Subradiance-protected excitation spreading in the generation of collimated photon emission from an atomic array

We show how an initial localized radiative excitation in a two-dimensional array of cold atoms can be converted into highly-directional coherent emission of light by protecting the spreading of the excitation across the array in a subradiant collective eigenmode with a lifetime orders of magnitude longer than that of an isolated atom. The excitation, which can consist of a single photon, is then released from the protected subradiant eigenmode by controlling the Zeeman level shifts of the atoms. Hence, an original localized excitation which emits in all directions is transferred to a delocalized subradiance-protected excitation, with a probabilistic emission of a photon only along the axis perpendicular to the plane of the atoms. This protected spreading and directional emission could potentially be used to link stages in a quantum information or quantum computing architecture.

Since the light-mediated resonant dipole-dipole interactions depend on the average atomic separation, the close proximity of the atoms can lead to a correlated optical response [15,16] that no longer obeys continuous medium descriptions of electrodynamics [17,18]. The quest for systems where the collective optical response could potentially be easily manipulated has resulted in the studies of strong light-atom coupling -and coupling of light with other dipolar resonators -in regular arrays .
Control, storage and transmission of collective excitations could play a key role in modular quantum architecture [53][54][55], consisting of individual quantum elements with coherent links [36,56]. Subradiant states [57], which decay more slowly than an isolated emitter, couple weakly to external fields and have posed a long-standing experimental challenge, with observations first emerging in two-and few-particle systems [58][59][60][61][62][63], and now also in larger ensembles [11,42]. Due to the isolation from the environment, subradiant modes have been shown to be useful in transport of excitations [64][65][66][67]. Recent work has explored light transport for closely spaced atoms in arrays with topological edge states [68,69] and in a onedimensional (1D) chain or ring [70][71][72].
We show here how an initial localized excitation at the center of a planar array of cold atoms, which will emit radiation in all directions, can be transported across the lattice and converted into highly directional emission. The initial localized excitation will have an overlap with several of the collective radiative many-atom excitation eigenmodes of the system. These collective modes arise from the effective dipole-dipole interactions between atoms, and will each have a collective resonance shift and a collective linewidth that differs from that of a single atom. As these eigenmodes have different linewidths, they will decay at different rates, and interference of the modes leads to the spatial spreading of the excitation across the lattice. After some time, the excitation will be in a very subradiant collective mode delocalized over the entire array. In this case coherence across the array comes from the collective nature of the many-atom eigenmode, rather than any driving field. After we release the excitation by controlling the atomic levels, the result is coherent and highly directional emission of a photon from the array.
In our model, we assume that a two-dimensional (2D) square array of cold atoms is prepared with a single atom per lattice site and with a single-photon excitation initially localized at the center of the lattice. Such excitations could be produced by coupling to a nearby emitter or by selectively controlling the atomic excitation. By choosing the initial state and the lattice spacing appropriately, this local excitation is transferred to one of two delocalized subradiant target modes, a mode with uniform in-phase out-of-plane polarization, and a mode where the phase of the out-of-plane polarization varies by π between neighboring atoms. Once the delocalized mode is established, Zeeman splitting can couple the uniform subradiant out-of-plane mode to a uniform in-plane mode that is strongly radiating, allowing the excitation to decay. We calculate the far-field radiation pattern of this emission which is highly collimated in the direction normal to the plane. Collimated emission has been achieved in plasmonic planar arrays after spreading of an initial excitation [73], but without an analogous procedure of transferring the excitation between weakly (dark) and strongly (bright) radiating states. An off-resonant pair of atomic layers has been proposed as a phased-array antenna for photons [36]. atomic transition (Fig. 1). The full quantum dynamics of the atomic system for a given initial excitation and in the absence of a driving laser follows from the quantum master equation for the many-atom density matrix ρ; whereσ + jν = (σ − jν ) † = |e jν g j | is the raising operator to the excited state ν on atom j. The single atom Zeeman splittings, ∆ ν , are given relative to the m = 0 transition frequency ω, with ∆ 0 = 0 and ∆ − = −∆ + ≡ ∆ = ω−ω ± , where ω ± are the frequencies of the m = ±1 transitions, respectively. The diagonal terms of the dissipative matrix γ (jj) νν = γ = D 2 k 3 /(6π 0 ) correspond to the single atom resonance linewidth with the reduced dipole matrix element D and k = 2π/λ = ω/c. The off-diagonal elements in the dissipation and interaction terms are given by the real and imaginary parts of where ξ = 6πγ/k 3 . Here, the dipole-dipole interaction between the atoms j and l with the orientations of the dipolesê ν andê µ at positions r j and r l , respectively, is determined by the dipole radiation kernel [74] where and the contact interaction term between the atoms has explicitly been removed [75] G αβ (r) = G αβ (r) + δ αβ δ(r) 3 .
For ideal point dipoles this contact interaction is inconsequential for the physics [76], and by assuming hard-core atoms it vanishes as the atoms cannot overlap. We assume that the Zeeman level splitting ∆ of the J = 1, m = ±1 states, is controllable, as could be obtained, e.g., by magnetic fields or by AC-Stark shifts of lasers or microwaves [77]. For the majority of what follows, we take ∆ = 0 and study the transfer and decay of an initial excitation in the absence of Zeeman splitting. In Sec. II 3 we show how turning on the Zeeman splitting ∆ couples a subradiant state, with very slow decay, to a rapidly decaying bright mode, allowing the photon to radiate away. Instead of solving the general quantum master equation for the full many-atom dynamics, we restrict ourselves to single-excitation systems determined by the initial state of precisely one electronic excitation and study their evolution. The dynamics of the single-excitation subspace decouples to give whereρ (jk) νµ = G|σ − jν ρσ + kµ |G are the matrix elements of ρ corresponding to the single excitation, |G is the state with all atoms in the ground state, and While dissipation will mix the single-excitation subspace with the ground state, the dynamics within the single-excitation subspace are coherent. Since we always assume a pure initial state, the single-excitation part of the density matrix retains the form where |Ψ(t) is a single-excitation state whose norm | Ψ(t)|Ψ(t) | 2 is not conserved due to the dissipation. This state can be expanded in terms of the atomic excitations Regarding single-particle expectation values, equations of motion can equivalently be written in terms of these single-excitation subspace amplitudes [78]. This is best expressed by rearranging the excitation amplitudes into a vector b 3j−1+ν = P (j) ν (ν = −1, 0, 1), and similarly arranging the evolution Hamiltonian into a 3N ×3N matrix νµ . Then the time evolution is described byḃ where H is a non-Hermitian matrix. This matrix equation describing the dynamics of the amplitudes of a single excitation state is formally equivalent to the equations of motion of N classical linear coupled dipoles. Thus we can interpret the results in terms of the decay of a single photon, or the decay of a classical coherent dipolar excitation. For a dipole P (j) ν of the jth atom, the scattered field at r reads [74] In order to understand the behavior of the system as it evolves away from the initial condition, we study the eigenvalues δ n + iυ n and eigenmodes v n of the coupling matrix H , where δ n = ω −ω n is the shift of the collective mode resonance from that of a single atom and υ n is the collective resonance linewidth.
Since H is not Hermitian, the eigenmodes will not be orthogonal, but they still form a basis. One can therefore uniquely express the excitation amplitudes as The amplitudes c n of the collective modes evolve independently and each satisfy the equation of motioṅ with solution

Excitation spreading
We now study numerically the evolution and decay of an initial localized excitation, examining how it spreads across the lattice and comes to occupy a target delocalized mode. In this section, we assume there is no Zeeman splitting (∆ = 0). Hence, the J = 0 → J = 1 transition is isotropic, with the symmetry broken only by the orientation of the plane of the lattice. In the following section we will look at the effect of Zeeman splitting in coupling different eigenmodes.
For simplicity, for our numerical analysis, we assume that the planar array of atoms is uniformly excited at time t = 0 at nine sites at the center of the array. While the excitation decays to zero in the long-time limit, it is clear from Eq. (14) that if there is appreciable initial excitation of a subradiant state with υ n γ, then this will dominate at intermediate times when other mode amplitudes have decayed. There are two collective eigenmodes of particular relevance for planar arrays, namely those with coherent uniform excitations of dipoles. The first is a uniform in-phase polarization which points perpendicular to the plane, along the x axis, and which we denote by P P . The second one, denoted P I , has a uniform in-phase polarization in-plane, here chosen to be along the y axis. The eigenmode P I directly couples to an incident plane wave propagating normal to the plane and is generally very easy to excite. The eigenmode P P can be directly coupled to P I by a symmetry breaking where induced Zeeman level shifts drive transitions between the modes [27,29]. Furthermore, these modes are useful in understanding the spectral response of the array, and can play the role of collective versions of dark and bright states [29] of a standard single-particle electromagnetic induced transparency (EIT) [79].
The eigenmode P P can be targeted with an appropriate initial excitation even when initially the system only exhibits a localized excitation. Here we also target another collective eigenmode, an antiferromagnetic excitation denoted by P AF , where each atom has a polarization in the normal direction along the x axis which is π out of phase with each of its nearest neighbors. This mode is of interest because it has quickly varying phase across the lattice, but can be reached from a localized initial excitation where the polarization of almost all the atoms is zero.
The resonance linewidth of each of these modes is shown in Fig. 2 as a function of lattice spacing for a 31 × 31 lattice. The uniform out-of-plane mode P P and the antiferromagnetic mode P AF can be strongly subradiant at the appropriate array spacings. The antiferromagnetic mode has maximum phase variation in the y and z directions, and so for periodic boundary conditions is located at the corner of the Brillouin zone with quasi-momentum q = (π/d, π/d). For small lattice spacing the quasi-momentum here is greater than the free-space wavevector, |q| > k, and the mode cannot decay [68]. While the linewidth is non-zero for a finite lattice, it is still much smaller than the linewidths of the uniform eigenmodes for d < ∼ λ/2. The uniform modes at q = 0 are not protected by momentum conservation; in the limit of an infinite array the uniform in-plane eigenmode only radiates exactly normal to the plane (the zeroth order Bragg peak) for any subwavelength lattice spacing [24,27,52]. For the P P mode, however, the dipoles oscillate along the x axis and so emit mainly in the lattice plane, where light is absorbed by other atoms and must undergo many scattering events to escape. This leads to this mode also being very subradiant for larger lattices, scaling with the number of atoms N as υ P /γ ≈ N −0.9 [27], with the mode becoming completely dark in the infinite lattice limit, lim N →∞ υ P = 0. The in-plane mode P I has a linewidth which is well described for d < λ by its large-N limit [24,29] υ I,∞ ≡ lim If we choose the initial excitation to match one of these eigenmodes in the center of the lattice, and to be zero everywhere else, we expect it to have a non-zero overlap with the collective mode. If this mode is sufficiently subradiant, then, as other modes decay, it will come to dominate and the initial localized excitation will quickly evolve into a coherent, subradiant state extended across the entire lattice. However, since H is not Hermitian, the eigenvectors are not orthogonal in general. We therefore use the definition for a measure of the occupation of an eigenmode v j in the current state b which has been shown to describe accurately the contribution of the collective mode populations in the decay dynamics of radiative excitations [27]. This measure can also be used to determine a target eigenmode of a finite lattice which most closely matches an ideal mode with equal amplitude on every site. We take this measure to be normalized at t = 0. For a 31 × 31 square lattice with a lattice spacing of d = 0.75λ and ∆ = 0, the uniform mode P P is very subradiant, with a linewidth (5 × 10 −4 )γ, as shown in Fig. 2. To target this mode, we start from an initial excitation which has P (j) x ≡ (P  Fig. 3 (a), where the mode occupation of the uniform perpendicular mode, L P , is plotted along with the sum of the occupations of all other eigenmodes, denoted by L . The dynamics can be understood by looking at the initial mode occupations, plotted in Fig. 3 (b). While several modes are initially occupied, the target mode, indicated by the red square, is the most subradiant. As the less subradiant modes decay rapidly, the total excitation initially falls off quickly. However, because the target mode is very subradiant, the population of this mode decays much slower, and after some time it becomes the dominant mode. At this point, light is effectively stored in a very subradiant state delocalized across the entire lattice.
For an appropriate choice of the spacing d = 0.55λ, the antiferromagnetic eigenmode is more subradiant than the  uniform perpendicular mode. Also the antiferromagnetic mode can be targeted by a matching initial excitation; here we take the alternating polarization P (j) x = ±1/3 on each of the central nine atoms, where the sign varies between nearest neighbors, and zero everywhere else. The initial excitation and the time dynamics are shown in Fig. 3 (c) and (d). As in the case of the uniform mode, the initial localized excitation tends to a delocalized state.
The importance of subradiance in letting the excitation spread across the lattice is illustrated by comparing an attempt to target the uniform in-plane mode P I . The result is shown in Fig. 3 (e) and (f). Because the excited modes are not significantly subradiant, the occupations decay on a much faster timescale. Even within this short time, however, the occupation of the target mode is never higher than that of all other modes.
The spatial spreading is illustrated in Fig. 4, which shows the absolute value of the excitation amplitude (which always points normal to the plane) across the lattice at various times, for initial in-phase and antiferromagnetic excitation at the center. The excitation quickly spreads across the lattice as each eigenmode component of the initial excitation decays at different rates. After about t = 500/γ, the excitation has settled into the most subradiant target mode.
In the case of both the modes P P and P AF , while the initial excitation is successfully transferred to the target mode, the final overall occupation is L ≈ 0.02 (representing the probability of generating a single delocalized photon), much less than the initial occupation which is normalized to one. This is due to the low occupation of the target mode in the initial state, which is approximately 0.04 of the total, as the initial excitation is localized to ≈ 1% of the atoms. This could be improved by starting with a larger excitation. The occupation of the target mode rises to 0.07 for an initial excitation of the central 16 atoms for example, and to 0.1 for an initial excitation of the central 25 atoms.

Photon release
In the previous section we described how an initially localized excitation can be transferred to a delocalized subradiant eigenmode occupation, extending over the entire array of atoms. As the photon then is effectively stored in a subradiant state, it decays slowly and little light is emitted. To release the photon, the Zeeman splitting can be turned on at the desired time. This can be understood by considering the two uniform eigenmodes of the system P I and P p . When we introduce a non-zero Zeeman splitting, these two modes are no longer eigenmodes, but are coupled to one another. The dynamics in the presence of Zeeman splitting can be understood by considering a two-mode model [27,29], which for a driven system represents a linearized version of the EIT equations for bright and dark states [79]. The two-mode model qualitatively captures many aspects of the dynamics of the full lattice for sufficiently large arrays, since the phasematching conditions of the other modes are not satisfied. We find that it can also illustrate how the excitation is transferred from one uniform mode to the other in the present case. The relevant physics is captured by the simple coupled two-mode dynamicṡ where δ P,I is the collective resonance shift and υ P,I the collective linewidth of the uniform out-of-plane and inplane modes, respectively. These two modes capture much of the physics because once the dipoles are all oscillating in phase, they will continue to do so even when Zeeman splitting is turned on and the direction of the effective magnetic field breaks the isotropy of the J = 0 → J = 1 transition. As seen from Eqs. (17a) and (17b), in the presence of Zeeman splitting, P P is not an eigenmode, and the dipoles start to rotate towards the plane. Applying the The angle θmax such that 99% of the integrated far field intensity is between 0 < θ < θmax, where θ = arctan ( k 2 y + k 2 z /kx), as a function of the number of atoms N . To compare different lattice sizes, the excitation is released when the target mode accounts for 90% of the population. (e) Far-field radiation for a 71 × 71 lattice, released when 90% of the excitation was in the mode PP , showing highly directional emission of the stored excitation. The intensity I rel is plotted in relative units scaled to a maximum of one.
splitting for a short time, the excitation can then be transferred to the mode P I , where each atom has approximately uniform in-phase polarization in the y direction, with much faster emission rate, allowing the excitation to quickly radiate away. We calculate the emitted light, given by the sum of all the atomic contributions from Eq. (11), in the far-field limit (r d, λ, where r is the distance from the source to the observation point) [74]. Figure 5 shows the results of Zeeman splitting being turned on at t = 800/γ for a short time. As soon as the splitting is turned on, the occupation P P of the outof-plane mode falls rapidly, while the occupation of the in-plane mode P I shows a corresponding rise. Since this mode has a much larger linewidth, the occupation then begins to fall quickly and, within a time ≈ 5/γ, the photon has been emitted from the lattice.
While emission from the initial excitation is omnidirectional, emission from the delocalized mode is highly collimated along the direction of the x axis, perpendicular to the lattice. We quantify this by the angle θ max such that integrating between 0 ≤ θ ≤ θ max and 0 ≤ φ ≤ 2π gives 99% of the total integrated forward-scattered intensity, where θ = arctan ( k 2 y + k 2 z /k x ) is the polar angle of the ray to the k x axis and φ = arctan (k y /k z ) the azimuthal angle in the k y k z plane. For the initial excitation, this angle is θ max = 0.997(π/2), i.e. the light is not collimated at all. However, when the excitation is released from the delocalized mode, the photon emission is highly directional, with θ max = 0.05(π/2). In this case the emission is equal in the forward and backward directions. Forward-only scattering could be achieved using two arrays offset in the x direction with a suitable phase shift [36], analogously to the directed radiation of antennas.
The far-field radiation pattern of the delocalized mode is that of a 2D diffraction grating, dominated by the central zeroth order Bragg peak (the higher order Bragg peaks do not exist because of the subwavelength lattice spacing). For lattices with a higher number of atoms, this central peak becomes sharper. To compare different lattice sizes we start with the same initial excitation on the central nine atoms, and let it evolve until the target uniform mode accounts for 90% of the remaining mode occupation. Increasing the number of atoms leads to a sharp drop in θ max , as shown in Fig. 5 (d). For the largest lattice (71 × 71), we find θ max = 0.02(π/2) which represents a photon wave-packet that is highly localized in k-space. For an infinite array, the propagation reaches the precise 1D limit propagating only in the x direction.
The process described in this section is reminiscent of the procedure to slow and store light within an atomic cloud using the standard single-particle EIT [80,81]. In our case it is the collective modes of the atomic array which act as the bright and dark states. To release the photon, the Zeeman splitting couples these states, playing the role of the coupling laser which is usually used to restore transparency. This allows the photon to radiate away while preserving spatial coherence.

The effects of position fluctuations
The atoms may not be perfectly localized, but rather their positions will fluctuate due to the finite size of the trap. We account for this by taking many individual realizations with fixed atom positions drawn randomly from a harmonic oscillator ground-state probability distribution for each lattice site with root-meansquare width l, and stochastically averaging over these realizations [21]. This procedure has been shown to reproduce a full quantum model exactly [75,82]. The result of spatial disorder is shown in Fig. 6, where we plot the surviving amplitude P = j,µ |P (j) µ | 2 as a function of time for varying fluctuation lengths. For increasing disorder, the excitation decays more quickly, as also observed in other subradiance-protected excitation transfer studies [72]. However, in these cases the life-time is still much longer than the corresponding case of an in-phase in-plane localized excitation with no disorder. This can be understood by looking at the distribution of the eigenmodes, weighted by their initial occupation L, across many stochastic realizations. This is compared in Fig. 6 to the distribution of the most subradiant modes which contribute up to 30% of the initial excitation of each realization, shown in green. Although disorder means that each realization has far fewer very subradiant modes, these few modes are consistently wellrepresented in the initial excitation.

III. CONCLUDING REMARKS
Subradiant states are isolated from the environment and therefore difficult to excite. Standard field excitation typically only results in a very small fraction of the total population to notably subradiant modes [11]. Transferring a more substantial population to slowly radiating states typically requires first the breaking of the eigenmode symmetry before the excitation, followed by restoration of the symmetry, such that the modes are temporarily made to interact with radiation [27]. Here we have demonstrated a probabilistic conversion. We have shown how a localized single-photon excitation of an atomic lattice, with omnidirectional emission, can spread out into a subradiant state which is delocalized across the whole lattice and is protected from decay. For suitable lattice spacing, this initial excitation can evolve into either a uniform mode, with the polarization of each atom in phase, or an antiferromagnetic mode, with the polarization of each atom π out of phase with its nearest neighbors. In the case of the uniform mode, we have shown how turning on a Zeeman splitting allows the excitation to be released as highly collimated directional emission.
Such an operation could form part of a quantum information or quantum computing architecture [36,54,83], coupling via short-range interactions to an input state, and coherently converting this local excitation into directional emission, and effective 1D propagation. This output could then be transferred via free space to another stage. Future work could identify collective quantum effects in higher-order correlations beyond the singleexcitation limit, and study the effect of such correlations on the emitted collimated light.

ACKNOWLEDGMENTS
We thank Chris Parmee for reading and commenting on the manuscript. We acknowledge financial support from EPSRC.