PHYSICAL REVIEW A 97, 023827 (2018)
Light scattering from an atomic array trapped near a one-dimensional nanoscale waveguide: A microscopic approach
V. A. Pivovarov,1 A. S. Sheremet,2,3 L. V. Gerasimov,4 V. M. Porozova,5 N. V. Corzo,2 J. Laurat,2 and D. V. Kupriyanov5,* 1Physics Department, St.-Petersburg Academic University, Khlopina 8, 194021 St.-Petersburg, Russia 2Laboratoire Kastler Brossel, Sorbonne Université, CNRS, PSL Research University, 4 place Jussieu, 75005 Paris, France 3Russian Quantum Center, Novaya 100, 143025 Skolkovo, Moscow Region, Russia 4Faculty of Physics, M.V. Lomonosov Moscow State University, Leninskiye Gory 1-2, 119991 Moscow, Russia 5Department of Theoretical Physics, St-Petersburg State Polytechnic University, 195251 St.-Petersburg, Russia
(Received 17 November 2017; published 16 February 2018)
The coupling of atomic arrays and one-dimensional subwavelength waveguides gives rise to interesting photon transport properties, such as recent experimental demonstrations of large Bragg reflection, and paves the way for a variety of potential applications in the field of quantum nonlinear optics. Here, we present a theoretical analysis for the process of single-photon scattering in this configuration using a full microscopic approach. Based on this formalism, we analyze the spectral dependencies for different scattering channels from either ordered or disordered arrays. The developed approach is entirely applicable for a single-photon scattering from a quasi-one-dimensional array of multilevel atoms with degenerate ground-state energy structure. Our approach provides an important framework for including not only Rayleigh but also Raman channels in the microscopic description of the cooperative scattering process.
DOI: 10.1103/PhysRevA.97.023827
I. INTRODUCTION with the evanescent field of the waveguide mode has important differences from the light scattering from atoms in free space. Efficient control of light-matter interaction at the single- The effects of Zeeman degeneracy of an atomic transition photon level is a central and challenging task for quantum should be taken into consideration for a relevant description optics and quantum information science [1–7]. At the fun- of the Raman channels, which at the present time has been damental level, this interaction manifests itself via the basic primarily studied in free space [37,38]. The problem becomes quantum electrodynamics processes of spontaneous emission, more complicated as the strong-field confinement provided absorption, and scattering of a single photon by a single atom. by the nanofiber imposes an inherent link between the local In free space, the efficiency of this interaction is limited by the polarization and propagation direction of light. In such a small atomic scattering cross section in comparison to the usual strongly nonparaxial regime, the spin and orbital angular large beam shining area. However, it can be greatly enhanced momentum of the guided light obey joint dynamics and cannot by placing a single atom in the vicinity of a subwavelength be independently considered. In particular, it leads to direction- waveguide due to the Purcell effect [8] or by considering large sensitive emission and absorption of the guided light by the atomic ensembles [9]. The combination of these approaches atoms [30,39–42]. has motivated recent experimental efforts towards the develop- All these experimental results and theoretical works demon- ment of novel platforms that integrate an atomic chain coupled strate that such a hybrid system is a versatile platform for with a dielectric nanoscale waveguide [10–16]. studying various cooperative effects emerging from both the One example of such platforms is the so-called subwave- collective atomic structuring and the waveguide-mediated length nanofiber [15–20]. Due to a large evanescent field, the long-range coupling between atoms. The internal correlations, guided light can be efficiently used for trapping atoms close to existing in a many-particle microscopic quantum entangled the surface [21–23] and interacting with them [24–27]. Recent state, can play an essential role in the cooperative scattering experiments include, for instance, the demonstration of optical process. In this context, an ab initio theoretical insight on the memories in this setting [28,29]. The trapping technique can problem of light-atoms interaction would provide a natural also be adjusted to arrange atoms in an optical lattice commen- extension of the convenient but sometimes insufficient self- surate or nearly commensurate with the resonant wavelength. consistent description, based on the density matrix approach Such capability enables one to investigate Bragg reflection and the Maxwell-Bloch formalism. [30,31] and long-range interactions [32]. To clarify the last statement, let us be more concrete and These recent experiments and supporting theoretical studies point out that any self-consistent description of a many-body [33–36] have shown that light scattering from atoms interacting problem conventionally assumes the physical equivalence in description of proximal particles and underestimates the role of internal correlations. But for dense atomic systems, the mutual correlations are an attribute of their exact microscopic dynam- *[email protected] ics, which can be manifestable when cooperative processes
2469-9926/2018/97(2)/023827(15) 023827-1 ©2018 American Physical Society V. A . P I VOVA ROV et al. PHYSICAL REVIEW A 97, 023827 (2018) of light spontaneous emission or scattering, generally associ- The latter requires infinitely long “interaction” time, τ → ated with the Dicke-type super- and subradiance phenomena, +∞.TheT matrix, considered as a function of arbitrary become sensitive to the spatial distribution of atomic dipoles complex energy argument E, is expressed as follows: and to the excitation conditions; see [7,43,44]. Such a situation 1 takes place when the cooperative phenomena are considered Tˆ (E) = Vˆ + Vˆ V,ˆ (2.3) for atoms confined and interacted with nanoscale structures. E − Hˆ For the subwavelength waveguide systems, this has motivated researchers to extend the Maxwell-Bloch formalism toward where a more rigorous Heisenberg-Langevin approach based on ˆ = ˆ + ˆ the effective Hamiltonian concept; see [33–36]. But for the H H0 V. (2.4) particular case of a single-photon scattering, the problem can Hˆ , Vˆ , and Hˆ are the unperturbed Hamiltonian, the interacting be described in the more natural formalism of the quantum 0 part, and the total system Hamiltonian, respectively. scattering theory. The present paper develops such a systematic For a particular case of a single-photon scattering on an microscopic theory of light scattering from an atomic array ensemble of N atomic dipoles, the interaction part is given by trapped near a nanofiber, taking into account the entire interac- tion dynamics together with the complete angular momentum N structure of the guided light and the degenerate energy structure (a) Vˆ =− dˆ · Eˆ (ra) + Hˆself . (2.5) of the atoms. a=1 The paper is organized as follows. In Sec. II, we first provide a general description of our microscopic approach of The first term accumulates partial interactions for each of an light scattering in a waveguide configuration and emphasize ath atomic dipole d(a) with an electric field Eˆ (r) at the point the differences with a similar process in free space. The of the dipole’s position. The second term is the dipole’s self- derivation details concerning the modification of the electric- energy part, which is mainly important for renormalization of field Green’s function near the nanofiber are given in Appendix the self-action divergencies [7,46,47]. The field operator can A. In Sec. III, we then present numerical simulations for light be expressed by a standard expansion in the basis of plane scattering from an array consisting of five atoms. Ordered waves as and disordered configurations are considered. The parameters + − of the waveguide mode used in the simulation are given in Eˆ (r) ≡ Eˆ ( )(r) + Eˆ ( )(r) Appendix B. Finally, Sec. IV concludes the paper. 1/2 2πhω¯ † = j i[e a eik·r − e∗a e−ik·r], (2.6) V j j j j II. THE SCATTERING PROBLEM REVISED j FOR A WAVEGUIDE CONFIGURATION where j ≡ k,α is the complex mode index with the mode wave In this section, we review the basic points of the quantum vector k, frequency ωj = ck, and α = 1,2 numerating two scattering problem. Normally introduced in terms of the orthogonal transverse polarization vectors ej ≡ ekα for each k. † transformation of plane waves, it is extended here to the Here, aj and aj are the annihilation and creation operators for specific configuration where the input and output states are the jth field’s mode in free space and the quantization scheme fundamental modes of the waveguide. includes the periodic boundary conditions in the quantization volume V = L3. A. Mathematical framework According to the standard assumptions of the time- dependent scattering theory in free space, the infinite “in- The mathematical framework of the quantum scattering teraction” time τ is physically associated with the duration problem is based on the concept of the asymptotic evolutionary with which the photon’s wave packet overlaps and passes the operator Sˆ that transforms the system states from infinite past | | scattering object. As far as the wave packet tends to approach ψ in to infinite future ψ out as a result of the interaction pro- the monochromatic state, its duration should approach infinity cess [45]. In the interaction representation, the corresponding and, as a consequence, this justifies the presence of the δ asymptotic transformation is given by function and energy conservation in Eq. (2.2). However, the i H τ − i Hτ i H τ transfer τ →+∞ should be done only in the final step |ψ = e 2¯h 0 e h¯ e 2¯h 0 |ψ ≡ Sˆ|ψ , (2.1) out in in of the derivation procedure together with constructing the where τ →+∞. The operator Sˆ can be represented as a matrix main physical characteristics of the process such as transition in a decoupled basis of two interacting subsystems, which we probability per unit time, scattering tensor, and cross section. specify as |φi for the initial and |φf for the final system states, The derivation is based on the arguments of time-dependent perturbation theory, where the field subsystem is considered as sin[(E − E )τ/2¯h] f i a plane-wave contribution into the states |φi and |φf posed Sfi = δfi − 2i Tfi(Ei + i0) Ef − Ei with periodic boundary conditions into a certain quantization volume with a length scale L thatisthesameasassumedin ⇒ δfi − 2πiδ(Ef − Ei ) Tfi(Ei + i0). (2.2) expansion (2.6). Because of it, for any wave packet, its longest The first term selects a noninteractive contribution and the duration τ is naturally limited by the time scale associated with interaction part constructs the T matrix, i.e., the transition the quantization length L and both of these parameters should amplitude between the states with the same energy, Ef = Ei . approach infinity consistently such that τ = L/c →+∞.
023827-2 LIGHT SCATTERING FROM AN ATOMIC ARRAY TRAPPED … PHYSICAL REVIEW A 97, 023827 (2018)
where we redefined f = i emphasizing thereby the physical equivalence of initial and final states for light scattering within the waveguide modes. The energy conservation Ei = Ei is fulfilled in the applied regularization and we shall further treat the S-matrix components given by Eq. (2.7) in the original physical meaning as the probability amplitudes between the initial and final states. As a next step, we keep the interaction operator in the form of Eq. (2.5), but reexpand the field operator (2.6)inthe complete basis of the guided and external modes, ˆ = 1/2 (s) − † (s)∗ +··· E(r) (2πhω¯ s ) i[bs E (r) ds D (r)] . s (2.8)
Here we selected only the contribution of the guided modes, FIG. 1. Light scattering from an atomic array trapped near a enumerated by the mode index s, for the electric field E(s)(r) subwavelength dielectric waveguide. The atoms, which are located and the displacement field D(s)(r) = (r) E(s)(r), where (r) at a distance ρ − a from the surface, are spin oriented along the is the spatially dependent dielectric permittivity of the entire waveguide and the incident light is in the left-handed polarized medium (free space and the dielectric nanofiber). The modes waveguide mode. The scattered light leaves the one-dimensional are specified by Eqs. (A6) and (A8) in Appendix A, and the channel in either the forward or backward direction. ellipsis in Eq. (2.8) denotes the contribution of external modes. The mode operators are given by
B. Light scattering within the waveguide ωfree 1/2 1 = k √ 3 (s)∗ · ik·r We now consider the case where the incident photon bs akα wg d r D (r) ekα e , ω V impinges on a cylindrical single-mode nanoscale dielectric k,α=1,2 s waveguide in its fundamental HE11 mode [48]. We focus on free 1/2 † ω 1 a specific process where both the incident and the scattered d† = a k √ d3re−ik·r e∗ · E(s)(r). s kα ωwg V kα photons belong to the fundamental mode, i.e., only forward k,α=1,2 s or backward scattering occurs inside the waveguide. Actually (2.9) there is a continuum of the waveguide modes propagating along the fiber as translational waves parameterized by a longitudinal wave number k and posed with a periodic boundary conditions The expansion (2.8) is identical to the basic definition (2.6) due into a longitudinal quantization segment of length L.The to orthogonality and completeness relations for the complete geometry of the scattering process is shown in Fig. 1. set of the guided and external modes. In expression (2.9), we For such a specific scattering geometry, Eq. (2.2) is valid additionally labeled the mode frequencies for distinguishing wg since it is based on a general perturbation-theory analysis ap- the field’s mode in the presence of the waveguide, ωs ≡ ωs , free ≡ = plied to any quantum system with continuous spectrum. How- and in free space, ωk ωj ck. ever, the scattering channel is now described by transmission Because of the difference in mode representation in terms and reflection probabilities instead of scattering cross section. of the electric and the displacement fields, the operators bs and † The singularity of the second term in the definition of the S ds are not Hermitian conjugated counterparts. Nevertheless, matrix can be naturally regularized by associating the initial they are candidates for, respectively, annihilation and creation and final states with the wave packets having longitudinal operators of a photon in a specific waveguide mode. It may L wg free length . In accordance with the physical interpretation of time seem that the difference between ωs and ωk prevents their τ as the duration with which the photon’s wave packets overlap commutation relations from fulfilling the standard bosonic and pass the scattering object, we have τ = L/vg →∞, where operators. But for the waveguide, designed as a dielectric vg = dω/dk is the group velocity for the fundamental mode nanofiber with diameter less than the wavelength of the guided free wg at frequency ω associated with either incident or scattered light, the free-space modes with ωk ωs mainly dominate photons. Here we neglect a possible difference in group in the overlapping integral in (2.9) such that the respective fac- velocities for the case of Raman-type scattering caused by tor vanishes in these expansions. Then we can safely accept that † transitions in the atomic spin subsystem. operators bs and ds obey the standard bosonic commutation Thus for the scattering within waveguide modes and in † rules, [b ,d ] = δ . To prove this statement, it is important to L →∞ s s s,s the limit , the singular form of relation (2.2) can be take into account that the mode of displacement field D(s)(r) has regularized as follows: transversal profile in the reciprocal k representation because of divD(s)(r) = 0. L For the considered process, in the case of a near-resonant Sii = δii − i Tii (Ei + i0), (2.7) hv¯ g scattering, the matrix element of the transition operator (2.3)
023827-3 V. A . P I VOVA ROV et al. PHYSICAL REVIEW A 97, 023827 (2018) can be disclosed by the following expansion: waveguide configuration. The inverse resolvent matrix can be √ expressed in the following form: Tgs,g s (E) = 2πh¯ ωs ωs N R˜ˆ −1(E) = E − hω¯ − (E), (2.13) (s) ∗ (s) 0 × d · D (rb) d · E (ra) n mb nma = b,a 1 n ,n where ω0 is the resonant atomic frequency and (E)istheself- ˜ energy part. This term is calculated via its relevant expansion × ...m ,n,m ...|Rˆ(E)|...m − ,n,m + ... , b−1 b+1 a 1 a 1 by a set of irreducible diagrams. It is expected to have smooth (2.10) dependence on its energy argument and, for near-resonant scattering, can be reliably approximated by substituting E = where ω and ω are the frequencies of incident and scattered s s hω¯ with the assumption that the ground-state energy E = 0 photons, respectively. The transition amplitude is intrinsically 0 g for a degenerate system of the atomic Zeeman sublevels. determined by the matrix element of the resolvent operator The self-energy part can be constructed by keeping the of the system Hamiltonian projected onto a collective atomic leading contributions in its diagram expansion. For each ath state with a single optical excitation, atom from the ensemble, there is the following single-particle 1 self-energy term: R˜ˆ(E) = Pˆ Rˆ(E) Pˆ ≡ Pˆ P.ˆ (2.11) E − Hˆ The projector Pˆ is given by N ˆ = | P m1,...,ma−1,n,ma+1,...,mN dω μ ν (E) ⇒ = { } = dn mdmniDμν (ra,ra; ω) a 1 mj ,j a n 2π m × m1,...,ma−1,n,ma+1,...,mN |×|0 0|field, (2.12) × 1 ≡ (a) nn(E), (2.14) and selects in the atomic Hilbert subspace the entire set of E − hω¯ − Em + i0 the states where any jth of N − 1 atoms populates a Zeeman sublevel |mj in its ground state and one specific ath atom (with where the internal wavy line expresses the causal-type vacuum a running from 1 to N and j = a) populates a Zeeman sublevel Green’s function of the chronologically ordered polarization |n of its excited state. The field subspace is projected onto its components of the field operators, ˜ vacuum state and operator Rˆ(E) can be further considered as iD(E)(r,r ; τ) = T Eˆ (r,t)Eˆ (r ,t ) , (2.15) a matrix operator acting only in atomic subspace. μν μ ν
In the representation of the T matrix by the expansion (E) (2.10), the selected specific product of matrix elements runs and Dμν (r,r ; ω) is its Fourier image defined by Eq. (A1)in all the possibilities when the incoming photon is absorbed by Appendix A. For the sake of generality, here we use covariant any ath atom and the outgoing photon is emitted by any bth notation for the vector (dipole) and tensor (Green’s function) atom of ensemble, including the possible coincidence a = b. components and omit the indication of sum over repeated tensor indices. In a particular case of Cartesian frame, we The initial atomic state is given by |g ≡|m1,...,mN and | ≡| will further simplify our notation and show all the tensor the final atomic state by g m1,...,mN , where atoms can populate all the accessible internal states. Thus for the components via subscribed indices. system consisting of many atoms with degenerate ground state, Unlike the similar calculations performed in free space, the there is an exponentially rising number of scattering channels. Green’s function (2.15) depends here on its spatial arguments Nevertheless, for most problems, such as quantum memories, separately and is strongly modified in the presence of the the elastic scattering channel is mostly important and can be waveguide; see Appendix A. As an important consequence, calculated once we find the resolvent operator (2.11). even a single-particle contribution to the self-energy depends As can be verified, the arbitrary parameter L vanishes when on the atom’s location and has an anisotropic matrix structure substituting (2.10)into(2.7) such that the S matrix becomes a in the basis of its excited states. The respective correction to the regular and fairly defined physical quantity in our calculation atomic energy structure as well as to its decay parameters has an scheme. Its matrix elements give us the quantum transition anisotropic structure and, in the general case of the degenerate amplitudes for observing the system in particular final states excited state, cannot be simply reduced to the radiative Lamb in the considered quasi-one-dimensional scattering process. shift and decay constant. The correct renormalization of the We now turn to the determination of the resolvent operator single-particle self-energy part (2.14) concerns both the near- via the Feynman diagram method in the perturbation-theory field self-action as well as the radiative interaction of the atom technique. with the quantized field. As explained in [37], in the first leading orders of the perturbation theory, the double-particle coupling dominates in C. The resolvent operator the cooperative correction to the self-energy part. The respec- Below we apply a microscopic calculation of the projected tive contribution is given by the sum of two diagrams where resolvent operator for an atomic system with degenerate the excitation transfer induced by the dipole-type interaction ground state. This approach was developed earlier in [7,37,38] (2.5) has different time ordering. The time-ordered transfer for light scattering in free space and we adapt it here to the of a single optical excitation from an atom a to an atom b is
023827-4 LIGHT SCATTERING FROM AN ATOMIC ARRAY TRAPPED … PHYSICAL REVIEW A 97, 023827 (2018) described by the following diagram: near a nanofiber. The geometry is shown in Fig. 1. The atoms have two energy levels with degenerate Zeeman structure of the ground state. We consider an array of -configured atoms with the minimal accessible number of quantum states, i.e., with angular momentum F0 = 1 in the ground state and F = 0 in the excited state. Thus we further associate the quantum indices m ≡ F0,M0 and n ≡ F,M = 0,0, where M0 and M are the Zeeman projections of the atomic spin angular dω μ ν (E) momentum of the ground and excited states, respectively. Such ⇒ d d iD (r ,r ; ω) 2π n m m n μν b a an energy and angular momentum configuration exists as a closed transition in the hyperfine manifold of 87Rb and we use × 1 ≡ (ab+) mn nm(E), (2.16) the spectral parameters of a rubidium atom in our estimates. − − − + ; E hω¯ Em Em i0 We assume the initial collective state of atoms as spin oriented and the time-antiordered transfer by the complementary dia- along the waveguide direction such that all the atoms populate gram only one Zeeman sublevel, F0 = 1,M0 = 1, which is relevant for the realization of quantum interfaces based on atomic systems.
A. The waveguide parameters In our calculations, we consider a subwavelength nanofiber with radius a and dielectric constant . The solution of the ho- dω μ ν (E) mogeneous Maxwell equations can be obtained by factorizing ⇒ d d iD (rb,ra; ω) 2π n m m n μν the mode functions of the waveguide in cylindrical coordinates as specified by the first line in Eq. (A6) (see [48] for derivation × 1 ≡ (ab−) mn;nm(E). (2.17) details). For the fundamental HE mode, its components can E + hω¯ − E − E + i0 11 n n be superposed in the set of three basic functions, E (ρ), E (ρ), ν ρ φ The vector components of the dipole matrix elements dmn and μ and Ez(ρ), which in turn are compiled by Bessel functions of dnm are related with atoms a and b, respectively. In the pole a different type (see Appendix B). ≈ = approximation E En hω¯ 0,theδ-function singularities In cylindrical coordinates, the mode components, specified dominate in the sum of spectral integrals (2.16) and (2.17) by the first line in Eq. (A6), are given by and the sum of both terms gives + − (±1k) (±1k) (ab) ≈ (ab ) + (ab ) E (ρ) = Eρ (ρ),E (ρ) =±Eφ(ρ), mn;nm(E) mn;nm(¯hω0) mn;nm(¯hω0) ρ φ (±1k) = 1 μ ν (E) Ez (ρ) Ez(ρ), (3.1) = d d D (r ,r ; ω ). (2.18) h¯ n m m n μν b a 0 and, in a Cartesian frame, the mode components, specified by The derived expression has a clear physical meaning. The the second line in Eq. (A6), are given by real component of the double-particle contribution to the self- energy part reproduces both the static interaction between two E (ρ) − iE (ρ) E (ρ) + iE (ρ) (±1k) = ρ √ φ + ρ √ φ ±2iφ proximal dipoles and radiative correction to the quasi-energy Ex (ρ,φ) e , structure for the distant dipoles. Its imaginary component is 2 2π 2 2π responsible for the cooperative dynamics of the excitation iE (ρ) + E (ρ) iE (ρ) − E (ρ) E(±1k)(ρ,φ) =± ρ √ φ ∓ ρ √ φ e±2iφ, decay in the entire radiation process. For long distances, when y 2 2π 2 2π the atomic dipoles are separated by the radiation zone, this E (ρ) (±1k) = √z ±iφ latter term describes radiation interference between any pair of Ez (ρ,φ) e . (3.2) two distant atoms, which weakly reduces with the interatomic 2π separation for interaction via external field modes. But for −1 a collection of atoms arrayed along the waveguide, there For a single-mode√ waveguide with a k and with the refrac- = → → is always strong cooperative interaction via the evanescent tion index n 1, the solution approaches to Eρ (ρ) − → field of the fundamental waveguide mode (see Appendix iEφ(ρ) with vanishing longitudinal component Ez(ρ) 0 A). Thus, in the considered quasi-one-dimensional scattering, such that the fundamental mode becomes independent of the the cooperative effects become extremely important and the azimuthal angle φ and can be reliably approximated by a scattering process becomes strongly dependent on a particular Gaussian fundamental mode propagating in free space. In configuration of the atomic array. this paraxial-type approximation, the two degenerate modes with σ =±1 transform to two orthogonal right-handed and left-handed polarized Gaussian modes, respectively. How- III. RESULTS: SCATTERING FROM A ever, in a more realistic situation, as considered here, with NANOFIBER-TRAPPED ATOMIC ARRAY a k−1 but n 1, all the components are competitive and In this section, we present the results of our numerical all of them mediate the excitation process in the atomic simulations for light scattering from an array of atoms trapped system.
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associated with both the dipole self-action and the radiation Lamb shift (incorrectly described in the dipole gauge), which can be safely renormalized [7]. Nevertheless, beyond its infinite part, (r) contains a finite and important correction to the energy shift induced by the dipole coupling with the waveguide. Here we avoid the nontrivial part of the calculation of this correction and will further assume that all the atomic dipoles are set in similar conditions such that the unknown quantity (r) can be associated with an energy shift identical for all dipoles. According to the expansion of the Green’s function in terms of contributions of the guided and external modes [see Eq. (A9) in Appendix A], the radiation decay rate γ (r) can be similarly expanded as
2 2 (E) (wg) (ext) γ (r) =− d Im D (r,r; ω0) = γ (r) + γ (r), h¯ 2 0 μμ (3.4) with 4ω (wg) = 0 2 | |2 +| |2 +| |2 γ (r) d0 [ Eρ (ρ) Eφ(ρ) Ez(ρ) ], hv¯ g (3.5) (ext) 2 2 (ext) γ (r) =− d Im D (r,r; ω0) , h¯ 2 0 μμ
where d0 denotes modulus of the dipole matrix element, which is the same for all optical transitions in the considered case, and we conventionally assume the sum over the repeated tensor − − FIG. 2. Functions iEρ (ρ) (dashed blue line), Eφ (ρ) (green index μ = x,y,z. Because of the axial symmetry, the decay rate line), and Ez(ρ) (red line, lower in the graphs) contributing to the depends only on the distance ρ of the atom from the z axis. The waveguide modes (3.1)and(3.2)fora = 200 nm and for vacuum waveguide contribution γ (wg)(r) is a result of exact substitution = wavelength λ0 780 nm (rubidium). The vertical shaded area (dash- of the respective contribution to the Green’s function, given by dotted bounded) indicates the waveguide. The upper panel displays Eq. (A10). By approximating D(ext) with (A16) and (A17), we = the mode structure for silica with refraction index n 1.45 and the can estimate the contribution of the external modes γ (ext)(r)as = lower panel relates to a dielectric medium with n 1.1. For this lower the following correction to the natural decay constant γ : panel, we show a Gaussian fit of the fundamental mode (dash-dotted 2ω line). (ext) ∼ − 0 2 | |2 +| |2 γ (r) γ d0 Eρ (ρ) Eφ(ρ) hc¯ + ∗ This important property of a nanofiber is illustrated by the 2Re iEρ (ρ)Eφ(ρ) . (3.6) plots shown in Fig. 2. For a nanofiber with a = 200 nm and As commented in Appendix A, the second term eliminates for the mode frequency taken at the rubidium wavelength λ = 0 spontaneous emission into those vacuum modes, which in the 780 nm, we plot three functions −iE (ρ), −E (ρ), and E (ρ), ρ φ z paraxial approach coincide with the waveguide modes. which can be set as real and which were calculated for two In Fig. 3, we show the results of our numerical simulations different refraction indices, n = 1.45 (silica) and n = 1.1(to for the rate of spontaneous decay γ (ρ) as a function of the radial follow the tendency to the paraxial asymptotic). For the latter position of the atom from the surface. The calculations, based case, we additionally show the Gaussian fit to the HE mode to 11 on Eqs. (3.5) and (3.6), are performed for silica, with mode follow how it reproduces the tail asymptote of the evanescent functions shown in the upper plot of Fig. 2, and compared with field outside the fiber. the exact result calculated via Fermi’s golden rule where the complete set of the external modes is kept [18,49]. As can be B. Single-atom scattering seen from the plotted graphs, the highest deviation is observed For a single atom with a nondegenerate upper state, its self- when the atom is located near the fiber surface. That indicates a energy part (2.14) can be expressed as significant contribution from the process of recurrent scattering to the Green’s function, which can be recovered via iterative ih¯ | = − solution of the scattering equation (A12) in Appendix A. (E,r) E=hω¯ 0 h¯ (r) γ (r), (3.3) 2 Nevertheless, at the distance comparable with the waveguide where we omitted unnecessary specification by quantum index radius a, the exact result is relevantly reproduced by estimate n ≡ (F = 0,M = 0) but parameterized the self-energy by the (3.6). It is noteworthy to point out that at the intermediate additional argument r that indicates the atom’s position. The distances, where ρ − a ∼ a, the simple sum of the natural real part (r) is diverging and should be incorporated into γ with the waveguide contribution overestimates γ (ρ) and the physical energy of the atom. The infinite energy shift is should be corrected as suggested by Eq. (3.6).
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1.0
0.8
0.6
R, T 0.4
0.2
0.0 0.2 0.1 0.0 0.1 0.2 1.00
FIG. 3. Spontaneous decay rate γ (ρ) of a rubidium atom located near the waveguide as a function of its distance ρ − a to the surface: 0.98 contribution of the fundamental mode (blue, lower curve); sum of contributions of the fundamental mode and natural decay γ (black, 0.96 dash-dotted curve); calculations based on Eqs. (3.4)and(3.5) (green T dashed curve); and exact result (red curve). The vertical shaded area (dash-dotted bounded) indicates the waveguide. The waveguide 0.94 parameters are the same as in the upper plot of Fig. 2.
0.92 The obtained parameters contribute to the resolvent operator 0.08 and S matrix as explained in Sec. II. The light scattering in a quasi-one-dimensional geometry can be described by coefficients of transmission T , reflection R, and losses L, 0.06 which are, respectively, given by
0.04 = = | |2
T T (ω) Si i , R,L i,k>0 2 0.02 R = R(ω) = |Sii | , (3.7) i,k<0 = = − − 0.00 L L(ω) 1 R(ω) T (ω), 1.0 0.5 0.0 0.5 1.0
where i ={σ =−1,k; M0 =+1} and the sum over i = Γ { } =± σ ,k ; M0 includes both polarization channels σ 1 and = ± three atomic transitions with M0 0, 1. The elastic forward- FIG. 4. The transmission T (dashed lines), reflection R (solid and backward-scattering channels are distinguished by the lines), and losses L = 1 − R − T (dash-dotted lines) calculated as = − signs of the longitudinal wave numbers k =+k and k =−k, a function of frequency detuning ω ω0 for light scattering respectively; see Eq. (A6). The wave number of an incident from a single rubidium atom trapped at distances ρ − a = 0.5 a (blue photon is parameterized by its frequency according to the thin lines) and ρ − a = a (red thick lines). The mode and atomic dispersion law of the waveguide, k = k(ω). In Fig. 4,weplot decay parameters are the same as in Figs. 2 and 3. The upper panel the single-atom transmission and reflection coefficients as a provides the spectra for the lossless scattering with light emission function of frequency detuning from atomic resonance, = into the waveguide mode only, while the lower plot corresponds to the complete scattering process. ω − ω0, for two different positions of the atom near a nanofiber: ρ − a = 0.5 a and ρ − a = a. The waveguide parameters are chosen the same as in the upper plot of Fig. 2. The upper plot of section, we study the scattering provided by a full chain of Fig. 4 shows the result of our calculations with the assumption atoms trapped along the waveguide. that the atom would emit the light into the waveguide mode only. For such an ideal lossless configuration, the balance R(ω) + T (ω) = 1 evidently fulfills in all the spectral domain. C. Light scattering from an atomic array For a complete scattering process, the graphs of the lower plot We consider the scattering process from a collection of also indicate a few-percent interaction of the atom with the atoms trapped along the waveguide in the geometry shown in evanescent field. The small but not negligible fraction of light is Fig. 1. In the case of several atoms, the scattering problem is mostly scattered into external modes and the scattering process described by the same set of parameters given by Eq. (3.7), is as strong as the atom is closer to the fiber surface. In the next with updated definitions of initial and final states. Initially
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1.0 shown in Fig. 1. Similarly to the single-atom case, the array is considered as located at two distances ρ − a = 0.5 a; a from the fiber surface. For the shortest distance, we additionally 0.9 indicate the partial contributions of the Rayleigh scattering channel, which leaves atoms at the initial Zeeman state, but either preserves mode polarization or can transfer the outgoing 0.8 T light into an orthogonal polarization mode. As it can be seen, the Rayleigh contribution dominates in the entire scattering process. As pointed out above, the Raman scattering mainly 0.7 results from a single spin flip, shared among all atoms of the ensemble, such that most of the atoms preserve their = = 0.6 initial population of the F0 1,M0 1 state. This can be 2 1 0 1 2 explained by approximate azimuthal symmetry of the complete 0.08 system (photon and atoms) in respect to collinear geometry of the forward scattering. The symmetry provides conservation for the total angular momentum such that the spin angular 0.06 momentum transfer physically implies the mechanism of spin exchange between the photon and atomic spin subsystem in the Raman process, which just results in a single spin 0.04 R flip. It is also noteworthy to point out that for the Rayleigh channel, the scattering into the orthogonal polarization mode, =+ =− 0.02 i.e., into σ 1 for forward or σ 1 for backward directions, is possible but quite small for both the transmission and reflection. Such scattering channel would be completely 0.00 forbidden in the paraxial approach. For the considered small 2 1 0 1 2 collection of atoms, the reflection is still weak and the losses Γ associated with incoherent scattering can be mainly estimated by deviation of the transmission coefficient from the level of ideal transparency. FIG. 5. The transmission T (upper plot) and reflection R (lower The transmission T (ω) and the reflection R(ω) spectra plot) calculated for an array of five ordered rubidium atoms trapped from the atomic array now demonstrate a clear signature at distances ρ − a = 0.5 a (blue thin lines) and ρ − a = a (red thick lines). The atoms are separated by a distance d = π/k = λwg/2. of cooperativity in the scattering process. By comparing the ρ − a = . a spectral dependencies of Figs. 4 and 5, we can see that the For 0 5 , we additionally indicate the partial contributions = of Rayleigh scattering channels with preserving mode polarization incoherent losses are enhanced by the factor of N 5in (dotted line) and with keeping both the polarization components of accordance with the natural tendency to the Beer-Lambert law. the outgoing light (dashed line). The solid curves show the total However, the reflection exhibits much stronger enhancement 2 contribution including Raman scattering channels. Other parameters roughly scaled by a factor of N = 25 and justifies the effect are the same as in Fig. 4. of coherent Bragg-type reflection. Such scaling is valid only for a small number of scatterers and far from the saturation limit. In the case of a mesoscopic system, the enhancement our system is prepared in a spin-oriented collective state such from an atomic array consisting of many atoms, showing strong that i ={σ =−1,k; M0 =+1(all atoms)} but the final state reflection and the effect of one-dimensional atomic mirror, has ={ { (a)}N } been recently observed [30,31]. Let us emphasize that in our can be any of i σ ,k ; M0 a=1 , where each of the N atoms can be redistributed onto an arbitrary Zeeman sublevel approach, this observation rigorously results from the ab initio (a) = ± calculation of the scattering process, which is based on the with M0 0, 1. The dimension of the Hilbert space for the resolvent operator as well as the number of scattering channels resolvent operator and the S-matrix formalism. Interpretation exponentially rises up with the number of atoms. However, as of the Bragg reflection via a semiclassical approach and the confirmed by our numerical simulations, for an array consisting transfer matrix formalism can be found in the above references. of a relatively small number of atoms, the Rayleigh channel All of the spectral dependencies are redshifted from the makes a dominant contribution to the scattering process. The atomic resonance and from the single-atom spectra presented contribution of the Raman process can be reliably estimated in Fig. 4. This is a consequence of the short-range static inter- by keeping in the output channel only a single spin flip equally action of atomic dipoles and is a precursor of the well-known shared among all atoms of the ensemble. Lorentz-Lorenz effect existing in infinite, homogeneous, and dense dielectric media. This type of interaction is naturally incorporated into our calculation scheme, as explained in the 1. Ordered atomic array previous section and detailed in Appendix A. Due to the First we simulate an ordered array of atoms. Figure 5 translational symmetry in the lattice-type and ordered atomic provides the parameters of the scattering process calculated for configuration, all of the presented spectra have Lorentzian- a system of five atoms with a fixed longitudinal separation, d = shaped monotonic profiles. Such a smoothed spectral behavior π/k = λwg/2 [half of the mode wavelength; see Eq. (A6)], as would be dramatically changed once we introduce disorder in
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1.00 microscopic description based on the structure of the resolvent operator of the system Hamiltonian (2.11). Indeed, a single- photon state of the electromagnetic field is an intrinsically 0.95 quantum microscopic object, which has no phase, is parameter- ized by a negative-valued Wigner function in its phase space, and cannot be fairly introduced in classical optics. Although 0.90 T the empirical description of the process in terms of the classical wave scattering (i.e., in terms of scattering of a weak coherent 0.85 light) with classical interpretation of interference paths may give a realistic estimate of the output transmission and reflec- tion spectra, it should be provided by the phenomenologically 0.80 defined elementary scattering parameters, namely, by a single- 2 1 0 1 2 atom decay constant and scattering amplitude. In this sense, the 0.008 performed calculations emphasize that the resolvent operator (2.11), transformed to its diagonal form, generates the set of unstable Dicke-type excited quantum entangled states, which 0.006 have strong internal and configuration-sensitive correlations, and which respond to the driving photon as one complex 0.004 quantum system. The parameters of the system are rigorously R defined by its self-energy part (effective Hamiltonian), intro- duced in Sec. II C. The microscopically calculated resolvent 0.002 operator correctly reproduces the complicated dynamics of the scattering process and its sensitivity to variation of the external conditions. 0.000 2 1 0 1 2 Γ IV. CONCLUSION In this paper, we have developed the principles of quan- FIG. 6. Reflection and transmission spectra as in Fig. 5, but here tum scattering theory toward the microscopic description of for a particular random configuration of the atoms in a disordered cooperative light scattering from an array of atoms trapped array. near a subwavelength dielectric waveguide. The developed approach is entirely applicable for a single-photon scattering in a quasi-one-dimensional geometry from atoms having multi- the atomic distribution, such as delocalization of the trapping level and degenerate energy structure. The basic mathematical potential wells due to finite temperature. attributes of the scattering process, namely the S matrix and resolvent operator, are linked to the transmission and reflec- 2. Disordered atomic array tion coefficients characterizing the propagation of the guided Figure 6 presents the modified spectra for the case of photon through the atomic chain. The crucial elements of the disordered configuration. The atoms have the same average performed calculation scheme are the self-energy part and the separation, d ∼ π/k = λwg/2, but are randomly and uniformly electric-field Green’s function and both are strongly affected distributed along the chain. The spectral dependencies, plotted by the presence of the waveguide. In the case of a nanofiber for a particular configuration, become quite sensitive to its that supports a single fundamental mode, these functions can variation because of internal correlations in the entire system. be analytically constructed within certain approximations as The cooperative effects are revealed in a different way and we have described. the scattering process is generally weaker. This is clearly In the context of a quantum interface and coherent control of seen in the backward-scattering channel whose outcome is an the signal light, atomic systems with a degenerate ground state order of magnitude smaller than in the case of ordered array, are especially interesting. For such systems, the dimension and is negligible when compared with the incoherent losses. of the Hilbert subspace of the quantum scattering equations Another important feature of the scattering process is the rises exponentially with the number of atoms. Therefore, in the complicated structure of the transmission spectra where several present study, we have restricted our numerical simulations to local minima and maxima appear. This is a signature of a a configuration of five -configured atoms with the minimal microcavity structure created by a disordered but cooperatively accessible number of quantum states, i.e., with angular mo- organized system composed of atomic scatterers in a one- mentum F0 = 1 in the ground state and F = 0 in the excited dimensional configuration [47]. In the case of disorder, the co- state. For this case, we have provided an important illustration operativity tends to an Anderson-type localization mechanism that includes not only Rayleigh but also Raman channels in the that suppresses the scattering process in the one-dimensional microscopic description of the cooperative scattering process geometry. in the correct vector model by keeping the complete angular It is important to point out that such a configuration- momentum structure of the guided light. sensitive interference for an impinging photon scattered from The atomic scatterers have been considered as distributed either ordered or disordered atomic chains has here a rigorous in either ordered or disordered arrays, and the theory predicts
023827-9 V. A . P I VOVA ROV et al. PHYSICAL REVIEW A 97, 023827 (2018) that the parameters of the scattering process strongly depend 1. Basic links with macroscopic theory on the distribution type. Our numerical simulations show that As proven in statistical physics (see [50]), the causal- in the case of an atomic chain structured as a one-dimensional type electric-field Green’s function considered in a spatial lattice with a period of half wavelength of the guided mode, region nearby a macroscopic object can be expressed by there is a significant enhancement of the backward scattering the retarded-type fundamental solution of the macroscopic and light reflection. Importantly, this clear manifestation of Maxwell equation, the Bragg diffraction in the scattering process is obtained here as result of ab initio description via the calculation of ∞ (E) =− iωτ Dμν (r,r ; ω) i dτ e TEμ(r,t) Eν (r ,t ) τ=t−t the resolvent operator and can be associated with a specific −∞ periodic entangled structure of its eigenstates created by ω2 an optical excitation. In the alternative situation where the = D(R)(r,r; |ω|). (A1) atoms are distributed randomly with random phase-matching c2 μν conditions for an optical excitation shared among the atoms, Here the integrand under the Fourier transform, introduced by we have observed strong dependence of the forward scattering Eq. (2.15) as an element of the diagram expansion of the self- and light transmission on a particular atomic configuration. The energy part, performs the expectation value of the time-ordered transmission spectrum has a nonmonotonic profile with several product between the exact field operators in the Heisenberg configuration-sensitive maxima and minima, which indicates representation, “dressed” by the interaction with the object. a certain precursor of the Anderson-type localization mech- The second line identifies this quantity as the retarded-type anism in the conditions of quasi-one-dimensional scattering fundamental solution of the macroscopic Maxwell equation geometry. (photon propagator in a medium [50]), The developed approach can be further generalized and applicable for the description of various light-matter interface ∂2 (R) − (R) protocols developed in ensembles consisting of a macroscopic Dμν (r,r ; ω) Dαν (r,r ; ω) number of atoms. Potentially this can be done because of ∂xμ∂xα the convenient approximate form of the electric-field Green’s ω2 + [1 + 4πχ(r)]D(R)(r,r; ω) = 4πhδ¯ δ(r − r), function, which we have found in the paper. The main difficulty c2 μν μν is in an exponentially uprising dimension of the Hilbert space in (A2) the case of a many-particle problem. But this difficulty could be overcome by involving mainly the proximate dipoles in where χ(r) is the spatially dependent dielectric susceptibility the calculation of the self-energy part, similarly to how it was of the medium. In the vacuum case, when χ(r) → 0, expres- demonstrated for atomic systems in free space; see [7,38]. sions (A1) and (A2) reproduce the direct relation between electric-field Green’s function and photon propagator in free space. But in the general case, they allow us to obtain the ACKNOWLEDGMENTS correction to the Green’s function due to a nearby macroscopic This work was supported by the RFBR Grants No. 15-02- object such as a dielectric waveguide. By neglecting dispersion, 01060 and No. 18-02-00265, the Emergence program from as we can assume if ω is varied in a narrow spectral domain Ville de Paris, the IFRAF DIM NanoK from Région Ile-de- near the reference atomic resonance frequency ω0,Eq.(A2) France, and the PERSU program from Sorbonne Université. can be rewritten directly for the positive frequency component (E) N.V.C. and A.S.Sh. were supported by the EU (Marie Curie of the causal-type Green’s function D just by adding the 2 2 Fellowship). The work was carried out with financial support factor ω /c in its right-hand side. from the Ministry of Education and Science of the Russian In this case, and for a transparent medium, the standard Federation in the framework of increase Competitiveness quantization scheme in free space can be straightforwardly Program of NUST “MISIS,” implemented by a governmental generalized by tracking boundary conditions on the object decree dated March 16, 2013, No. 211. surface. Expansions (2.8) and (2.9) in the main text construct the field components as Schrödinger operators reexpanded in the complete set of modified basic operators for the creation APPENDIX A: THE ELECTRIC-FIELD GREEN’S and annihilation of a photon either in the modes confined with FUNCTION NEAR A SUBWAVELENGTH the waveguide or in the delocalized external modes. The ex- DIELECTRIC WAVEGUIDE pectation value of the time-ordered product for such modified field operators in the Heisenberg representation automatically In this appendix, we consider how the microscopic Green’s reproduces the basic relations (A1) and (A2). function of the electric field is modified in the presence of a The obtained differential equation for the causal-type subwavelength dielectric waveguide. The waveguide, designed Green’s function can be further transformed to the following as a tiny nanoscale optical fiber (“nanofiber”) produced from a integral form: transparent dielectric medium (silica), can support propagation (E) = (0) − of only one (confined in the transverse direction) fundamental Dμν (r,r ; ω) Dμν (r r ; ω) mode, conventionally named the HE mode. We aim to find 11 the relevant correction to the microscopic Green’s function − 3 (0) − (E) d r Dμα(r r ; ω)χ(r )Dαν (r ,r ; ω), outside the nanofiber, which is associated with the modified h¯ structure of the field’s modes. (A3)
023827-10 LIGHT SCATTERING FROM AN ATOMIC ARRAY TRAPPED … PHYSICAL REVIEW A 97, 023827 (2018) where the Green’s function of the freely propagating field (not can always fit it to the value, which provides a homogeneous modified by the waveguide) is given by solution of such modified Eq. (A5)inform(A6) for arbitrary k. ∞ For the interaction with atoms, the evanescent field fre- (0) iωτ (0) =− Dμν (R; ω) i dτ e TEμ(r,t) Eν (r ,t ) τ = t − t quency ω is expected to be quite close to the atomic frequency −∞ R = r − r ω0. The possible variations of k from k(ω) can be scaled by variation of frequency detuning as ω/c, where ω ω,ω |ω|3 2 |ω| 0 =−h¯ i h(1) R δ by many orders of magnitude. In this case, the eigenfunctions c3 3 0 c μν of the integral operator for any acceptable k have eigenvalue k X X 1 |ω| very close to “one” and, being considered on a macroscopic + μ ν − δ ih(1) R , (A4) R2 3 μν 2 c scale of a finite waveguide, are practically indistinguishable from the waveguide modes (A6) for the same k. Excitation where the superscript zero index emphasizes that the averaging of these quasiresonant waveguide modes makes a consider- (1) is done in the basis of the plane-wave modes. Here, hL (...) able correction to the electric-field Green’s function near the with L = 0,2 are the spherical Hankel functions of the first nanofiber. kind. In the integral equation (A3), we have added an auxiliary → parameter 1 for resolving a possible conflict with the 2. Contribution of the waveguide modes Fredholm alternative. = Consider Eq. (A2) for the causal-type Green’s function and Indeed, 1 is an eigenvalue of the integral operator in Eq. (A3) and the respective eigenfunction (integrable in the in the case when both the spatial arguments r and r are located transverse plane) is given by the solution of the homogeneous outside the fiber, wave equation, ∂2 ω2 D(E)(r,r; ω) − D(E)(r,r; ω) + D(E)(r,r; ω) μν αν 2 μν ∂2 ω2 ∂xμ∂xα c E(s)(r) − E(s)(r) + [1 + 4πχ(r)]E(s)(r) = 0, μ α 2 μ 2 ∂xμ∂xα c ω = 4πh¯ δμν δ(r − r ). (A7) (A5) c2 as can be straightforwardly verified by applying the wave Let us specify the waveguide modes, defined by Eqs. (A5) and (A6), by the following normalization condition: operator to Eq. (A3). This is a signature of a resonance contribution of the waveguide modes, determined as solutions 3 (s)∗ · (s) (localized in the transverse plane) of Eq. (A5), to the Green’s d r (r) E (r) E (r) function. 3 (s)∗ (s) If χ(r) ∼ const inside the fiber, then due to azimuthal and ≡ d r D (r) · E (r) = δss , (A8) translational symmetry the solution can be factorized in the following product: where (r) = 1 + 4πχ(r) is the dielectric permittivity and D(s )(r) = (r) E(s )(r)isthes mode of the displacement field. 1 (s) = (σk) √ iσφ ikz Eq (r) Eq (ρ) e e (Cylindric), As explained above, the integral over the z variable is bounded 2πL L →∞ (A6) by the quantization segment and implies periodic 1 E(s)(r) = E(σk)(ρ,φ) √ eikz (Cartesian), boundary conditions and quasidiscrete wave number k. μ μ L The formal solution of Eq. (A7) can be constructed via an expansion of the Green’s function in the series of the where the longitudinal wave number k and azimuthal quantum complete basis set for all the cylindrical modes, i.e., given by number σ =±1 are the mode parameters incorporated in the waveguide modes (A6) and by the infinite set of the external the entire mode index, s = σ,k. The solution is obtained delocalized modes. Thus the Green’s function is given by the in cylindrical coordinates r → ρ,φ,z, but with the vector sum of two contributions, projection defined in respect to either cylindrical basis with = = (E) = (wg) + (ext) q ρ,φ,z (first line) or to Cartesian frame with μ x,y,z Dμν (r,r ; ω) Dμν (r,r ; ω) Dμν (r,r ; ω). (A9) (second line). For the sake of convenience [see normalization condition(A8) below], we pose a longitudinal wave inside a The first term specifies the contribution of the waveguide certain quantization segment of length L with periodic bound- modes and outside the fiber, where the modes of displacement ary conditions and consider k = 2π/L × (any integer) as a and electric fields coincide, and is given by 2 quasidiscrete variable. The mode frequency and longitudinal 4πhω¯ ∗ = D(wg)(r,r ; ω) = E(s)(r) E(s) (r ). (A10) wave number are connected via the dispersion relation ω μν ω2 − ω2 + i0 μ ν s s ωs ≡ ωk, which depends on the waveguide parameters, such that each particular frequency ω determines only one specific The contribution of the external modes, denoted by the second wave number, k = k(ω). term in (A9), can be similarly obtained, but, by keeping all the The next observation is that there is a continuum family of modes, it makes it a quite cumbersome and hardly applicable eigenfunctions of integral operator contributing to Eq. (A3), result for evaluating the cooperative resolvent operator in a which has formed (A6) and corresponds to a family of many-particle problem. Nevertheless, as we show below, in the eigenvalues = k ∼ 1 varied with k. This can be justified case of a single-mode nanofiber, we can modify the integral by the fact that with slightly varying , we can change the equation (A3) to another form, which suggests convenient dielectric susceptibility of the waveguide, χ(r) → χ(r), and approximation for D(ext).
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3. Contribution of the external modes and coincides with the expansion (A11) with the vacuum In expansions (A9) and (A10), the spatial argument r can be Green’s function substituted in the right-hand side. The second taken even inside the fiber by accepting the Green’s function term in Eq. (A14) gives the contribution of a single scattering as a fundamental solution of the wave equation in its general from a spatial inhomogeneity in the dielectric permittivity, i.e., form (A2). Then, due to the orthogonality of the basis functions gives the simplest perturbative estimate for the reflection from associated with the waveguide, we can select the contribution the waveguide. of external modes via the following identity: The solution of the scattering equation in the integral form (A12) suggests a realistic estimate of the function D(ext) in (ext) = (E) − (s) Dμν (r,r ; ω) Dμν (r,r ; ω) Eμ (r) the considered assumptions and simplifications. As a zero approximation, we can accept s ∗ × d3r D(s) (r )D(E)(r ,r ; ω). (A11) (ext) ≈ ˜ (0) α αν Dμν (r,r ; ω) Dμν (r,r ; ω), (A16) The second term actually subtracts the contribution (A10), per- whose validity is explained below. For the propagating modes, formed here in the projected form. If the waveguide is excited the evanescent field typically has transverse scale sufficiently on frequency ω, it would be effectively responding on those broader than the radiation wavelength. Then the second term modes whose frequencies ω ≡ ω are close to ω. Having also s k in Eq. (A15) can be approximately evaluated for points r and in mind that the frequency ω is quasiresonant to the reference r separated by a distance comparable to the wavelength or atomic frequency ω , we can expect that only a limited number 0 shorter, and the complete Green’s function (A9) can be given of near-resonant modes with ω ∼ ω ∼ ω would make a k 0 in the following closed form: meaningful contribution in (A11). As was pointed out above, these modes, considered on a macroscopic scale, of the finite (E) ≈ (wg) + (0) − waveguide approximately coincide with the eigenfunctions of Dμν (r,r ; ω) Dμν (r,r ; ω) Dμν (r r ; ω) ∼ 2 the integral operator in Eq. (A3) with eigenvalue k 1. 4πhω¯ ∗ − E(s)(r) E(s) (r), By substituting expansion (A11) into the integral equation ω2 − c2k2 + i0 μ ⊥ν (A3) and by taking = 1, it can be transformed to the s following form: (A17) (ext) ≈ (0) − Dμν (r,r ; ω) Dμν (r r ; ω) (s)∗ where in the last subtracting term E⊥ (r ) denotes the vector ∗ 1 projection of E(s) (r ) on the plane transverse to the z axis, + d3r K (r,r ; ω)D(ext)(r ,r ; ω), (A12) h¯ μα αν i.e., to the waveguide direction; see Eq. (3.2). It is given by with the following modified kernel of the integral operator: the leading (in paraxial limit) contributions in the right-hand side of (3.2) with the azimuthal angular-dependent terms =− (0) − Kμα(r,r ; ω) Dμα(r r ; ω)χ(r ) omitted. The obtained result has a quite natural physical interpre- − (s) (s)∗ Eμ (r) Dα (r ). (A13) tation. If a pointlike dipole source emits light near a single- s mode waveguide of subwavelength transverse scale, then the The sign of “approximately equal” in Eq. (A12) means that here respective fundamental solution of the Maxwell equation has we have equated the waveguide modes with the eigenfunctions more or less a similar structure as in the vacuum case. Then, part of the original integral operator (A3) and have neglected the of the emitted modes in the paraxial approach coincide with differences in the eigenvalues such that k ≈ 1. In this assump- the waveguide modes, but in the presence of the waveguide tion, the integral operator with kernel (A13) eliminates the these modes have the dispersion law ωs = ωk = ck, which waveguide contribution (A10) and, as a consequence, justifies is different from the dispersion relation in free space; this replacing D(E) → D(ext) in the right-hand side of Eq. (A12). difference should be taken into consideration. The respective The inhomogeneous integral equation (A12) fulfills the correction can be seen by comparing the waveguide contri- resolving conditions of the Fredholm theorem (see Ref. [51]) bution (A10) with the last term in the expression for the and its solution for a nanofiber implies rapidly converging complete Green’s function (A17). In the paraxial approach, iterative expansion. In the first iteration step, we have the subtracted contribution, given by this last term, eliminates the emission into the vacuum mode coinciding, in paraxial D(ext)(r,r; ω) ≈ D˜ (0)(r,r; ω) μν μν limit, with the waveguide modes, whose contribution is already 1 correctly incorporated into Eq. (A10). The approximation, − d3rD(0) (r−r; ω)χ(r)D(0)(r−r; ω), h¯ μα αν performed by Eqs. (A16) and (A17), seems sufficient for a dipole source distant from the fiber on a length comparable (A14) with its transverse scale. Nevertheless, it becomes insufficient where the first term is given by for a dipole located just near the fiber surface since it com- pletely ignores any corrections associated with the reflection ˜ (0) = (0) − − (s) Dμν (r,r ; ω) Dμν (r r ; ω) Eμ (r) of the source wave from the fiber. The respective corrections s could be recovered via an iterative solution of Eq. (A12) × 3 (s)∗ (0) − and their simplest estimate is given by the second term d r Dα (r )Dαν (r r ; ω), (A15) in Eq. (A14).
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APPENDIX B: THE WAVEGUIDE MODES Bessel and Hankel functions of the first kind (see [48]): We consider the waveguide-mode equations in the cylin- ik Eρ (ρ) ∝ [(1 − u)J0(κρ) − (1 + u)J2(κρ)], drical frame. By substituting Eq. (A6) into Eq. (A5)forthe 2κJ1(κa) =± longitudinal z component for both types of σ 1 polariza- k tion modes, we have Eφ(ρ) ∝− [(1 − u)J0(κρ) + (1 + u)J2(κρ)], 2κJ1(κa) ∂2 1 ∂ σ 2 U (σk)(ρ) + U (σk)(ρ) + κ2 − U (σk)(ρ) = 0, ∝ 1 ∂ρ2 ρ ∂ρ ρ2 Ez(ρ) J1(κρ), (B4) J1(κa) (B1) with κ = κin as the real quantity for ρ
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