Quantum Optics in Maxwell's Fish Eye Lens with Single Atoms and Photons
QQuantum Optics in Maxwell’s Fish Eye Lens with Single Atoms and Photons
J. Perczel,
1, 2
P. K´om´ar, and M. D. Lukin Physics Department, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Physics Department, Harvard University, Cambridge, MA 02138, USA (Dated: May 31, 2018)We investigate the quantum optical properties of Maxwell’s two-dimensional fish eye lens at thesingle-photon and single-atom level. We show that such a system mediates effectively infinite-rangedipole-dipole interactions between atomic qubits, which can be used to entangle multiple pairsof distant qubits. We find that the rate of the photon exchange between two atoms, which aredetuned from the cavity resonances, is well described by a model, where the photon is focused toa diffraction-limited area during absorption. We consider the effect of losses on the system andstudy the fidelity of the entangling operation via dipole-dipole interaction. We derive our resultsanalytically using perturbation theory and the Born-Markov approximation and then confirm theirvalidity by numerical simulations. We also discuss how the two-dimensional Maxwell’s fish eye lenscould be realized experimentally using transformational plasmon optics.
PACS numbers: 42.50.Ex 03.67.Bg 42.50.Dv
I. INTRODUCTION
Maxwell’s two-dimensional fish eye is an optical lenswith remarkable imaging properties. Light emitted from any point inside the lens refocuses at the antipodal pointon the opposite side of the lens. Since J. C. Maxwell’soriginal work that studied ray optics inside the lens [1],the properties of the fish eye have been analyzed in avariety contexts, including electromagnetic waves [2, 3],scalar waves [4], quantum mechanics [5] and supersym-metry [6].More recently, it was proposed that Maxwell’s fish eyelens may have the ability to perfectly refocus electromag-netic waves emerging from a point source [7–9], therebyovercoming the diffraction limit [10]. The idea of per-fect imaging with Maxwell’s fish eye has generated vig-orous debate [11–38] . It has focused on how the pres-ence of a point-like detector, placed at the focus point,changes the image formed and whether perfect imagingis an artifact of the detector. On the one hand, it hasbeen argued that the presence of the detector, which canabsorb the incoming radiation, is necessary to form aperfect image [7–9, 11–14]. On the other hand, concernshave been raised that the detector itself would contributeelectromagnetic waves to the image formed, giving rise tothe apparent subwavelength focus point [14–17]. Subse-quently, the discussion about perfect imaging has shiftedto finding a simple and realistic model for such detec-tors [18–24]. More recently, it was suggested that perfectimaging may be possible when operating very close tothe resonances of the fish eye lens [35–38].In this paper, we study the imaging properties ofMaxwell’s two-dimensional (2D) fish eye lens at thesingle-photon level using single atoms. In particular, weassume that both the source and the detector of the pho-ton are individual atoms and thus no ambiguity arisesregarding their fundamental properties. One atom, ini-tially in its excited state, emits the photon and the secondatom, initially in its ground state, absorbs the photon,
FIG. 1. (color online) Light rays propagating within the in-finite 2D fish eye lens trace out perfect circles (dashed redlines). If a mirror of radius R is introduced (black circle),the trajectories remain closed (solid red lines). All light raysemerging from an arbitrary point within the lens (green dot)refocus at the antipodal point (blue star). The color code andthe inset show the spatial variation of the refractive index asa function of the radius, where we assume that n = 1 inEq. (1). For r > R the refractive index of the fish eye dipsbelow 1. storing it in a metastable state for fluorescent readout.This is conceptually the simplest model for a source anda detector [37].We model the 2D lens as an effective photonic cav-ity filled with an inhomogeneous dielectric material andsolve for the atom-photon dynamics inside the lens. Sincethe rate of photon exchange between the atoms is set bythe local electric field strength, the atomic dynamics is a a r X i v : . [ qu a n t - ph ] M a y sensitive indicator of the electric field distribution of thephoton during absorption. In particular, we find the thephoton exchange rate between the two atoms, which aredetuned from the cavity resonances, is well described bya simple model, which assumes that the photon is focusedto a diffraction-limited area during absorption.We also analyze the capabilities of the fish eye to en-hance the interaction between distant atoms. In particu-lar, we show that the dipole-dipole interaction mediatedby the fish eye lens is effectively infinite in range. Thisinfinite-range interaction is a consequence of the uniquefocusing properties of the fish eye lens and is analogousto the infinite-range interactions mediated by quasi-1Dwaveguides, which have been the subject of extensive re-search in recent years in the context of hollow [39, 40],plasmonic [41–43], microwave [44–46] and dielectric [47–52] waveguides. Within this model, we quantitativelyevaluate entangling operations and discuss a realistic ex-perimental realization.This paper is organized as follows. In Section II we dis-cuss the general formalism behind our work and derivethe dipole-dipole interaction mediated by the lens be-tween atoms. In Section III we discuss the entanglementof atoms within the lens. In Section V we discuss a pos-sible physical realization of the 2D fish eye using trans-formational plasmon optics. Key insights of our work aresummarized in Section VI. II. GENERAL FORMALISM
In this section we describe the general formalism be-hind our calculations for exploring the quantum opticalproperties of the system and calculate the dipole-dipoleinteraction between atoms placed inside the lens.
A. Maxwell’s Fish Eye Lens
The two-dimensional fish eye lens is a dielectricmedium of infinite size with refractive index [8] n ( r ) = 2 n r/R ) , (1)where r = (cid:112) x + y , R is the natural length scale of theproblem and n ≥ n = 1 for all numerical calculations in this paper.In the limit of geometric optics, light rays propagate inperfect circles (Fig. 1, dashed circles). All rays emittedfrom a single point inside the lens ultimately meet at theantipodal point. For | r | > R the refractive index variesbetween n and 0, which is difficult to achieve in practice.Thus the lens is modified by placing a mirror around thecircle of radius | r | = R (black circle in Fig. 1). In thepresence of the mirror the trajectories still remain closed(solid red lines in Fig. 1) [7]. The 2D fish eye can be realized for electromagneticwaves in a thin disk of radius R with a dielectric mate-rial of radially varying refractive index given in Eq. (1),which is constant along the ˆ z direction. If all surfacesof the disc are surrounded with mirrors, the TE modeswith the lowest frequency will realize the ideal dynamicsof the 2D fisheye [9, 30]. To ensure that a source radi-ating at frequency ω excites only the lowest TE modesof the lens, the thickness of the disk, b , is chosen suchthat the relevant optical frequency of the source is muchsmaller than πc/b , where c is the speed of light in vac-uum [30, 53]. Later, we consider a realistic realizationof the two-dimensional fish eye with surface plasmons,where the transverse confinement arises naturally fromthe confinement of the plasmons to the metal-dielectricinterface [54, 55]. B. Hamiltonian
We model the atoms as two-level systems with groundand excited states denoted by | g (cid:105) and | e (cid:105) , respectively.The Hamitonian describing the evolution of the systemcomposed of the two atoms and the fish eye modes isgiven by H = H atom + H field + V, (2)where the atoms evolve according to H atom = (cid:126) ω (cid:80) i =1 , | e i (cid:105) (cid:104) e i | and the evolutionof the electromagnetic field is described by H field = (cid:80) l,m (cid:126) ω l a † l,m a l,m , where a l,m is the anni-hilation operator of an eigenmode of the lens labelledby ( l, m ). The interaction of the two atoms with theelectromagnetic field is given by V = − (cid:80) i =1 , d i · E ( r i ),where d i = d z ( σ † i + σ i )ˆ z with σ i = | g i (cid:105) (cid:104) e i | and d z isthe z -component of the dipole moment of the e → g transition of the atom, E ( r i ) is the electric field operatorat position r i within the lens, and we neglect variationsof the field over the size of the atoms. The two atoms arepositioned at r and r (see Fig. 2(a)). Note that Eq. (2)describes a closed lossless system composed of the lensand the two atoms with no coupling to free-space modes.Later we will consider how photon loss from the fish eyemodes affects our results. C. Quantization in the Fish Eye Lens
We follow the quantization scheme of Glauber andLewenstein [56] to write down the expression for thequantized electromagnetic field E ( r i ) of the lens E ( r i ) = i (cid:88) l,m (cid:18) (cid:126) ω l ε (cid:19) / [ a l,m f l,m ( r i ) − a † l,m f ∗ l,m ( r i )] , (3) FIG. 2. (color online) (a) Schematic depiction of the two dipoles embedded in the fish eye cavity, which is surrounded by mirrorson all sides. (b) Spectrum of the cavity ( ω l = (cid:112) l ( l + 1) c/ ( R n ), l = 1 , , . . . ) in the absence ( κ = 0) and the presence ( κ (cid:54) = 0)of losses. The atomic resonant frequency ω is tuned between two resonances of the cavity. (c) Strength of the dipole-dipoleinteraction δω ( r , r ) / Γ between two atoms for four different lens radii: (i) R = 4 . λ , (ii) R = 8 . λ , (iii) R = 11 . λ and (iv) R = 14 . λ , assuming lens thickness b = λ/
10 and Γ = d z ω / (3 πε (cid:126) c ). The lens radii are chosen such that thetransition frequency of the atoms ω = 2 πc/λ lies halfway between the resonances of the lens ( l = 1 , , . . . ). In particular, wechose the order parameters (i) ν = 30 .
5, (ii) ν = 50 .
5, (iii) ν = 70 . ν = 90 .
5, where ν = ( (cid:112) π ( R n /λ ) + 1 + 1)and n = 1. The atom on the left is positioned exactly λ away from the mirror, whereas the position of the second atom issweeped. The strength of the interaction peaks λ away from the opposite mirror surface with a height that is independent ofthe radius of the lens and the interatomic distance. (d) Enlarged view of the dipole-dipole interaction near the antipodal point,showing that the width of the peak is approximately λ/ where f l,m are the classical eigenmodes of the cavity thatare solutions of the wave equation n ( r ) ω l,m c f l,m ( r ) − ∇ × [ ∇ × f m,l ( r )] = 0 , (4)subject to the transversality condition ∇ · (cid:2) n ( r ) f l,m ( r ) (cid:3) = 0 , (5)together with the boundary condition that f l,m · ˆ z = f l,m · ˆ φ = 0 at | r | = R due to the pres-ence of the mirror. The position-dependent refractiveindex n ( r ) is given by Eq. (1). The solutions of Eq. (4)and Eq. (5) can be chosen to form an orthonormal setsatisfying (cid:90) V d r n ( r ) f l,m ( r ) · f ∗ l (cid:48) ,m (cid:48) ( r ) = δ ll (cid:48) δ mm (cid:48) , (6)where the integral is performed over the quantization vol-ume V .Solving these equations, the lowest TE modes of thefish eye take the following form f l,m ( r, φ ) = (cid:115) bR n Y ml (cid:18) arccos (cid:18) | r | − R | r | + R (cid:19) , φ (cid:19) ˆ z, (7)where Y ml ( θ, φ ) are the spherical harmonic func-tions, φ = arccos( x/ | r | ) is the azimuthal angle as-sociated with position r and the eigenfrequencies are ω l = c (cid:112) l ( l + 1) / ( R n ). The modes f l,m are la-belled with the rescaled wavenumber l = 1 , , . . . and the angular momentum index m , where m ( l ) = − ( l − , − ( l − , ... , ( l −
1) is enforced bythe boundary condition f l,m ( R , φ ) = 0. The discretespectrum of the fish eye is schematically shown inFig. 2(b). The number of degenerate states increaseslinearly with l , since (cid:80) m ( l ) l . D. Photon transfer between two atoms viadipole-dipole interaction
In this section, we investigate the resonant transfer ofa photon between two atoms via the dipole-dipole inter-action, the strength of which we denote by δω .In quantum optics, the most fundamental model forphoton emission and detection assumes that one atom isinitially in its excited state | e (cid:105) , while the second atom isin its ground state | g (cid:105) . When the system evolves coher-ently in time, the excited atom (virtually) emits the pho-ton and after time t int ∼ π/ (2 δω ) the second atom fullyabsorbs the photon as its atomic population is transferredto the excited state | e (cid:105) [43, 57].Furthermore, by making use of additional metastablestates | s i (cid:105) with i = 1 , | e i (cid:105) via the time-dependent classical control pulse Ω i ( t ) (suchthat Ω i (cid:29) δω ), the photon transfer can be performed ina controlled, realistic scheme [58–60]. In particular, byadjusting Ω ( t ) and Ω ( t ), the photon transfer can beinitiated via the excitation of | e (cid:105) and, as the photon isreabsorbed, the atomic population of the second atomcan be transferred to the metastable state | s (cid:105) . Then,by switching off Ω ( t ), reemission into the cavity can beprevented. From the metastable state the photon can beread out using standard fluorescence techniques [61, 62].This completes the detection of the photon.In a standard quantum optical setting, the dipole-dipole interaction between two atoms with level spac-ing ω between ground | g i (cid:105) and excited states | e i (cid:105) inany environment can be expressed in terms of the clas-sical Green’s function components G αβ ( r , r , ω ) (with α, β = x, y, z ) through the following expression [63–66] δω ( r , r ) = d z ω (cid:126) ε c Re { G zz ( r , r , ω ) } , (8)where we assume that the two atoms are located at r and r and their dipole moments d z are orientedalong the z -axis. Note that the real (imaginary) partof the Green’s function G zz ( r , r , ω ) has the simpleinterpretation of being the z -component of the in-phase(out-of-phase) component of the electric field generatedat position r within the lens due to the presence of a z -oriented point-like dipole at position r radiating atfrequency ω . FIG. 3. (color online) Schematic depiction of a realisticscheme for the photon transfer between the two atoms. Thefirst atom emits the photon, while the second atom fully ab-sorbs it. By applying classical time-dependent control pulsesΩ ( t ) and Ω ( t ), the transfer can be initiated and the photoncan be captured in the metastable state of the second atom,from which the photon can be read out using fluorescencetechniques. We note that, when the classical Green’s function ofa problem is analytically known, it is typically a sim-ple matter to evaluate Eq. (8) and find the dipole-dipoleinteraction between atoms. However, for the fish eyethere is debate about what Green’s function correctlydescribes the imaging process. The subtlety of the issuearises from the fact that the fish eye, which models theclosed surface of the sphere, is inherently a closed systemfrom which radiation cannot escape in the absence oflosses and detectors [7–9]. As mentioned previously, theaccurate mathematical modeling of detectors has beena key focus of the discussion regarding perfect imaging[11, 12, 14–34, 37, 38].Here, since we model both the ‘source’ of the radia-tion and the ‘detector’ as atoms, the exact expression forthe dipole-dipole interaction can be obtained from the standard quantum optical master equation [67], whereno ambiguity arises in the derivation of the results. Fur-thermore, as we show below, the expression obtained forthe dipole-dipole interaction from the master equationexactly matches one of the two Green’s functions dis-cussed extensively in the fish eye literature, allowing usto directly use Eq. (8), which substantially simplifies nu-merical calculations.The quantum optical master equation in the Born-Markov approximation, which governs the evolution ofthe atoms inside the lens, takes the following form in theinteraction picture [67] d ˜ ρdt = − (cid:126) (cid:90) ∞ dτ Tr (cid:104) ˜ V ( t ) , (cid:104) ˜ V ( t − τ ) , ˜ ρ ( t ) ⊗ | (cid:105)(cid:104) | (cid:105)(cid:105) , (9)where | (cid:105)(cid:104) | is a projector onto the vacuum state of thelens (i.e. no photons in the lens) and the trace is impliedover all photonic Fock states of the lens, (cid:80) n (cid:104) n | ... | n (cid:105) , and˜ ρ ( t ) = e iH atom t/ (cid:126) ρ ( t ) e − iH atom t/ (cid:126) , (10)and˜ V ( t ) = e i [ H atom + H field ] t/ (cid:126) V ( t ) e − i [ H atom + H field ] t/ (cid:126) . (11)In Eq. (9), the Born approximation was performed bywriting the density matrix for the system in the form˜ ρ ( t − τ ) ⊗ | (cid:105)(cid:104) | , which amounts to neglecting correla-tions between the atoms and the electromagnetic modesof the lens [67]. The Markov approximation was madeby replacing ˜ ρ ( t − τ ) by ˜ ρ ( t ), which is based on the as-sumption that the atom-field correlation time is negligi-bly short compared to the time scale on which the systemevolves [67]. The Markov approximation allowed us toself-consistently extend to infinity the upper limit of theintegration with respect to dτ . We confirm the validityof the Born-Markov approximation in Section IV.After performing the trace over the modes of the fisheye lens, we need to evaluate the following standard in-tegral (cid:90) ∞ dτ e − i ( ω l ∓ ω ) τ = πδ ( ω ∓ ω l ) ± iP ω ∓ ω l , (12)where δ ( x ) stands for the Dirac delta and P f ( x ) denotesthe principal value component of the function f ( x ). Sincethe spectrum of the fish eye modes (which act as thereservoir for the atoms) is discreet, the Dirac delta andthe principal value do not contribute away from reso-nances and we may simply replace the right-hand sideof Eq. (12) with ± i/ ( ω ∓ ω l ). More specifically, in theabsence of any mechanism for photon loss that wouldbroaden the energy levels, the atoms experience no spon-taneous decay or cooperative emission when their tran-sition frequency does not coincide with the resonant fre-quencies of the lens. The master equation then describesthe fully coherent, lossless evolution of the atoms andtakes the form dρdt = 1 i (cid:126) [ H at , ρ ] − i (cid:88) i,j =1 , i (cid:54) = j δω ( r i , r j ) [ σ † i σ j , ρ ] , (13)where the dipole-dipole interaction between the atoms isgiven by δω ( r i , r j ) = d z (cid:126) ε (cid:88) l,m ω l ω l − ω f ∗ l,m ( r i ) f l,m ( r j ) , (14)where the fish eye modes f l,m ( r ) are given by Eq. (7) andthe summation runs over all eigenmodes of the fish eye.Given the summation over an infinite number of modes,it is difficult to work directly with the expression givenin Eq. (14) and it is desirable to replace it with a simple,closed-form expression.As shown in Appendix A, the right-hand side ofEq. (14) can indeed be replaced by an expression of thesame form as Eq. (8) using a Green’s function, where theGreen’s function is given by the following expression G zz ( r , r , ω ) = − P ν ( ξ ( α , α )) − P ν ( ξ ( α , /α ∗ ))4 b sin( πν ) , (15)where P ν is the Legendre function of (non-integer) order ν = ( (cid:112) π ( R n /λ ) + 1 − ν depends on the atom frequency ω throughthe free-space wavelength λ = 2 πc/ω and the order pa-rameters with integer values ( ν = 1 , , . . . ) correspondto the resonances of the lens. We have also defined ξ ( α , α ) = ( | ζ ( α , α ) | − / ( | ζ ( α , α ) | + 1) and ζ ( α , α ) = ( α − α ) / ( α α ∗ + 1), with α j = r j R e iφ j ,where ( r j , φ j ) are the cylindrical coordinates of thepositions of the two atoms ( j = 1 ,
2) within the lens. InEq. (15) the second term on the right hand side accountsfor the presence of the mirror at | r | = R , ensuring thatthe electric field goes to zero [7]. This Green’s functionwas first derived in Ref. [13], and is obtained from thecanonical equation of the dyadic Green’s function in thepresence of a single source term [7, 9, 12]. This Green’sfunction has been used previously to describe the staticelectric field distribution inside the lens for the casewhen a diffraction-limited image forms at the antipodalpoint in the presence of a classical source and in theabsence of a ‘drain’ [12, 15].Using Eq. (8) and Eq. (15), the dipole-dipole interac-tion can be calculated in a straightforward manner withinthe lens. In Fig. 2(c) we plot the strength of the dipole-dipole interaction between two atoms. The position ofthe first atom is fixed exactly one wavelength away fromthe mirror and the position of the second atom is var-ied across the lens. We plot the interaction strength forfour different radii of the fish eye. As Fig. 2(c) shows,the strength of the dipole-dipole interaction peaks at theantipodal point, exactly one wavelength away from themirror.As noted at the start of this section, in quantum opticsthe strength of the dipole-dipole interaction sets the rateat which a photon can be resonantly transferred fromone atom to the other. Physically, this exchange ratedepends on the strength of the photon field at the lo-cation of the second atom that absorbs the photon. Ingeneral, the smaller the volume the photon is focused to, the larger the field strength gets. Thus, the dipole-dipoleexchange rate depends sensitively on the area the photonis focused to. Fig. 2(d) provides an enlarged view thatshows the dipole-dipole interaction rate – and thus theelectric field strength – experienced by the second atomnear the antipodal point [68]. The width of the peak isapproximately λ/
2, suggesting that the photon is focusedto a diffraction-limited area at the location of the secondatom. These results for the rate of photon transfer arenumerically confirmed in Section IV.Fig. 2(c) also shows that the height of the peak remainsconstant as the radius of the fish eye and, therefore, thedistance between the two atoms is increased. The pho-ton emitted by an atom anywhere within the cavity getsrefocused at the antipodal point regardless of the size ofthe lens. Such infinite range dipole-dipole interaction isa well-known feature of quasi-1D waveguides [39–52]. In-tuitively, the 2D fish eye lens acts as quasi-1D systemdue to the fact that the lens mimics the propagation oflight on the surface of a sphere [8]. Just as in 1D lightis confined to propagate along a single axis without dis-persion, the same way light emitted from a point on the2D surface of a sphere is constrained to propagate alongthe geodesics of the sphere and refocuses at the antipodalpoint without any dispersion.The functional form of the dipole-dipole interactioncan also be understood analytically by considering theasymptotic behavior of the Green’s function near thesource and image points. In particular, note thatthe source and image points in the lens correspond to ξ ( α , α ) = − ξ ( α , /α ∗ ) = +1 respectively [7].As ξ → − P ν ( ξ ) → sin( νπ ) π (cid:34) log (cid:18) ξ (cid:19) + F ( ν ) (cid:35) , (16)where we have defined the function F ( ν ) = γ + 2 ψ ( ν + 1) + π cot( νπ ) . (17)Here γ is Euler’s constant and ψ is the digamma func-tion. In addition, when ξ → P ν ( ξ ) → G zz ≈ − b sin( πν ) . (18)This shows that the absolute value of the Green’s func-tion is maximized when the frequency falls half-way be-tween two resonances such that ν = m + 0 .
5, where m ∈ N . Furthermore, this expression also shows thatthe height of the peak at a given frequency only dependson the transverse confinement of the modes b and is in-dependent of the lens radius R . Finally, we note thatEq. (18) also shows that the dipole-dipole interaction isindependent of where we place the atoms within the lensas long as they are situated at antipodal points. E. Spontaneous and cooperative decay of atoms
In all calculations so far, we assumed that the fish eyelens is completely isolated from its surrounding environ-ment and the photon cannot leak out of the cavity. Here,we next consider the situation when the lifetime of theeigenmodes of the fish eye are finite e.g. due to the imper-fection of the mirrors and dissipation in the dielectrics.We account for the gradual loss of photons from the fisheye modes by modifying the Hamiltonian in Eq. (2) witha non-Hermitian term of the following form H field = (cid:88) l,m (cid:126) ( ω l − iκ ) a † l,m a l,m , (19)where 2 κ sets the rate of decay from the modes, whichis assumed to be frequency-independent in the range ofinterest. The decay of the cavity modes broadens the dis-crete energy levels of the fish eye, creating a continuousspectrum, as shown schematically in Fig. 2(b).With this modification, we can re-derive the masterequation from Eq. (10). We evaluate the following inte-gral (cid:90) ∞ dτ e − i ( ω l ∓ ω ) τ e − κτ = 1 i ( ω l ∓ ω ) + κ , (20)and after neglecting the off-resonant decay terms [67] weobtain the master equation in the following form dρdt = 1 i (cid:126) [ H at , ρ ] − i (cid:88) i,j =1 , i (cid:54) = j δω ( r i , r j ) [ σ † i σ j , ρ ] − (cid:88) i,j =1 , i (cid:54) = j Γ( r i , r j ) (cid:18) σ i ρσ † j − (cid:110) σ † i σ j , ρ (cid:111)(cid:19) , (21)where the rate of decay is given byΓ( r i , r j ) = d z (cid:126) ε (cid:88) l,m κ f ∗ l,m ( r i ) f l,m ( r j )( L + l + L − l ) , (22)and the modified dipole-dipole interaction is given by δω ( r , r ) = d z (cid:126) ε (cid:88) l,m ω l f ∗ l,m ( r ) f l,m ( r ) (cid:0) D + l + D − l (cid:1) , (23)where we have defined L ± l = ∓ ω l κ + ( ω l ± ω ) and D ± l = ω l ± ω κ + ( ω l ± ω ) . (24)Since we are now including losses in the system, the ex-cited states of the two atoms can irreversibly decay intothe eigenmodes of the lens and leave the cavity, lead-ing to non-zero single atom decay γ ( r i ) = Γ( r i , r i ) (with i = 1 ,
2) and cooperative decay γ coop ( r , r ) = Γ( r , r ).The single atom decay γ describes how quickly an ex-citation decays from state | e (cid:105) of an individual atom tothe fish eye modes, whereas the cooperative decay γ coop governs the coherent joint emission of the two atoms intothe modes leading to super ( γ + γ coop ) and subradiantdecay ( γ − γ coop ) of the symmetric and anti-symmetricsuperpositions of the two atoms, respectively [67].As for the lossless case, it is desirable to find closed-form expressions to replace the expressions that involveinfinite summations on the right-hand side of Eq. (22)and Eq. (23). As shown in Appendix A, the decay ratesand the dipole-dipole interaction can be expressed usingthe Green’s function of Eq. (15) in the following formΓ( r i , r j ) = 2 d z (cid:126) ε c Im { ( ω + iκ ) G zz ( r i , r j , ω + iκ ) } , (25)and δω ( r i , r j ) = d z (cid:126) ε c Re { ( ω + iκ ) G zz ( r i , r j , ω + iκ ) } . (26)These simple, analytic expressions provide a convenientway to calculate the quantum optical properties of atomsinside the lossy fish eye lens and to study the atomicdynamics.We also note that when κ (cid:28) ω , Eq. (25) and Eq. (26)can be approximated asΓ( r i , r j ) ≈ d z ω (cid:126) ε c Im { G zz ( r i , r j , ω + iκ ) } , (27)and δω ( r i , r j ) ≈ d z ω (cid:126) ε c Re { G zz ( r i , r j , ω + iκ ) } . (28)Eq. (27) and Eq. (28) suggest an alternative way of ac-counting for the loss of photons from the modes of thefish eye. In particular, it can be shown (see AppendixA) that G zz ( r i , r j , ω + iκ ) is the Green’s function of thefish eye lens with the following complex refractive index˜ n ( r ) = n ( r )(1 + iα ) , (29)where α = κ/ω , (30)and n ( r ) is given by Eq. (1). Therefore, the loss of pho-tons from the modes of the fish eye can also be thought toarise from material absorption in the dielectric [7]. Thisis a key observation, which allows us to associate a κ value with material absorption and, therefore, treat alllosses that contribute to photon decay from the fish eyemodes in a unified manner. In particular, even if differ-ent loss processes are present, e.g. material absorptionand leakage through the mirror, we can still associate a κ value with each of these processes and calculate the totaldecay rate via κ total = κ abs + κ mirror , (31)which can be substituted into Eq. (27) and Eq. (28) tocalculate the relevant atomic properties in the lossy lens.This will be particularly useful when we consider a possi-ble physical realizations of the fish eye lens with plasmons(see Section V).Furthermore, we can also find how ν , Γ and δω scalewith α for system parameters of interest. First, wenote that 16 π ( R /λ ) (cid:29)
1, whenever λ (cid:46) R . Assum-ing α (cid:28)
1, to first order in α we find that ν ≈ πR λ (1 + iα ) . (32)Assuming that Re[ ν ] = m + 0 . m ∈ N (whichcorresponds to tuning the atomic frequency between tworesonances), from Eq. (18) we obtain that, to lowest or-der in α , the following approximation holds at the imagepoint ( r = − r ) G zz ( r , − r , ω + iκ ) ≈ − b sin( πν ) ≈ ∓ b (1 + (2 π R α/λ ) ) , (33)where the choice of sign ∓ depends on whether m is evenor odd. This is a purely real quantity and, therefore,from Eq. (27) and Eq. (28) we find that the cooperativedecay is given by γ coop = Γ( r , − r ) ≈ , (34)and the dipole-dipole interaction takes the form δω ( r , − r ) ≈ ∓ d z ω (cid:126) ε c b (1 + (2 π R α/λ ) ) . (35)Finally, we can find the single atom decay rate γ by substituting r i = r j into Eq. (27) and substitutingEq. (16) and Eq. (32) into Eq. (15). We find that toleading order in α the following approximation holds γ = Γ( r , r ) ≈ d z ω (cid:126) ε c π R αbλ . (36) III. ENTANGLEMENT OF ATOMS
Structures that mediate long-range dipole-dipole in-teractions are of significant interest in quantum infor-mation processing, as such interactions make it possibleto entangle [40] and perform deterministic phase gatesbetween distant atoms [43]. In this section, we charac-terize the potential of the fish eye to entangle distantatomic quits. We focus on the simple case of a single ex-citation being exchanged between two atoms due to thedipole-dipole interaction. In what follows, for simplicitywe assume that the two atoms are located at antipodalpoints (i.e. | r | = | r | and φ = φ + π ) and, therefore, γ = Γ( r , r ) = Γ( r , r ).In the absence of a driving field, the no-jump evolu-tion of the system can be described by a non-Hermitian effective Hamiltonian of the form [70] H = ( (cid:126) ω − iγ ) | e , e (cid:105)(cid:104) e , e | + ( δω − i ( γ + γ coop ) / | + (cid:105)(cid:104) + | + ( − δω − i ( γ − γ coop ) / |−(cid:105)(cid:104)−| , (37)where we have defined |±(cid:105) = ( | e , g (cid:105) ± | g , e (cid:105) ) / √
2, andrecall from the previous section that γ coop = Γ( r , r )and δω ( r , r ) stand for the cooperative decay anddipole-dipole interaction of the atoms, respectively. Notethat the overall decrease of population in Eq. (37) due tothe non-Hermitian terms reflects the gradual loss of thephotonic excitation from the cavity. FIG. 4. (color online) Excitation probability of two atomswithin the cavity as a function of time. Initially, atom 1 isexcited and atom 2 is in its ground state. As the systemevolves, the two atoms repeatedly exchange a photon via thedipole-dipole interaction. The photon gradually decays fromthe cavity modes, leaving the atoms in their ground states. Afully entangled state with maximal fidelity is formed at t = π/ (4 δω ) (see arrow). The plot was obtained for R = 3 . λ with a cavity loss rate of α = κ/ω = 5 × − , assuming thatthe two atoms are located at two antipodal points within thelens such that | r | = | r | = 0 . R and φ = φ + π . Assuming that at t = 0 the two atoms are in the state | ψ (0) (cid:105) = (cid:12)(cid:12) e , g (cid:11) = ( | + (cid:105) + |−(cid:105) ) / √
2, the time evolution ofthe atomic wavefunction is governed by | ψ ( t ) (cid:105) = 1 √ (cid:16) e − i [ δω − i ( γ + γ coop )] t | + (cid:105) + e − i [ − δω − i ( γ − γ coop )] t |−(cid:105) (cid:17) , (38)which, upon substitution, yields | ψ ( t ) (cid:105) = C + ( t ) | e , g (cid:105) + C − ( t ) | g , e (cid:105) , (39)where | C ± ( t ) | = e − γt γ coop t ) ± cos(2 δωt )] . (40)The expressions | C + | and | C − | give the excitation prob-ability of atom 1 and atom 2, respectively, as a functionof time. In Fig. 4 we plot the excitation probability ofthe two atoms as a function of time. As the plots shows,the photon is coherently exchanged a number of timesbetween the two atoms before it gradually decays fromthe cavity modes. FIG. 5. (color online) Error (1 − F ) of the entanglingoperation between two qubits located at two antipo-dal points within the lens ( | r | = | r | = 0 . R and φ = φ + π ). (a) Error of the entangling operationas a function of the cavity loss rate α = κ/ω forfour different lens sizes R ∈ { . , . , . , . } λ ,where the R /λ ratio was chosen such thatRe( ν ) = ( (cid:112) π ( R /λ ) + 1 + 1) ∈ { . , . , . , . } .The error increases as the losses and lens radii in-crease. (b) Error of the entangling operation for a fixedloss rate α = 5 × − as a function of the detuning∆ ν = Re( ν ) − ν center , where ν center ∈ { . , . , . , . } .Error is plotted for the same four lens radii as in (a). Theerror increases with radius and as the frequency approachesone of the resonances. Analytic (numerical) results areshown with solid (dotted) lines. Good agreement is obtainedbetween analytic and numerical data, confirming the validityof the Born-Markov analysis. During time evolution, the state | ψ ( t ) (cid:105) will havemaximal overlap with the maximally entangled state | ξ (cid:105) = ( | e , g (cid:105) − i | g , e (cid:105) ) / √ | C + ( t ) | = | C − ( t ) | ,which happens when 2 δωt ≈ π + mπ , where m ∈ Z . Sincein the presence of losses the fidelity decreases over time,we choose m = 0. Thus, the time needed to reach themaximal overlap with the entangled state is t = π/ (4 δω )(see arrow in Fig. 4) and the maximum fidelity of the en-tanglement operation will be F = (cid:12)(cid:12) (cid:104) ξ | ψ ( t ) (cid:105) (cid:12)(cid:12) = exp (cid:16) − π (cid:12)(cid:12)(cid:12) γδω (cid:12)(cid:12)(cid:12)(cid:17) cosh (cid:16) π (cid:12)(cid:12)(cid:12) γ coop δω (cid:12)(cid:12)(cid:12)(cid:17) . (41)Eq. (41) gives a simple, analytic expression for the fidelityof the entangling operation in terms of γ = Γ( r i , r i ), γ coop = Γ coop ( r i , r j ) and δω ( r i , r j ), which can be evalu-ated analytically through Eq. (25) and Eq. (26). Here,the key figure of merit is the ratio β = δω/ ( γ + γ coop ). Ifthe frequency of the atoms is chosen to lie half-way be-tween two resonances of the fish eye (see Fig. 2(b)), thesingle atom decay γ and the cooperative decay γ coop aresmall and the dipole-dipole interaction dominates [70].Intuitively, in the absence of losses ( γ = γ coop = 0), thefidelity of the entangling operation is 1.In Fig. 5(a) we plot the error in the entangling oper-ation (1 − F ) for four different lens radii as a functionof α , where α = κ/ω = 1 /Q is the inverse of the cavityQ-factor, characterizing the ratio of the lifetime of theeigenmodes of the lens to the frequency of the excitation.For all lens sizes, the position of the two atoms is fixed attwo antipodal points such that | r | = | r | = 0 . R and φ = φ + π . The ratio of the lens radius to the transitionwavelength ( R /λ ) was chosen such that the real partof the order parameter ν = ( (cid:112) π ( R /λ ) + 1 + 1)associated with the atomic frequency falls half-way be-tween two resonances of the fish eye for all four lens radii(i.e. Re( ν ) = q + 0 . q ∈ { , , , } , wherenote that for Re( ν ) = 1 , , . . . the transition frequency ω is resonant with one of the eigenenergies ω l of thelens). Clearly, the error increases with increasing α andincreasing R (i.e. increasing interatomic distance). Themaximal value of the error is 0 .
5, which is reached when β becomes so small that the initial state has the highestfidelity ( F = |(cid:104) ξ | ψ (0) (cid:105)| = 0 . α = 5 × − for the same four lens radii as in (a) and thesame antipodal atomic positions. The error is now plot-ted as a function of the detuning ∆ ν = (Re( ν ) − ν center ),where ν center = q + 0 . q ∈ { , , , } . Clearly,the error is minimal half-way between the resonances andincreases as the frequency approaches the resonances.To gain further insight, we assume that the atomicfrequencies lie between two resonances of the lens andobtain the scaling of the fidelity with system parame-ters by substituting Eq. (34), Eq. (35) and Eq. (36) intoEq. (41). We obtain the following simple expression F = e − π R α/λ . (42)In Fig. 6 we plot the fidelity of the entangling operationas a function of the lens radius using both the exact ex-pression in Eq. (41) and the analytic approximation in FIG. 6. (color online) Maximum fidelity of the entanglementoperation as a function of the lens radius. The fidelity wasevaluated at discrete values of R /λ that correspond to tuningthe atomic frequency half-way between two resonances, i.e.Re[ ν ] = m + 0 .
5, where m is an integer. The red line markedwith circles was obtained from the exact analytical expressionin Eq. (41), whereas the blue line marked with squares wasobtained from the approximate expression in Eq. (42). Goodagreement is obtained between the two curves. The loss ratewas assumed to be α = κ/ω = 5 × − . Eq. (42). Very good agreement is observed between thetwo curves.Finally, we note that the fish eye lens could be usedfor entangling many pairs of atoms simultaneously. Asthe radius of the fish eye is increased, the dipole-dipoleinteraction at all points further than λ/ R > λ , the dipole-dipole interaction at the antipo-dal point is an order of magnitude larger than anywhereelse in the cavity (see Fig. 2(c)). Thus, by placing numer-ous pairs of atoms into the cavity simultaneously, theycan be entangled pairwise, without substantial interac-tion between the different pairs. IV. VALIDITY OF THE BORN-MARKOVAPPROXIMATION
In our derivation of Eq. (14), Eq. (22) and Eq. (23)we made use of the Born-Markov approximation, whichpresupposes that the environment is large and the cor-relation time of the environment is very short comparedto the evolution of the atomic states [67]. Since in ourformalism the role of the ‘environment’ is played by themodes of the finite cavity, the validity of these assump-tions needs to be evaluated carefully.In order to verify the validity of the above results,we numerically solve the Sch¨odinger equation, where theHamiltonian is given by Eq. (2) together with the non-Hermitian term introduced in Eq. (19). The form of V isconsiderably simplified when the two atoms are placed attwo antipodal points within the lens such that | r | = | r | and φ = φ + π . In this case the in-phase combina-tion of the atomic dipole moments ( d z ( σ + σ ) / √ l = 1 , , . . . )and the out-of-phase combination of the dipole moments( d z ( σ − σ ) / √ l = 2 , , . . . ) of the fish eye (see Appendix B). Thisreduces the size of the Hilbert space, making it possibleto efficiently simulate the system while including a largenumber of the eigenmodes of the lens with frequenciesclose to ω . We further restrict the Hilbert space to haveat most a single excitation in the system.We numerically determine the time-evolution, start-ing from the state | ψ (0) (cid:105) = | e , g (cid:105) via the operator U ( t ) = exp[ − iHt/ (cid:126) ]. To obtain the maximum fidelity ofthe entangling operation, the overlap of the time-evolvedatomic state is calculated with the maximally entangledstate ( | e, g (cid:105) − i | g, e (cid:105) ) / √
2. In Figs. 5(a) and 5(b) we plotthe numerically obtained values for the error (1 − F ) (dot-ted lines) for different lens radii as a function of losses andatom frequencies, respectively. Even though the analyt-ical results were derived using the Born-Markov approx-imation and neglecting retardation [57], good agreementis obtained between the analytic results and numericaldata. This confirms the validity of the analytical formal-ism described in previous sections. V. POSSIBLE EXPERIMENTAL REALIZATION
A promising way to realize the fish eye lens is via trans-formational plasmon optics [54, 55]. The idea behind thisapproach is to engineer an effective refractive index dis-tribution for surface plasmon polaritons by depositinga layer of high-index dielectric on top a 2D silver sur-face (see Fig. 7). By varying the height of the dielectriclayer on the surface, the effective refractive index seen bythe plasmons can be changed. In particular, when thereis no dielectric on top of the silver, the effective refrac-tive index seen by the plasmons is close to 1, whereasin the presence of a thick dielectric layer, the effectiveplasmonic refractive index will be close to the refractiveindex of the dielectric itself. Through this experimen-tal technique, complex spatially-varying refractive indexprofiles can be engineered [54]. Crucially, the behaviorof plasmons in a plasmonic lens with a particular refrac-tive index profile closely mimics the predicted behaviorof classical light rays in the corresponding 2D lens. Thiscorrespondence between 2D classical lenses and quasi-2Dplasmonic lenses was theoretically established in Ref. [54]and experimentally confirmed for the nanoscale Luneb-urg and Eaton lenses [55].We expect that the plasmonic version of the nanoscalefish eye lens could be experimentally realized analogouslyto the Luneburg and Eaton lenses. A dielectric layer ofvarying height could be deposited on a flat silver sur-face while the lens is surrounded by a circular mirror(see Fig. 7(b)). To explore the quantum optical proper-ties of the fish eye, atom-like color defects in diamond0
FIG. 7. (color online) Physical realization of the fish eye lensusing transformation plasmon optics. (a) Effective refractiveindex n ( d ) = Re { ˜ n ( d ) } created as a function of the height ofthe dielectric d deposited on the silver surface. The insetshows the material losses χ ( d ) = Im { ˜ n ( d ) } as a function ofthe dielectric height d . (b) Schematic depiction of the plas-monic fish eye lens. The two emitters are embedded in thedielectric. The height of the dielectric varies across the lens,which creates the effective refractive index distribution of thefish eye lens. The lens is surrounded by mirrors from all sides(the front part of the mirror has been removed to show theinterior). could be used as quantum emitters. Subwavelength po-sitioning and coherent manipulation of such color defectshas been experimentally demonstrated previously [71–74]. Recently, the entanglement of two silicon-vacancy(SiV) color defects inside a nanoscale cavity was alsodemonstrated [75].For illustration, we provide here an estimate of the en-tanglement fidelity of two atoms inside a particular ex-ample of a plasmonic fish eye lens. We assume that thelens operates at 406 . λ SiV = 737nm. Furthermore, we assume that the lenshas a radius of R = 1 . λ SiV , which ensures that theSiV resonance falls between two resonant modes of cav-ity (Re( ν ) = 10 . (cid:15) m = − .
23 + 0 . i and gives rise to plasmonic prop-agation distances on the order of ∼ λ Siv . It is alsoassumed that there is a thin ( ∼ − | r | = | r | = 0 . R and φ = φ + π , as schematically shown in Fig. 7(b).Due to their proximity to the silver surface, the two ˆ z -polarized emitters will couple strongly to the surface plas-mons, which are tightly confined to the metal-dielectricinterface.The spatially varying refractive index n ( r ) of the fish(Eq. (1) with n = 1) could be experimentally realizedby depositing a dielectric of permittivity (cid:15) d = 3 . n d = √ (cid:15) d =1 .
9) was chosen such that the effective index can reach 2,but a dielectric with even higher index (such as diamondwith (cid:15) diamond = 5 .
76) was avoided to ensure that theplasmons are not confined unnecessarily tightly to thesilver surface, which would give rise to significantly higherohmic losses.The direct relationship between the height of the di-electric layer d and the resulting (complex) refractive in-dex ˜ n ( d ) = n ( d ) + iχ ( d ) can be obtained from the follow-ing implicit equation [54]tanh( k d (cid:15) d d ) = − k air k d + k d k m k d + k air k m , (43)where k air = (cid:113) (˜ nk ) − k , (44) k d = (cid:112) (˜ nk ) − ε d k ε d , (45)and k m = (cid:112) (˜ nk ) − ε m k ε m , (46)where k = 2 π/λ SiV and in our calculation we ignored,for simplicity, the presence of the diamond layer, as itdoes not significantly modify the effective index seen bythe plasmons as long as the diamond layer is much thinerthan the transverse confinement of the plasmons, whichis on the order of a wavelength.Fig. 7(a) shows the real part n ( d ) and imaginary part χ ( d ) (inset) of the complex refractive index ˜ n ( d ) seenby the plasmons as the thickness of the dielectric d isvaried. The effective refractive index increases monoton-ically with the thickness of the dielectric layer. Since therefractive index of the fish eye increases radially inward(see Eq. (1)), the dielectric layer in the fish eye lens takesa conical shape as shown in Fig. 7(b).From the imaginary part of the effective refrac-tive index χ ( d ), we can estimate the average pho-ton loss rate due to ohmic losses via the relation κ abs ( r ) /ω = χ ( r ) /n ( r ) (see Section II E). Since this loss1rate varies significantly across the lens, we numericallyaverage χ ( r ) /n ( r ) over the radius of the lens and obtainthe averaged quantity κ abs ( r ) /ω ≈ × − . This is theleading order contribution to the photon loss.Photons can also be lost from the lens by leaking outthrough the mirror. Assuming that the reflectivity of themirror is r , we can estimate the loss rate κ mirror /ω . Inthe absence of other loss mechanisms, the photon wouldbounce off the mirror ∼ /t times before being lost,where t = 1 − r . The time interval between two bouncesis approximately (2 R ) / ( c/ ¯ n ), where R is the radius ofthe lens, c is the speed of light in vacuum and ¯ n is theaverage index of refraction in the lens. Thus the lifetimeof the photon due to the finite mirror reflectivity is τ mirror ∼ κ mirror ∼ R c/ ¯ n t . (47)Making the conservative estimate that r = 0 .
95, we ob-tain the following loss rate κ mirror ω ∼ π n λ t R ∼ × − , (48)where we have used n ( r ) = 1 .
57, which is obtained bynumerically averaging the refractive index over the radiusof the lens. Note that this shows that the losses due to thefinite reflectivity of the mirror are an order of magnitudesmaller than the ohmic losses.Next, we consider emission into free space γ . Inthe close proximity of a metal surface, the rate of de-cay of the emitter into plasmonic modes γ can signifi-cantly exceed the rate of emission into free-space modes γ = d z ω / (3 πε (cid:126) c ) [41, 77]. Here, we take the Purcellfactor to be η = γ/γ ≈
3, which is the approximate valuefor a z-oriented dipole 10-15 nm away from a flat silversurface emitting radiation at 737nm [78]. Furthermore,we also make the conservative estimate that the emissionto free space is reduced by a factor of two due to thepresence of the silver surface [79]. In order to account forthe presence of this additional decay channel, we need tomake the replacement γ → γ + γ / F = e − π ( η ) R λ α . (49)Note that this equation holds only if the atomic frequen-cies fall half-way between two resonances and the atomsare placed at two antipodal points anywhere in the lens.Substituting R /λ SiV = 1 . α = ( κ abs + κ mirror ) /ω =3 . × − and η = 3 into Eq. (49), we obtain that thefidelity of the entangling operation would be approxi-mately F = 80%. We note that this fidelity could befurther improved by utilizing the adiabatic passage of adark state in a Raman scheme [58]. VI. CONCLUSION
In conclusion, we have investigated the single-photondynamics of atoms inside the fish eye lens. We demon-strated that the lens mediates long-range interactions be-tween distant emitters. The dipole-dipole interaction hasan infinite range, limited only by the decay rate of thecavity modes. Furthermore, our results show that the fisheye focuses a single photon to a diffraction-limited areaduring the exchange of a photon between two antipodalatoms, whose frequency is tuned between two resonancesof the cavity. We derived closed-form expressions for thedecay rates and dipole-dipole interaction of atoms in thepresence of losses and studied the fidelity of entanglingoperations. We confirmed the validity of our analysis,which relied on the Born-Markov approximation, by nu-merically solving the Schr¨odinger equation. Finally, weproposed a possible realization for the fish eye lens us-ing tranformational plasmon optics and silicon-vacancycenters that could open up the fish eye for practical ap-plications.
VII. ACKNOWLEDGEMENTS
We thank U. Leonhardt, E. Shahmoon, M. Mezei, S.Horsley, K. Lalumi`ere, W. M. R. Simpson, A. Levy,S. Choi, D. Wild, and M. Kan´asz-Nagy for stimulat-ing discussions. We acknowledge funding from the MIT-Harvard CUA, NSF and the Hungary Initiatives Foun-dation.
APPENDIXAppendix A: Derivation of a closed-form expression for the dipole-dipole interaction
In this section we show that the single-source Green function derived by Leonhardt [7, 12] for the 2D fish eye lenscan be written as a sum over the eigenmodes of the lens (Eq. (7)). This result enables us to connect the Green’sfunction to the expressions obtained for the atomic properties from the master equation treatment.2
1. Green’s function of the 2D fish eye
The single-source Green’s function of Maxwell’s 2D fish eye (of radius R , thickness b and refractive index profile n ( r ) = n r/R ) ) is a solution of the following equation( ∂ α ∂ ν − δ αν ∂ η ∂ η ) G νβ ( r , r (cid:48) , ω ) − ε ( r ) ω c G αβ ( r , r (cid:48) , ω ) = δ αβ δ ( r − r (cid:48) ) , (A1)where α, β, µ, ν = x, y, z and summation is implied over repeated indices and ε ( r ) = n ( r ) = (cid:2) n / (1 + ( r/R ) ) (cid:3) isthe position-dependent electric permittivity. When b is chosen such that ω (cid:28) πc/b , only the lowest TE polarizedmode of the fish eye can be excited and the electric field is invariant along the z -axis ( ∂ z E ( r ) = 0). The explicitexpression for the zz -components of the Green’s function (Eq. (15)) is then given by G zz ( r , r , ω ) = F ( α , α ) − F ( α , /α ∗ ) , where F ( α , α ) = − P ν ( ξ ( α , α ))4 b sin( πν ) , (A2)where P ν is the Legendre function of (non-integer) oder ν , ν = 12 (cid:34)(cid:114) ω c R n + 1 − (cid:35) / ∈ Z and ξ ( α , α ) = | ζ ( α , α ) | − | ζ ( α , α ) | + 1 , where ζ ( α , α ) = α − α α α ∗ + 1 , and α j = r j R (cid:124)(cid:123)(cid:122)(cid:125) ρ j e iφ j ( j = 1 , .
2. Virtual coordiantes
The stereographic transformation [8] r j (cid:55)→ r j − R r j + R = ρ j − ρ j + 1 =: cos θ j , (A3)can be used to map any point ( r, φ ) on the real plane to a point ( θ, φ ) on the surface of a virtual sphere (where φ isthe same value in both coordinate systems). Using this transformation, we can simplify the definition of the Green’sfunction ζ ( α , α ) = ( ρ cos φ − ρ cos φ ) + i ( ρ sin φ − ρ sin φ ) ρ ρ cos( φ − φ ) + 1 + iρ ρ sin( φ − φ ) | ζ ( α , α ) | = ρ + ρ − ρ ρ cos( φ − φ )( ρ ρ ) + 1 + 2 ρ ρ cos( φ − φ ) ξ ( α , α ) = ρ + ρ − ( ρ ρ ) − − ρ ρ cos( φ − φ ) ρ + ρ + ( ρ ρ ) + 1 == − (cid:20)(cid:18) ρ − ρ + 1 (cid:19) (cid:18) ρ − ρ + 1 (cid:19) + (cid:18) ρ ρ + 1 (cid:19) (cid:18) ρ ρ + 1 (cid:19) cos( φ − φ ) (cid:21) == − (cid:2) cos θ cos θ + sin θ sin θ cos( φ − φ ) (cid:3) = − cos θ , where θ is the spherical distance beetween two points, ( θ , φ ) and ( θ , φ ), on the surface of a unit sphere, sincecos θ = x x = sin θ cos φ sin θ sin φ cos θ · sin θ cos φ sin θ sin φ cos θ = cos θ cos θ + sin θ sin θ cos( φ − φ ) . (A4)3Similarly, ζ ( α , /α ∗ ) = ( ρ cos φ − ρ cos φ ) + i ( ρ sin φ − ρ sin φ ) ρ ρ cos( φ − φ ) + 1 + i ρ ρ sin( φ − φ ) , | ζ ( α , /α ∗ ) | = ρ + ρ − ρ ρ cos( φ − φ )( ρ ρ ) + 1 + 2 ρ ρ cos( φ − φ ) ,ξ ( α , /α ∗ ) = ρ + (cid:16) ρ (cid:17) − ( ρ ρ ) − − ρ ρ cos( φ − φ ) ρ + (cid:16) ρ (cid:17) + ( ρ ρ ) + 1 == − (cid:20)(cid:18) ρ − ρ + 1 (cid:19) (cid:18) − ρ ρ + 1 (cid:19) + (cid:18) ρ ρ + 1 (cid:19) (cid:18) ρ ρ + 1 (cid:19) cos( φ − φ ) (cid:21) == − (cid:2) cos θ cos( π − θ ) + sin θ sin( π − θ ) cos( φ − φ ) (cid:3) = − cos θ (cid:48) , where, now, θ (cid:48) is the spherical distance between the points ( θ , φ ) and ( π − θ , φ ).Now, we can write the Green’s function as G zz ( r , r , ω ) = − P ν ( − cos θ ) − P ν ( − cos θ (cid:48) )4 b sin( πν ) . (A5)
3. Expansion in Spherical harmonics a. Expansion with respect to l The full set of Legendre polynomials, P l , form a complete, orthogonal basis on the space of smooth [ − , → R functions. This allows us to expand the Legendre function P ν in terms of the Legendre polynomials P l P ν ( x ) = ∞ (cid:88) l =0 c l P l ( x ) , where c l = 2 l + 12 +1 (cid:90) − dx P l ( x ) P ν ( x ) . (A6)According to Abramowitz & Stegun, Section 8.14 [80] +1 (cid:90) − dx P η ( x ) P ν ( x ) = 2 π πη ) sin( πν )[ ψ ( η + 1) − ψ ( ν + 1)] + π sin( πν − πη )( ν − η )( ν + η + 1) , (A7)where ψ is the digamma function and which expression, in case of η = l ∈ N , simplifies to +1 (cid:90) − dx P l ( x ) P ν ( x ) = 2 π ( − l sin( πν ) ν ( ν + 1) − l ( l + 1) , if l ∈ N . (A8)This means that P ν ( x ) = sin( πν ) π ∞ (cid:88) l =0 ( − l l + 1 ν ( ν + 1) − l ( l + 1) P l ( x ) , (A9)and we can write the Green’s function as G zz ( r , r , ω ) = − πb ∞ (cid:88) l =0 ( − l l + 1 ν ( ν + 1) − l ( l + 1) (cid:2) P l ( − cos θ ) − P l ( − cos θ (cid:48) ) (cid:3) . (A10)4 b. Expansion with respect to m According to the addition theorem of spherical harmonics, P l ( x · x ) = P l (cos θ ) = 4 π l + 1 + l (cid:88) m = − l Y m ∗ l ( θ , φ ) Y ml ( θ , φ ) , (A11)where the spherical harmonics are defined by Y ml ( θ, φ ) = (cid:115) l + 14 π ( l − m )!( l + m )! P ml (cos θ ) e imφ , (A12)where P ml are the associated Legendre polynomials.By using this theorem, and the property that P l ( − x ) = ( − l P l ( x ), we can write P ( − cos θ ) = ( − l π l + 1 + l (cid:88) m = − l Y m ∗ l ( θ , φ ) Y ml ( θ , φ ) ,P ( − cos θ (cid:48) ) = ( − l π l + 1 + l (cid:88) m = − l Y m ∗ l ( θ , φ ) Y ml ( π − θ , φ ) (cid:124) (cid:123)(cid:122) (cid:125) ( − l − m Y ml ( θ ,φ ) ,P ( − cos θ ) − P ( − cos θ (cid:48) ) = ( − l π l + 1 + l (cid:88) m = − l (cid:104) − ( − l − m (cid:105) Y m ∗ l ( θ , φ ) Y ml ( θ , φ ) . The expression inside the square brackets is zero if l and m have the same parity, and 2, if they have different parity.The set of m values for which the corresponding term is non-zero is M l = {− ( l − , − ( l − , . . . , ( l − , ( l − } .Using this notation, we can write the Green’s function as G zz ( r , r , ω ) = − b ∞ (cid:88) l =0 (cid:88) m ∈ M l Y m ∗ l ( θ , φ ) Y ml ( θ , φ ) ν ( ν + 1) − l ( l + 1) (A13)
4. Expansion in cavity modes
Recall from Eq. (7) that the TE eigenmodes of Maxwell’s fish eye with radius R and width b are f l,m ( r ) = (cid:115) bR n (cid:115) l + 14 π ( l − m )!( l + m )! P ml (cid:18) r − R r + R (cid:19) e imφ = (cid:115) bR n ( − m Y ml ( θ, φ ) , (A14)where r and φ are polar coordinates of r and cos θ = r − R r + R . They satisfy the orthonormality condition, δ l,l (cid:48) δ m,m (cid:48) = R (cid:90) dr r π (cid:90) dφ b (cid:90) dz n ( r ) f ∗ l,m ( r, φ ) f l (cid:48) ,m (cid:48) ( r, φ ) . (A15)The corresponding (partially degenerate) eigenfrequencies are ω l,m = ω l = cR n (cid:112) l ( l + 1) =: ck l . (A16)Now, we can write the Green’s function in Eq. (A13) as G zz ( r , r , ω/c ) = − R n ∞ (cid:88) l =0 (cid:88) m ∈ M l f ∗ l,m ( r ) f l,m ( r ) ν ( ν + 1) − l ( l + 1)= − ∞ (cid:88) l =1 (cid:88) m ∈ M l f ∗ l,m ( r ) f l,m ( r )( ω/c ) − k l , (A17)5where we used the connection between ω l and l , and ω and ν . Using Eq. (8), we can then write the dipole-dipoleinteraction within the fish eye in the following form δω ( r , r ) = d z ω (cid:126) ε c Re { G zz ( r , r , ω ) } = d z ω (cid:126) ε c ∞ (cid:88) l =1 (cid:88) m ∈ M l f ∗ l,m ( r ) f l,m ( r ) k l − ( ω /c ) . (A18)We note that this decomposition of the Green’s function in terms of the eigenmodes of the fish eye is a particularexample of Fredholm’s theorem [81].
5. Comparison with master equation results
Recall that the dipole-dipole interaction obtained from the master equation (see Eq. (14)) has the form δω ( r , r ) = d z (cid:126) ε (cid:88) l,m ω l ω l − ω f ∗ l,m ( r ) f l,m ( r ) . (A19)Making use of the transformation ω l ω l − ω = (cid:20) ω ω l − ω (cid:21) , (A20)and the dipole-dipole interaction becomes δω ( r , r ) = d z (cid:126) ε (cid:88) l,m f l,m ( r ) f l,m ( r ) + ω (cid:88) l,m f ∗ l,m ( r ) f l,m ( r ) ω l − ω . (A21)Since the modes f l,m form a complete basis, i.e. (cid:88) l,m f ∗ l,m ( r ) f l,m ( r ) = δ (3) ( r − r ) , (A22)the first term inside the square brackets does not contribute if r (cid:54) = r , thus δω ( r , r ) = d (cid:126) ε ω c (cid:88) l,m f ∗ l,m ( r ) f l,m ( r ) k l − ( ω /c ) , if r (cid:54) = r , (A23)which is identical to Eq. (A18). This shows that the right-hand side of Eq. (14) can indeed be replaced by Eq. (8)and Eq. (15).More generally, using Eq. (A20) and Eq. (A22) we can express Eq. (A17) in the form d (cid:126) ε ω c G zz ( r , r , ω ) = d (cid:126) ε (cid:88) l,m ω l f ∗ l,m ( r ) f l,m ( r ) (cid:18) ω + ω l − ω − ω l (cid:19) , (A24)from which it is straighforward to show that d (cid:126) ε ( ω + iκ ) c G zz ( r , r , ω + iκ ) = d (cid:126) ε (cid:88) l,m ω l f ∗ l,m ( r ) f l,m ( r ) (cid:26) ( ω l + ω ) κ + ( ω l + ω ) + ( ω l − ω ) κ + ( ω l − ω ) (A25) − i (cid:18) κκ + ( ω l + ω ) − κκ + ( ω l − ω ) (cid:19)(cid:27) , (A26)which allows us to express the decay rates Eq. (22) and the dipole-dipole interaction (Eq. (23)) in the presence oflosses in the following closed formΓ( r i , r j ) = 2 d z (cid:126) ε c Im { ( ω + iκ ) G zz ( r i , r j , ω + iκ ) } . (A27)6and δω ( r i , r j ) = d z (cid:126) ε c Re { ( ω + iκ ) G zz ( r i , r j , ω + iκ ) } . (A28)We note that G zz ( r i , r j , ω + iκ ) is the solution of the following equation( ∂ α ∂ ν − δ αν ∂ η ∂ η ) G νβ ( r , r (cid:48) , ω + iκ ) − n ( r ) ( ω + iκ ) c G αβ ( r , r (cid:48) , ω + iκ ) = δ αβ δ ( r − r (cid:48) ) , (A29)which can be thought of as the dyadic equation for the fish eye lens with the complex refractive index˜ n ( r ) = n ( r )(1 + iκ/ω ), since n ( r ) ( ω + iκ ) = n ( r ) (1 + iκ/ω ) ω = ˜ n ( r ) ω . (A30)Noting that for κ (cid:28) ω the following approximations holdΓ( r i , r j ) ≈ d z ω (cid:126) ε c Im { G zz ( r i , r j , ω + iκ ) } and δω ( r i , r j ) ≈ d z ω (cid:126) ε c Re { G zz ( r i , r j , ω + iκ ) } , (A31)we find that photon loss from the modes of the fish eye of the form of Eq. (19) can simply be modeled with thecomplex refractive index profile ˜ n ( r ). Appendix B: Numerical Solution of the Schr¨odinger Equation
In this Appendix we describe an efficient way to numerically solve the Schr¨odinger Equation while including thetwo atoms and the modes of the fish eye in the dynamics.
1. Hamiltonian a. Electric dipole interaction of a single atom
Recall that the electric dipole coupling of a single atom, placed at r i , to the electromagnetic field modes of the fisheye is given by V = − d i · E ( r i ) , where d i = d z ˆ z (cid:0) σ † i + σ i (cid:1) , where σ i = | g i (cid:105)(cid:104) e i | , (B1)where d z is the magnitude of the transition dipole moment between the two states of the atom | e i (cid:105) and | g i (cid:105) , whoseenergy difference is (cid:126) ω . Substituting Eq. (3) into Eq. (B1) and neglecting the counter-rotating terms in V , we arriveat V RWA = (cid:88) l,m id (cid:115) (cid:126) ω l bR ε (cid:2) a l,m σ † Y l,m ( θ, φ ) − a † l,m σY l,m ( θ, φ ) (cid:3) , (B2)where n = 1 was assumed. b. Two atoms Assuming that there are two identical atoms positioned at r and at r , the interaction term takes the form V RWA ( r ) + V RWA ( r ). The total Hamiltonian then becomes H = (cid:126) ω σ † σ + (cid:126) ω σ † σ + (cid:88) l,m (cid:126) ω l a † l,m a l,m + (cid:88) l (cid:126) g l (cid:34) σ † (cid:88) m a l,m Y l,m ( θ , φ ) + σ † (cid:88) m a l,m Y l,m ( θ , φ ) (cid:35) + h.c. , (B3)where g l = id z √ (cid:126) bR ε √ ω l .7The diagonal terms (cid:126) ω (cid:16) σ † σ + σ † σ + (cid:80) l,m a † l,m a l,m (cid:17) , simply give a constant energy shift to all eigenvectors inthe subspace of interest and can, therefore, be subtracted from the Hamiltonian. The modified Hamiltonian thentakes the form H/ (cid:126) = (cid:88) l δ l (cid:88) m a † l,m a l,m + (cid:88) l g l (cid:34) σ † (cid:88) m a l,m Y l,m ( θ , φ ) + σ † (cid:88) m a l,m Y l,m ( θ , φ ) (cid:35) + h.c. , (B4)where δ l = ω l − ω . c. Opposite positions If the two atoms are placed at opposite positions ( θ = θ = θ and φ = φ = − φ ), then we can write the interactionpart of H as V / (cid:126) = (cid:88) l g l (cid:88) m (cid:104) σ † a l,m y l,m + σ † a l,m ( − m y l,m + h.c. (cid:105) , (B5)where y l,m = Y l,m ( θ, φ ). Here we used that Y l,m ( θ, − φ ) = ( − m Y l,m ( θ, φ ). Since the summation of m goes over m = − l + 1 , − l + 3 , . . . l − , l − m and l always have opposite parity, and we can pull out ( − m = ( − l +1 fromthe summation, giving V / (cid:126) = (cid:88) l g l (cid:104) σ † + ( − l +1 σ † (cid:105) (cid:88) m a l,m y l,m + h.c. (B6)We define an incomplete set of new modes, A l = (cid:80) m a l,m y l,m N l , N l = (cid:88) m | y l,m | , [ A l , A † l ] = 1 . (B7)The normalization factor can be calculated as follows. (cid:88) m ∈ M | Y l,m ( θ, φ ) | = + l (cid:88) m = − l − ( − l − m | Y l,m ( θ, φ ) | , where M = {− l + 1 , − l + 3 , . . . l − , l − } . (B8)Recall the sum rule: + l (cid:88) m = − l Y ∗ l,m ( θ , φ ) Y l,m ( θ , φ ) = 2 l + 14 π P l (cos θ ) , (B9)where θ is the angle between point 1 and 2 on the unit sphere, and P l is the l th Legendre polynomial. We useEq. (B9) to evaluate the two series: + l (cid:88) m = − l | Y l,m ( θ, φ ) | = 2 l + 14 π P l (1) = 2 l + 14 π , (B10) + l (cid:88) m = − l ( − l − m | Y l,m ( θ, φ ) | = + l (cid:88) m = − l Y ∗ l,m ( π − θ, φ ) Y l,m ( θ, φ ) = 2 l + 14 π P l (cid:0) cos( π − θ ) (cid:1) , (B11)where we used that ( − l − m Y l,m ( θ, φ ) = Y l,m ( π − θ, φ ) and that the angle between point ( θ, φ ) and point ( π − θ, φ )is θ = π − θ .Using these new A l modes, the interaction can be written as V / (cid:126) = (cid:88) l g l N l (cid:104) σ † + ( − l +1 σ † (cid:105) A l + h.c. (B12)Modes A l with different l parity couple to different combination of the two atoms. Let us define σ o = σ + σ √ , σ e = σ − σ √ , (B13)8and write V / (cid:126) = (cid:88) l ∈ odd G l ( A l σ † o + A † l σ o ) + (cid:88) l ∈ even G l ( A l σ † e + A † l σ e ) , (B14)where G l = √ g l N l = (cid:115) d z c (cid:126) bR ε (cid:115) (2 l + 1) (cid:112) l ( l + 1)4 π (cid:2) − P l (cos( π − θ )) (cid:3) . (B15)
2. Numerical analysis a. Hilbert space
We are interested in the dynamics of a single excitation, i.e. we truncate the Hilbert space to H = Span (cid:8) | e (cid:105)| g (cid:105)| vac (cid:105) (cid:124) (cid:123)(cid:122) (cid:125) | a (cid:105) , | g (cid:105)| e (cid:105)| vac (cid:105) (cid:124) (cid:123)(cid:122) (cid:125) | b (cid:105) , {| g (cid:105)| g (cid:105) A † l | vac (cid:105) (cid:124) (cid:123)(cid:122) (cid:125) | l (cid:105) : l = 1 , , . . . l max } (cid:9) , (B16)and separate it into two subspaces H o = Span (cid:8) σ † o | g (cid:105)| g (cid:105)| vac (cid:105) (cid:124) (cid:123)(cid:122) (cid:125) | o (cid:105) , { A † l | g (cid:105)| g (cid:105)| vac (cid:105) (cid:124) (cid:123)(cid:122) (cid:125) | l (cid:105) : l = 1 , , , . . . } (cid:9) (B17) H e = Span (cid:8) σ † e | g (cid:105)| g (cid:105)| vac (cid:105) (cid:124) (cid:123)(cid:122) (cid:125) | e (cid:105) , { A † l | g (cid:105)| g (cid:105)| vac (cid:105) (cid:124) (cid:123)(cid:122) (cid:125) | l (cid:105) : l = 2 , , , . . . } (cid:9) , (B18)each of which is governed by its own Hamiltonian block. b. Hamiltonian The following H o , H e act as two independent blocks on H o and H e . H o = (cid:88) l ∈ odd (cid:104) δ l A † l A l + G l ( A l σ † o + A † l σ o ) (cid:105) = (cid:88) l ∈ odd (cid:2) δ l | l (cid:105)(cid:104) l | + G l ( | o (cid:105)(cid:104) l | + | l (cid:105)(cid:104) o | ) (cid:3) (B19) H e = (cid:88) l ∈ even (cid:104) δ l A † l A l + G l ( A l σ † e + A † l σ e ) (cid:105) = (cid:88) l ∈ even (cid:2) δ l | l (cid:105)(cid:104) l | + G l ( | e (cid:105)(cid:104) l | + | l (cid:105)(cid:104) e | ) (cid:3) , (B20)where δ l = cR (cid:104)(cid:112) l ( l + 1) − (cid:112) l ( l + 1) (cid:105) , (B21)where l stands for the atomic frequency, i.e. ω = cR (cid:112) l ( l + 1) and G l = (cid:115) d z c (cid:126) bR ε (cid:115) (2 l + 1) (cid:112) l ( l + 1)4 π (cid:2) − P l (cos( π − θ )) (cid:3) . (B22) c. Results: Time series We start the system from | ψ (0) (cid:105) = | e (cid:105)| g (cid:105)| vac (cid:105) = | a (cid:105) = | o (cid:105) + | e (cid:105)√ , evolve it with U ( t ) = exp[ − iHt/ (cid:126) ], to get | ψ ( t ) (cid:105) = 1 √ (cid:2) e − iH o t/ (cid:126) | o (cid:105) + e − iH e t/ (cid:126) | e (cid:105) (cid:3) = 1 √ (cid:2) (cid:88) j (cid:104) φ o,j | o (cid:105) e − i Ω o,j t | φ o,j (cid:105) + (cid:88) k (cid:104) φ e,k | e (cid:105) e − i Ω e,k t | φ e,k (cid:105) (cid:3) , (B23)9where Ω o,j , | φ o,j (cid:105) and Ω e,k , | φ e,k (cid:105) are eigenvalues and eigenstates of H o / (cid:126) and H e / (cid:126) , respectively.Finally, we note that the numerical results shown in Fig. 5 are independent of the atomic parameters and thethickness of the lens as only ratios of the dipole-dipole interaction, spontaneous decay and cooperative decay areconsidered (each of which is proportional to the square of the prefactor (cid:112) d z c/ (cid:126) bR (cid:15) ). 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