Tunable super- and subradiant boundary states in one-dimensional atomic arrays

Efficient manipulation of quantum states is a key step towards applications in quantum information, quantum metrology, and nonlinear optics. Recently, atomic arrays have been shown to be a promising system for exploring topological quantum optics and robust control of quantum states, where the inherent nonlinearity is included through long-range hoppings. Here we show that a one-dimensional atomic array in a periodic magnetic field exhibits characteristic properties associated with an effective two-dimensional Hofstadter-butterfly-like model. Our work points out super- and sub-radiant topological edge states localized at the boundaries of the atomic array despite featuring long-range interactions, and opens an avenue of exploring an interacting quantum optical platform with synthetic dimensions. Efficient manipulation of quantum states is a key step towards applications in quantum information, quantum metrology, and nonlinear optics. In this paper, the authors show that 1D atomic arrays in a periodic magnetic field can realize an effective 2D Hofstadter-butterfly-like model with synthetic dimensions exhibiting super- and sub-radiant topological edge states despite featuring long-range interactions.

A tomic arrays refer to an ensemble of atoms where the interaction of atoms and photons takes place 1 . The lightatom coupling in atomic arrays exhibits fundamental physical phenomena, including facilitating the long-range coherent interactions and promoting the collective radiative loss 2 . Recent advances in assembling highly ordered onedimensional (1D) and two-dimensional (2D) atomic arrays provide unique platforms for exploring the light-matter interaction in quantum optics [3][4][5] . The interference in the emitted optical field leads to remarkable optical properties such as the subradiant state [6][7][8][9] , a high reflection of radiation [10][11][12] , the efficient storage and retrieval for quantum memory 13 , and topologically-protected edge states 14,15 , which show important applications toward quantum information processing, quantum metrology, and nonlinear optics 16,17 .
Topological physics is of fundamental importance where physical characteristics are robust against microscopic variation of system details [18][19][20] . Related phenomena can be explored by engineering the Hamiltonian of an atomic or optical system [21][22][23] . Such approach shows a great potential toward quantum simulation of topological matter. In particular, atomic or optical systems provide a novel fundamental way of manipulating quantum states of the light, such as robust photon transport in photonic systems [24][25][26][27] and non-reciprocal transport in hot atomic gas 28,29 . Recently, it has been shown that atomic arrays hold a promise for studying topological quantum optics, where the inherent nonlinearity brings a natural way to explore the physics including long-range hoppings 14,15,[30][31][32] .
Robust single-photon super-and subradiant states hold a significant promise for applications related to quantum storage and quantum information. In this paper, we investigate 1D atomic arrays subjected to a spatially periodic magnetic field. The spatial phase of the magnetic field is an external parameter, and can be used to map one momentum dimension in a 2D topological system 33,34 . Therefore, 1D atomic arrays with the synthetic momentum dimension manifests important optical features associated with 2D systems. Systems with synthetic dimensions simplify experimental design and enable capabilities of manipulating atomic quantum states or photons along the synthetic dimension [33][34][35][36][37][38] . By changing the periodicity of the magnetic field, we show that the 1D atomic arrays exhibit a butterfly-like spectrum, which has not been discussed in the 2D atomic arrays under a uniform magnetic field 14,30 . Such spectrum, associated with the open quantum optical system involving long-range hoppings along the synthetic dimension, exhibits features which are dramatically distinct from the spectrum in the 1D photonic model [39][40][41] . For a finite 1D atomic array, the system supports pairs of topological boundary states with opposite circular polarizations, exhibiting super-or subradiance dependent on the magnetic field distribution and atomic excitation frequency, which is found to be persevering with the small disorder involved in simulations. The subradiant state localized at the boundary of atomic arrays provides a potential application toward robust quantum storage against small perturbations. The results discussed here show a route toward the possibility of exploring longrange interacting topological physics in quantum optical system with synthetic dimension.

Results
Formalisms. We propose the experimental arrangement consisting of a 1D array of N atoms which are aligned along the y direction with the spacing a. Each atom (labeled by n and located at y n ) has a V-type internal level structure with the ground state g n and excited states ± n j i ¼ Çð x n j i± i y n Þ= ffiffi ffi 2 p , where the transition between g n and ± n j i is coupled with the right (left) circularly polarized light. Here xðyÞ j i refers to the state polarized along the xðyÞ direction. The degeneracy of the excited states is broken by the presence of an external magnetic field B n Bðy n Þ along the z axis (see Fig. 1a).
We consider the dynamics of single-excited atoms coupled to free-space modes of the radiation field. After integrating out radiation modes under the dipole approximation, one obtains the non-Hermitian effective Hamiltonian 12,14,30 where ω 0 ¼ k 0 c ¼ 2πc=λ is the atomic transition frequency with the wave vector k 0 and the wavelength λ, γ 0 is the atomic decay rate in the free space, sgnð ± Þ ± , and μB n gives the Zeeman shift for the nth atom with the magnetic moment μ. G αβ ðy n À y m Þ is the free-space dyadic Green's function describing the electric field at y n emitted by the atom located at y m . By using the Green's Fig. 1 Schematic of a 1D atomic array in a periodic magnetic field. a The atomic array subjected to the external magnetic field B n , where each atom has a V-type atomic level structure with non-degeneracy excited states ± n j isplit by the magnetic field. b The equivalent tight-binding lattice model with on-site potentials ± B n and photon-mediated long-range hoppings. Black solid, red dotted and blue dashed lines label the nearest-neighbor, next-nearest-neighbor and triatomic spacing hoppings, respectively, indicating dipole-dipole interactions and collective dissipations. function in Cartesian basis 7,30,31,42 , one has where r ¼ jy n À y m j.
The atomic system under investigation is an effective tightbinding lattice model (see Fig. 1b). The photon-mediated longrange hoppings amplitude is described by coefficients in the last term of the Hamiltonian, where the real part describes photonmediated dipole-dipole interaction potential between the nth and mth atoms, while the imaginary part denotes the collective dissipative rate of the two atoms.
To construct the spectrum of the non-Hermitian Hamiltonian in Eq. (1), we take the linear combination of single-excited states ψ j i ¼ P n ðC n;þ þ n j iþ C n;À À n j iÞ, where C n; ± is the amplitude of the wave function for the nth atom with the ± polarization. The spectrum can be calculated by using the time-independent where l is a non-zero integer. E ω À iγ=2 is the complex eigenvalue, in which ω denotes the self-energy of the collective atomic excitation and γ is the collective decay rate of the system. We consider a spatially periodic magnetic field where B 0 is the field amplitude, the parameter b controls the periodicity of the magnetic field, and ϕ is the modulation phase. By applying the magnetic field with different spatial shapes along the y direction, one has the control of parameters b and ϕ. Here ϕ provides an additional degree of freedom to our system serving the purpose of the synthetic dimension, so the system can be explored by exploiting the parameter-dependency of the Hamiltonian 33,34 . In such a synthetic space, b gives the effective magnetic flux while ϕ denotes a synthetic momentum dimension 40,43 . Hence we can study the physics associated with an open 2D system with long-range couplings under the effective magnetic flux.
Spectra of the non-Hermitian Hamiltonian. We plot the spectrum projected over the synthetic momentum dimension ϕ while varying b in Fig. 2. The parameters are a ¼ 0:1λ and μB 0 ¼ 10γ 0 14 , and spectrum is computed by following the method by Hofstadter 44 for the atomic array with N ¼ 150 under a periodic boundary condition and also choosing b as rational numbers from 0 to 1. Hoppings in Eq. (2) are long-range but decay as 1=r, we then make truncation in the calculation to get rid of the divergence in the spectrum. One can see the resulting butterfly-like spectrum (ω), which exhibits multiple bulk bands and gaps for each b. As compared with the original Hofstadterbutterfly bandstructure 44 and also the butterfly-like spectrum in the 1D photonic model 39 , the spectrum in Fig. 2 shows the open and close of the bandgap with the difference in subtleties while b is varying even with the presentence of the long-range non-Hermitian couplings in the atomic arrays. Fig. 2 The numerically calculated spectrum. The spectrum ω is projected over the synthetic momentum dimension ϕ, which is the modulation phase, versus the parameter b in the external magnetic field with the spacing a ¼ 0:1λ and the magnetic field amplitude μB 0 ¼ 10γ 0 . ω 0 denotes the atomic transition frequency and γ 0 denotes the atomic decay rate in the free space. Red line corresponds to b ¼ ffiffiffi The parameter b can be externally adjusted by controlling the magnetic field. Once it is irrational, the effective magnetic flux is incommensurate with the lattice and the system exhibits a quasicrystal structure 40 . We set b ¼ ffiffi ffi 5 p =10, indicated by the red line in Fig. 2. In Fig. 3a, the spectrum of the lattice is plotted under an open boundary condition with N ¼ 101 against the modulation phase ϕ in the external magnetic field. As a consequence of the presence of the magnetic field which breaks the time-reversal symmetry of the system, one can see that there is a fractal set of band gaps, and, inside each gap, it exhibits pairs of topological boundary states. Moreover, the striking characteristic of this structure is that the collective decay rate (γ) is changing for different ω and ϕ, covering the range from 0 to $ 7:5γ 0 . The destructive interference in the atom-photon interaction leads to suppressed radiative loss for certain states, corresponding to subradiant states with decay rate below the single-atom emission rate γ 0 . In particular, the collective decay rates γ for boundary states in two larger gaps show different physical features. The boundary states inside the upper gap has γ > γ 0 , corresponding to superradiant modes with enhanced collective emission, while the boundary states inside the lower gap are subradiant because γ is smaller than γ 0 . Moreover, the collective decay rate is also changing along each boundary state when one varies the parameter ϕ, as shown in Fig. 3b, c. In particular, for the subradiant boundary states in the lower gap, as indicated in Fig. 3c, γ changes from $ 0:6γ 0 to $ 0:1γ 0 , showing a significant suppression of the spontaneous emission. Furthermore, the lifetimes of boundary states are influenced by the choice of the parameter b. For instance, for the case b ¼ ð ffiffi ffi 3 p À 1Þ=2, the boundary states inside the two large gaps are both subradiant (see Supplementary Note 1).
The aforementioned boundary states are localized on the left or right boundary of the lattice with a combination of þ j i and À j i excited states. As an example, in Fig. 4, we plot the intensity distributions of boundary states from the eigenfunction analysis versus the position of the atom n for ± j i excited states labeled by A2 and B2 in Fig. 3a, which correspond to superradiant and subradiant states, respectively. The superradiant boundary state A2 is located mainly at the À j i excited state on the leftmost atom, while a small portion of the intensity is distributed at þ j i on other atoms near the left boundary due to the hoppings between the two excited states on different atoms. We denote the boundary state A2 by LÀ then. Similarly, the subradiant boundary state B2 is located mainly at the þ j i excited state on the leftmost boundary, and hence is labeled by Lþ. The two boundary states corresponding to the same ϕ but different excitation frequency, so one can selectively excite either the superradiant or subradiant boundary states for a given external magnetic field.
Other boundary states labeled by A1, A3, A4 in the upper gap and B1, B3, B4 in the lower gap, as shown in Fig. 3a, give R À (mainly distributed at À j i on the right boundary), R þ (mainly distributed at þ j i on the right boundary), L þ and R þ , R À , L À , respectively. One therefore can selectively prepare a super-or subradiant state with either right or left circular polarization by a choice of ϕ as well as the excitation frequency of the source field, which we will demonstrate in numerical simulations. In each gap, the boundary states located at the same boundary with opposite polarization excitations exhibit slopes of edge states being both either positive or negative, while the boundary states at different boundaries support edge states having slopes with the opposite sign.
Simulations. We further perform numerical simulations with solving the time-dependent Schrödinger equation H ψðtÞ j i¼ id ψðtÞ j i=dt, where ψðtÞ j i¼ P n ðC n;þ ðtÞ þ n j iþ C n;À ðtÞ À n j iÞ and H is defined in Eq. (1). The physical system is chosen to be the same as that explored in Fig. 4, where n ¼ 1; ; 101, b ¼ ffiffi ffi 5 p =10, and ϕ=2π ¼ 0:15. A gaussian-shape source field Ω s ðtÞ is applied to excite both ± j i states on the leftmost atom with a tunable frequency detuning Δω. The Rabi frequency of the source field is chosen to be small enough so that the excitation of the system follows mainly on the collective decay from the system rather than the Rabi oscillation. In Fig. 5a, we show the simulated dynamics of intensity distributions jC n;þ ðtÞj 2 and jC n;À ðtÞj 2 for n from 1 to 20, with Ω s ðtÞ having the temporal envelope in Fig. 5b and Δω=γ 0 ¼ À7:21, which aims to excite the subradiant boundary state B2 in Fig. 4. The simulation result shows that the excitation is localized mostly on the intensity jC 1;þ j 2 , while a small portion of the intensity is extended to jC 3;À j 2 . jC n; ± ðtÞj 2 for n > 20 is also nearly zero. Furthermore, Fig. 5 Simulation results of intensity distributions. a Normalized intensity distributions jC n;þ ðtÞj 2 and jC n;À ðtÞj 2 on the nth atom versus the time t, where the source field is applied on the leftmost atom, having the frequency detuning Δω=γ 0 ¼ À7:21 with the temporal envelope Ω s ðtÞ in b. γ 0 denotes the atomic decay rate in the free space. b Evolutions of jC 1;þ ðtÞj 2 and jC 3;À ðtÞj 2 under a fitting with I 0 ¼ jC 1;þ ðγ 0 t ¼ 9Þj 2 , respectively. c jC n;þ ðtÞj 2 and jC n;À ðtÞj 2 versus t, where the source field has the frequency detuning Δω=γ 0 ¼ 8:15. d Evolutions of jC 1;À ðtÞj 2 and jC 3;þ ðtÞj 2 under a fitting with I 0 ¼ jC 1;À ðγ 0 t ¼ 9Þj 2 , respectively. e-h Simulation results similar to a-d but the location of each atom being shifted at a random number within 5% along the y-axis in the atomic array. Fig. 5b shows evolutions of jC 1;þ ðtÞj 2 and jC 3;À ðtÞj 2 . jC 1;þ ðtÞj 2 is fitted with the decay function e À0:28γ 0 t from γ 0 t ¼ 9, when the source field passes, so the affect from the source excitation is mostly prevented. From the plot, one can see that the tail of jC 1;þ ðtÞj 2 follows the decay at a collective rate 0:28γ 0 . Hence the subradiant boundary state B2 demonstrated in the simulation is consistent with the eigenfunction analysis in Fig. 4. Such selectively prepared subradiant state localized at the boundary of the arrays, is robust against small variations of the system. It therefore shows a potential for the robust quantum storage, which is of great importance for quantum device applications.
We then perform the simulation using Ω s ðtÞ with Δω=γ 0 ¼ 8:15, and plot results in Figs. 5c, d. jC 1;À j 2 is mostly excited, which decays at a rate $ 1:71γ 0 t in the fitting. The simulation result shows the superradiant boundary state A2 in Fig. 4. From this set of simulations, we show that, such super-or subradiant boundary state is localized at the boundary of atomic arrays. Moreover, by tuning the frequency detuning Δω of the source field, one indeed can selectively excite the subradiant state with the right circular polarization or the superradiant state with the left circular polarization.
To show that the proposed system is robust against small disorder, we perform simulations with the same atomic arrays in Figs. 5a-d, but the location of each atom being shifted at a random number within δ ¼ 5% along the y-axis. The corresponding simulation results are plotted in Figs. 5e-h, where one can see that the features of subradiant/superradiant boundary state still exhibit.
The excitation of boundary states depends on the choice of Δω. For the same system in Figs. 5a-d, we can also excite the bulk state by choosing Δω=γ 0 ¼ À10, where the intensities of the excitation jC n; ± ðtÞj 2 evolve toward the middle of the atomic array. On the other hand, further increase of disorder can destroy the subradiant/superradiant boundary state. We find that, for disorder with δ ¼ 20%, the features in the excitation dynamics corresponding to subradiant/superradiant phenomena from couplings between atoms in the entire array no longer persist (see Supplementary Note 2). Moreover, not only pulsed source can excite the desired boundary states, we notice that the continuous-wave (CW) laser can also be used to prepare such boundary states.

Discussion
The proposed system is experimentally feasible. For example, an atomic array with the subwavelength-scale lattice spacing can be realized by using bosonic strontium 45,46 . The transition between triplet states 3 P 0 and 3 D 1 of atom 84 Sr gives emission at the wavelength λ ¼ 2:6 μm. One can use the optical lattice formed by lasers at 412.8 nm to trap the atoms, which achieves a subwavelength lattice spacing a ¼ 206:4 nm, i.e., a=λ % 0:08 45 . Inhomogeneous magnetic field is widely used to produce spinorbit couplings in the condensed matter systems 47,48 . The magnetic field in Eq. (4) can be implemented by a variety of experimental technologies which have been proposed to construct magnetic lattices [49][50][51][52][53] . One can use either a CW or pulsed laser at a frequency resonant with the boundary state L ± (R ± ) inside the bandgap to excite the ± j i state of the atom located at the left (right) boundary. The emission of such super-or subradiant boundary state is localized at the boundary of atomic arrays with the enhanced or suppressed collective decay rate.
In summary, we have investigated 1D atomic arrays subjected to the spatially periodic magnetic field, which supports the non-Hermitian lattice model with long-range hoppings. The phase in the magnetic field serves as an external parameter, which gives a synthetic momentum dimension. In atomic arrays with longrange hoppings, the existence of topological edge states is not generally valid 15 . In the open system proposed here, we consider a synthetic space including one spatial dimension and one synthetic momentum dimension. By carefully selecting parameters, we show that this synthetic space exhibits pairs of topological boundary states, which holds fundamentally different physics from the 1D atomic arrays with a non-zero Zak phase 31 . These boundary states are localized at the boundary of atomic arrays and exhibit super-or subradiance with right or left circular polarization, which can be robust against small disorder in simulations. Our results show potential applications toward manipulating atomic emission at the single-photon level in atomic arrays with the concept of synthetic dimensions, which is important for the quantum memory and quantum information, and also leads potential implications for quantum sensors 54,55 and super-resolution spectroscopy 56 by using the robust singlephoton superradiant states. The study of quantum optics with synthetic dimensions opens a route of possibly exploring topological phenomena in versatile higher-dimensional strong-interacting open quantum systems in the future.

Data availability
The data that support the findings of this study are available from the corresponding author on request.