first, which describes theevolution of the noninteracting system, reads

Hfree ¼ pωpa†aþ pω0σ

†σ (1)

where a and σ = |Væ Æv| (a† and σ† = |væ ÆV|) are theannihilation (creation) operators for the graphene plas-mon and the excited state of the emitter, respectively.The couplingbetween the twoexcitations is describedbythe second term

Hint ¼ pg(a†σþ aσ†) (2)

where g is a coupling constant that determines thestrength of the interaction. The last term accounts forthe effect of the laser

Hext ¼ pΩeiωLta† þ pΩeiωLta (3)

which operates at frequencyωL with an intensity definedin terms of the Rabi frequency Ω. In this expression, wehaveneglecteddirect pumpingof the emitter because itscross section is generally several orders of magnitudesmaller than the one of the graphene plasmon.We incorporate dissipation via the master equation

for the density operator42 F· = L (F), where the Liovillian43

L is given by

L (F) ¼ i

p[F, H]þ L 0(F)þ L p(F) (4)

Figure 1. Description of the graphene-nanodisk/emitter combined system under study. (a) A quantum emitter is placed10 nmabove a doped graphene nanodisk of diameterD and Fermi energy EF. The disk can support localized surface plasmonsof frequency ωp and lifetime Γp

1. The quantum emitter is described as a two-level system (ground state |Væ and excited state |væ,transition frequency ω0, excited-state lifetime Γ0

1). The dipole moment of the quantum emitter, taken to be parallel to the disk,interacts with the near field of the graphene plasmon with effective coupling strength g. The entire system is illuminated by anexternal laser of frequencyωL and intensity quantified in termsof the Rabi frequencyΩ. (b) Energy level diagramof the interactingsystemforω0=ωp. Theenergydiagramsof (a) correspond to thenoninteractingdiskandemitter, inwhich theenergy level spacingis constant. In contrast, the spacings between the interacting levels in (b) are different as a result of the coupling between theemitter and the graphene plasmon. This interaction introduces anharmonicity and nonlinearity in the optical response.

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The first term of this expression contains the Hamiltonianand determines the coherent evolution of the system,while the Lindblad terms43 L i(F) = Γi/2[2ciFci† Fci†ci ci†ciF], with i = 0, p, account for the damping of both theexcited emitter state (c0 = σ) and the graphene plasmon(cp = a).

RESULTS AND DISCUSSION

Plasmon Blockade. Apart from the coupling to exter-nal light, H is the well-known JaynesCummingsHamiltonian,44 which has the same ground state asHfree

|ψ0æ = |V0æ, and whose excited states |ψn(æ are pairs oflinear combinations of |vn 1æ and |Vnæ, where n is theplasmon occupation number, with eigenenergies

En( ¼ p nωp þ δ

2(

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiδ2

4þ ng2

s0B@

1CA (5)

where δ = ω0 ωp (see Supporting Information). Ineach pair of states, the sum of the number of plasmonand emitter excitations is n. As we show in Figure 1b,the interaction of the plasmon with the two-levelemitter results in a nonuniform distribution of statesknown as JaynesCummings ladder,45 in which thestates of each pair are separated by an energy deter-mined by the emitter-plasmon coupling g, the detuningδ, and the quantum occupation number n. The resultingnonlinear dependence of the excitation energy on n isequivalent to anharmonicity and has an important con-sequence: an external light source tuned to be resonantwith the plasmon frequency ωp becomes increasinglymore detuned as the system climbs the energy ladderand n increases. The detuning with respect to theuncoupled plasmon frequency is given by (E(nþ1)( En()/p ωp =(((δ2/4þ (nþ 1)g2)1/2 (δ2/4þ ng2)1/2),which is maximum for δ = 0 (i.e., ω0 = ωp). An increasingdetuning obviously limits the possibility of generating anarbitrary number of plasmons. We refer to this behavioras plasmon blockade, in direct analogy to the so-calledphoton blockade observed in cavity QED experiments.31

It is important to notice that the nonlinear behaviorassociated with the plasmon blockade is only effectivein systems forwhich the coupling g dominates over thelosses Γp and Γ0. In general, plasmonic losses areseveral orders of magnitude larger than quantumemitter losses (Γp . Γ0), and thus the condition forobservation of an efficient plasmon blockade amountsto g/Γp . 1. This is known as strong-coupling regime,which is difficult to reach using plasmons supported bystandard metallic nanostructures.1624 However, thecoupling strength g/Γp associated with the dipolarplasmon of a graphene nanodisk of diameter D =100 nm reaches values in the 1.53 range for moder-ate Fermi energies EF = 0.20.6 eV, and it decreases forlarger D. Here, we assume typical values for the quan-tum emitter natural decay rate Γ0 = 5 107 s1, the

nanodisk diameters D = 100200 nm, and the Fermienergies EF = 0.20.6 eV, for which the plasmon energyand width take values pωp ≈ 0.10.25 eV and pΓp ≈13.5meV(seeMethods formoredetailson thecalculationof ωp and Γp and their dependence on D and EF).

Figure 2 shows some of the main signatures ofplasmon blockade in the nanodisk-emitter systems.In what follows, we present results for steady-state

Figure 2. Plasmon blockade. (a) Population of the differentstates of a graphene-nanodisk/emitter system for twodifferent values of the external laser intensity under con-tinuous illumination conditions. Results for interacting (g/Γp = 2.55, red) and noninteracting (g/Γp = 0, black) systemsare contrasted. We assume resonant coupling (ω0 =ωp) andtune the laser frequency to the |ψ0æ f |ψ1þæ transition. Forthis laser frequency, only |ψnþæ states are significantlypopulated in the interacting system. (b) Average numberof plasmons in the graphene nanodisk Np as a function oflaser intensity for different disk diameters D and Fermienergies EF (solid curves) under the same conditions as in(a). The uncoupled system (dashed line) shows a linearincrease of Np with the laser intensity. In contrast, thecoupled system (solid curves) exhibits a clear nonlinearbehavior for |Ω|2/Γp

2 > 102. The nonlinearity is morepronounced for larger graphene-nanodisk/emitter couplingg (cf. red, blue, and green curves). (c) Absorption crosssection of the graphene-nanodisk/emitter coupled systemnormalized to the value of the uncoupled configuration as afunction of laser intensity. A departure from the linearregime is observed for large laser intensities.

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conditions under continuouswave illumination. In partic-ular, Figure 2a shows the population of different statepairs n under stationary conditions. We consider twodifferent illumination intensities (|Ω|2 = 0.01Γp

2 and|Ω|2 = Γp

2), take ω0 = ωp, and tune the laser frequencyto be resonant with the |ψ0æ f |ψ1þæ transition. Theblack bars are the results for the uncoupled system (i.e.,in the absence of the quantum emitter), while the redbars refer to the interacting system with a couplingconstant g = 2.55Γp. For low illumination intensities(upper part), the ground state (n = 0) takes most of theweight, while the first excited states (n = 1) are onlymarginally populated. In contrast, for high intensity(lower part), the population of the system is distributedamong different excited states but with very differentprofiles for configurationswith andwithout coupling. Forthe uncoupled system, the population follows a broadPoisson distribution centered around n = 4, but for thecoupled system, thepopulation ismainly concentrated inthe ground (n= 0) and singly excited (n= 1) states. This isa direct consequence of the nonlinearity and anharmoni-city associated with the plasmon blockade, which pre-vents efficient plasmon excitation beyond n = 1.

In Figure 2b, we show the average number ofplasmons Np = Æa†aæ supported by the graphenenanodisks as a function of external illumination inten-sity for different values of EF and D (solid curves). Thecorresponding calculated coupling constant (g, seeMethods), plasmon frequency, and decay rate (ωp

and Γp) are shown in the text insets throughoutFigure 2b,c with color codes standing for the selectedvalues of EF and D. Obviously, the uncoupled system(g = 0) presents a linear dependence on intensity, Np =4|Ω|2/Γp

2 (dashed curve, plotted for reference). In con-trast, a significant departure from this linear behavior isobserved above |Ω|2/Γp

2≈ 102 in the coupled system asa signature of plasmon blockade (solid curves). As ex-pected, the nonlinearity increases with g. In the smallintensity limit, the linear behavior is recovered for anyvalue of g, as Np is well below 1 (e.g., for |Ω|2/Γp

2 < 102).However, even in this limit, the plasmon blockade se-verelymodifies the plasmon statistics, aswe showbelow.

The normalized absorption cross section is shownin Figure 2c as computed by assuming that the domi-nant dissipation channel is inelastic plasmon decay,and that the radiative part represents only a smallfraction, e1%, of the total decay rate.46 In this limit,the absorption cross section can be written as σabs =pωpΓpNp/I, where I |Ω|2 is the laser intensity (seeMethods for the explicit dependence of I on |Ω|2). It isconvenient to normalize the cross section to that of theuncoupled graphene disk σabs,0 as

σabs

σabs, 0¼ Np

4

Γ2p

jΩj2 (6)

As shown in Figure 2c, the normalized absorptiondecreaseswith increasing intensityandthenanodisk-emitter

system behaves as a saturable absorber, again as aresult of plasmon blockade, which ismore pronouncedfor larger g (cf. green and blue curves).

Nonclassical Plasmons. Besides the plasmon blockadeeffect, the coupling between a graphene plasmon anda quantum emitter can also produce strong modifica-tions in the plasmon statistics, which we characterizethrough the equal-time second-order correlation func-tion, defined as47 g(2)(0) = Æa†a†aaæ/Np

2. This quantitybecomes 1 for Poissonian distributions (also known ascoherent states), which also characterize photons in acoherent laser source. In contrast, g(2)(0) > 1 is asso-ciated with thermal distributions (super-Poissonianstatistics), leading to bunching of photon pairs emergingfrom a thermal source (similar bunching is expected inthermally excited plasmons in uncoupled nanodisks).However, it is only for g(2)(0) < 1 (sub-Poissonian statistics,leading to antibunching48) that a quantum description ofphotons or plasmons is needed, as a classical modelcannot produce these values of the correlation function.Indeed, g(2)(0) = 0 would mean the existence of a singleplasmon state.

For graphene nanodisk plasmons, g(2)(0) can bedirectly measured from the photons emitted via radia-tive plasmon decay.26 Although less than one in ahundred plasmons decays by emitting a photon,33

even for the smallest intensities of the external illumi-nation under consideration the resulting photon fluxshould allow for experimental verification of the non-classical, quantumplasmonic regime: for instance, for amoderate incident intensity Ω = 0.1Γp, the rate ofphoton generation ≈ NpΓp,rad ≈ 4 104Γp ≈ 109 s1

is significant.Figure 3 shows g(2)(0) as a function of laser inten-

sity for the different graphene nanodisks discussed inFigure 2. We assume that the laser frequency is

Figure 3. Nonclassical plasmon states. We show the equal-time second-order correlation function g(2)(0) = Æa†a†aaæ/Np

2

versus laser intensity for different disk diameters D andFermi energies EF (solid curves) under the same conditionsas in Figure 2. We also show g(2)(0) = 1 (dashed line), asobtained for the uncoupled system. For finite coupling(solid curves) and relatively low intensity, g(2)(0) lies below1 and decreases with increasing g/Γp, thus revealing thecreation of nonclassical plasmon states. For large intensity,g(2)(0) increases and eventually becomes larger than 1.

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resonant with the |ψ0æf |ψ1þæ transition, andω0 =ωp.In the absence of a quantum emitter (uncoupledsystem), we have g(2)(0) = 1 (dashed line), so thatplasmons are generated in coherent states, which aretypical of coherent classical systems. When coupling isswitched on (solid curves), g(2)(0) drops below 1, andthus, nonclassical plasmon states are generated. Further-more, theminimum value that is reached depends on thecoupling strength, decreasing with increasing g. As thelaser intensity grows, g(2)(0) approaches the classicalthreshold g(2)(0) = 1 and eventually rises above this value.This behavior can be understood by examining Figure 2a,in which we see that small values of |Ω|2/Γp

2 produce anegligible population of states above n = 1, thereforeresulting in sub-Poissonian statistics. In contrast, for largerintensities, the population of n > 1 states increases, thusresulting in the super-Poissonian behavior shown inFigure 3 despite the plasmon blockade.

Tunability of Plasmon Blockade. A great advantage ofthe plasmons supported by the graphene nanodisks isthe possibility of controlling their properties by tuningthe Fermi energy. This can be achieved, for instance, byintroducing electrostatic doping,40 thus allowing usthe tuning of both the plasmon energy and its couplingwith the quantum emitter, and by extension, themodulation of the plasmon blockade effect. This isillustrated in Figure 4a, which shows the normalizedabsorption cross section defined by eq 6 as a functionof the graphene Fermi energy for different values ofthe laser intensity. The excitation energy of the quan-tum emitter is set to 0.16 eV, which coincides with theplasmon energy for EF = 0.4 eV. As in the remainder ofthis work, the laser frequency is chosen to be resonantwith the |ψ0æ f |ψ1þæ transition.

At small intensities (red curve), the normalized ab-sorption cross section approximately follows the samebehavior as in the uncoupled system (dashed line), witha slight departure for low Fermi energies. When |Ω|2

increases (blue and green curves), the plasmon blockadeeffect produces significant nonlinearity. The nonlinearityis stronger for EF below the resonance condition EF =0.4 eV. This behavior is directly connected to the evolu-tion of |ψn(æ states with detuning δ: these states arelinear combinations of |vn 1æ and |Vnæ, but absorptiondominates in the latter because it contains a largernumber of plasmons. More precisely, the external illumi-nation is tuned to induce transitions from the groundstate to |ψ1þæ. For EF > 0.4 eV (i.e., g < |δ| and δ < 0), onefinds that |V1æ has stronger weight in |ψ1þæ (see Support-ing Information), thus leading to higher absorption. Incontrast, for EF < 0.4 eV, where g < |δ| and δ > 0, theweight of |V1æ is smaller and absorption is reduced. Thisexample clearly illustrates that it is possible to enhanceplasmon blockade by detuning the plasmon energy,which of course can be controlled via the Fermi energyof the graphene nanodisk, providedwe have coupling toa quantum emitter.

The enhancement of the plasmon blockade alsoinfluences the statistics of the generated plasmons, asshown in Figure 4b, where g(2)(0) is plotted as afunction of the Fermi energy for the same conditionsas in Figure 4a. The dashed line represents the un-coupled system (classical states). In the coupled system(solid curves), we again observe two different regions:for large values of EF, the system approaches theclassical limit (g(2)(0) f 1) due to the reduction in theefficiency of plasmon blockade. In contrast, for small EF,the plasmon blockade is enhanced, giving rise to anti-bunching (g(2)(0) < 1). Clearly, the antibunching increaseswhen the Fermi energy becomes smaller than EF = 0.4 eV,where the plasmon and the laser are on resonance. Asexpected, this effect is more pronounced for smallerillumination intensities (see also Figure 3).

Interestingly, there is an optimum value of EF forwhich g(2)(0) reaches a minimum as a result of theinterplay between the plasmon-emitter coupling g and

Figure 4. Tunability of graphene-nanodisk/emitter systems.(a) Normalized absorption cross section of a graphene-nanodisk/emitter system as a function of graphene Fermienergy for different values of the external laser intensity.The nanodisk diameter is D = 100 nm. The emitter has fixedexcitation energy pω0 = 0.16 eV, which coincides with theplasmon energy pωp for EF = 0.4 eV. We tune the laserfrequency to be resonant with the |ψ0æ f |ψ1þæ transition.The cross section is normalized to that of the uncoupledsystem. At low laser intensity (red curve), the cross sectionswith andwithout coupling are very similar. As |Ω|2 increases(blue and green curves), the cross section becomes stronglynonlinear for small EF and essentially linear for large EF. Thetransition between these two regimes is sharper for largerlaser intensity. (b) Equal-time second-order correlationfunction g(2)(0) under the same conditions as in (a). Forlarge intensity, g(2)(0) is found to be close to the classicalvalue g(2)(0) = 1. When the intensity decreases, we againobserve two different regimes: for small EF, the systemexhibits sub-Poissonian statistics associated with the exis-tence of nonclassical plasmons, while for larger EF, a classi-cal behavior is recovered.

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the detuning δ. When the latter increases for EF < 0.4 eV,the state |ψ1þæ approaches |v0æ, thus preventing higher nstates to be populated, which results in lower g(2)(0).However, at the same time, g decreases, therefore reduc-ing the anharmonicity of the energy ladder, which con-tributes to increasing the value of g(2)(0). The trade-offbetween these two effects results in anoptimumvalue ofEF at which g(2)(0) presents a minimum.

CONCLUSIONS

In summary, we have fully characterized the non-linear optical response of a graphene nanodisk coupledto a quantum emitter. Because of the extraordinarilylarge lifetimes of graphene plasmons, this system iscapable of reaching the strong-coupling regime,where a plasmon blockade effect emerges naturallyin its optical response. This effect has two importantconsequences: (i) the optical absorption cross section

is dramatically modified and becomes strongly non-linear for large illumination intensities; and (ii) thecorrelation of the generated plasmons exhibits aclear nonclassical behavior, as we demonstrate bystudying the equal-time second-order correlationfunction g(2)(0). More precisely, nonclassical plasmo-nic states (i.e., states for which g(2)(0) < 1) are gener-ated due to the plasmon blockade, an effect thatcould be possibly detected by measuring correla-tions of photons emitted by radiative decay. Inter-estingly, we show that it is possible to control theoptical nonlinearity of the graphene nanodisk/emit-ter system by tuning the plasmonic spectrum of thegraphene nanodisk via electrostatic doping. Ourwork opens new paths for the design of novel quan-tum plasmonic devices with applications rangingfrom active metamaterials to quantum informationprocessing.

METHODSOptical Response of Graphene Nanodisks. The optical response

of graphene nanodisks has been characterized in previousworks.33,46 Here we follow the same methodology, consistingof rigorously solving Maxwell's equations using the boundaryelement method (BEM).49 For our purposes, the nanodisks areparametrized as thin films of thickness t with rounded edgesand dielectric function ε= 1þ 4πiσ/ωt. We use t= 0.5 nm, whichis well converged with respect to the t f 0 limit for the diskdiameters under consideration. The conductivity σ(ω) is takenfrom the local limit of the random-phase approximation for anextended graphene sheet50,51 (see ref 33 and its SupportingInformation for more details). Furthermore, we assume anintrinsic relaxation time τ = μEF/evF

2, where vF ≈ c/300 is theFermi velocity and μ = 10 000 cm2/(V 3 s) is the measured dcmobility.37 We extract the plasmon energies pωp and widthspΓp by fitting the plasmon peak of the numerically calculatedabsorption spectra to a Lorentzian. This classical description hasbeen proved52 to be accurate enough and well reproduced byab initio calculations for the dimensions under consideration,with nanodisk diameters larger than 40 nm.

The coupling constant g requires a more elaborate treatment.In the classical limit, the decay rate of a quantum emitter placedclose to the graphene nanodisk is given by the expression1

Γ ¼ Γ0 þ 2pImfμ0 3 Eindg (7)

where Γ0 is the free-space decay rate, μ0 is the quantum emitterdipole moment, and Eind is the field self-induced by the emitterdue to the presence of the graphene nanodisk. We obtain Γthrough the calculation of the self-induced field generated by adipole and evaluated at its own position. This calculation, whichinvolves the rigorous solution of Maxwell's equations, is per-formed using BEM. Now, it is clear that this value of Γ must beequal to the emitter decay rate computed from the quantummodel in the limit of weak coupling (i.e., g/Γpf 0). In such limit,we have |ψ1æf |v0æ, and therefore Γ≈ ImE1 becomes (seeSupporting Information)

Γ Γ0 þ 4g2

Γp(8)

and finally, we obtain the coupling parameter g by comparingeqs 7 and 8.

Figure S1 in the Supporting Information shows the relevantgraphene nanodisk parameters as a function of disk diameter

D and Fermi energy EF. The plasmon energies lie in the0.100.25 eV range for the diameters and Fermi energiesconsidered in this work, while the decay rates lie in the 1.03.5meV range. It is interesting to note that the plasmondecay rateis almost independent of diameter. Finally, the values of thenormalized coupling constant g/Γp lie always above 1, indicatingthat the graphene-nanodisk/emitter systems under considerationare in the strong-coupling regime. Furthermore, for the para-meters used in this paper, the crossover from the strong to theweak coupling regimes takes place when the separation betweenthequantumemitter and thegraphenenanodisk is on theorder ofa few tens of nanometers. The preservation of the strong-couplingregime to such large separations, which is a consequence of thenarrowness and strengthof graphene plasmons,makes it possibleto use optical or electrostatic trapping to place the quantumemitter in close proximity to the graphene nanostructure, thusavoiding the undesired effects of coupling to a substrate orelectronhole excitations of the carbon sheet that can be anefficient source of quantum decoherence. Alternatively, a passivedielectric spacer could be used in which case the graphene diskparameters would have to be tuned slightly to accommodate forthe plasmon red shift and reduced emitter coupling strength g.

Plasmon Dipole Moment and Rabi Frequency. The polarizability ofa graphene disk can be accurately modeled using the followingexpression46

R(ω) ¼ 3c3Γp, rad2ω2

p

1ω2

p ω2 iΓpω3=ω2p

(9)

where ωp is the plasmon frequency, and Γp,rad and Γp are theradiative and the total plasmon decay rates, respectively.Furthermore, the polarizability associated with the plasmonresonance can also be written as33

R(ω) μ2ppωp pω ipΓp=2

(10)

where μp is the dipolemoment of the graphene plasmon, whichcan be determined by comparing this expression with eq 9 forω = ωp. From here, we obtain

μ2p ¼ 34pΓp, rad

c

ωp

!3

(11)

The Rabi frequency is defined as

Ω ¼ μpE

p(12)

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where E is the electric-field amplitude corresponding to anexternal illumination intensity I = [c/(2π)]E2. Using eq 11, we canrelate the Rabi frequency to the intensity as

I ¼ 2cp3π

Γ2p

Γp, rad

!ω3

p

c3Ω

Γp

!2

(13)

We need to know ωp, Γp, and Γp,rad in order to evaluate thisexpression. The first two of these parameters can be obtained asexplained in the previous section, and we find Γp,rad by fittingthe extinction cross section calculated from BEM to σext = (4πω/c)ImR(ω), with R(ω) given by eq 9.

Figure S2a in the Supporting Information shows the plas-mon radiative decay rate normalized to the total decay rate as afunction of disk diameter D for three different values of theFermi energy (EF = 0.2, 0.4, and 0.6 eV). Radiative decay system-atically amounts for less than 1% of the total decay. Finally, weshow in Figure S2b the Rabi frequency calculated from eq 13using these results for Γp,rad. Incidentally, the Rabi frequenciesconsidered in Figure 2 of this article lie in theΩ = 0.01Γp 5Γp

range, which corresponds to moderate laser intensities extend-ing from I = 96 W/m2 to I = 1.3 108 W/m2.

Acknowledgment. This work has been supported by theSpanish MICINN (MAT2010-14885 and Consolider NanoLight.es) and the European Commission (FP7-ICT-2009-4-248909-LIMA and FP7-ICT-2009-4-248855-N4E). A.M. acknowledges finan-cial support through FPU from the SpanishME. P.N. acknowledgessupport from the Robert A. Welch Foundation under Grant C-1222and theCenter for Solar Photophysics, an Energy Frontier ResearchCenter funded by the U.S. Department of Energy.

Supporting Information Available: We provide additionalinformation of eigenstates and eigenenergies of the JaynesCummings Hamiltonian, together with figures of the relevantparameters of our model. This material is available free ofcharge via the Internet at http://pubs.acs.org.

REFERENCES AND NOTES1. Novotny, L.; Hecht, B. Principles of Nano-Optics; Cambridge

University Press: New York, 2006.2. Pasquale, A. J.; Reinhard, B. M.; Negro, L. D. Engineering

PhotonicPlasmonic Coupling in Metal NanoparticleNecklaces. ACS Nano 2011, 5, 6578–6585.

3. Zhao, J.; Frank, B.; Burger, S.; Giessen, H. Large-Area High-Quality PlasmonicOligomers FabricatedbyAngle-ControlledColloidal Nanolithography. ACS Nano 2011, 5, 9009–9016.

4. Maier, S. A. Plasmonics: Fundamentals and Applications;Springer: New York, 2007.

5. Juluri, B. K.; Chaturvedi, N.; Hao, Q.; Lu, M.; Velegol, D.;Jensen, L.; Huang, T. J. ScalableManufacturing of PlasmonicNanodisk Dimers and Cusp Nanostructures Using Salting-Out Quenching Method and Colloidal Lithography. ACSNano 2011, 5, 5838–5847.

6. Duan, H.; Hu, H.; Kumar, K.; Shen, Z.; Yang, J. K. W. Directand Reliable Patterning of Plasmonic Nanostructures withSub-10-nm Gaps. ACS Nano 2011, 5, 7593–7600.

7. Chen, H.; Shao, L.; Ming, T.; Woo, K. C.; Man, Y. C.; Wang, J.;Lin, H.-Q. Observation of the Fano Resonance in GoldNanorods Supported on High-Dielectric-Constant Sub-strates. ACS Nano 2011, 5, 6754–6763.

8. Van Dorpe, P.; Ye, J. Semishells: Versatile Plasmonic Nano-particles. ACS Nano 2011, 5, 6774–6778.

9. Halas, N. J.; Lal, S.; Chang, W.-S.; Link, S.; Nordlander, P.Plasmons in Strongly Coupled Metallic Nanostructures.Chem. Rev. 2011, 111, 3913–3961.

10. Aubry, A.; Lei, D. Y.; Maier, S. A.; Pendry, J. B. PlasmonicHybridization Between Nanowires and a Metallic Surface:A Transformation Optics Approach. ACS Nano 2011, 5,3293–3308.

11. Alvarez-Puebla, R. A.; Liz-Marzán, L. M.; García de Abajo,F. J. Light Concentration at the Nanometer Scale. J. Phys.Chem. Lett. 2010, 1, 2428–2434.

12. Taylor, R. W.; Lee, T.-C.; Scherman, O. A.; Esteban, R.;Aizpurua, J.; Huang, F. M.; Baumberg, J. J.; Mahajan, S.Precise Subnanometer Plasmonic Junctions for SERSwithinGold Nanoparticle Assemblies Using Cucurbit[n]uril “Glue”.ACS Nano 2011, 5, 3878–3887.

13. Ozbay, E. Plasmonics: Merging Photonics and Electronicsat Nanoscale Dimensions. Science 2006, 311, 189–193.

14. Banerjee, P.; Conklin, D.; Nanayakkara, S.; Park, T.-H.; Therien,M. J.; Bonnell, D. A. Plasmon-Induced Electrical Conductionin Molecular Devices. ACS Nano 2010, 4, 1019–1025.

15. Zheng, Y. B.; Yang, Y.-W.; Jensen, L.; Fang, L.; Juluri, B. K.;Flood, A. H.; Weiss, P. S.; Stoddart, J. F.; Huang, T. J. ActiveMolecular Plasmonics: Controlling PlasmonResonanceswithMolecular Switches. Nano Lett. 2009, 9, 819–825.

16. Zhang, W.; Govorov, A. O.; Bryant, G. W. Semiconductor-Metal Nanoparticle Molecules: Hybrid Excitons andthe Nonlinear Fano Effect. Phys. Rev. Lett. 2006, 97,146804.

17. Artuso, R. D.; Bryant, G. W. Optical Response of StronglyCoupled Quantum Dot-Metal Nanoparticle Systems: Dou-ble Peaked Fano Structure and Bistability.Nano Lett. 2008,8, 2106–2111.

18. Savasta, S.; Saija, R.; Ridolfo, A.; Di Stefano, O.; Denti, P.;Borghese, F. Nanopolaritons: Vacuum Rabi Splitting with aSingle Quantum Dot in the Center of a Dimer Nanoanten-na. ACS Nano 2010, 4, 6369–6376.

19. Wu, X.; Gray, S. K.; Pelton, M. Quantum-Dot-Induced Trans-parency in a Nanoscale Plasmonic Resonator. Opt. Express2010, 18, 23633–23645.

20. Artuso, R. D.; Bryant, G. W. Strongly Coupled QuantumDotMetal Nanoparticle Systems: Exciton-Induced Trans-parency, Discontinuous Response, and Suppression asDriven Quantum Oscillator Effects. Phys. Rev. B 2010, 82,195419.

21. Ridolfo, A.; Di Stefano, O.; Fina, N.; Saija, R.; Savasta, S.Quantum Plasmonics with QuantumDotMetal Nanopar-ticle Molecules: Influence of the Fano Effect on PhotonStatistics. Phys. Rev. Lett. 2010, 105, 263601.

22. Kyas, G.; May, V. Density Matrix Based Microscopic Theoryof Molecule MetalNanoparticle Interactions: Linear Absor-bance and Plasmon Enhancement of Intermolecular Excita-tion Energy Transfer. J. Chem. Phys. 2011, 134, 034701.

23. Manjavacas, A.; García de Abajo, F. J.; Nordlander, P. N.Quantum Plexcitonics: Strongly Interacting Plasmons andExcitons. Nano Lett. 2011, 11, 2318–2323.

24. Zhang, W.; Govorov, A. O. Quantum Theory of the Non-linear Fano Effect in Hybrid Metal-Semiconductor Nano-structures: The Case of Strong Nonlinearity. Phys. Rev. B2011, 84, 081405.

25. Chang, D. E.; Sorensen, A. S.; Demler, E. A.; Lukin, M. D. ASingle-Photon Transistor Using Nanoscale Surface Plas-mons. Nat. Phys. 2007, 3, 807–812.

26. Akimov, A. V.; Mukherjee, A.; Yu, C. L.; Chang, D. E.; Zibrov,A. S.; Hemmer, P. R.; Park, H.; Lukin, M. D. Generation ofSingle Optical Plasmons in Metallic Nanowires Coupled toQuantum Dots. Nature 2007, 450, 402–406.

27. Kolesov, R.; Grotz, B.; Balasubramanian, G.; Stöhr, R. J.;Nicolet, A. A. L.; Hemmer, P. R.; Jelezko, F.; Wrachtrup, J.Wave-Particle Duality of Single Surface Plasmon Polari-tons. Nat. Phys. 2009, 5, 470–474.

28. Reithmaier, J. P.; Sek, G.; Löffler, A.; Hofmann, C.; Kuhn, S.;Reitzenstein, S.; Keldysh, L. V.; Kulakovskii, V. D.; Reinecke,T. L.; Forchel, A. Strong Coupling in a Single Quantum Dot-Semiconductor Microcavity System. Nature 2004, 432,197–200.

29. Hennessy, K.; Badolato, A.;Winger,M.; Gerace, D.; Atatüre,M.;Gulde, S.; Fält, S.; Hu, E. L.; Imamoglu, A. Quantum Natureof a Strongly Coupled Single Quantum Dot-Cavity System.Nature 2007, 445, 896–899.

30. Imamoglu, A.; Woods, H. S. G.; Deutsch, M. Strongly Inter-acting Photons in a Nonlinear Cavity. Phys. Rev. Lett. 1997,79, 1467–1470.

31. Birnbaum, K. M.; Boca, A.; Miller, R.; Boozer, A. D.; Northup,T. E.; Kimble, H. J. Photon Blockade in an Optical Cavitywith One Trapped Atom. Nature 2005, 436, 87–90.

ARTIC

LE

MANJAVACAS ET AL. VOL. 6 ’ NO. 2 ’ 1724–1731 ’ 2012

www.acsnano.org

1731

32. Fink, J. M.; Göppl, M.; Baur, M.; Bianchetti, R.; Leek, P. J.;Blais, A.; Wallraff, A. Climbing the JaynesCummingsLadder and Observing Its Nonlinearity in a Cavity QEDSystem. Nature 2008, 454, 315–318.

33. Koppens, F.H. L.; Chang,D. E.; García deAbajo, F. J. GraphenePlasmonics: A Platform for Strong LightMatter Interac-tions. Nano Lett. 2011, 11, 3370–3377.

34. Jablan, M.; Buljan, H.; Soljacic, M. Plasmonics in Grapheneat Infrared Frequencies. Phys. Rev. B 2009, 80, 245435.

35. Vakil, A.; Engheta, N. Transformation Optics Using Graph-ene. Science 2011, 332, 1291–1294.

36. Nikitin, A. Y.; Guinea, F.; García-Vidal, F. J.; Martín-Moreno, L.Fields Radiated by a Nanoemitter in a Graphene Sheet.Phys. Rev. B 2011, 84, 195446.

37. Novoselov, K. S.; Geim, A. K.; Morozov, S. V.; Jiang, D.;Zhang, Y.; Dubonos, S. V.; Grigorieva, I. V.; Firsov, A. A.Electric Field Effect in Atomically Thin Carbon Films.Science 2004, 306, 666–669.

38. Novoselov, K. S.; Geim, A. K.; Morozov, S. V.; Jiang, D.;Katsnelson, M. I.; Grigorieva, I. V.; Dubonos, S. V.; Firsov,A. A. Two-Dimensional Gas of Massless Dirac Fermions inGraphene. Nature 2005, 438, 197–200.

39. Bonaccorso, F.; Sun, Z.; Hasan, T.; Ferrari, A. C. GraphenePhotonics and Optoelectronics. Nat. Photonics 2010, 4,611–622.

40. Chen, C. F.; Park, C. H.; Boudouris, B. W.; Horng, J.; Geng, B.;Girit, C.; Zettl, A.; Crommie, M. F.; Segalman, R. A.; Louie,S. G.; et al. Controlling Inelastic Light Scattering QuantumPathways in Graphene. Nature 2011, 471, 617–620.

41. Mak, K. F.; Sfeir, M. Y.; Wu, Y.; Lui, C. H.; Misewich, J. A.; Heinz,T. F. Measurement of The Optical Conductivity of Graph-ene. Phys. Rev. Lett. 2008, 101, 196405.

42. Ficek, Z.; Tanas, R. Entangled States and Collective Non-classical Effects in Two-Atom Systems. Phys. Rep. 2002,372, 369–443.

43. Meystre, P.; Sargent, M., III. Elements of Quantum Optics;Springer-Verlag: Berlin, 1990.

44. Jaynes, E.; Cummings, F. Comparison of Quantum andSemiclassical Radiation Theories with Application to theBeam Maser. Proc. IEEE 1963, 51, 89–109.

45. Laussy, F. P.; del Valle, E.; Schrapp, M.; Laucht, A.; Finley, J. J.Climbing the Jaynes-Cummings Ladder by Photon Count-ing; arXiv:1104.3564v2.

46. Thongrattanasiri, S.; Koppens, F. H. L.; García de Abajo, F. J.Total Light Absorption in Graphene; arXiv:1106.4460v1.

47. Loudon, R. The Quantum Theory of Light; Oxford UniversityPress: Oxford, 2000.

48. Kimble, H. J.; Dagenais, M.; Mandel, L. Photon Antibunch-ing in Resonance Fluorescence. Phys. Rev. Lett. 1977, 39,691–695.

49. García de Abajo, F. J.; Howie, A. Retarded Field Calculationof Electron Energy Loss in Inhomogeneous Dielectrics.Phys. Rev. B 2002, 65, 115418.

50. Wunsch, B.; Stauber, T.; Sols, F.; Guinea, F. Dynamical Polar-ization of Graphene at Finite Doping. New J. Phys. 2006, 8,318.

51. Falkovsky, L. A.; Varlamov, A. A. Space-Time Dispersion ofGraphene Conductivity. Eur. Phys. J. B 2007, 56, 281.

52. Thongrattanasiri, S.; Manjavacas, A.; García de Abajo, F. J.Quantum Finite-Size Effects in Graphene Plasmons. ACSNano 2012, DOI: 10.1021/nn204780e.

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Supporting information for:

Plasmon Blockade in Nanostructured Graphene

A. Manjavacas,∗,† P. Nordlander,‡ and F. J. García de Abajo∗,†,¶

Instituto de Química-Física Rocasolano - CSIC, Serrano 119, 28006 Madrid, Spain, and

Department of Physics and Astronomy, M.S. 61, Rice University, Houston, Texas 77005-1892

E-mail: a.manjavacas@csic.es; j.g.deabajo@csic.es

Eigenstates and eigenenergies of the Jaynes-Cummings model

The Jaynes-Cummings HamiltonianS1

H = h[ωpa†a+ω0σ

†σ +g

(a†

σ +aσ†)]

(1)

is diagonal in the basis set formed by states

|ψ0〉= |↓ 0〉

and

|ψn±〉= An± |↓ n〉+Bn± |↑ n−1〉 ,∗To whom correspondence should be addressed†Instituto de Química-Física Rocasolano - CSIC, Serrano 119, 28006 Madrid, Spain‡Department of Physics and Astronomy, M.S. 61, Rice University, Houston, Texas 77005-1892¶Currently on sabbatical at Optoelectronics Research Centre, University of Southampton, Southampton SO17 1BJ,

UK

S1

where

An± =

√ng√

ng2 +

(δ

2 ±√

δ 2

4 +ng2)2

,

Bn± =

δ

2 ±√

δ 2

4 +ng2√ng2 +

(δ

2 ±√

δ 2

4 +ng2)2

, (2)

δ = ω0−ωp, and n = 1, . . . ,∞. These states have energies

En± = h

(nωp +

δ

2±√

δ 2

4+ng2

). (3)

It is worth exploring some limits of these expressions. In particular, when the emitter and the

plasmon are on resonance (i.e., δ = 0), we have An± = Bn± = 1/2. However, when δ > 0 and

g/δ → 0, we have An+,Bn−→ 0 and An−,Bn+→ 1, while for δ < 0 and g/δ → 0, we obtain the

opposite behavior, An+,Bn−→ 1 and An−,Bn+→ 0.

If we introduce decay rates for the plasmon and the quantum emitter, the energies of the Jaynes-

Cummings model become complex. Such energies can be obtained by diagonalizing the Liouvil-

lianS2 given in Eq. (4) of the main paper. By doing so, we obtain

En± = h

nωp +δ

2− i

(2n−1)Γp +Γ0

4±

√(iΓ0−Γp

4− δ 2

4

)2

+ng2

. (4)

The real part of these energies determine the position of the different states in the Jaynes-Cummings

ladder, while the imaginary parts represent the corresponding decay rates. Notice however that we

solve the full density-matrix equations to obtain the results presented in the main paper.

S2

Graphene nanodisk parameters

0.2 0.3 0.4 0.5 0.6 40 80 120 160 200EF (eV) D (nm)

EF = 0.4 eV

3

2

1

g/Γ p (

eV)

(f)

0.28

0.21

0.14ωp (

eV)

D = 100 nm0.20

0.15

ωp (

eV)

(a) (b)

Γ p (m

eV)

4

3

2

1 (c)

1.70

1.65

1.60

Γ p (m

eV)

(d)

2.5

2.0

3.0

g/Γ p (

eV)

(e)

Figure S1: Graphene nanodisk parameters. The left column shows the model parameters (plasmonenergy hωp, plasmon decay rate Γp, and coupling constant g) as a function of Fermi energy EFfor fixed diameter D = 100 nm, while the right column corresponds to fixed Fermi energy EF =0.4 eV and varying diameter. All calculations are performed for the lower-order dipolar plasmonsupported by the graphene nanodisk.S3,S4

S3

Relation between external light intensity and Rabi frequency

|Ω|2/Γp2

80 120 160 200D (nm)

Γ p,ra

d / Γ

p

5·10-3

1·10-2

010-4 10-110-210-3 100 101

102

108

106

104

I (W

/m2 )

EF = 0.4 eV D = 100 nmEF = 0.6 eV D = 100 nmEF = 0.4 eV D = 200 nm

EF = 0.2 eV EF = 0.4 eV EF = 0.6 eV

(b)(a)

Figure S2: (a) Radiative decay rate of the dipolar graphene-nanodisk plasmon normalized to thetotal decay rate and calculated as a function of disk diameter D for three values of the Fermi energyEF . (b) Intensity versus Rabi frequency for the three different sets of Fermi energies and diametersconsidered in the main text.

References

(S1) Meystre, P.; III, M. S. Elements of Quantum Optics; Springer-Verlag: Berlin, 1990.

(S2) Laussy, F. P.; del Valle, E.; Schrapp, M.; Laucht, A.; Finley, J.J.; arXiv:1104.3564v2.

(S3) Koppens, F. H. L.; Chang, D. E.; García de Abajo, F. J. Graphene plasmonics: A platform for

strong light-matter interactions. Nano Lett. 2011, 11, 3370–3377.

(S4) Thongrattanasiri, S.; Koppens, F. H. L.; García de Abajo, F. J. arXiv:1106.4460v1.

S4