Radiative Toroidal Dipole and Anapole Excitations in Collectively Responding Arrays of Atoms

Radiative toroidal dipole and anapole excitations in collectively responding arrays of atoms

K. E. Ballantine∗ and J. Ruostekoski† Department of Physics, Lancaster University, Lancaster, LA1 4YB, United Kingdom (Dated: June 23, 2020) A toroidal dipole represents an often overlooked electromagnetic excitation distinct from the standard electric and magnetic multipole expansion. We show how a simple arrangement of strongly radiatively coupled atoms can be used to synthesize a toroidal dipole where the toroidal topology is generated by radiative transitions forming an effective poloidal electric current wound around a torus. We extend the protocol for methods to prepare a delocalized collective excitation mode consisting of a synthetic lattice of such toroidal dipoles and a non-radiating, yet oscillating charge- current configuration, dynamic anapole, for which the far-field radiation of a toroidal dipole is identically canceled by an electric dipole.

The concept of electric and magnetic multipoles vastly dipole and non-radiating anapole moments, even though simplifies the study of light-matter interaction, allowing individual atoms only exhibit electric dipole transitions. the decomposition both of the scattered light and of the The method is based on simple arrangements of atoms, charge and current sources [1, 2]. While the far-field ra- where the toroidal topology is generated by radiative diation can be fully described by the familiar transverse- transitions forming an effective poloidal electric current electric and -magnetic multipoles, the full characteriza- wound around a torus, such that an induced magneti- tion of the current requires, however, an additional series zation forms a closed circulation inside the torus. The that is independent of electric and magnetic multipoles: toroidal and anapole modes can be excited by radially dynamic toroidal multipoles [3–6]. These are extensions polarized incident light and, in the case of the anapole, of static toroidal dipoles [7] that have been studied in with a focusing lens. The resulting anapole excitation nuclear, atomic, and solid-state physics, e.g., in the con- shows a sharp drop in the far-field dipole radiation, de- text of parity violations in electroweak interactions [8–10] spite having a large collective electronic excitation of the and in multiferroics [11]. Often obscured and neglected in atoms associated with both electric and toroidal dipole comparison to electric and magnetic multipoles due to its modes. Such a configuration represents stored excitation weakness, the toroidal dipole can have an important re- energy without radiation that is fundamentally different, sponse to electromagnetic fields in systems of toroidal ge- e.g., from subradiance [47]. We extend the general prin- ometry [12]. Dynamic toroidal dipoles are actively stud- ciple to larger systems, and show in sizable arrays how ied in artificial metamaterials that utilize such designs, this collective behavior of atoms allows us to engineer a with responses varying from microwave to optical part delocalized collective radiative excitation eigenmode con- of the spectrum [13–18]. Crucially, an electric dipole to- sisting of an effective periodic lattice of toroidal dipoles. gether with a toroidal dipole can form a non-radiating dy- Such an array is then demonstrated to exhibit collective namic anapole [19–22], where the far-field emission pat- subradiance, with the narrow resonance line sensitively tern from both dipoles interferes destructively, so the net depending on the lattice spacing and manifesting itself emission is zero. as a Fano resonance in the coherently transmitted light. Light can mediate strong resonance interactions be- Utilizing cold atoms as a platform for exploring tween closely-spaced ideal emitters and an especially pris- toroidal excitation topology has several advantages, as tine system exhibiting cooperative optical response is it naturally allows for the toroidal response at optical that of regular planar arrays of cold atoms with unit frequencies, with emitters much smaller than a reso- occupancy [23–36]. Subradiant linewidth narrowing of nant wavelength, and avoids dissipative losses present in transmitted light was now observed in such a system plasmonics or circuit resonators forming metamaterials. formed by an optical lattice [37]. In the experiment the Moreover, the atomic arrangement can form a genuine whole array was collectively responding to light with the quantum system, and our analysis is valid also in the atomic dipoles oscillating in phase. Furthermore, the lat- single photon quantum limit. tice potentials can be engineered [38], and a great flex- To demonstrate the formation of an effective toroidal ibility of optical transitions is provided by atoms such dipole and anapole in an atomic ensemble, we briefly de- as Sr and Yb [39]. Also several other experimental ap- scribe the radiative coupling between cold atoms. For proaches exist to trap and arrange atoms with single-site simplicity of presentation, we analyze coherently driven control [40–46]. case, but the formalism is also valid in the quantum Here we propose how to harness strong light-mediated regime of single photon excitations [48]. interactions between atoms to engineer collective radia- We consider atoms at fixed positions, with a J = 0 → tive excitations that synthesize effective dynamic toroidal J 0 = 1 transition, and assume a controllable Zeeman 2

∗ 0 ξeˆσ ·G(rj −rk)eˆσ0 for (j, σ) 6= (k, σ ), where the dipole ra- (a) (b) (j) diation kernel G(r) gives the ﬁeld 0Es (r) = G(r−rj)dj from a dipole moment dj at rj [1]. The diagonal ele- (j) (j) (j) r ment H3j+σ−1,3j+σ−1 = ∆σ +iγ, where ∆σ = δσ +∆ consists of an overall laser detuning ∆ = ω − ω0 from a (j) the single-atom resonance ω0, plus a relative shift δσ of T x ikT each level. The dynamics follows from eigenvectors vn and eigenvalues δn + iγn of H giving the collective level d y z shifts δn and linewidths γn [56]. m d The limit of low light intensity corresponds to linear regime of oscillating atomic dipoles. The analogous quan- FIG. 1. (a) Geometry of a toroidal dipole unit cell consist- tum limit is that of a single photon that experiences no ing of a number of four-atom squares of width a, arranged nonlinear interactions, as at minimum, two simultaneous in a circle of radius r, with atomic dipoles indicated by the arrows. A circulating electric polarization on the surface of a photons are required for interactions. Regarding one- torus leads to magnetic dipoles forming a closed loop inside body expectation values, the equations of motion for the (j) a torus (inset). These in turn contribute to a toroidal dipole dipole amplitudes Pσ are indeed precisely the same [57] moment through the center of the entire unit cell. (b) Geom- as those for single-photon excitation amplitudes that are etry of an anapole unit cell. Adding two atoms to the center radiatively coupled between the atoms by H [48]. with induced dipole moments in the x direction generates an To illustrate the role of toroidal multipoles, we con- electric dipole which destructively interferes with the toroidal dipole (inset). sider the far-ﬁeld scattered light from a radiation source decomposed into vector spherical harmonics [1],

∞ l (j) X X (j) (j) Es = αE,lmΨlm + αB,lmΦlm , (1) l=0 m=−l

that allows us to represent it as light originated from a set of multipole emitters at the origin, with l = 1 representing dipoles, l = 2 quadrupoles, etc. However, while the magnetic coefficients αM,lm are due to magnetic multipole sources with transverse current r×J 6= 0, the electric coefficients αE,lm can arise from two different types of polarization; electric and toroidal multipoles. FIG. 2. Projections of geometry of toroidal dipole and These contributions can be calculated directly from the anapole unit cells in (a) The yz and (b) the xy plane. Shaded induced polarizations. Taking atom j to be fixed at po- atoms at y, z = 0 are absent for toroidal dipole unit cell but sition rj, the induced displacement current density is present for anapole unit cell. (c) Alternative structure for re- J (r) = −iωD P P(j)δ(r − r ). Inserting this in the alizing a toroidal dipole or anapole excitation with all atoms σ j σ j in the xy plane the response of which is shown in Fig. S3 [48]. standard multipole decomposition for an arbitrary distribution of currents [6] gives for the total electric and mag- P P netic dipoles d = j dj and m = −(ik/2) j(rj × dj), splitting of the J 0 = 1 levels, generated, e.g., by a pe- respectively, and for the toroidal dipole riodic optical pattern of ac Stark shifts [49]. The dipole ik (j) X 2 P T = − (rj · dj) rj − 2r dj . (2) moment of atom j is dj = D j Pσ eˆσ, where D de- 10 j (j) j notes the reduced dipole matrix element, and Pσ and eˆσ the polarization amplitude and unit vector associated The magnitude of the far-field electric dipole compo- with the |J = 0, m = 0i → |J 0 = 1, m = σi transition, re- P 2 1/2 2 nent, |αE,1| ≡ [ m |αE,1m| ] ∝ k |p|/(4π0) then de- spectively. The collective response of the atoms in the pends on the combination p = d + ikT [22]. We have limit of low light intensity [23, 50–54] then follows from checked that in our numerics corrections beyond the long- ˙ (j) b = iHb + f, where b3j−1+σ = Pσ and the driving wavelength approximation of Eq. (1) are negligible. ∗ f3j−1+σ = i(ξ/D)eˆσ ·0E(rj), with the incident light field We now turn to the design and preparation of a collec- of amplitude E(r) = E0eîn exp (ikx) [55], polarization eîn, tive toroidal dipole. Even for atoms exhibiting electric and frequency ω = kc. Here ξ = 6πγ/k3 depends on dipole transitions, their collective excitation eigenmodes 2 3 the single-atom linewidth γ = D k /(6π0~). The ma- can be utilized in synthesizing radiative excitations, e.g., trix H describes interactions between different atoms due with magnetic properties [49, 58]. The toroidal dipole, as to multiple scattering of light, with H3j−1+σ,3k−1+σ0 = illustrated in the inset of Fig. 1(a), consists of a poloidal 3

FIG. 3. Excitation of a toroidal dipole T as a multipole decomposition of (a) atomic dipoles [in dimensionless units of DE0/(~γ)]. (b) far-ﬁeld radiated power consisting of dipole contribution P1 and the remaining power of all other con- 2 2 tributions (in units of Iin/k , where Iin = 2c0|E0| is the incoming intensity), as a function of the laser detuning from the atomic resonance, with r = 0.2λ and a = 0.08λ, as deﬁned in Fig. 1(a).

FIG. 4. Collective excitation of a 12 × 12 square array of electric current wound around a torus, such that mag- toroidal dipole unit cells with spacing d = 0.85λ (a = 0.1λ, netic dipoles form a closed loop, reminiscent of vortex r = 0.2λ). (a) Multipole decomposition of atomic dipoles [in current, pointing along a ring around the center of the units of DE0/(~γ)] for a single central unit cell, excited by lin- torus. We approximate this geometry using squares of early polarized light; (b) decomposition of the dipole contribution to the far-field radiated power, and the sum of all other four atoms [see Fig. 1(a)]. This is possible, since an iso- 2 contributions (in units of Iin/k ); (c) collective linewidth υ of lated square has a collective excitation eigenmode with the uniform toroidal dipole eigenmode as a function of d; (d) 2 2 the dipoles oriented tangentially to the center of the coherent transmission T = |E| /|E0| of light through the ar- square [49]. While electric dipoles of the atoms average ray, where E is the total field amplitude. to zero on each square, they generate a magnetic dipole moment normal to the plane of the square. Arranging several of these squares in a circle, with each aligned per- be matched by radial polarization eîn = eˆρ, where eˆρ pendicular to the circumference, leads to the magnetic points outwards in the yz plane from the center of the dipole moments winding around the center, as illustrated toroidal dipole. The multipole decomposition of the lo- in the inset, creating a toroidal dipole pointing in the x cal excitation in Fig. 3(a) displays a strong response of direction. The projections of this geometry in the yz the toroidal dipole, as well as a weaker electric dipole re- and xy planes are shown in Fig. 2(a,b). A general choice sponse. Figure 3(b) shows the decomposition of the far- R 2 of parameters could be realized with independent optical field power P = 2c0 |E| dA integrated over a closed 2 tweezers. However, in the case that r = (n+1/2)a for in- surface into the dominant dipole component P1 ∝ |αE,1| , teger n, the ensemble could also be formed by selectively which does not distinguish between contributions from d populating sites on a bilayer square lattice, as indicated and T, as well as the remaining sum of all other contri- by the grey grid. butions. At the toroidal dipole resonance ∆ = −40.7γ We demonstrate this by an example calculation of four the occupation of the collective eigenmode is ≈ 99% [48]. such squares, with r = 0.2λ and a = 0.08λ [Fig. 1(a)], re- Here we take four squares, distributed evenly on a ring sulting in altogether 48 collective excitation eigenmodes. around the center, to form the toroidal dipole moment, This corresponds to a bilayer configuration of atoms, con- but similar results can be achieved with a minimum of fined to selective sites of a square lattice with lattice con- only two. As illustrated in Fig. 2(c), with two squares stant 0.08λ. We find a collective eigenmode exhibiting a centered at, e.g., ±ryˆ having opposite chirality dipole strong toroidal dipole, with only a weak radiative cou- orientation a toroidal dipole moment can also be achieved pling due to subradiant resonance linewidth υ = 0.2γ. while all atoms lie in the single xy plane [48]. The scattered light from this eigenmode is dominated by We next consider a planar square lattice in the yz plane 2 |αE,1| with > 99% of the radiated power coming from with each unit cell as in Fig. 1(a). Due to radiative in- this contribution, which can be decomposed locally into teractions, for a subwavelength-spaced lattice, the entire toroidal and electric dipoles with |T|/|d| = 2.2. At larger system responds as a coherent, collective entity, with de- lattice spacings, the toroidal dipole contribution can get localized collective excitation eigenmodes extending over even more dominant. the array. In particular, there is a collective eigenmode To excite the toroidal dipole mode, we consider a plane which corresponds to a uniform excitation of a toroidal wave propagating in the x direction. The toroidal sym- dipole at each site. However, this mode cannot be ex- metry of the mode inhibits coupling to a drive field with cited by radially polarized light as it would require the uniform linear polarization. Instead, the symmetry can symmetry to be broken around the center of each individ- 4

such that the net radiation vanishes when d = −ikT. Despite having no emission, the anapole state has a non- zero energy, and a vector potential which cannot be fully eliminated by gauge transformation [19, 62]. We show that the collective radiative excitations of strongly coupled atoms can form a dynamic anapole by adding a pair of atoms to the toroidal dipole configuration of Fig. 1(a), in the same bilayer planes of the existing atoms, that FIG. 5. Excitation of a non-radiating dynamic anapole, with then synthesizes a coherent superposition of electric and a multipole decomposition of (a) atomic dipoles [in units of toroidal dipoles [Fig. 1(b)]. The inset illustrates how the DE0/(~γ)], with p = d + ikT, and (b) the far-field scattered contribution of the electric dipole moment at the origin light from all contributions, the total dipole component with to the total dipole moment p points in the opposite di- intensity proportional to |p|2, and the sum of all other contri- rection to that of the toroidal dipole. 2 butions (in units of Iin/k ) as a function of the overall laser Again, we illustrate the case of four squares, dis- frequency detuning for a = 0.08λ, r = 0.2λ. tributed evenly on a ring around the center but similar results can be achieved with a minimum of only two. ual unit cell. Instead, we use uniform linear polarization, As illustrated in Fig. 2(c), adding two central atoms at with eîn = eˆy, but vary the atomic level shifts within ±(a/2)xˆ, results in an anapole excitation while all atoms each atom of the unit cell independently that are then lie in the xy plane [48]. repeated across the array on each unit cell. We numer- The anapole can be excited by a radially polarized ically optimize the toroidal dipole moment on a single plane-wave focused through a lens with high numerical unit cell to find these level shifts numerically. aperture, leading to a longitudinal field in the x direc- The corresponding toroidal and electric dipole excitation along the beam axis which excites the central two tions are shown in Fig. 4(a). Despite the presence of the atoms [48]. This field is calculated via the standard electric dipole, the toroidal dipole is the dominant com- Richard-Wold diffraction integral [63] with a numerical ponent at ∆ = −17γ where the ratio |T|/|d| = 3.3 is aperture of 0.7. The resulting multipole decomposition of at its maximum. The dipole radiation is compared to atomic dipoles [Fig. 5(a)] displays a strong excitement of the intensity of all other contributions to the scattered both the electric and toroidal dipole. However, the com- light in Fig. 4(b), showing that all other modes are also bination of these dipoles, p = d + ikT, is much weaker. suppressed at this detuning. The total scattered intensity, along with the decompo- This excitation closely corresponds to an eigenmode, sition into the dipole contribution and that of all other delocalized across the entire array, consisting of a repe- multipoles, is shown in Fig. 5(b), indicating a near total tition of the poloidal dipole excitation on each unit cell, cancellation of the scattered light. and forming an effective lattice of coherently oscillating In conclusion, we have shown how strong light- toroidal dipoles. The linewidth of the collective mode mediated dipolar interactions between atoms can be [Fig. 4(c)] narrows strongly as the unit cell spacing de- harnessed to engineer collective radiative excitations creases. that synthesize an effective dynamic toroidal dipole or The transmitted light through the array can be calcu- anapole. In both cases the toroidal topology is gener- lated by adding the scattered light from each individual ated by radiative transitions forming an effective poloidal atom to the incoming light. At a positon√ζxˆ from a uni- electric current wound around a torus. In a large lattice < form lattice of area A, when λ ∼ ζ A the electric we show how to engineer a collective strongly subradi- field of the light transmitted in the forward direction is ant eigenmode consisting of an effective periodic lattice given by [33, 59–61] of toroidal dipoles that exhibits a narrow Fano transmis- ik sion resonance. ikζ X ik(ξ−xj ) 0E = 0E0eˆye + [dj − eˆx · djeˆx] e . We acknowledge financial support from EPSRC. 2A j (3) 2 2 The transmission T = |E| /|E0| shown in Fig. 4(d) displays narrow Fano resonances at the frequencies of toroidal and electric dipole excitations. 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K. E. Ballantine and J. Ruostekoski Department of Physics, Lancaster University, Lancaster, LA1 4YB, United Kingdom (Dated: July 28, 2020)

We include additional details of how the formalism em- tween the atoms are given by the real and imaginary part ployed in the main text for atomic transitions relates to of eﬀective oscillating currents and nanoresonator systems. (jl) (jl) (jl) We present a brief description of the mathematical model Ωνµ + iγνµ = ξGνµ , (S3) which applies both in the limit of low light intensity with (jk) coherent drive and to a single photon excitation in the ab- where Gνµ = eˆν · G(ri − rj)eˆµ is the dipole radiation sence of drive. We extend the discussion of the toroidal kernel that, acting on a dipole d at the origin, produces dipole eigenmode to the occupation of the mode. Ad- the familiar expression [S8] (with ˆr = r/|r|) ditional details of how the proposal could be realized ( experimentally are considered. Finally we give results dδ(r) k3 eikr G(r)d = − + (ˆr × d) × ˆr for simpliﬁed toroidal dipole and anapole unit cells with 3 4π kr atoms only in a single 2D plane. ) i 1 − [3ˆr (ˆr · d) − d] − eikr , (kr)2 (kr)3 SI. LIGHT-ATOM INTERACTIONS (S4) SI.A. Quantum system of atoms and light and ξ = 6πγ/k3.

In the main text we describe the excitation of an effective toroidal dipole and anapole in a coherently driven SI.B. Low light intensity limit atomic ensemble by the equations b˙ = iHb+f where b is a vector of atomic dipole amplitudes and f is proportional In the limit of low drive intensity we neglect terms con- to the amplitude of the driving field. Here we describe taining products of two or more excited state amplitudes, the quantum model of atoms interacting with light in or one or more excited state amplitudes multiplied by the the length gauge, obtained by the Power-Zienau-Woolley drive field [S4]. In our system, this amounts to neglect- transformation [S1–S4]. In the limit of low light intensity, + − ing terms hσjν σ`µi (along with higher order correlators of the system with a single electronic ground state can be ± + exactly described by a classical model of coupled dipoles σjν ) and E0 hσmi. The excited state population is negligible while the ground state population is invariant. The driven by coherent light with field E(r) [S5, S6]. The full − many-excitation dynamics is described by the many-body only non-trivial equation is for hσjν i and Eq. S1 reduces quantum master equation for the density matrix [S7]; to ˙ X X (jl) h + − i b = iHb + f, (S5) ρ˙ = i [Hjν , ρ] + i Ωνµ σˆjν σˆlµ, ρ j,ν jlνµ(l6=j) − (S1) in terms of the vector b3j+ν−1 = hσjν i, the matrix X (jl) − + + − + − + γνµ 2ˆσlµρσˆjν − σˆjν σˆlµρ − ρσˆjν σˆlµ , (j) (jk) (jk) jlνµ H3j+ν−1,3k+µ−1 = ∆ν δνµδjk+Ωνµ (1−δjk)+iγνµ , (S6) with the square brackets representing commutators and

(j) + − and the drive f3j+ν−1 = i(ξ/D)eˆν · E(rj). Hjν = ∆ν σˆjν σˆjν ξ ξ (S2) + eˆ · E (r )ˆσ+ + eˆ∗ · E∗(r )ˆσ− , D ν 0 0 j jν D ν 0 0 j jν SI.C. Single photon excitation where σ+ = |ei hg| is the raising operator for atom j jν jν j The dynamics of a single photon excitation, decaying from its ground state to its excited state with m = ν. (j) in the absence of drive, can also be described by the same Here, ∆ν is the detuning of each level from resonance. (jl) formalism. Without drive, there is decay from the single- The Hermitian interaction terms Ωνµ and collective dis- excitation state to the ground state, but the remaining (jl) sipation terms γνµ due to the dipole-dipole coupling be- single-excitation state remains pure, and the dynamics 2

(a) (b)

FIG. S1. (a) Overlap of each eigenmode of a toroidal dipole FIG. S2. Longitudinal component |Ex| and radial compo- unit cell (ordered by the collective resonance linewidth) with nent |Eρ| of the focused radially polarized beam used to excite an ideal toroidal dipole mode with poloidal polarization. (b) the anapole, as a function of the distance ρ = py2 + z2 from Occupation measure L of the toroidal dipole collective eigen- T the beam axis in the focal plane x = 0. Here the incoming mode and sum L0 of the occupation of all other modes in the field before focusing is E = eˆ and the NA is 0.7. steady state response, as laser detuning is varied, for r = 0.2λ, ρ a = 0.08λ. collective radiative excitation eigenmodes of the atoms, is linear in the amplitude of this state. It is this lin- and the corresponding eigenvalues δj + iγj [S11], where ear response without saturation which provides identical γj denotes the collective resonance linewidth and δj the single-particle amplitudes to those of the low light inten- collective resonance line shift from the single-atom res- sity limit of coherently driven atoms [S9]. onance. While these eigenvectors are not orthogonal in We again start from the full atomic equations of motion general, they do form a basis, and the state can be ex- P Eq. S1, but now assume that the initial state consists of a panded at all times as b(t) = i ci(t)vi. The occupation pure, single-photon excitation. Then the density matrix of each eigenmode can then be defined as [S12] at later times can be written as |vT b|2 L = i . (S9) i P T 2 ρ = |Ψi hΨ| + pg |Gi hG| , (S7) j |vj b| where |Ψi is a state consisting of exactly one excitation, For a single-photon excitation, each eigenmode will decay |Gi is the state with all atoms in their ground state, and individually with pg is the probability that the excitation has decayed. In this case there is no incoherent mixing between the ex- cj(t) = exp [(iδj − γj)t], (S10) cited and ground state. While dissipation means that the with c (0) determined by the initial state. norm of |Ψi is not conserved, the dynamics within the j The toroidal dipole unit cell has a collective eigenmode single-excitation subspace is coherent, and it can always exhibiting a strong toroidal dipole as discussed in the be expanded in terms of the individual atomic excitations main text. We can similarly measure the overlap of this X mode with an ideal toroidal dipole eigenmode v , de- |Ψi = P(j)(t)σ ˆ+ |Gi , (S8) T ν jν picted in Fig. 1 of the main text, with poloidal polariza- j,ν T 2 P T 2 tion. This is given by Ci = |vi vT | / j |vj vT | . This (j) overlap is illustrated for each eigenmode in Fig. S1(a) with amplitudes Pν (t). For single-particle expectation values, the dynamics can equally be written in terms of showing that there is a unique collective mode which ap- (j) proximates the ideal poloidal polarization mode. The these amplitudes. In terms of the vector b3j+ν−1 = Pν , occupation L of this eigenmode, as well as the sum of ˙ T we have again b = iHb [S10], formally equivalent to occupations of all other modes, is plotted in Fig. S1(b) the equations describing the low light intensity limit in as a function of laser detuning, with LT > 0.99 at reso- the absence of drive. The identical description is due nance. The maximum occupation in Fig. S1 corresponds to the fact that in each case the relevant dynamics in- to the maximum toroidal dipole excitation of Fig. 3 in cludes only a part of the density matrix which evolves the main text. linearly without saturation, with the collective response (jl) being determined by the dipole propagation kernel Gνµ . SIII. CLASSICAL ANALOGUE OF POLARIZATION DISTRIBUTION SII. COLLECTIVE TOROIDAL EIGENMODE In this work, we have synthesized collective radiative The collective dynamics of the atomic ensemble is de- excitations of atoms that produce a toroidal dipole and termined by the eigenvectors vj of H, representing the an anapole. This is achieved by a simple arrangement of 3 atoms that experience strong light-mediated interactions. dipole with a magnetic moment dµ = du(q/(2dt))rφ × In particular, the toroidal dipole is generated by the ori- ∆r = du(qρ2dφ/(2dt))zˆ such that the total magnetic mo- entation of the atomic transitions due to the dipole-dipole ment µ = R dµ = (dφρ2/2)(q/dt)zˆ is simply the current interactions forming an effective poloidal electric cur- times the area of the circle within the angle dφ. rent wound around a torus. Each atom produces an ef- While this description applies to a continuous distribu- fective electric dipole generated by quantum-mechanical tion of dipoles located at rφ + u∆r (or n discreet dipoles electric-dipole-allowed transitions in electronic orbitals. without taking the limit n → ∞ in Eq. S11), it can be ap- Here we provide a classical analogue description of this proximated with a discrete series of a smaller number of quantum-mechanical process, illustrating how effective dipoles located at fixed positions rj. The corresponding polarization and current densities originate from such P (j) polarization density Pσ(r, t) = exp (−iωt)D Pσ δ(r− atomic dipole transitions. This allows us to make com- j rj) induced by a driving field with frequency ω leads parisons with systems of nanoparticles and solid-state ar- to a current density J = −iωP. When these discrete tificial metamaterial resonators that have been used in currents are arranged to point tangentially to a circle, studies of electromagnetic multipole radiation. they approximate a closed loop of current. The magnetic dipole moment appearing in the scattering cross section is then [S13] m = µ/c = 1/(2c) R d3r r × J(r). Similarly, SIII.A. Classical effective currents from trapped poloidal current distributions are approximated by cur- atoms rents from time dependent point dipoles giving a toroidal dipole moment [S14] Classically, we describe the interactions of light with atoms trapped in a ring-shaped pattern, represented by a 1 Z T = d3r r(r · J) − 2r2J . (S13) sequence of oscillating charges. Consider a ring in the xy 10c plane with radius ρ, and let φ be the azimuthal angle in the plane. Suppose there is a flow of charge on the ring Numerically, we include all the light-mediated dipole- and consider an infinitesimal segment where a negative dipole interactions between the atoms and calculate the charge −q moves from a point rφ at angle φ a distance collective excitation eigenmodes for the coupled dipoles ∆r = −ρ dφφˆ along the ring in a time dt, leaving a pos- (see Sec. SII). One of the eigenmodes then corresponds to itive charge q stationary at rφ. The charge distribution the poloidal current configuration of the toroidal dipole can be replaced by n dipoles with spacing ∆r/n such that [Fig. S1(a)]. This also applies to the case of an array of the positive charge of each dipole overlaps with the neg- several toroidal dipoles, as discussed in the main text. ative charge of the next, leaving only the original charges at each end. Then the polarization density in the limit n → ∞ is [S3] SIII.B. Nanoparticles and solid-state resonators

n−1 X ∆r p + 1/2 The classical description of the effective current due to P = lim (−q) δ r − rφ + ∆r n→∞ n n atomic dipole transitions, given in the previous section, p=0 allows us to make comparisons with systems of nanopar- 1 Z ticles and resonators in artificial metamaterials that form = − du q∆rδ(r − rφ − u∆r), 0 multipole radiation sources. The most dramatic dif- (S11) ference is the quantum-mechanical nature of the opti- giving a continuous polarization density consisting of cal interaction for the case of atoms, where the atomic point dipoles dP = −q∆rδ(r−rφ −u∆r)du = qρ dφ δ(r− transitions are determined by the precise resonance fre- ˆ rφ − u∆r)duφ located at rφ + u∆r and pointing in the quency and the quantum-mechanical Wigner-Weisskopf direction φˆ. These point dipoles describe the response resonance linewidth [S3]. Atoms also form truly point- of single atoms, whose size is negligible compared to the like electric dipoles. optical wavelength. In the limit of low light intensity, the atoms interacting The current density associated with the change in the with incident light can be considered as a linear classical electric charge distribution from each point dipole is coupled-dipole system (see Sec. SI SI.B). Small nanopar- given by ticles [S15, S16] or circuit resonators [S17, S18] are fre- quently approximated as effective coupled point dipoles 1 in electromagnetic fields. Nanoparticles with their elec- dJ = [−q(r + ∆r ) − (−q)r ]δ(r − r − u∆r)du, dt φ φ φ φ tric dipoles (provided that the higher-order multipole (S12) contributions are negligible) forming a closed ring can dφ then exhibit a mode that behaves as an effective magnetic = qρ δ(r − r − u∆r)duφ,ˆ dt φ dipole [S15], analogously to the earlier classical charge distribution description of the atoms. In resonant LCR which is simply equal to dP/dt. This circulating current, circuits in metamaterials, the current oscillations form for a sufficiently small circle, creates an effective magnetic effective electric and magnetic dipoles that radiatively 4 couple with the current oscillations of the other circuits. are loaded deterministically into a Mott-insulator state If each circuit is modeled as a pointlike particle, and with one atom per site [S21–S23], and the desired geom- higher-order multipole contributions of each circuit are etry achieved by removing excess atoms on a site-by-site negligible, the dynamics corresponds to a coupled-dipole basis [S24]. system [S17, S18]. Further techniques have also been developed to pro- As a comparative example, a nanorod can be approxi- vide deeply subwavelength features in optical trapping mated as an effective point dipole [S19], where the polar- potentials. For example, coupling three atoms in a Λ ization density is to first order assumed to be a uniform configuration via a strong control field with Rabi fre- distribution of point dipoles throughout the volume of quency Ωc(y, z) = Ωc sin (ky) sin (kz) and a weak probe the rod. Depending on the geometry of the problem and field with Rabi frequency Ωp leads to a trapping poten- the thickness of the nanorods, all other excitation modes, tial which depends on the ratio Ωc(y, z)/Ωp, which varies such as radial or azimuthal currents are then in this ap- rapidly in a small region close to the nodes of Ωc(y, z), proximation ignored. For a single nanorod of radius a for Ωp/Ωc 1 [S25]. Alternatively, internal degrees of and length H at the origin, and aligned along the z axis, freedom can be exploited [S26]. Lattices with periodic- the resulting polarization distribution is ity less than half the wavelength of the control field can also be engineered by stroboscopicaly shifting the lat- Q(t) ˆ tice at high frequency such that the atoms experience P = 2 zΘ(a − ρ)Θ(H/2 − z)Θ(H/2 + z), (S14) πa a time-averaged potential with higher periodicity than where Θ is the Heaviside function, ρ = px2 + y2, and the instantaneous potential at any one time [S27, S28]. Q(t) is a generalized coordinate whose derivative Q˙ (t) Finally, optical tweezers provide an alternative means to describes current oscillations in the nanorod. In con- design arbitrary potentials [S29–S31] with single-site control [S32, S33]. trast to atomic dipoles, although for H . λ this polarization density can be approximated by a point dipole Atoms such as Sr and Yb are particularly suitable for 33 P = Q(t)Hzˆδ(r), this approximation easily breaks down subwavelength trapping. Sr has a transition between 3 3 for interacting nanorods that are too closely separated the P0 state and the triply degenerate D1 state [S34] compared with the resonant wavelength [S19]. with wavelength λ = 2.6µm and linewidth Γ = 2.9 × 105/s. The magic wavelength for these states gives an optical lattice with spacing d = 206.4nm, less than λ/10. SIV. EXPERIMENTAL CONSIDERATIONS While most of the examples we consider require varying only the overall laser detuning, the data shown in For most of the examples discussed in the main text Fig. 4 of the main text requires the individual atomic we choose r = (n + 1/2)a for integer n, with the result level shifts of different atoms to be controlled. This could that the atoms lie on selected sites of a square lattice be achieved by ac Stark shifts [S35] from standing wave with lattice constant a (see main text Fig. 2). Four in- lasers offset from the trapping lattice such that different tersecting beams, two pairs of propagating beams in the sites experience different intensities. y and z directions respectively, can be used to produce such a lattice with confining potential SV. ANAPOLE EXCITATION h y z i V (r) = sE sin2 π + sin2 π , (S15) R a a To excite the anapole we use a tightly focused radially 2 2 2 where ER = π ~ /(2ma ) is the lattice recoil energy and polarized beam. The focusing leads to a longitudinal s is a dimensionless constant which determines the depth component in the x direction on the beam axis which di- of the lattice [S20]. An additional potential can con- rectly drives the x component of polarization on the two fine the atoms in the x = ±a/2 planes. This could be atoms at the center of the anapole. Off axis, a combi- an identical pair of counter-propagating beams in the x nation of radial and longitudinal polarization couple to direction, or a simpler double-well potential. Locally, the toroidal dipole mode of the remaining atoms. The each atom experiences a harmonic trapping potential amplitude of the longitudinal and radial components of P 2 2 √ V (r) = (m/2) ωµ(∆rµ) where ωy,z = 2 sER/~, ωx the field in the x = 0 focal plane are shown in Fig. S2 as is the harmonic trapping frequency in the x direction, a function of the radial distance ρ from the beam axis. and ∆r is the displacement from the center of the lattice site. Then the atom at site rj has a Wannier function φj(r) = φ(r − rj) given by SVI. IN-PLANE TOROIDAL DIPOLE AND 1 y2 + z2 x2 ANAPOLE φ(r) = exp − − , (S16) (π3l4l2)1/4 2l2 2l2 x x An experimentally even simpler realization of both the −1/4 p with width l = as /π and thickness lx = ~/(mωx). toroidal dipole and the anapole with atoms confined only The atoms can be increasingly strongly confined by in- to a single plane can be achieved by removing the atoms creasing the trapping strength s. Experimentally, atoms for which z 6= 0, as shown in Fig. 2(c) of the main 5

(a) (b) text, leaving two squares with opposite chirality polarization. These squares will have magnetic dipole moments pointing in the zˆ and −zˆ directions, respectively, leading to zero net magnetic moment but contributing to a toroidal dipole. The resulting multipole decomposition of the atomic dipoles and the far-field scattered light is shown in Fig. S3(a,b). A strong toroidal dipole is present, although the far-field scattered light in this case also shows a comparable contribution from higher order (c) (d) modes. Again this toroidal dipole can interfere with a net electric dipole moment on two atoms at ±(a/2)xˆ to form an anapole. The resulting multipole decomposition is shown in Fig. S3(c,d). There is strong suppression of the total dipole moment and a significant dip in the scattered light, although again the contributions of other multipole moments, especially quadrupoles, is stronger than the case with four squares presented in the main text. These contributions of higher-order multipole mo- FIG. S3. Excitation of toroidal dipole and dynamic anapole ments could be further suppressed by adding more atomic with two magnetic dipole moments such that all atoms are squares, 8, 16, etc. in the single xy plane. Multipole decomposition for toroidal dipole unit cell of (a) atomic dipoles [in units of DE0/(~γ)], and (b) the far-field scattered light, separated into the total dipole component, and the remaining sum of all other con- 2 tributions (in units of incident light intensity Iin/k ). Mul- tipole decomposition for the single-plane dynamic anapole of (c) atomic dipoles [in units of DE0/(~γ)], with p = d + ikT, and (d) the far-field scattered light, separated into the total dipole component, and the remaining sum all other contribu- 2 tions (in units of Iin/k ). In all cases r = 0.2λ and a = 0.08λ.

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