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Optical waveguiding by atomic entanglement in multilevel atom arrays

A. Asenjo-Garcia1, H. J. Kimble2, and D. E. Chang3,4 1Physics Department, Columbia University, New York, NY 10027, USA

2Norman Bridge Laboratory of Physics MC12-33, California Institute of Technology, Pasadena, CA 91125, USA

3ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain and

4ICREA-Instituciò Catalana de Recerca i Estudis Avançats, 08015 Barcelona, Spain (Dated: December 20, 2019)

The optical properties of sub-wavelength arrays of atoms or other quantum emitters have attracted significant interest recently. For example, the strong constructive or destructive interference of emitted light enables arrays to function as nearly perfect mirrors, support topological edge states, and allow for exponentially better quantum memories. In these proposals, the assumed atomic structure was simple, consisting of a unique electronic ground state. Within linear optics, the system is then equivalent to a periodic array of classical dielectric particles, whose periodicity supports the emergence of guided modes. However, it has not been known whether such phenomena persist in the presence of hyperfine structure, as exhibited by most quantum emitters. Here, we show that waveguiding can arise from rich atomic entanglement as a quantum many-body effect, and elucidate the necessary conditions. Our work represents a significant step forward in understanding collective effects in arrays of atoms with realistic electronic structure.

Realizing efficient atom-light interactions is a ma- jor goal in quantum optics. Due to the intrinsically weak coupling between photons and atoms in free space, atomic ensembles have risen as one of the workhorses of the field, as the interaction probability with photons is enhanced due to the large number of atoms in the cloud [1]. Atomic ensembles have broad potential ap- plications, which include, among others, photon storage and retrieval [1–3], few-photon non-linear optics [4–7], and metrology [8–10]. The fidelity of an atomic ensem- ble in carrying any of these applications is fundamentally limited by the so-called optical depth, which is a product of the interaction probability between a single atom and a photon in a given optical mode and the total number of atoms. While the important role of optical depth is ubiquitously stated in literature [11–14], the underlying arguments in fact rely on one crucial assumption: that the atoms do not interact with each other and thus that photon emission happens at a rate given by that of sin- gle atoms. It is clear, however, that this approximation breaks down when atoms are close to each other, as pho- ton emission is a wave phenomenon and interference and multiple scattering effects will be relevant at short dis- tances.

In dense and ordered atomic arrays [15–23], strong constructive or destructive interference of light emitted by excited atoms allows one to exceed the fidelities pre- dicted by these simple optical depth arguments in appli- cations [24]. For example, it has been theoretically shown that interference can impact communication and metrol- ogy applications: it enables both an exponential improve- ment in the fidelity of a quantum memory [24, 25] and an improvement of the signal-to-noise ratio in optical lat- tice clocks [26, 27]. More generally, interference in arrays can give rise to exotic phenomena [28–36], which have no counterpart in disordered atomic gases. These in-

clude perfect reflection of light [37, 38] or the existence of guided topological edge states of light in two-dimensional arrays [39, 40]. In these previous theoretical works, the atoms were as-

sumed to have a unique electronic ground state. For two level atoms, and within the single-excitation manifold, multiple scattering enables a process where an excited atom i can interact and exchange its excitation with an- other atom j in its ground state [shown in Fig. 1(a)]. The resulting dynamical equations are exactly equivalent to N classical polarizable dipoles interacting via their radi- ated fields. In particular, it is well known that ordered arrays of dielectric particles can support lossless guided modes [41–43]. Within the context of infinite atomic arrays, waveguiding manifests itself in the form of per- fectly “subradiant” single-excitation states with zero de- cay rate [24, 32], a key idea underlying the previously proposed phenomena. In reality, though, most atoms display a rich hyperfine structure – which arises from the coupling between the total electron and nuclear angular momenta – and have more than one ground state. Given the growing body of theoretical and experimental liter- ature about atomic arrays, it is critical to understand the underlying physics of collective optical phenomena for atoms with non-trivial internal structure. The complexity introduced by hyperfine structure is il-

lustrated in Fig. 1, where one can see that light-mediated dipole-dipole interactions generally do not allow the atomic dynamics to be confined to a two-level subspace. In particular, even if atoms are initialized in such a sub- space, emitted photons can drive other atoms out of the two level manifold, as photons do not have a uniform po- larization in space. Once an atom is excited out of this subspace, the possibility to decay into unoccupied ground states cannot be suppressed by interference. Thus, even for a single excitation, the mechanism of subradiance, if

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FIG. 1. Illustration of the break-down of the two-level atom picture of dipole-dipole interactions, due to atomic hyperfine structure. (a) Illustration of photon-mediated interactions between two two-level atoms, with unique ground and excited states. (b) Schematic of a 1D array of multi-level atoms extended along the z-direction. The atoms considered have two ground states {|0〉 , |1〉} with Zeeman quantum numbers mg = {−1/2, 1/2}, and four excited states {|2〉 , |3〉 , |4〉 , |5〉}, with quantum numbers me = {−3/2,−1/2, 1/2, 3/2}, respectively. The angular-momentum quantization axis lies parallel to the orientation of the chain. The transitions are coupled by photons of different polarization (depicted by different colors), such that me −mg = {0,±1}, for polarizations {π, σ∓}, respectively. (c) Illustration of the breakdown of a two-level subspace and subradiance. It is assumed that all atoms are initially in ground state mg = −1/2, with the exception of atom i, which decays from a stretched state emitting a photon. While atom i necessarily ends up in state mg = −1/2, the emitted photon does not have a spatially uniform polarization. In particular, in a geometry that is not purely 1D, the emitted photon could drive another atom k out of the stretched two-level subspace (here illustrated by absorption of a σ+ photon). Once atom k is outside the two-level subspace, the excited state can decay into an unoccupied state (illustrated here by emission of a π-photon) at the rate of a single, isolated atom, which is not affected by collective effects.

it exists, could involve some many-body phenomenon. Indeed, the condition for subradiance to exist has al- ready been investigated in the “Dicke” limit [44], where all atoms are located at a single point and thus effectively interact with a single, common electromagnetic mode. Interestingly, it was found that subradiance required a specific entanglement structure within the ground state manifold.

In this manuscript, we tackle the problem of collective effects in extended arrays of atoms with hyperfine struc- ture. In particular, using a generalized “spin model” de- scribing dipole-dipole interactions in the presence of hy- perfine structure, we identify and analyze different classes of subradiant single-excitation states in a 1D atomic ar- ray. We find that the classical waveguiding effect still un- derlies the vast majority of subradiant states. Here, over large spatial regions, atoms in the array essentially live within a two-level subspace, interrupted by local “defect” states or domain walls that divide “phase separated” re- gions. However, we also describe a new, truly many-body mechanism, where waveguiding is enabled through rich, long-range entanglement within the ground-state man- ifold, and we elucidate the necessary conditions for its existence. These results are an important step forward in understanding collective effects in atoms with realistic electronic structure.

I. SPIN MODEL AND MINIMAL TOY ATOM

Here, we introduce a spin model to describe the photon-mediated quantum interactions between atoms. We will consider atoms whose electronic structure con- sists of a ground- and excited-state manifold, with to- tal hyperfine angular momenta quantum numbers Fg and Fe, respectively. A complete basis can be obtained by labeling states according to the projection of an- gular momentum along the z-axis, |Fg/emg/e〉, where mg/e ∈ [−Fg/e, Fg/e] are Zeeman sublevels. We thus can describe the state of an atom by its quantum num- bers, i.e., |Feme〉 if it is excited or |Fgmg〉 if it is in the ground state manifold. The ground and excited states couple to light via well-defined selection rules, such that me = mg + q, with q = {0,±1} denoting the units of angular momentum that can be transferred by a photon. We can define an atomic raising operator that depends on q as

Σ̂†iq = Fg∑

mg=−Fg

Cmg,qσ̂ i Femg−q,Fgmg , (1)

where