A plasmonic nanoantenna based triggered single photon source

J. Straubel,1 R. Filter,2 C. Rockstuhl,1, 3 and K. Sªowik1, 4, ∗

1Institute of Theoretical Solid State Physics, Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany2Institute of Condensed Matter Theory and Solid State Optics, Abbe Center of Photonics,

Friedrich-Schiller-University Jena, Max-Wien-Platz 1, 07743 Jena, Germany3Institute of Nanotechnology, Karlsruhe Institute of Technology, 76021 Karlsruhe, Germany

4Institute of Physics, Faculty of Physics, Astronomy and Informatics,Nicolaus Copernicus University, Grudziadzka 5, 87-100 Torun, Poland

Highly integrated single photon sources are key components in future quantum-optical circuits.Whereas the probabilistic generation of single photons can routinely be done by now, their triggeredgeneration is a much greater challenge. Here, we describe the triggered generation of single photonsin a hybrid plasmonic device. It consists of a lambda-type quantum emitter coupled to a multimodeoptical nanoantenna. For moderate interaction strengths between the subsystems, the description ofthe quantum optical evolution can be simplied by an adiabatic elimination of the electromagneticelds of the nanoantenna modes. This leads to an insightful analysis of the emitter's dynamics,entails the opportunity to understand the physics of the device, and to identify parameter regimesfor a desired operation. Even though the approach presented in this work is general, we considera simple exemplary design of a plasmonic nanoantenna, made of two silver nanorods, suitable fortriggered generation of single photons. The investigated device realizes single photons, triggered,potentially at high rates, and using low device volumes.

PACS numbers: 73.20.Mf, 32.80.Qk 42.50.Nn

I. INTRODUCTION

The fabrication of nanoscopic systems has made greatadvancement during the last two decades. These ad-vancements allowed to enter new regimes of light-matter-interaction, in which eects distinctive for cavity quan-tum electrodynamics could be realized at the nanoscale.This aim can be achieved in the nearest future by exploit-ing on-chip integrated plasmonic nanoantennas, acting asopen cavities coupled to adjacent quantum systems. Themost prominent example of physical eects that have al-ready been realized in this context is the Purcell enhance-ment of the radiative decay rate of molecules or quantumdots adjacent to plasmonic nanoantennas (for a reviewplease see Ref. 1). The change in the transition rate dueto a modied local density of electromagnetic states hasbeen a subject of intensive studies on both experimental[27] and theoretical grounds [810]. Nanoantennas andnanowires that mediate the interaction between multipleemitters have also been discussed in numerous scenar-ios [8, 11], including entanglement generation protocols[1214].

The key asset of nanoantennas is to tailor the lightin the near- and far-eld upon request. This includesshaping eld distribution [1517], directing light emis-sion [1820], and tuning resonance frequencies [21, 22] -all possibly with polarization-sensitive geometries. Thescattered eld can even possesses quantum properties,such as nonclassical statistics [2329] or entanglement[3032], if suitable quantum emitters are exploited assources. Such emitters, usually molecules and quantumdots, can be positioned at close vicinity of nanoantennasusing the state-of-the-art techniques [3335]. Due to the

(b)

1❭

κ1

2❭

e❭κ2

(e1)γsp.em(e2)γsp.em

γ12

1eγdephγ2eγdeph

γ12γdeph

(a)

Fig. 1. (a) Scheme of the system under consideration: Atwo-modal nanoantenna coupled to a lambda-type quantumemitter. (b) Jablonski diagram of the quantum emitter.Two-headed arrows indicate coherent, reversible transitions,while single-headed arrows correspond to incoherent, one-waytransfer. Waivy arrows stand for dephasing, i.e. decay of theo-diagonal elements of the density matrix.

broad character of the resonances in the optical domain,matching transition frequencies of the quantum emitterscan be achieved with both dipolar, or even higher-ordernanoantenna modes [36].

In most of so-far proposed scenarios, quantum emitterswere approximated as two-level systems. In such systemsthe transition probabilities to other energy eigenstatesare assumed to be negligible. Considering richer energycongurations may, however, enable more complex dy-namics. This allows promising eects such as nonlineartwo-photon interactions [37].

However, in the context of interactions with quantumsystems, there is still a distinct gap between what cav-ity quantum dynamics is able to predict as fundamentaleects at the nanoscale and what can be achieved exper-imentally. With this paper our goal is to provide a stoneto bridge this gap. We use a theoretical description ofa carefully designed nanoantenna coupled to a realisticquantum system to yield new features that the are hardly

2

observable with the isolated systems alone. We proposeto exploit lambda-conguration quantum emitters cou-pled to bimodal nanoantennas for the conversion of lightat the single-photon level. The unique advantage is theability to achieve a triggered emission of a single-photon,i.e. the emission happens upon request [38]. It will beshown that the extent of the deterministic character islimited by the nanoantenna eciency; being however aparameter that can be engineered.

A lambda-type emitter can represent a molecule or aquantum dot. Its conguration is given by two low- andone higher-energy state [Fig. 1(b)]. Electric dipole tran-sitions are assumed to be forbidden between the lowerstates, but allowed otherwise.

Suitable molecules have successfully been positioned inthe vicinity of, or even adsorbed at a nanosurface, in nu-merous experiments in the context of surface-enhancedRaman scattering (for a review see Ref. 39). Also,lambda-type and similar structures have recently beenproduced with self-assembled InAs quantum dots [4043]. The exemplary nanoantenna design that we con-sider in this work is adjusted to support resonances inthe frequency range and of polarization characteristics,corresponding to transitions in such quantum dots. Nat-urally, the nanoantenna can be redesigned for other tar-get quantum systems.

In general, the nanoantenna does not necessarily re-quire a specic geometry. On the contrary, various typesof nanoantennas can be exploited, e.g. made from metal-lic or dielectric materials or even nanoantennas madefrom graphene [10, 44, 45]. It is only important thatthe nanoantenna shall support multiple resonances, i.e.at least two. The resonances shall be sustained at the twotransition frequencies of the lambda-type quantum emit-ter. For the simplied analytical description presentedbelow, the nanoantenna should be weakly coupled to thequantum emitter. Then, multiple exchanges of energyamong the subsystem are unlikely. This approximationis well justied for typical nanoantennas - it requires spe-cial eort to design a nanoantenna that operates beyondthis limit [11, 46, 47]. Furthermore, we provide impor-tant gures of merit as denite analytical results for asimplied description of the eects of such hybrid quan-tum systems. Our approach to model the interaction ofthe nanoantenna and the lambda-type quantum systemwithin the adiabatic limit is very general. However, wediscuss an example of a metallic nanoantenna designedespecially for the purpose of triggered single photon gen-eration with lambda-type quantum emitters.

Following discussions in the prior work [38], the ratiobehind the design is the following. To achieve a trig-gered emission of a photon, a driving eld is applied tothe nanoantenna. The drive is resonant with the rsttransition of the lambda system. The drive excites thequantum emitter from its ground to its excited state. Inthe following, the emitter may relax to one of the two

lower states - each of these processes is enhanced due tothe coupling to a suitable resonance of the nanoantenna.If the emitter returns to its ground state, the procedurequickly repeats due to the ongoing drive, i.e. typicallyat the nanosecond time scales. Otherwise, the desiredsingle photon is generated in the transition to the otherlow-energy metastable state. This photon is then scat-tered into the far-eld or absorbed by the nanoantenna attime scales of a few femtoseconds [48]. Once the lambdasystem is in this second low-energy metastable state, theprocesses stop. We have witnessed the triggered emis-sion of a single photon. The source might be continouslyused afterwards if a mechanism is applied that resets thequantum emitter into its ground state. This mechanismcan be the slow internal decay from the metastable to theground state, but it can also be enforced with, e.g., anoptical pump. In this way the investigated device acts asan integrated source of triggered single photons, at ratescontrolable by the applied pump. The description of thefunctionality of such plasmonically enhanced triggeredsingle photon source is at the heart of this contribution.

The paper is organized as follows. In Section II we in-troduce theoretical tools to explore the dynamics of thehybrid system of a quantum emitter and light modes sup-ported by the nanoantenna. In Section III we describe indetail a specic design of a nanoantenna that couples in-dependently to the two transitions of the quantum emit-ter. This simple design can be tuned in a straightforwardmanner to the target quantum system. Next, in SectionIV, a systematic analysis of the triggered emission of sin-gle photons is performed. The inuence of numerous pa-rameters of the hybrid system is investigated and clari-ed. In Section V, we identify a transition between twodistinguished regimes of the system's dynamics. Whiledecreasing the driving eld strength, the source loses itstriggered character and the time of emission becomesrandom. The paper is summarized in Section VI. Inthe following Appendix A, we compare our proposal toa selection of other triggered single photon sources. InAppendix B, we develop a simplistic eective picture byadiabatically eliminating the electromagnetic elds. Thisprovides an intuitive tool that hugely simplies numeri-cal simulations in the weak-coupling regime and grants amuch better study of the physics. What is crucial, it al-lows to obtain the coupling constants between the eldsand the quantum emitter. Finally, the adiabatic pictureleads to necessary and sucient conditions for an ecienttriggered single-photon emission.

II. THEORETICAL DESCRIPTION OF THE

HYBRID SYSTEM

In this Section we introduce the system and discussin detail the time-reversible processes described by itsHamiltonian, and various decay and decoherence mecha-

3

nisms accounted for via the Lindblad formalism.The system consists of a quantum emitter in a lambda

energy conguration [Fig. 1(b)]. The two low-energystates |1〉 and |2〉 are commonly referred to as "spinstates". They may have dierent energies ~ω|1〉 and~ω|2〉. We assume that state |1〉 is the ground state ofthe quantum emitter, in which the system is initiallyprepared. The excited state |e〉 has higher energy ~ω|e〉.We additionally assume that the transitions between thestates |1〉 (|2〉) and |e〉 are signicantly faster than the di-rect transition between states |1〉 and |2〉. For instance,the rst pair of transitions could be dipole-allowed, andthe latter electric-dipole-forbidden.The quantum emitter is coupled to a nanoantenna. Its

optical response is characterized by scattering and ab-sorption spectra. Here, we assume them to be well ap-proximated by two Lorentzian resonances in the spectralregion of interest. Since we will work at the single-photonlevel, we will apply the cavity-quantum-electrodynamicsapproach introduced in Ref. 49 and represent the reso-nances as two quantum-mechanical bosonic modes, withannihilation operators a1 and a2. We assume that theyare not directly coupled to each other, i.e. they are spec-trally well separated or correspond to orthogonal polar-izations. The modes are centered at frequencies ω1 andω2, respectively. Due to frequency separation or due topolarizations, mode j (j = 1, 2) couples only to the dipole-allowed transition between the state |j〉 and the excitedstate |e〉. The dipole-forbidden transition between thespin states is additionally assumed to be spectrally far-detuned from any resonance supported by the nanoan-tenna. The nanoantenna is illuminated by a laser of fre-quency ωL, that directly drives mode 1. The coupling tomode 2 shall be negligible. All these requirements willbe satised in excellent agreement by the explicit designwe later suggest.The corresponding Hamiltonian, in the frame rotating

with the frequency of the driving eld, reads:

H/~ =(ω|e〉 − ωL

)σee +

∑j

ω|j〉σjj (1)

+∑

j

(ωj − ωL) a†j aj

+∑

j

(κja†jσje + κ?j σejaj

)+(

Ωa†1 + Ω?a1

),

where σkl ≡ |k〉〈l| is a ip operator between the quantumemitter's states. The rst line in the above Hamiltoniancorresponds to the free evolution of the quantum emit-ter. The second line represents the free dynamics of theeld modes. The last line contains two terms: the rstone stands for the coupling of the nanoantenna modeswith the corresponding transitions in the quantum emit-ter. The coupling strength κj corresponds to the rateat which excitation is exchanged between the quantumemitter and mode j of the nanoantenna. A way to evalu-

ate κj for a given nanoantenna design will be described inSection III. The last term of the Hamiltonian representsthe drive acting on mode 1 of the nanoantenna with thestrength Ω, as described above. The value of Ω is pro-portional to the polarizability of the nanoantenna and tothe electric eld of the applied driving laser, as discussedin Ref. 11. Please note that we have neglected a directcoupling of the drive to the quantum emitter. This is anapproximation rather than an additional requirement orcomplication to the scheme. Due to generally large polar-izabilities of nanoantennas, the plane wave drive couplesto them signicantly more eciently than to quantumdots. Since the dot is position in the nanoantenna hotspot, the eld which it experiences is dominated by thestrong contribution scattered by the nanoantenna.The state of the full system (the quantum emitter and

the two modes) is given by its time-dependent densitymatrix ρ(t). The size of the density matrix is in prin-ciple innite, but in our calculations we obtained stableresults by restricting the Hilbert space to up to 10 pho-tons in mode 1, and 5 photons in mode 2. The evolutionof the system is governed by the Lindblad-Kossakowskiequation [50, 51]:

ρ(t) = −i/~ [H, ρ(t)] +∑

p

γpLCp[ρ(t)] , (2)

where the Lindblad superoperators:

LCp[ρ(t)] = Cpρ(t)C†p −

1

2

(C†pCpρ(t) + ρ(t)C†pCp

)(3)

stands for various incoherence mechanisms in our system.They are represented by the corresponding operators Cp

and rates γp. We consider a number of processes thatwe will describe in the following. We will additionallyprovide orders of magnitude corresponding to the specicnanoantenna design that we discuss in Section III, andfor possible quantum systems.Usually, the fastest process in the hybrid system cor-

responds to losses by the nanoantenna. We consider ra-diative (scattering of photons into the far-eld) and non-radiative contributions (absorption in the metal nanoan-tenna). Losses in mode j are given by the rate Γj =

Γ(j)rad + Γ

(j)nonrad. For metallic nanoantennas, the typical

order of magnitude is 1013 − 1014 Hz (see Section III,or Ref. 52). For dielectric nanoantennas, the scatteringrate is of the same order but it dominates over absorp-tion [53]. The scattering and absorption processes aredescribed by Lindblad operators Laj [ρ(t)].Spontaneous emission of the bare quantum emitter

from the excited state to the state |j〉 can be includedvia the term Lσje

[ρ(t)]. In this way, the correspondingdecoherence is naturally taken into account. The sponta-

neous emission rate is given by γ(ej)sp.em.. Usually, this pro-

cess is much slower than scattering and absorption in the

nanoantenna modes: γ(ej)sp.em. ≈ 107 − 109 Hz [42, 54]. As

4

we will show, the corresponding time-scale may be com-parable to the time-scale at which nanoantenna-inducedprocesses take place in the system. Therefore, this eectcannot be neglected.

A similar process Lσ12[ρ(t)] is related to nonradia-

tive population transfer from the metastable state |2〉 tothe ground state |1〉 (direct radiative transfer is dipole-forbidden). Normally, the corresponding intrinsic trans-fer rate γ12 is an order of magnitude smaller than theabove-described spontaneous emission rate [42]. How-ever, it can be boosted by pumping techniques.

The last process has to be taken into account if semi-conductor quantum dots are considered as the quantumemitters. Pure decoherence or dephasing, i.e. decay ofthe coherence between states |k〉 and |l〉 in the quantumemitter, represented in an o-diagonal element of its den-sity matrix, is given by the Lindblad term Lσkk−σll

[ρ(t)].In this process, the quantum-mechanical coherence is de-stroyed due to a coupling of the system to a uctuatingreservoir. In general, the impact of this process growswith the size of the quantum system. In the solid whichforms the quantum dot, the most important source ofdephasing is usually the phononic bath: the system mayacquire a random phase shift due to uctuating latticevibrations. The dephasing rate between the excited andone of the low-energy levels γje

deph varies for room tem-

peratures between 109 − 1012 Hz [41, 5558]. This is atleast two orders of magnitude smaller than scattering andabsorption in the nanoantenna, but can be comparableto coupling strengths κj and Ω. The dephasing rate be-tween the spin states is smaller γ12

deph ≈ 108 Hz [41]. Fora comprehensive discussion of quantum dephasing on anintroductory or an advanced level, please see respectivelythe Refs. 59 and 60.

To summarize this section, we introduced here thetheoretical tools within the Lindlad-Kossakowski formal-ism to describe the dynamics of a lambda-type quantumemitter in a vicinity of a nanoantenna that supports twoorthogonal modes. In Appendix B, we perform an adi-abatic elimination of the eld that leads to a simpliedpicture valid in the regime of large nanoantenna scatter-ing or absorption rates. This will be the regime in whichthe hybrid device will be operated.

In the following section, we propose a nanoantennathat supports modes in agreement with the previouslyformulated requirements. The nanoantenna will supporttwo modes that are orthogonal in polarization and shiftedin frequencies. In Section IV, we will show that couplinga lambda-type quantum emitter to such nanoantenna willresult in triggered single photon emission at nanosecondtime-scales, if a suciently strong drive is applied.

1.55 1.6 1.65 1.70

1

2

3

Psca [a rb. units ]s

Pabs [a rb. units ]

160nm

250nm

20nm

20nm

75nm

250nm

125nm

0nm

0nm 150nm

+1

-1

0

ℜ(E

90°)

[ar

b. u

nits

]

75nm

250nm

125nm

0nm

0nm 150nm

+1

-1

0

ℜ(E

90°)

[ar

b. u

nits

]

1.55 1.6 1.65 1.70

2

4

(a) (b)

(c)

(d) (e)

Fig. 2. (a) Detailed scheme of the nanoantenna describedin Section III. The quantum emitter is placed at the posi-tion indicated by the arrows. Their orientations correspondto four orientations of the dipole moment that were used tosimulate the scattering (b) and absorption (c) power spec-tra of the nanoantenna: blue line indicates a dipole emitterparallel to the longer nanorod, green, yellow, and red linescorrespond to the dipole rotated by 30, 60 and 90 degrees,respectively, as shown in panel (a). (d,e) Field distributionaround the nanoantenna in the plane 5 nm above the surfaceof the nanorods: (d) due to a dipole source at ω1 parallel tothe longer nanorod; (e) due to a dipole source at ω2 parallel tothe shorter nanorod. The component parrallel to the sourceorientation is shown.

III. NANOANTENNA

In this section we describe a specic plasmonic nanoan-tenna that serves for the purpose of mode conversion andof the triggered generation of single photons. By simu-lating absorption and scattering spectra using a classi-cal Maxwell solver, we calculate the related loss ratesand the coupling constants to adjacent quantum emit-ters with predened dipole moments. This methodologyis standard and has been previously applied to describesingle resonances [31, 47].

The proposed nanoantenna consists of two silvernanorods, perpendicular to each other [Fig. 2(a)], em-bedded in glass with a permittivity of ε = 2.25. Forsimplicity, the rods are modelled as cuboids. Their geo-metrical parameters, i.e. lengths of 250 and 160 nm andheights and widths of 20 nm each, have been adjusted tomatch with their resonances the infrared spectral regionat which the transitions appear in the emitters described

5

in Refs. 4043. Naturally, the resonances can be tunedaccording to the spectral properties of the specic quan-tum emitters. The spectral position of the two modes andtheir overlap can be inuenced by, e.g., a modication ofthe lengths of the two nanorods and their ratio. We areinterested in the optical response of such nanoantennasubject to an excitation by a classical dipole positionedat a point equidistant from the tips of both nanorods,with a distance of 13.5 nm. For a xed frequency, suchclassical dipole approximates the behaviour of a corre-sponding transition of the quantum emitter.To simulate the scattering and absorption spectra of

the nanoantenna, we solve the classical scattering prob-lem in the frequency domain. The driving frequenciesof the dipole source are xed within a spectral rangeof interest. For this purpose, we have used the COM-SOL Multiphysics simulation platform. The relative per-mittivity of silver based on the experimental data fromRef. 61 has been considered. Postprocessing consists ofcalculating the absorbed Pabs (ω) and scattered Psca (ω)powers. The former can be obtained by integration ofthe resistive losses:

Pabs (ω) =

ˆV

j (r, ω) ·E (r, ω) dV , (4)

where the volume V is restricted to the bulk of thenanoantenna. The scattered power is given by an in-tegration of the outward normal Poynting vector over aclosed surface A:

Psca (ω) =

ˆA

S (r, ω) · dA. (5)

In Fig. 2(b), the simulated spectra Pabs (ω) andP s

sca (ω) are presented. Here, P ssca (ω) is the power re-

lated to the Poynting vector of the eld scattered bythe nanoantenna. The blue (red) line corresponds tothe dipole moment of the emitter oriented horizontally(vertically). Green and yellow lines represent interme-diate orientations. Clearly, the nanoantenna supportstwo orthogonally polarised modes. They are centered atω1 = 2π×2.70×1014 Hz and ω2 = 2π×2.50×1014 Hz, re-spectively. As follows from Fig. 2(d) and (e), both modesdier in symmetry of their eld distribution and we canconsider mode 1 as associated with the longer, and mode2 - with the shorter nanorod. This means that, to a goodapproximation, with the proper orientation of the dipolesource we can address each mode individually. The per-pendicular arrangement of the nanorods implies distinctradiation patterns into perpendicular directions for thetwo modes of interest. This entails the opportunity todistinguish the emitted photons by frequency, polariza-tion, and propagation direction.To estimate the radiative and nonradiative loss rates

by the nanoantenna, we tted the scattering and absorp-tion spectra, respectively, with Lorentzian line shapes.

As a result, we obtained: Γ(1)rad = Γ

(1)nonrad = 6.8×1013 Hz,

Γ(2)rad = 1.0 × 1014 Hz, Γ

(2)nonrad = 2.2 × 1014 Hz. Please

note that the second mode is less pronounced in the ab-sorption spectra since it is spectrally actually very broad.Its large width causes the second resonance to be ratherat, i.e. barely visible in the absorption spectrum.To estimate the coupling constant κj, we exploit the

resonant version of the adiabatic expression (A4) fromAppendix B:

1 + η(j) 4|κj|2

Γjγ(ej)sp.em

=P t

sca (ωj)

P 0sca (ωj)

. (6)

On the right-hand side, we have expressed the same quan-tity as the ratio of the power related to the Poynting vec-tor of the total eld P t

sca (ωj) and the one correspondingto the Poynting vector of the illumination eld P 0

sca (ωj).Please note that Purcell enhancement [62, 63] is oftenunderstood as radiative enhancement of a localized pointsource into the far-eld [3]. In general this diers by

the nanoantenna eciency η(j) ≡ Γ(j)rad/

(Γ

(j)rad + Γ

(j)nonrad

)from the actual decay rate enhancement of the quantumemitter, which in turn can be used to estimate the cou-pling constant. For this reason we include the eciencyη(j) of the mode j into the expression (6). An eciencydeviating from unity accounts for the share of the energyextracted from the quantum emitter that is dissipated bythe nanoantenna and will not be scattered into the far-eld. The spontaneous emission rate of a bare emittercan be expressed by the Weisskopf-Wigner formula [64]

γ(ej)sp.em =

(ω|e〉 − ω|j〉

)3√ε|dej|2

3πε0~c3,

with the vacuum speed of light c and the transition dipolemoment of dej = 6× 10−29 Cm. This leads to a rate con-sistent with experimental values [42, 54]. With these as-sumptions, the coupling constants κj can be estimated tobe almost equal, κ1 = 5.73×1011 Hz and κ2 = 5.76×1011

Hz respectively. Please note that such coupling con-stants are huge with respect to those typically obtained inconventional cavities, and therefore allow to address thequantum emitter with light extremely fast. Nevertheless,the ratio of the estimated coupling constants to the lossrates supports the approach that we have adopted here:the nanoantenna losses are indeed dominant, and there-fore the adiabatic eective expression (A4), that servesas a basis to estimate the coupling constants, is well jus-tied and valid.With this exemplary design at hand, we can proceed

to investigate the dynamics of the system modelled bythe eective approach.

IV. TRIGGERED PHOTON EMISSION

In this section, we will study the dynamics of the emis-sion of a photon in mode 2 for a lambda-type quantum

6

0 1050

0.5

1.0

statestatestate

(b)

(d)

0 10

1e-6

2e-6

(c)1.0

0 1050

3e-5

6e-5

50

ngen

0

0.5

ngen

0

(a)

0 1050

0.5

1.0

(e)51 2 3 4

5

1

2

3

4 10

10-12

10-11

-10

P g (0)(2)

Fig. 3. Dynamics of a system consisting of a lambda-typequantum emitter coupled to a nanoantenna (described in Sec-tion III): (a) mean photon number in mode 1; (b) probabilitydistribution of the quantum dot; (c) mean photon number inmode 2 (blue solid line), and number of generated photonsngen proportional to its integration (purple dashed line) for

the case of neglected transfer rates γ(e2) = γ(12) = 0; (d)

ngen for included spontaneous emission rate γ(e2)sp.em. = 108 Hz

(blue solid line), and for additionally included transfer rate

γ(12) = 107 Hz (red dashed line). (e) Glauber second order

correlation function at zero time delay g(2)(0) calculated formode 2 in function of the drive Ω and the pump P proves thesingle-photon character of the generated eld.

emitter coupled to a nanoantenna described above. Wewill show that for a driving eld Ω of sucient inten-sity and duration, a triggered single photon emission isachieved. First, we will investigate the dynamics of thesingle photon generation, where for simplicity we willstart with a continous drive. Later, we emphasize thetriggered character by studying the response to a pulsedexcitation.

In Fig. 3, we plot the dynamics of the emission process,given by the eective Lindblad equation (A5), derivedin Appendix A. It is in good qualitative and quantita-tive agreement with the one obtained from the full Lind-blad equation (2). In both cases, we have used the freelyavailable QuTiP2 toolbox in Python [65] for calculations.Mode 1 of the nanoantenna is driven by a resonant lasereld Ω = 5×1011 Hz. To assure the excitation of mode 1by the external drive, a possibility would be to use a planewave polarized perpendicular to both nanorods. The di-

rection of propagation should be at 92 degrees relative tothe short nanorod and at −2 degrees relative to the longnanorod. These slight deviations from an arrangementthat adheres to the two nanorods are benecial to sup-press the excitation of mode 2 by more than 10 dB. More-over, the quantum emitter is suitably oriented such thatthe transition dipole moments of both transitions matchthe nanoantenna geometry. Both nanoantenna modesare assumed to be resonant with the quantum emitter'stransition frequencies: ω|e〉 − ω|1〉 = ω1, ω|e〉 − ω|2〉 = ω2.For this simulation we take into account small, but real-istic dephasing rates γ1e

deph = γ2edeph = 10γ12

deph = 109 Hz[41, 5558]. We include spontaneous emission from the

excited state to state |1〉 at rate: γ(e1)sp.em. = 108 Hz, but

set the spontaneous emission to state |2〉 and the popu-

lation transfer between the spin states γ(e2)sp.em., γ12 = 0.

Their inuence will be discussed later.The quantum emitter is initially set in its ground state|1〉. Both nanoantenna modes are assumed to be in vac-uum states. After switching on the drive, the mean num-ber 〈n1〉 = 〈a†1a1〉 of photons in mode 1 quickly reachesits stationary value [Fig. 3(a)], which is a result of atrade-o between driving and scattering and absorption.Note that the stationary value 〈n1〉steady ≪ 1. This isdue to huge losses characterizing metallic nanoantennas.However, this small value, that corresponds to a veryrare presence of photons, is enough to drive the transi-tion between states |1〉 and |e〉 of the quantum emitter[Fig. 3(b)]. As the excited state gets populated, transferto state |2〉 with a photon emission into mode 2, becomes

possible (here 〈n2〉 ≡ 〈a†2a2〉 is a very accurate approxi-mation of the probability of a single photon in this mode).State |2〉 is the nal state of the quantum emitter, asthe mechanisms that could in principle drive a transitionfrom state |2〉 to any other state, are very inecient.The population transfer γ12 due to internal collisions inthe quantum emitter, has been neglected for this calcula-tion. Due to huge values of scattering/absorption of thenanoantenna, the just-emitted photon quickly leaves thecavity, preventing re-absorption [Fig. 3(c)].To conrm that such dynamics corresponds indeed to

an emission of a single photon, we dene the followingfunction:

ngen (t) = Γ2

ˆ t

0

dt′〈a†2a2〉 (t′) . (7)

For nanoantennas operating in the weak-coupling regime,this quantity can be interpreted as mean number of pho-tons generated in mode 2 by the time t. In the absenceof transfer between the spin states γ(12) = 0, ngen isequal to the probability of photon emission. In the ideal-ized case of negligible free-space spontaneous emission tostate |2〉, the quantum emitter can only make the transferfrom |e〉 to |2〉 by emitting a photon in the desired mode

2, and therefore ngent→∞−→ 1 [Fig. 3(c), purple dashed

7

line]. However, for γ(e2)sp.em. > 0, the spontaneous emission

channel opens, and the probability of photon emissiondrops. The blue solid line in Fig. 3(d) corresponds to

a realistic γ(e2)sp.em. = 108 Hz. This demonstrates that in

case of metallic nanoantennas the correction to the prob-ability due to spontaneous emission is rather small: thesteady-state value of ngen (t→∞) ≈ 0.96. (It might beof much greater importance for dielectric nanoantenas,whose Purcell enhancement is signicantly smaller.) Fi-nally, we include the process of internal population trans-fer between the spin states γ(12) = 107 Hz. Now it is pos-sible that after the successful generation of a photon inmode 2, the quantum emitter is reset to its ground stateand the whole procedure repeats. As a result, the steady-state value of 〈a†2a2〉 (t→∞) > 0: it is about 1% of itsmaximal value for the set of data analyzed here. Anotherphoton may be generated in mode 2, and therefore afterthe steady state is reached, ngen grows linearly with a

slope equal to 〈a†2a2〉 (t→∞) [Fig. 3(d), red dashed line].It is interesting to note that for our set of data the inu-

ences of γ(e2)sp.em. and γ

(12)sp.em. happen to approximately can-

cel each other around the time when the system reachesits steady state.For completeness, we analyse the zero-delay second-

order correlation function for mode 2: g(2)(0) =

〈(a†2

)2

(a2)2〉/〈a†2a2〉2. Its value is strictly equal to zero

for single-photon Fock states, and can grow if a contri-bution from higher-number states is considerable. Forclassical light g(2)(0) = 1. For the scenario proposed inthis work the correlation function is equal to zero to avery good approximation, for physically realistic valuesof the drive Ω and the pump P [Fig. 3(e)]. First of all,this is because the nal state of the dot is a metastableone, preventing two subsequent photon generation acts.In principle, a second act of photon generation might beenabled, if a fast pump P from the nal state |2〉 to theinitial state |1〉 was applied to the lambda system, im-mediately recharging it after the rst successful photongeneration. However, even if such pump is applied, thesecond order correlation function will not grow, becauseof another reason: the time necessary to produce a pho-ton is much longer than its scattering/absorption time1/Γ2, with Γ2 κj,Ω. In other words, the rst pho-ton is either emitted or absorbed before another one isproduced.The photon generation in mode 2 is thus complete. It

is an on-demand process, i.e. it happens almost withcertainty. However, it must be clearly stated that thephoton may be afterwards emitted into the far-eld witha probability equal to the nanoantenna eciency η(2), orabsorbed otherwise. The emission is triggered, i.e. ittakes place in a short temporal window after the drivingeld Ω has been applied, i.e. at nanosecond time-scales.To emphasize the triggered character of the emission,

we analyze a pulsed excitation scheme, where a sequence

(a)

(b)

(c)

statestatestate

400 80

2

0

1

400 80

1.0

0

0.5

400 80

2e-6

0

1e-6

0

3

2

1

ngen

Fig. 4. A sequence of single photons emitted in the pulsed ex-citation scheme: (a) time-dependent drive Ω(t) and incoher-ent pump between the metastable states P (t); (b) dynamics ofthe population distribution of the quantum emitter; (c) mean

number 〈a†2a2〉 of photons in mode 2 (red) and of generatedphotons ngen (black line).

of driving pulses of Gaussian temporal proles is ap-

plied: Ω (t) = Ω0

∑i exp

[− (t− ti)2

/2τ2], of amplitude

Ω0 = 2 THz, widths τ = 1 ns, centered at t1 = 4 ns,t2 = 28 ns, t3 = 64 ns, respectively. In between, Gaus-sian pump pulses are applied that transfer the populationfrom the metastable state |2〉 to the ground state |1〉:P (t) = P0

∑i exp

[−(t− ti

)2/2τ2

], of the same ampli-

tude P0 = 2 THz, same widths and centers at t1 = 16 nsand t2 = 40 ns [Fig. 4(a)]. We include the pump asan additional incoherent term P (t)Lσ12

[ρqd(t)

]. All the

other parameters are the same as above, and all decayand dephasing channels in the quantum emitter are in-cluded. Figure 4(b) illustrates how the population of thequantum emitter is ipped when the pulses are applied,leading with high probability to subsequent emissions ofsingle photons directly after each application of a driv-ing pulse [Fig. 4(c)]. The nal value of ngen ≈ 2.94, andwould be equal to 3 were the decay and dephasing inthe quantum emitter neglected. This demonstrates thatthese processes play only a marginal role.

In this section we have discussed in detail the perfor-mance of the proposed nanoantenna coupled to a lambda-type quantum emitter, as a triggered source of single pho-tons. Below, we will show that the driving eld Ω is thecrucial parameter that controls the quality of the lightsource. Only for sucient and reasonably large values,

8

2000 400

2e-6

4e-6

0

(a) (b)

1000x

0.5

00 0.2 0.4

1.0

statestatestate

Fig. 5. (a) As in Fig. 3(c), but for a weaker Ω = 2× 1010 Hz(blue line, multiplied 1000 times) and stronger Ω = 2 × 1012

Hz (red line) drive. (b) Dependence of the stationary pop-ulation distribution in the quantum emitter on the drive Ω,for the nanoantenna parameters corresponding to the designof Section III, and quantum-emitter parameters given at thebeginning of Section IV.

a triggered single photon emission is achieved. For toosmall values, the emission becomes random and loosesits triggered character. The eective description intro-duced in Section VI will provide conditions that need tobe fullled for an ecient, triggered emission of a singlephoton.

V. IMPACT OF THE DRIVE

For the calculation in Fig. 3, we have assumed a cer-tain constant value of Ω = 5 × 1011 Hz, that is strongenough for an ecient emission. As an indicator of suc-cessful single-photon generation we have taken the prob-ability close to 1 of the quantum dot being transferred tostate |2〉. The eciency of emission is, however, a com-plicated function of the parameters of the nanoantenna,location and orientation of the quantum dot, and of thedriving eld intensity. In a given experiment, the lattercan still be tuned once the nanoantenna is produced andthe quantum emitter positioned. To illustrate the im-pact, we have plotted in Fig. 5(a) the dynamics of theeld in mode 2 for a substantially weaker [Ω = 2 × 1010

Hz, blue line] and stronger [Ω = 2 × 1012 Hz, red line]drive. In the former case, the excitation turns out to betoo weak to successfully ip the quantum emitter into itsstate |2〉 and prevent its come-back to the initial state,leading to an emission of a series of photons at randomtimes rather than a single triggered emission. We willrefer to this case as the random-emission regime. Inthe latter case, the photon is eciently generated, with atemporal shape that shows multiple peaks (their numberincreasing with the driving eld), leading to a time-bintype emission [66].

To identify a threshold the drive needs to overcome forecient triggered photon generation, we have exploitedthe eective formalism described in the Appendix. Wehave obtained a condition, which reads for the resonance

δ1 = 0:

|Ω|≫ Γ1

√Γ2γ12

2|κ1κ2|

(|κ1|2

Γ1+|κ2|2

Γ2

), (8)

where we have assumed |κj|2/Γj ≫ γ(ej)sp.em., γ

ijdeph. For a

sketch of the derivation, please see Appendix B.The right-hand-side of the expression above is equal to

approximately Ωthreshold = 4 × 1010 Hz for the nanoan-tenna design discussed in Section III. In Fig. 5(b), weplot the stationary population distribution of the quan-tum emitter as a function of the drive Ω, for the nanoan-tenna parameters given in Section III, and quantum emit-ter dephasing rates dened at the beginning of this sec-tion. The result was obtained in the eective formalism.Please note that the value of Ωthreshold indicates approxi-mately the driving eld strength for which the stationarypopulation is equally distributed between the spin states|1〉 and |2〉. However, to fulll the condition (8), thedrive needs to be signicantly stronger than Ωthreshold.This is in accordance with Fig. 5(b): the requirement ofρ22 ≈ 1 is fullled for drives at least an order of magni-tude larger. Otherwise, if the driving eld is too weak,a signicant part of the stationary population remainsin the ground state |1〉, from which it can be re-excited,leading to a sequence of photon emission acts in mode 2at random moments. The device in such modus operandimight be considered for continuous conversion betweennanoantenna modes via the quantum emitter.Please note that even though the condition in Eq. (A9)

has been derived in the eective formalism, the systemdoes not fail to act as a single photon source for verystrong drives, for which the eective formalism itself isno longer valid. This can be clearly seen from Fig. 3(e)and its extensions to even larger Ωs and P s, which hasbeen obtained within the full Hamiltonian picture.In this section we have exploited the eective formal-

ism to formulate an analytical condition of successfulsingle-photon generation in the desired mode: switchingbetween two working regimes of triggered and random-time photon emission may be possible by tuning the driv-ing eld intensity. For strong driving elds, a time-binsingle photon can be produced.

VI. CONCLUSIONS

The scope of this work expands beyond the regime ofsingle-mode nanoantennas, that is usually considered inthe context of coupling to quantum emitters. Here, themultiresonant character of the nanoantenna spectrum isexploited for the purpose of inter-modal conversion. Thisis possible due to a coupling to a lambda-type quantumemitter. Specically, we have considered the design of anL-shaped plasmonic nanoantenna, with a spectrum rep-resented by two independent modes, centered at the tran-sition frequencies of the quantum emitter. An analysis

9

of these spectra has provided the parameters responsiblefor the dynamics of the coupled system.We have applied the eective description within the

adiabatic approximation to nd a quantitative distinc-tion between two regimes of the dynamics, depending onthe intensity of the drive. A driving eld strong enoughleads to a desired, triggered single-photon generation ina well-dened, short temporal window, directly after theapplication of the drive. Depending on the pump, suchtriggered emission may achieve high rates. A weakerdrive results in a regime of continuous inter-modal con-version, with emission of subsequent quanta at randommoments in time.The problem of multimodal nanoantennas coupled to

possibly multilevel quantum emitters should be furtherinvestigated in diverse contexts. Potential generaliza-tions involve an analysis of light properties other thanquantum statistics, e.g., entanglement in degrees of free-dom related to particular modes. Furthermore, we haveset the grounds for an important generalization to thecase of a quantum-emitter transition coupled to morethan one spectrally detuned nanoantenna mode. It willbe the subject of our work in the nearest future.

APPENDIX A: PARAMETER TABLE AND

COMPARISON

To give an overview over the most important parame-ters and most expressive properties of our proposed singlephoton generation scheme, we assembled these informa-tion and listed them in a tabular form. Furthermore inTable I we juxtapose these data in opposition to bothpreviously proposed single photon sources utilizing sim-ilar principles [38] as well as state-of-the-art experimen-tal single photon sources [43, 67, 68]. Please note thatRef. 43 is also based on lambda-type quantum dots.The comparison of the dierent parameters and prop-

erties displayed in Table I claries both advantages anddrawbacks of making use of a metal nanoantenna in com-bination with a lambda-type quantum emitter. Firstof all, the anticipated zero-delay second-order correla-tion function g(2)(0) is extremely small due to the weak-coupling regime our source operates at: The cavity emis-sion rate can be increased by multiple orders of magni-tude in comparison to other single photon sources us-ing the proposed setup. Additionally, the single photonrate as well as the cavity coupling constant exceed theperformance data of alternative single photon generationschemes. The major disadvantage to all these benetsis the single photon extraction eciency, which suersnaturally from the Ohmic losses of the metal nanoan-tenna. But even though these undesired, unavoidablelosses are considered to be the dominating detriment tousing metallic structures that support surface plasmonpolaritons, the resulting extraction eciency decreases

only by a factor of 3 compared to the theoretical op-timum without any Ohmic losses [38]. This is in so farsurprising as that one of the alternative experimental sin-gle photon generation setups possesses an even slightlylower eciency, even though its operation is based on avery dierent physical mechanism.Finally, the proposed plasmonic single photon genera-

tion scheme provides further advantages. While the ex-citation of the lambda-type quantum emitter can be re-alized with classical resonant laser illumination, the trig-ger can be initiated by a resonant laser pulse, that ispurely classical too. This allows the dynamic, externalcontrol of the single photons emission with smallest pos-sible eort. Unlike some of the alternative single photonsources, the proposed scheme neither requires any π-pulseillumination nor any kind of cryostat. These should beobvious benets regarding the experimental realization.And lastly the proposed setup is also at least one order ofmagnitude smaller in volume than any of the other singlephoton generation setups. This is why we consider it avery promising candidate for any kind of integrated pho-tonic device relying on the triggered emission of singlephotons.

APPENDIX B: EFFECTIVE HYBRID SYSTEM

DESCRIPTION

The goal of this section is to derive an eective de-scription of the investigated system within the adiabaticapproximation. The reduced size of the eective Hilbertspace hugely reduces the numerical simulation costs. Aneven more important benet is that this picture allows toobtain the coupling constants κj. Moreover, it drasticallysimplies analytical calculations, so that it becomes fea-sible to derive the condition (8) on ecient single-photongeneration.If a part of the physical system undergoes a decay at

a rate much larger than other characteristic parametersof the dynamics, it can be eectively eliminated fromthe evolution [69, 70]. This is due to a mismatch in thetime scales for the fast-decaying part and the remain-ing part of the system. With such elimination, the sizeof the physical problem can be drastically reduced. Inthe system investigated here, it is the electromagneticeld that undergoes huge scattering and absorption andwhich evolves at much shorter time scales. To obtainthe eective description for the evolution of the quan-tum emitter, we integrate the equations of motion of theeld operators in the Heisenberg picture and plug the re-sulting adiabatic expression back into the equations de-scribing the evolution of the quantum emitter. In thelast step, we transform the resulting equations back tothe Schrödinger picture. A similar elimination has beenexplicitly described in Refs. 14 and 64.Such procedure leads to a set of eective Bloch equa-

10

Table I. List of important parameters and expressive properties and comparison to other single photon sources.

parameter / proposed design Kuhn et al. [38] Ding et al. [67] Rambach et al. [68] Vora et al. [43]

property theory theory experiment experiment experiment

2nd order

correlation g(2)(0) ∼ 10−11 − 0.009 − 0.14

single photon

emission rate(∼ 3.3× 108 Hz

)a ∼ 1.5× 106 Hz 3.7× 106 Hz 1.2× 108 Hz 1.5× 105 Hz

cavity emission

rate ∼ 1.0× 1014 Hz ∼ 2π × 1.5× 106 Hz − 2.4× 108 Hz 4.6× 1010 Hz

cavity coupling

constant 238.2 µeV 0.013 µeV − − 57 µeV

single photon

extraction eciency (25.0%)b (73.6%)b 66% (23.2%)b −

external classical excitation pulsed excitation pumped by SHG-

excitation or & pulse to reset & pulse to reset excitation cavity driven by pulsed laser

control scheme to initial state c to initial state by π-pulses CW-laser excitation

additional variable trigger cryogen-free multiple electro-

experimental pulse length for bath cryostat optic modulators closed-cycle

requirements full functionality − (4.2 . . . 30) K & frequency locks helium cryostat

approx. volume L-shaped antenna, high nesse cavity, micropillar, SPDC cavity, photonic crystal,

of setup ∼ 0.003 µm3 ∼ 0.01 µm3 ∼ 177 µm3 > 1012 µm3 ∼ 0.4 µm3

a limited by spontaneous emission rateb assuming a single photon detector with 80% eciencyc additional level of control: dierent behavior depending on choice of external drive and external trigger pulse

1❭

Ωeff

2❭

e❭

γeffγ1γeffγ2

γ12

γ1eγdeph

γ2eγdeph

γ12γdeph

Fig. 6. The Jablonski diagram of the quantum emitter in theeective picture. Please compare to Fig. 1(b).

tions for the three-level quantum emitter solely. The ef-fective Hamiltonian describes a coupling to a single co-herent eld driving the transition between the groundand the excited state [Fig. 6]:

Heff/~ =(ωeff|e〉 − ωL

)σee + ω|1〉σ11 + ω|2〉σ22 (A1)

+Ωeffσe1 + Ωeff?σ1e,

where

ωeff|e〉 = ω|e〉 +

∑j=1,2

|κj|2 (ωj − ωL)

(ωj − ωL)2

+ (Γj/2)2, (A2)

is the energy of the excited state, shifted due to thenanoantenna eld in a process analogous to the vacuum-induced Lamb-shift, and

Ωeff =−iκ?1Ω

i (ω1 − ωL) + Γ1/2, (A3)

is the eective driving eld.Coupling to nanoantenna modes, originally coherent,

leads in the eective picture to a modication of the pop-ulation decay rates of the quantum emitter:

γeffj = γ(ej)

sp.em. +|κj|2Γj

(ωj − ωL)2

+ (Γj/2)2. (A4)

The process is eectively incoherent and accounted for ina Lindblad term, because the nanoantenna modes decayfast enough to prevent any back action, i.e. a reabsorp-tion of the emitted photon. At this point we would liketo point out, that Purcell enhancement and modicationof population decay rate concur in case of nanoantennasthat do not suer from Ohmic losses [see Eq. (6)].The adiabatic dynamics can be found by solving the

eective Lindblad-Kossakowski equation:

ρqd(t) = −i/~[Heff , ρqd(t)

]+∑

p

γpLeffCp

[ρqd(t)

], (A5)

11

0.2

0.4

0.6

0.04e3 8e3

full Hamiltonianeffective Hamiltonian

0 4e3 8e300

1e-4

2e-4

06 120

0.05

0.10

050 1000

0.05

0.10

0.15

50 10000

3e-3

6e-3

6 1200

0.1

0.2

(a) (b)

(c) (d)

(e) (f)

Fig. 7. Evolution of the population in the state |e〉 (left) andphoton number in mode 2 (right) in the full-Hamiltonian andeective description. (a,b) Very good agreement for smallγ12,Ω, κj, (c,d) qualitative, but not quantitative agreementin an intermediate case, (e,f) breakdown of the adiabatic ap-proximation for values of parameters comparable to Γj. Ex-act values of parameters chosen for simulations are providedin the main text.

where ρqd(t) is the density matrix of the quantum emit-ter. The eective Lindblad term includes the modiedpopulation decay rates γeff

j , as well as unchanged spin

population decay γ12, and dephasing rates γijdeph.

Additionally, we can track the excitation dynamics ofthe nanoantenna, following an adiabatic expression forthe mean number of photons in each of the two modes:

〈a†1a1〉(t) =|κ1|2ρee(t) + [Ωκ?1ρ1e(t) + Ω?κ1ρe1(t)] + |Ω|2

(ω1 − ωL)2

+ (Γ1/2)2 ,

〈a†2a2〉(t) =|κ2|2ρee(t)

(ω2 − ωL)2

+ (Γ2/2)2 . (A6)

In Fig. 7, we test the validity of the eective descrip-tion, for simplicity assuming real κj and Ω parameters.The evolution of the excited state |e〉, as well as the pho-ton number in the desired target mode 2, calculated inthe eective picture, are compared to those obtained fromthe full Lindblad-Kossakowski equation (2). For the va-lidity analysis, we have chosen the population of state|e〉, because the dierence between the results obtainedin the two pictures is mostly manifested in this popula-tion. For this test, resonances in both transitions and

of the driving eld have been assumed, and decay anddecoherence mechanisms of bare quantum emitters areneglected.We nd almost perfect agreement as long as the cou-

pling, pump, and drive is two orders of magnitude smallerthan the nanoantenna loss rates, i.e. κj = γ12 = 0.01Γj,Ω = 0.05Γj [Fig. 7(a,b)]. A qualitative agreement, withslight dierences in numbers, is obtained for parametersenhanced by an order of magnitude: κj = γ12 = Ω =0.1Γj [Fig. 7(c,d)]. If loss rates of the nanoantenna arecomparable to other parameters κj = 2γ12 = 2Ω = Γj,the validity of the adiabatic approximation breaks down[Fig. 7(e,f)] and predictions in the eective descriptiondo not properly predict the exact evolution.Similar conclusions have been reached while modify-

ing other parameters, including dephasing rates, driv-ing eld, and detunings. In general, we have conrmedthat the eective description accurately describes the dy-namics when the scattering and absorption rates of thenanoantenna dominate among the parameters character-izing the system. This is in full agreement with the as-sumption of mismatched time-scales, made at the begin-ning of this section to justify the adiabatic elimination.We will now exploit the eective formalism to derive

the threshold (8), that the drive needs to overcome forecient transfer of the quantum dot into the desired -nal state |2〉, which corresponds to a triggered photongeneration.From stationary eective Bloch equations, given by

ρqd = 0, we derive the stationary population of state|2〉: ρqd

22 = (1 + ξ)−1

, where ρqd stands for the station-ary density matrix of the quantum dot in the eectivepicture. We have denoted

ξ(Ωeff

)=γ12

γeff2

[2 +

(γeff

1 + γeff2

) (Γ2

1e + 4δ21

)4Γ1e|Ωeff |2

], (A7)

with Γ1e = γeff1 + γeff

2 + γ12deph + 4γ1e

deph + γ2edeph, and δ1 =

ωeff|e〉−ωL−ω|1〉. For an ecient transfer of the quantum

dot into its state |2〉, we must have

ξ(Ωeff

)≪ 1. (A8)

This can be understood as a condition that the drivingeld should fulll. Typically, the ratio γ12/γeff

2 ≈ 0.1in free space, and is even smaller in the presence of ananoantenna. Therefore, condition (A8) can be regardedas always fullled if the driving eld is strong Ω → ∞.For weak elds, we can simplify the above condition

to |Ωeff |2 ≫ γ12

4γeff2

(γeff

1 + γeff2

)Γ1e, where a resonance

δ1 = 0 was assumed. In terms of original parameters,the condition of ecient state transfer can be approxi-mately expressed in Eq. (8):

|Ω|≫ Γ1

√Γ2γ12

2|κ1κ2|

(|κ1|2

Γ1+|κ2|2

Γ2

), (A9)

12

where we have considered the case of a nanoantenna-induced decay rate dominant over other decay and de-coherence mechanisms in the quantum dot |κj|2/Γj ≫

γ(ej)sp.em., γ

ijdeph.

ACKNOWLEDGEMENTS

The study was supported by a research fellowshipwithin the project Enhancing Educational Potentialof Nicolaus Copernicus University in the Disciplinesof Mathematical and Natural Sciences (project no.POKL.04.01.01-00-081/10.) and the Karlsruhe School ofOptics and Photonics (KSOP). Partial support by theGerman Science Foundation (within SPP 1391 UltrafastNano-optics) is acknowledged.

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