Optical waveguiding by atomic entanglement in multilevel atom arrays

Here, we introduce a spin model to describe the photon-mediated quantum interactions between atoms. We consider atoms whose electronic structure consists of a ground- and excited-state manifold, with total hyperfine angular momenta quantum numbers $F_g$ and $F_e$, respectively. A complete basis can be obtained by labeling states according to the projection of angular momentum along the $z$ axis, $\left|F_{g/e}\hspace{0.17em}{m}_{g/e}\right.⟩$, where $m_{g/e}\in \left[-F_{g/e},F_{g/e}\right]$ are Zeeman sublevels. We thus can describe the state of an atom by its quantum numbers, i.e., $\left|F_e\hspace{0.17em}{m}_e\right.⟩$ if it is excited or $\left|F_g\hspace{0.17em}{m}_g\right.⟩$ if it is in the ground-state manifold. The ground and excited states couple to light via well-defined selection rules, such that $m_e=m_g+q$, with $q=\left\{0,\pm 1\right\}$ denoting the units of angular momentum that can be transferred by a photon. We can define an atomic raising operator that depends on $q$ aswhere ${\widehat{\sigma }}_{F_e{m}_g-q,F_g{m}_g}^i={\left|F_e\hspace{0.17em}{m}_g-q\right.⟩}_i{\left.⟨F_g\hspace{0.17em}{m}_g\right|}_i$ is the atomic coherence operator between the ground and excited states of atom $i$. This operator conveniently groups all of the possible transitions which transfer $q$ units of angular momentum along $z$. These are weighted by the Clebsch–Gordan coefficientswritten here in terms of a Wigner 3j symbol, which reflect the different strengths that the possible transitions couple to light of a given polarization.

In each subspace $F_z$, it is first helpful to consider the full set of possible basis states. For example, within the single-excitation manifold, the maximum allowed value of $F_z$ is given by $F_z^{\text{max}}=\pm |F_e+\left(N-1\right)F_g|=\pm \left(N/2+1\right)$, which is achieved when one atom is excited in the state $\left|2\right.⟩$, with $m_e=-3/2$, while all other atoms are in the ground state $|0\rangle$, with $m_g=-1/2$. These states are connected through a ${\sigma }^{-}$ transition (that is, $q=1$). Since ${\mathcal{H}}_{\text{eff}}$ is block diagonalizable, $\left|F_z\right|=F_z^{\text{max}}$ thus corresponds to an effective 2-level atom subspace.

Numerically, we diagonalize ${\mathcal{H}}_{\text{eff}}$ within this subspace as a function of atom number $N$ and for selected different lattice constants $d$. For each $N$ and $d$, we then find the eigenstate with the minimum decay rate $\mathrm{\Gamma }$, which we plot in Fig. 3A. In agreement with our previous results (25), we find subradiant states if the interatomic distance $d$ is such that $d<{\lambda }_0/2$, ${\lambda }_0$ being the wavelength of the resonant transition. The decay rate of the most subradiant eigenstate scales as $\mathrm{\Gamma }/{\mathrm{\Gamma }}_0\sim 1/N^3$, as also found previously for 2-level atoms. We calculate the field intensity radiated by the most subradiant eigenstate $\left|\psi \right.⟩$ by numerically finding the expectation value of the intensity operator $I\left(\mathbf{\text{r}}\right)=⟨\psi |{\widehat{\mathbf{\text{E}}}}^{-}\left(\mathbf{\text{r}}\right)\cdot {\widehat{\mathbf{\text{E}}}}^{+}\left(\mathbf{\text{r}}\right)|\psi ⟩$, according to Eq. 7. The intensity pattern in Fig. 3B demonstrates strong emitted intensity at the ends of the chain and absence of intensity in the middle, as would be expected for a finite-size waveguide.

As previously remarked, the guiding effect is entirely classical. Because the ground state is unique and there is only one excitation in the system (i.e., $N$ degrees of freedom), one can alternatively deduce this state by considering $N$ coupled harmonic oscillators. In that case, the stationary states under full master equation evolution are coherent states $\left|{\psi }_{\xi }^{ho}\right.⟩=\left|{\alpha }_1{e}^{-i{\omega }_{\xi }t}\right.⟩\left|{\alpha }_2{e}^{-i{\omega }_{\xi }t}\right.⟩\dots \left|{\alpha }_N{e}^{-i{\omega }_{\xi }t}\right.⟩$. Here, the coherent-state amplitude ${\alpha }_i\propto c_i$ of harmonic oscillator $i$ is proportional to the spin wavefunction amplitude of the corresponding single-excitation spin eigenstate $\left|{\psi }_{\xi }\right.⟩={\sum }_i{c}_i{\sigma }_{20}^i{\left|0\right.⟩}^{\otimes N}$ found earlier. These coherent-state amplitudes evolve in time as $e^{-i{\omega }_{\xi }t}$ with a frequency ${\omega }_{\xi }$ corresponding to the spin-state eigenvalue. These harmonic oscillator states physically describe classical resonant dipole arrays, interacting with each others’ radiated fields. In particular, it is well-known that an infinite array can support guided modes. Starting from such states, we can apply an operator $\mathcal{P}$ that projects these states into the single-excitation manifold, which then exactly reproduces the single-excitation spin eigenstates $\left|{\psi }_{\xi }\right.⟩$. This implies that the resulting physics can be understood purely classically—even if the spin eigenstates themselves are technically entangled—as the entanglement simply arises from the projection at the end. As we will see later on, there are entangled states that do not support any analogy with those of classical dipoles.

The waveguiding concept can be additionally confirmed by considering an infinite chain. In this case, we can diagonalize the Hamiltonian using spin waves with well-defined momentum $k$ along $z$ and readily find ${\mathcal{H}}_{\text{eff}}={\sum }_{q=-1}^1{\tilde{\mathcal{H}}}_{\text{q}}$, wherewith ${Ŝ}_{k,q}^{†}=N^{-1/2}{\sum }_j{e}^{\mathrm{\text{i}}kdj}{\widehat{\mathrm{\Sigma }}}_{jq}^{†}$ andIn Eqs. 10a and 10b, $\widetilde{\mathbf{\text{G}}}\left(k\right)={\sum }_j{e}^{-\mathrm{\text{i}}kdj}\mathbf{\text{G}}\left(z_j\right)$ is the discrete Fourier transform of the free-space Green’s tensor. Eigenstates in the single-excitation manifold can be generated by applying a spin raising operator $S_{k,1}^{†}$ to the product ground state ${\left|0\right.⟩}^{\otimes N}$. For $k>{\omega }_0/c$—such that the spin wavevector exceeds the wavevector of free-space radiation—one finds that ${\mathrm{\Gamma }}_{k,q}=0$, indicating the decoupling of the spin wave from radiation fields and thus the guided nature of these excitations. We note that the ability to restrict dynamics to a 2-level subspace is unique to a 1D chain, as in higher dimensions the Hamiltonian of Eq. 6 is not restricted to $q=q\prime$. This motivates the deeper investigation of subradiance in subspaces of other total $F_z$, to find a true many-body mechanism.

Reducing the angular momentum projection by one unit increases the complexity of the problem. This is manifest in the larger size of the Hilbert subspace, of dimension $dim\left(\mathcal{H}\right)=N^2$. The basis states are of 2 types: 1) defect states where one atom is excited in $\left|2\right.⟩$ ($m_e=-3/2$), $N-2$ atoms are in $\left|0\right.⟩$ ($m_g=-1/2$) (reminiscent of the 2-level subspace of the previous subsection), and one defect atom is in the ground state $|1\rangle$ ($m_g=1/2$) and 2) states where all atoms except one are in the ground state $\left|0\right.⟩$, and the excited state involves level $|3\rangle$ ($m_e=-1/2$), a state without maximal angular momentum projection. From the considerations of the section Spin Model for Multilevel Atoms, it is already clear that any highly subradiant states must exhibit the following features. First, because the basis states containing excited-state $\left|3\right.⟩$ have an allowed transition (via a $q=1$ photon) to state $|1\rangle$, which is unoccupied, such states will decay into $|1\rangle$ at the full strength ${\mathrm{\Gamma }}_0/3$ of a single, isolated atom. Thus, these basis states must constitute a vanishingly small weight of a subradiant state. Second, for basis states composed of a single atom $j$ in state $|1\rangle$, one can conclude that coherent interactions with an atom $i$ in state $|2\rangle$ allow for an exchange $\left|2_i\hspace{0.17em}{1}_j\right.⟩\to \left|0_i\hspace{0.17em}{3}_j\right.⟩$, while we argued already that state $|3\rangle$ is undesirable. Thus, a subradiant state must find some mechanism to “hide” the atom in state $|1\rangle$ from the dynamics.

This intuition agrees with our numerical findings. For the most subradiant eigenstates, the population of state $\left|3\right.⟩$ decreases as ${\sum }_j⟨{\widehat{\sigma }}_{33}^j⟩\sim 1/N^3$; i.e., in the thermodynamic limit the most subradiant eigenstate is mostly formed by defect states and is of the formFig. 4A shows the spatial profile of the population in levels $\left|1\right.⟩$ and $\left|2\right.⟩$ of the most subradiant eigenstate of a chain of $N=30$ atoms. The population in the excited state $\left|2\right.⟩$ is spatially smooth, as it is distributed among all atoms. On the other hand, the population in the defect level $\left|1\right.⟩$ is localized in a superposition between both edges of the chain (in particular, the defect atom is the third one, counting from both edges). The population of level $\left|3\right.⟩$ is spatially correlated with that of $\left|1\right.⟩$, as $⟨{\sigma }_{33}^j⟩\propto ⟨E^{-}\left({\mathbf{\text{r}}}_j\right)E^{+}\left({\mathbf{\text{r}}}_j\right)⟩⟨{\sigma }_{11}^j⟩$ (SI Appendix). The population in this level is also localized around the third atom, but negligible in the scale of Fig. 4A. As shown in Fig. 4B, the decay rate of the most subradiant state with atom number scales as $\mathrm{\Gamma }/{\mathrm{\Gamma }}_0\sim 1/N^3$. Fig. 4C shows the scaling of the field intensity at the position of the central atom of the chain versus that at the defect atom. The latter scales as $I\sim 1/N^3$. This confirms our previous intuition that the defect atom is efficiently “hidden” from the rest of the chain. In particular, the low intensity seen by the defect atom ensures that it cannot be efficiently excited to level $\left|3\right.⟩$ and that the large decay rate of this excited state does not destroy subradiance. The precise spatial location of the defect atom depends on microscopic details and changes depending on the length of the chain (it appears just at the edge for short chains and in atoms farther away from the boundaries for longer chains), but the precise location does not qualitatively alter the physics (SI Appendix).

Furthermore, when $F_z=\left|F_z^{max}\right|-n_d$ is further reduced (with $n_d\ll \left|F_z^{max}\right|$), the most subradiant states are of similar character, with long chains of “2-level” atoms interrupted by $n_d$ defects, whose potentially detrimental effects are reduced as they are largely decoupled (SI Appendix). Importantly, the character of such states, dominated by the waveguiding of effective 2-level atoms, does not yet point to any nontrivial mechanism for subradiance.

For an even number of atoms, there is a subspace with zero angular-momentum projection. This manifold is the largest one, with dimension that scales exponentially with atom number. In this manifold, we have approximately the same number of atoms in each of the degenerate ground states, allowing for many different combinations. We are able to find 2 different kinds of subradiant states: phase-separated states and symmetric states. In the following, we study these 2 families in detail.

To summarize, we have shown that arrays of atoms with hyperfine structure can support highly subradiant, waveguiding states, and we have elucidated the conditions for their existence. In contrast to the classical effect that occurs in atoms with a single ground state, here, the waveguiding is fundamentally enabled by rich, many-body correlations within the ground-state manifold.

Having shown the existence of true waveguiding states, one interesting question going forward is how novel phenomena or applications, previously identified for simple atoms, can be encoded into these highly entangled states. For example, it would be interesting to investigate whether such states support more powerful quantum memory protocols or whether generalizations of such states exist in higher dimensions, such as to support topological edge states. In the case of higher dimensions, while numerics will likely be highly challenging, a promising approach could be the generalization of toy models such as Eq. 13. It would also be interesting to investigate multiexcitation subradiant states, e.g., to see whether they exhibit the same “fermionic” correlations as multiexcitation states in atoms with simple (2-level) structure (25).

An important associated question is how the necessary many-body entanglement in the ground-state manifold can be generated in the first place. We speculate that the same correlated dissipation processes encoded in the dipole–dipole interactions of Eq. 3 might be used to generate the necessary entanglement, such as through the steady state obtained under constant driving (i.e., correlated optical pumping).

More generally, the insight developed in this work could give rise to broader opportunities. For example, under constant driving, one could investigate whether correlated dissipation can give rise to useful many-body correlations within the ground-state manifold, such as for quantum-enhanced metrology (8, 9, 59–61). It would also be interesting to more systematically understand the forms and range of entanglement that can arise, as the Green’s function is varied (e.g., through a dielectric structure) and/or the atomic level structure is altered. Finally, as an optical phenomenon, it is intriguing that we have identified a mechanism for waveguiding that explicitly relies on entanglement. It would be interesting to more broadly search for additional mechanisms of waveguiding and other optical effects (62), which can emerge only through quantum correlations.

A.A.-G.’s work was supported as part of Programmable Quantum Materials, an Energy Frontier Research Center funded by the US Department of Energy, Office of Science, Basic Energy Sciences, under Award DE-SC0019443. D.E.C. acknowledges support from Fundación Ramon Areces, Fundacio Privada Cellex, Spanish Ministry of Economy and Competitiveness, through the Severo Ochoa Programme SEV-2015-0522; Plan Nacional Grant “Atom-light interactions as a quantum spin model (ALIQS),” funded by the Ministry of Science, Innovation and Universities, State Research Agency and European Regional Development Fund; Centres de Recerca de Catalunya (CERCA) Programme/Generalitat de Catalunya, European Research Council Starting Grant "Frontiers of Quantum Atom-Light Interactions," and Agència de Gestió d’Ajuts Universitaris i de Recerca Grant 2017 SGR 1334. H.J.K. acknowledges funding from the Office of Naval Research (ONR) Grant N00014-16-1-2399, the ONR Multidisciplinary University Research Initiative (MURI) Quantum Opto-Mechanics with Atoms and Nanostructured Diamond Grant N00014-15-1-2761, and the Air Force Office of Scientific Research MURI Photonic Quantum Matter Grant FA9550-16-1-0323.