Lamb shift multipolar analysis
Emmanuel Lassalle, Alexis Devilez, Nicolas Bonod, Thomas Durt, Brian Stout
LLamb shift multipolar analysis
Emmanuel Lassalle, Alexis Devilez, Nicolas Bonod, Thomas Durt, and Brian Stout ∗ Aix Marseille Univ, CNRS, Centrale Marseille, Institut Fresnel, Marseille, France (Dated: November 5, 2018)It is now well established that radiative decay of quantum emitters can be strongly modified bytheir environment. In this paper we present an exact — within the weak-coupling approximation —multipole expression to compute the Lamb (frequency) shift induced by an arbitrary set of resonantscatterers on a nearby quantum emitter, using multi-scattering theory. We also adopt a Quasi-Normal Mode description to account for the line shape of the Lamb shift spectrum in the near-fieldof a plasmonic nanosphere. It is then shown that the Lamb shift resonance can be blue-shifted as thesize of the nanoparticle increases, suggesting that nanoparticles may be used to tune this resonantinteraction. Finally, a realistic calculation of the Lamb shift is made for a dimer configuration.
I. INTRODUCTION
Control of the decay properties of quantum emitters via modifications of their local electromagnetic environ-ment is being actively pursued due to the rich perspec-tives it offers for both fundamental and practical appli-cations [1]. In the weak-coupling regime, the exponentialdecay in time of the excited state is characterized by thedecay rate, for which it is well known that it can beeither enhanced [2] or inhibited [3] by the local electro-magnetic (EM) environment. With the convergence ofcommunities such as near-field optical microscopy, semi-conductors, plasmonics, and metamaterials, engineeringthe quantum vacuum allows tailoring the decay rate inunprecedented ways [4, 5]. A less often discussed effectof spontaneous emission is that the surrounding environ-ment also induces level shifts of the excited atomic states,resulting in a frequency-shift for the emitted photons, incomparison with the bare resonance frequency. This isthe so-called
Lamb shift , which originally refered to levelshifts of atoms in free space [6, 7], also called radiativefrequency-shift or Casimir-Polder frequency-shift. Thiseffect has been theoretically studied in the case of perfectreflectors [8], partially reflecting surfaces [9, 10] and pho-tonic crystals [11–13]. Multipole formulas of the Lambshift have been derived in the case of a dielectric mi-crosphere without [14, 15] and with [16] absorption, andfor dielectric or metallic prolate spheroids [17]. How-ever, there is no such formulas in multi-scattering config-urations, except in the case of two-dimensional photoniccrystals [18].In this article, we derive — using the generalized Mietheory [19, 20] — a multipole formula for the Lamb shiftof a quantum emitter induced by an arbitrary set of scat-terers. This formula is exact within the weak-couplingapproximation and does not take into account non-localeffects which come into play for emitter - particle dis-tances below one nanometer [21].Section II justifies the use of a classical formalism tostudy the Lamb shift induced by the presence of mat-ter by showing, in the weak-coupling approximation, its ∗ [email protected] equivalence to the fully quantum result. An exact mul-tipole formula for the Lamb shift is then derived in sec-tion III and illustrated in section IV by computing theLamb shift in the vicinity of a silver nanosphere, where wealso show that the spectral line shape of the Lamb shiftcan be accounted for in the context of a “Quasi-NormalMode” description. In section V, we study the influenceof the nanoparticle’s size on the environmentally inducedLamb shift, and we predict a displacement of the emit-ter’s Lamb shift resonance as the size of the nanoparticlechanges. Finally, as a practical calculation, we computethe Lamb shift in the case of a dimer nanoantenna. II. ENVIRONMENTALLY INDUCED LAMBSHIFTA. Classical approach
An excited two-level atom with transition frequency ω and natural linewidth γ can be modeled by a har-monically oscillating point dipole , whose electric dipolemoment p ( r , t ) obeys, in the case of small damping ( γ (cid:28) ω ) [22]: d p ( r , t ) dt + γ d p ( r , t ) dt + ω p ( r , t ) = q m E s ( r , t ) , (1)where { ω , γ , q , m } are the characteristics of the clas-sical dipole (the natural frequency of the oscillator, thedamping constant in the homogeneous background, thecharge and the mass respectively) and E s ( r , t ) is thefield scattered by the environment at the dipole position r . Adopting the following ansatz : (cid:26) p ( r , t ) = p e − iΩ t E s ( r , t ) = E s ( r , ω ) e − iΩ t where (cid:26) Ω = ω + ∆ ω − i γ ∆ ω = ω − ω , (2) with γ and ω respectively indicating the new decay rateand resonance frequency, together with the weak-couplingapproximation in a classical context, q m | E s | (cid:28) ω | p | , (3)one finds the following expression for the frequency-shiftof the light emitted by the dipole due to the environment a r X i v : . [ phy s i c s . op ti c s ] A p r [22]:∆ ωγ (cid:12)(cid:12)(cid:12)(cid:12) ω = − π(cid:15) ε b k × | p | × Re( p ∗ · E s ( r , ω )) , (4)where k = n b ( ω /c ) is the wave-number of the nonab-sorbing homogeneous background medium of refractiveindex n b = √ ε b . In this classical picture, one can seefrom Eq. (4) that the environment contribution to thefrequency-shift is due to the dipole interacting with itsown electric field scattered back by the environment.To link this expression with the quantum one, one canderive the dipole fields using the Green-function formal-ism (for the sake of simplicity, we consider the dipoleemitter to be in vacuum: ε b = 1). The field producedat r by a point dipole located at r and with naturalfrequency ω is [22]: E ( r , ω ) = ω µ ↔ G ( r , r , ω ) · p , (5)where ↔ G denotes the dyadic Green tensor. By separat-ing the Green tensor into an “unperturbed” ↔ G plus a“scattering” ↔ G s contributions [22], ↔ G = ↔ G + ↔ G s , (6)Eq. (4) can be cast in terms of the scattering Green ten-sor: ∆ ωγ (cid:12)(cid:12)(cid:12)(cid:12) ω = − πcω × u p · Re( ↔ G s ( r , r , ω )) · u p , (7)with u p being the unit vector in the direction of the dipolemoment: p = p u p . B. Quantum approach
In a quantum approach, the excited two-level atom ismodeled by its state vector | e (cid:105) , and its interaction withthe electromagnetic field is represented by an interactionHamiltonian ˆ H I . The weak-coupling approximation in aquantum context consists of considering that the matrixelements of the interaction Hamiltonian are small com-pared to those of the non-interacting Hamiltonian ˆ H .Therefore, the energy level shift ∆ E of the excited atomicstate is calculated by using the usual perturbation the-ory to second order in the perturbation ˆ H I . Besides, byusing the fluctuation-dissipation theorem, one can showthat the energy-shift of the first excited state | e (cid:105) of barefrequency ω is [10]:∆ E | ω = − ω π(cid:15) c p i p j P (cid:20)(cid:90) + ∞ d ω Im ( G ij ( r , r , ω )) ω − ω (cid:21) , (8)where P denotes the principal value of the integral, p = (cid:104) g | ˆ p | e (cid:105) (ˆ p being the dipole moment operator and | g (cid:105) the ground state vector) is the transition dipole matrix element, and G ij is the previous classical Green tensor(let us note that the notation ↔ G used in [10] is the fieldsusceptibility that we call ↔ F , and which is related to theGreen tensor by ↔ F ( r , r (cid:48) , ω ) ↔ ω µ ↔ G ( r , r (cid:48) , ω )). By us-ing the Kramers-Kronig relations for the Green tensor,and separating as previously the Green tensor into twocontributions, one can cast the frequency-shift resultingfrom the energy level shift induced by the presence ofmatter, in the form:∆ ω | ω = − ω (cid:126) (cid:15) c p i p j Re (( G s ) ij ( r , r , ω )) + QC . (9)Except for the non-resonant quantum correction termQC which is negligibly small [16], this expression hasthe same form as the classical formula provided that onenormalizes by the quantum decay rate in free space γ = ω | p | π(cid:15) (cid:126) c , (10)because the normalization eliminates the dependency on p and provides a safe link between quantum and classicalformalisms.Thus, in the weak-coupling regime , the quantum treat-ment gives the same result as the classical treatmentwhen considering the normalized frequency-shift — thatwe will call Lamb shift in the following — between theground state and the first excited state (to consider otheratomic levels, the classical treatment and the two-levelatom model fail, and one must refer to the generalformula derived in [10]). Note that for an absorbing medium, characterized by an imaginary part of its per-mittivity, this equivalence still holds, because on onehand, in the classical approach developed in terms of theGreen tensor, the permittivity can become complex, andin a quantum context, the link between the ground-statefluctuations of the electric field and the classical Greentensor remains the same [16, 23, 24]. III. MULTIPOLE FORMULA FOR THE LAMBSHIFT
Now we move to the derivation of the exact multipoleformula for the Lamb shift induced by an arbitrary set ofresonant scatterers on a nearby quantum emitter. Onecan see from Eq. (4) that the Lamb shift induced by thesurrounding environment is embodied in the field scat-tered by the environment E s , which can be calculatedfrom the scattering part ↔ G s of the total Green tensorthrough Eqs. (5) and (6). The determination of ↔ G s isthus the chief obstacle to the calculation of the Lambshift. From a classical viewpoint, the scattering Greentensor ↔ G s must take into account the multiple scatteringof the incident radiation from all the scatterers. There-fore, for the purpose of calculation, it is advantageousto express the scattering Green tensor in terms of themultiple-scattering T-Matrix [25], where the T-Matrix isdefined in operator notation as ↔ G s = ↔ G ↔ T ↔ G = ↔ G N (cid:88) i =1 ,j =1 ↔ T ( i,j ) ↔ G , (11)and has been split into N operators ↔ T ( i,j ) (that rep-resent all multiple-scattering events from a multiple-scattering viewpoint [19]), i and j being the particle la-bels, and N the total number of scatterers.In order to calculate the ↔ T ( i,j ) operators, we will makeuse of the multipolar fields — also called multipolarmodes or multipoles — which are a set of basis EM modesthat are especially useful in describing EM scattering forparticles with spherical symetries [26]. We will denote amultipolar field as | Ψ q,n,m (cid:105) , each mode being specifiedby three discrete numbers: q accounts for the parity ofthe field, and q = 1 for a magnetic mode and q = 2for an electric mode; n = 1 , , ..., ∞ and will be calledthe ”multipolar order”; and m = − n, ..., n and will becalled the ”orbital number”. Explicit representations ofthese modes can be found in [26], and here the fields andoperators will be expressed in the basis of the multipolarfields satisfying the outgoing boundary conditions (calledthe Hankel multipolar fields in [26]), that we will note M nm ( k r ) for the magnetic modes ( q = 1) and N nm ( k r )for the electric modes ( q = 2) in the real space represen-tation.The ↔ T ( i,j ) operators are then expressed in the multi-pole basis [25]: ↔ T ( i,j ) = 2 (cid:88) q,q (cid:48) =1 ∞ (cid:88) n,n (cid:48) =1 n (cid:88) m = − n n (cid:48) (cid:88) m (cid:48) = − n (cid:48) | Ψ q,n,m (cid:105) T ( i,j ) q,n,m ; q (cid:48) ,n (cid:48) ,m (cid:48) (cid:104) Ψ q (cid:48) ,n (cid:48) ,m (cid:48)| , (12) and can be calculated from the infinite dimensional T ( i,j ) matrices, that can be rendered finite by truncating themultipolar order n to some finite dimension n cut (thechoice of n cut for which the summation with respect tothe multipolar order n converges will depend on particlesize and interaction strengths). Several methods exist forcalculating the T ( i,j ) matrices, and we use the analyticalbalancing techniques detailed in [20] and implementedin an in-house code used for the numerical simulationsof this article. Once the on-shell T ( i,j ) matrices havebeen determined, one can compute the expression of theelectric field E s ( r , ω ) scattered by the environment byemploying Eq. (11) in Eqs. (6) and (5): E s ( r , ω ) = i p kω (cid:15) c N (cid:88) i,j =1 (cid:104) [ M ( k r i ) , N ( k r i )] t T ( i,j ) H ( j, f (cid:105) , (13) where [ M , N ] is a column matrix composed of the M nm and N nm functions, f represents the dipolar source anddenotes a column matrix containing the emitter coeffi-cients in the multipole space, and H ( j, is the irregulartranslation-addition matrix between the emitter positionat r and the position of particle j (for more details, seethe derivation of Eq. (19) in [25]). Finally, the expression of E s ( r , ω ) can be utilized inEq. (4) to obtain the multipole expression for the normal-ized Lamb shift induced by the presence of N scatterers:∆ ωγ = 3 π × Im N (cid:88) i,j =1 f t H (0 ,i ) T ( i,j ) H ( j, f . (14)In the case of a single particle ( N = 1), Eq. (14) takesthe form: ∆ ωγ = 3 π × Im (cid:16) f t H (0 , tH (1 , f (cid:17) , (15)where t is the single-particle T-Matrix. In the case ofa spherical Mie scatterer, t is a diagonal matrix com-posed of the Mie coefficients of the sphere (given in Ap-pendix VII A), and Eq. (15) is then equivalent to ex-pressions previously derived for a single sphere [15, 16].Exact analytical expressions of the first two multipolarcontributions to the Lamb shift can be found in Ap-pendix VII B. IV. MULTIPOLAR ANALYSISA. Multipole contributions to the Lamb shift
Let us first calculate the Lamb shift in the case of a sil-ver nanosphere of radius a = 20 nm in vacuum ( n b = 1).Based on Eq. (15), we compute using an in-house codethe Lamb shift of a quantum emitter radially orientedand located at a distance d = 10 nm from the nanopar-ticle, as a function of the bare transition wavelength λ = 2 πc/ω (black curve in Fig. 1). We analyze thisLamb shift spectrum by plotting separately the differ-ent multipolar contributions (plotted in colors in Fig. 1: n = 1 corresponds to the contribution of the dipolarmode, n = 2 to the contribution of the quadrupolar modeand so on). One can thus see that in the near-field ofthe nanoparticle, the total Lamb shift is due to the con-tribution of several multipolar modes and the fact thatthe dipole approximation to model the response of thenanoparticle (corresponding to the red curve in Fig. 1)fails to account for the Lamb shift. In other words, in thenear-field region, the atom couples to several plasmonmodes of the silver nanoparticle (see also [27]), whichgives rise to the complex pattern of the Lamb shift spec-trum.In order to account for the spectral line shape, we willmake use of the analytical expressions of the dipolar andquadrupolar contributions derived in Appendix VII B, inthe case of a radially oriented dipole. In the non-retardedregime kd (cid:28) n = 1) reducesto, ∆ ω ⊥ γ = 92 1( kd ) Im[ a ] + O (cid:0) ( kd ) − (cid:1) , (16) − − ∆ ω / γ λ [nm] totaln=1n=2n=3n=4n=5 FIG. 1. Numerical simulations of the total Lamb shift ∆ ω (black curve) and its multipolar contributions n = 1 , , , , λ = 2 πc/ω for a perfect electric dipole emitter with radialorientation and located at d = 10 nm from a silver nanospherewith a = 20 nm radius (red arrow). The Lamb shift is normal-ized to the dipole’s decay rate in free space γ . The refractiveindex of the homogeneous background is n b = 1. A Drude-Lorentz model for the silver permittivity is used according to[28]. The total Lamb shift is computed by taking n cut = 10. while the quadrupolar contribution ( n = 2) reduces to,∆ ω ⊥ γ = 4052 1( kd ) Im[ a ] + O (cid:0) ( kd ) − (cid:1) , (17)where the subscript ⊥ indicates a dipole perpendicularto the particle surface (radially oriented), and a ( a ) isthe electric dipolar (quadrupolar) Mie coefficient whoseexpression can be found in Appendix VII A. The expla-nation of the spectral behavior of the Lamb shift is thusfound in the imaginary part of the Mie coefficient. InFig. 2, we plot the modulus ((b) and (e)) and phase ((c)and (f)) of the electric dipolar and quadrupolar Mie co-efficients a and a respectively as a function of the exci-tation wavelength, together with the first two multipolarcontributions n = 1 and n = 2 of Fig. 1 ((a) and (d)in Fig. 2 plotted with the same color code). One cansee that the inflection point of the Lamb shift spectrum(around 376 nm for n = 1 and 358 nm for n = 2) corre-sponds to a resonance maximum of the modulus of theassociated Mie coefficient accompanied by a strong phasechange (the resonance of the Mie coefficients around250 nm is a spurious resonance peculiar to the model ofpermittivity used [29]). This clearly shows the multipolarorigin of the plasmon resonance enhanced Lamb shift. B. Quasi-normal mode description
Another interpretation of the shape of the Lamb shiftspectrum can be given using a Quasi-Normal Mode(QNM) description [30] (also called “Resonant State” ex-pansions). By expanding the scattered field E s ( r , ω ) in − − ∆ ω / γ | a | − a r g ( a ) λ [nm] − − ∆ ω / γ | a | − a r g ( a ) λ [nm] FIG. 2. (a) and (d): Lamb shift dipolar (red curve) andquadrupolar (green curve) contributions of Fig.1 (same colorcode) normalized by γ . (b) and (e): Modulus of the as-sociated electric dipolar (red curve) and quadrupolar (greencurve) Mie coefficients a and a as a function of the excita-tion wavelength λ . (c) and (f): Argument of the associatedelectric dipolar (red curve) and quadrupolar (green curve) Miecoefficients a and a as a function of λ . A Drude-Lorentzmodel for the silver permittivity is used according to [28]. Eq. (4) onto a small set of QNMs of the plasmonic res-onator as in [31], we obtain:∆ ωγ (cid:12)(cid:12)(cid:12)(cid:12) ω (cid:39) (cid:88) α A α (cid:18) ω (cid:48) α ω (cid:19) ω (cid:48)(cid:48) α ω (cid:48) α − ω ( ω (cid:48) α − ω ) + ω (cid:48)(cid:48) α + B α ( ω ) , (18)where ω α = ω (cid:48) α + i ω (cid:48)(cid:48) α is the complex frequency ofthe QNM labeled α , while A α is a dimensionless fac-tor and B α ( ω ) a function of ω (for the qualitativeanalysis which follows, we will consider it as constant: B α ( ω ) ≡ B α ). An equivalent expression in term of thewavelength is obtained by extending the relation between ω and λ to complex numbers. Adopting λ α ≡ πc/ω α ,where λ α = λ (cid:48) α + i λ (cid:48)(cid:48) α is the complex wavelength asso-ciated with the complex frequency, ω α = ω (cid:48) α + i ω (cid:48)(cid:48) α , wefind:∆ ωγ (cid:12)(cid:12)(cid:12)(cid:12) λ (cid:39) (cid:88) α − A α (cid:18) λ (cid:48) α (cid:101) λ (cid:19) λ (cid:48)(cid:48) α λ (cid:48) α − (cid:101) λ ( λ (cid:48) α − (cid:101) λ ) + λ (cid:48)(cid:48) α + B α , (19)where (cid:101) λ ≡ | λ α | /λ . Note that Eqs. (18) and (19)are generally valid for any resonator shape, and showsthat the total Lamb shift can be given by the sum ofindependent contributions of the QNMs.For a spherical Mie resonator, the QNMs are the mul-tipolar modes, labeled by three numbers { q, n, m } , whoseassociated complex eigenfrequencies ω q,n,m are the polesof the Mie coefficients [32]. In order to find the QNMresonances in play in the previous configuration, we onlylook at the poles of the electric Mie coefficients, becausethe dipole emitter is radially oriented and therefore onlycouples to electric modes (see discussion at the end ofAppendix VII B). This consists in solving the transcen- − − − − − − − − − − − − − − − − ∆ ω / γ λ [nm] totalQNMs FIG. 3. Comparison between numerical simulations and analytical calculations of the Lamb shift. (a): Total Lamb shift ofFig. 1 (black curve) compared to the Lamb shift calculated with Eq. (21) using the five QNM resonances displayed in Table I(dotted curve). (b) to (f): Fit of each multipole contribution of Fig. 1 (full lines, same color code) with the correspondingQNM contribution calculated with Eq. (21) (dotted lines) with A n and B n the fitting parameters (displayed in Table I). dental equation (see Eq. (A1)):( ε s /ε b ) j n ( k s a ) ξ (cid:48) n ( ka ) = ψ (cid:48) n ( k s a ) h n ( ka ) , (20)where all the functions and parameters are defined inAppendix VII A. First note that Eq. (20) does not de-pend on m , which means that multipolar modes with thesame multipolar order n but different orbital number m are degenerate ( i.e. have the same eigenfrequency ω n ).Therefore, the Lamb shift in Eq. (19) can be expressedas a sum running on the multipolar order n ,∆ ωγ (cid:12)(cid:12)(cid:12)(cid:12) λ (cid:39) (cid:88) n − A n (cid:18) λ (cid:48) n (cid:101) λ (cid:19) λ (cid:48)(cid:48) n λ (cid:48) n − (cid:101) λ ( λ (cid:48) n − (cid:101) λ ) + λ (cid:48)(cid:48) n + B n . (21)For each n , we find one solution ω n of Eq. (20) corre-sponding to the dominant pole, whose associated com-plex wavelength λ n is given in Table I for n = [1; 5] (westill take the same Drude-Lorentz model for the permit-tivity of the silver nanosphere [28] as for the previousnumerical simulations). The corresponding A n and B n terms are left as free parameters and they are set byfitting each multipole contribution n of Fig. 1 with theformula − A n (cid:16) λ (cid:48) n (cid:101) λ (cid:17) λ (cid:48)(cid:48) n λ (cid:48) n − (cid:101) λ ( λ (cid:48) n − (cid:101) λ ) + λ (cid:48)(cid:48) n + B n , in Fig. 3 (b)-(f). The discrepancy out of resonance that can be seenin Fig. 3 (b)-(f) is due to the fact that Eq. (18) is validonly in the vicinity of the resonance frequencies ω α andthat we ignored the ω dependency of B α . The valuesof the A n and B n parameters that result from the fit aregiven in Table I for n = [1; 5]. Note that the value ofthe amplitude A n decreases as n increases, showing thatthe resultant coupling between the emitter and the QNMresonance n is less and less important.In Fig. 3 (a), we compare the Lamb shift given byEq. (21) using the five QNM resonances n = [1; 5] withthe previous total Lamb shift calculated by computing Eq. (15) (black curve in Fig. 1). We can see that the an-alytical formula Eq. (21) based on the QNM resonancesof the plasmonic resonator qualitatively reproduces theLamb shift resonance when only a few dominant reso-nances are taken into account, but the convergence couldbe further improved by increasing the number of QNMresonances (see also [33, 34] where it is shown that afew set of QNM resonances is enough to reproduce thescattering properties of a particle). Moreover, this sim-ple analytical formula clearly evidences that the Lambshift resonance results from the coupling of the quantumemitter to the resonant modes of the nanoparticle.Finally, it is interesting to note that this resonant cou-pling induces a positive Lamb shift ∆ ω = ω − ω > repulsive van der Waals potential as long as the atomremains in its excited state, which was shown experi-mentally with excited cesium atoms in the presence of asapphire surface [35, 36]. TABLE I.
QNM complex wavelengths and fitting pa-rameters. n λ n (nm) A n B n . .
5i 95 . − .
62 358 . .
0i 57 . − .
23 353 . .
1i 38 . − .
14 351 . .
2i 24 . − .
15 350 . .
2i 15 . − . V. PREDICTIONS ABOUT THE LAMB SHIFTA. Blue-shift of the resonance
In this section, we show how the size of the nanoparti-cle affects the position of the Lamb shift resonance. Westill consider the case of a silver nanosphere. We plotin Fig. 4 the normalized Lamb shift as a function ofthe transition wavelength for different particle radii (fulllines). The asymptotic case of a planar surface is alsoplotted (dashed line) according to the following expres-sion [9, 10]: ∆ ω ⊥ γ = − k | ε s | − | ε s + 1 | d , (22)which is valid in the non-retarded regime and for an emit-ter oriented perpendicular to the surface. In this case, thedipole emitter couples to the surface plasmon mode whichcomes from the infinite density of states of the high ordermodes (around λ (cid:39)
340 nm for a planar silver surface).In sharp contrast with a nanosphere characterized bya purely dipolar response, we predict in the near-fieldof the nanosphere a blue -shift of the Lamb shift reso-nance as the radius of the nanosphere increases (see Fig.4). To understand this feature, let us recall that as theradius increases, each plasmon resonance is red-shiftedand the dipole emitter couples to higher-order multi-poles [27]. The displacement (blue-shift) of the Lambshift resonance then results from the interference be-tween these different modes. Therefore, this effect willonly exist if the dipole emitter is located in the near-fieldof the nanoparticle, so that it will be able to excite sev-eral modes and to get this interference effect, resultingthen in a blue-shift of the resonance.Thus, it can be observed in Fig. 4 that in the near-field of the nanoparticle, a precise engineering of this res-onant coupling between the quantum emitter and theplasmon resonances is possible. For instance, the transi-tion wavelength at which the Lamb shift is suppressedis λ = 363 nm > λ = 357 nm > λ = 350 nm >λ = 342 nm > λ = 339 nm for the radii a = 2 . a = 5 nm, a = 10 nm, a = 50 nm and the case of theplanar silver surface respectively. The tuning of this in-teraction is of current interest [37, 38], and we suggestthat thanks to their highly tunable optical properties,metallic nanoparticles can also be used to tune and shapethe Lamb shift of a nearby quantum emitter through acontrol of their geometry, but also spatial organizationand environment, which can all be investigated throughEq. (14). B. Gold dimer nanoantenna
In order to make a realistic calculation of the Lambshift, let us now consider a gold dimer with a dipoleemitter located at the center of the nanogap. This con-figuration is now experimentally realizable using DNA − − − − ∆ ω / γ λ [nm] a=2.5nma=5.0nma=10.0nma=50.0nmplanar surface FIG. 4. Numerical simulations of the normalized Lamb shift∆ ω/γ as a function of the transition wavelength λ for a per-fect electric dipole emitter with radial orientation and locatedat d = 5 nm from a silver nanosphere (red arrow), for differ-ent radii a (full lines). The asymptotic case of a planar silversurface (Eq. (22)) is also plotted (dashed line). The refractiveindex of the homogeneous background is n b = 1. A Drude-Lorentz model for the silver permittivity is used according to[28]. The Lamb shift is computed by taking n cut = 10 exceptfor the case a = 50 nm where n cut = 50 in order to converge. templates [39, 40]. To compute the Lamb shift, we takethe parameters corresponding to [41]: the nanoparticlesradius is 40 nm, the nanogap is 6 nm, and the effective re-fractive index surrounding the nanoparticles is n eff = 1 . Alexa Fluor 647 dye, whichpresents an emission peak around λ = 670 nm with40 nm width; its total decay rate in the homogeneoussolution is measured at γ = 2 .
63 ns − [42].The Lamb shift spectrum of such a configuration witha dipole emitter of parallel orientation is shown in Fig.5. At λ = 670 nm, the normalized Lamb shift computedwith Eq. (14) is ∆ ω/γ = − outside of therange of the radiative linewidth, and therefore suitablefor direct observation (the numerical simulations — notshown here — give a radiative decay rate enhancement γ r /γ = 1700 at λ = 670 nm). In order to find the Lambshift of the dye, one needs to multiply the value given bythe numerical simulations by the reference quantum yield φ = 0 .
08 in open solution ( i.e. without the antenna):∆ ω = φ × ( − × γ . The corresponding shift in termsof wavelength is given by the following formula (valid if∆ ω/ω (cid:28) λ/λ = − ∆ ω/ω where ∆ λ = λ − λ with λ the new wavelength of the emitted photon. Thus,for the Alexa Fluor 647 dye, the relative shift is ∆ λ/λ =3 . × − , corresponding to a shift ∆ λ = 2 . λ (cid:39) . − − − ∆ ω / γ λ [nm] FIG. 5. Numerical simulations of the normalized Lamb shift∆ ω/γ as a function of the transition wavelength λ for aperfect electric dipole emitter with parallel orientation andlocated in the center of a gold dimer antenna of radius 40 nmand 6 nm gap (red arrow). The refractive index of the ho-mogeneous background is n b = 1 .
5. A Drude-Lorentz modelfor the gold permittivity is used according to [28]. The Lambshift is computed by taking n cut = 40. VI. CONCLUSION
In this paper, we derived an exact multipole formula,Eq. (14), to compute the Lamb shift induced by an ar-bitrary set of resonant scatterers on a nearby quantumemitter. In the case of a single silver nanoparticle, our nu-merical simulations show that the dipole approximationfails to account for the total Lamb shift spectrum in thenear-field region, and that one must include higher mul-tipolar contributions. We furthermore adopted a Quasi-Normal Mode description of this phenomenon, which pro-vides a more physically intuitive understanding of theinduced Lamb shift as resulting from the coupling be-tween the quantum emitter and the resonances of thenanoparticle, and shows that the total Lamb shift can begiven by the sum of the independent resonance contribu-tions. These formulas also predict a displacement of theLamb shift resonance in the near-field to higher frequen-cies (blue-shift). Finally, a calculation of the Lamb shiftin a physically realistic configuration indicates that a di-rect detection may be possible for fluorescent moleculesembedded in a gold dimer nanogap.
VII. APPENDIXA. Mie coefficients
In this Appendix, we give the expressions of the Miecoefficients in a slightly different way then in [45] (wherethey are called the scattering coefficients). By intro-ducing ε s ( µ s ) and ε b ( µ b ) as the relative permittivity(permeability) of the sphere and the homogeneous back-ground respectively, k s = (cid:112) ε s ( ω ) ω/c and k = √ ε b ω/c ,the Mie coefficients of a sphere of radius a take the form: a n = ( ε s /ε b ) j n ( k s a ) ψ (cid:48) n ( ka ) − ψ (cid:48) n ( k s a ) j n ( ka )( ε s /ε b ) j n ( k s a ) ξ (cid:48) n ( ka ) − ψ (cid:48) n ( k s a ) h n ( ka ) (A1)for the electric Mie coefficient of order n , and b n = ( µ s /µ b ) j n ( k s a ) ψ (cid:48) n ( ka ) − ψ (cid:48) n ( k s a ) j n ( ka )( µ s /µ b ) j n ( k s a ) ξ (cid:48) n ( ka ) − ψ (cid:48) n ( k s a ) h n ( ka ) (A2)for the magnetic Mie coefficient of order n , where j n ( x )and h n ( x ) are respectively the spherical Bessel functionsand the first-type (outgoing) spherical Hankel functions,and ψ n ( x ) and ξ n ( x ) are the Ricatti-Bessel functions de-fined as: ψ n ( x ) ≡ xj n ( x ) (A3) ξ n ( x ) ≡ xh n ( x ) . (A4) B. Analytical expressions of the dipolar andquadrupolar Lamb shift
In this Appendix, we derive from Eq. (15) analyticalexpressions for the Lamb shift dipolar and quadrupolarcontributions for a sphere. We consider the sphere placedin the + z direction with respect to an electric dipoleemitter oriented either perpendicular to the surface ofthe sphere (orbital number m = 0, dipole moment ori-ented on the z axis) or parallel to the surface ( m = 1,dipole moment oriented on the x axis). Due to sphericalsymmetry, the T-Matrix of the single sphere is a diagonalmatrix t composed of the Mie coefficients of the spheremultiplied by −
1. With a quadrupolar assumption [46]: t = − Diag( a , a , b , b ) , (B1)with a ( a ) the electric dipolar (quadrupolar) Mie co-efficient and b ( b ) the magnetic dipolar (quadrupolar)Mie coefficient defined in Appendix VII A, f = [ e , , , t , (B2)with e the incident electric dipole coefficient, and H (0 , = A ,m, ,m A ,m, ,m B ,m, ,m B ,m, ,m A ,m, ,m A ,m, ,m B ,m, ,m B ,m, ,m B ,m, ,m B ,m, ,m A ,m, ,m A ,m, ,m B ,m, ,m B ,m, ,m A ,m, ,m A ,m, ,m , (B3)where A n,m,n (cid:48) ,m (cid:48) ( B n,m,n (cid:48) ,m (cid:48) ) the coupling coefficientfrom the electric (magnetic) multipole order n with or-bital number m , to the multipole order n (cid:48) with orbitalnumber m (cid:48) . Note that H (1 , is the same as H (0 , withall the B coefficients multiplied by −
1. Employing theexpressions of the coefficients A and B calculated in [46]in Eq. (15), one gets for an electric dipole oriented per-pendicular to the particle surface ( m = 0):∆ ω ⊥ γ = 92 Im (cid:20) a e kd ( kd ) (1 − i kd ) (cid:21) (B4)for the dipolar contribution and∆ ω ⊥ γ = −
910 Im (cid:20) a e kd ( kd ) (cid:0) − − kd ) − kd ) (cid:1) (cid:21) (B5)for the quadrupolar contribution. In the case of an elec-tric dipole emitter oriented parallel to the particle surface( m = 1), the dipolar and quadrupolar contributions to the Lamb shift read:∆ ω (cid:107) γ = 98 Im (cid:20) a e kd ( kd ) (cid:0) − kd ) − kd ) + 2i( kd ) + ( kd ) (cid:1)(cid:21) −
98 Im (cid:20) b e kd ( kd ) (i + ( kd )) (cid:21) (B6)∆ ω (cid:107) γ = −
158 Im (cid:20) a e kd ( kd ) (cid:0)
6i + 6( kd ) − kd ) − ( kd ) (cid:1) (cid:21) + 158 Im (cid:20) b e kd ( kd ) (cid:0) − kd ) − ( kd ) (cid:1) (cid:21) (B7)It is interesting to note in the case of a dipole emitterwith parallel orientation the presence of the magnetic Miecoefficients b and b , which traduce the cross-couplingbetween the electric dipole emitter and the magnetic mul-tipole resonances. This is not the case for a dipole per-pendicularly oriented whose multipolar Lamb shift con-tributions only depends on the electric Mie coefficients,since the magnetic field produced by an electric dipole isnull along the dipole axis. ACKNOWLEDGEMENTS
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