Optimal subradiant spin wave exchange in dipole-coupled atomic ring arrays

The subwavelength array of quantum emitters provides an ideal platform for exploring rich many-body dynamics, such as super- and subradiance. In this paper, we explore the dynamics of spin wave exchange between two dipole-coupled atomic ring arrays. Subradiant spin waves lead to low-loss and high efficiency of ring-to-ring transfer. The optimal subradiant spin wave exchange occurs at appropriate separations between coplanar rings, despite the fact that the energy transfer efficiency is monotonically enhanced (in the regime ⩽λ0/2 ) as the rings’ separation decreases. However, the spin wave will scatter due to the dephasing mechanism of close-by atom pairs, as the separation of two rings is too small. With the increase in the number of atoms on the ring, the subradiant shielding effect also strengthens, leading to a shorter distance for the transfer of spin waves. We investigate the rotation of one of the rings and find that the optimal spin wave exchange corresponds to the scenario where the line connecting the two nearest atoms of the two rings aligns with the center of the circle. Moreover, we study the influence of transition dipole moment orientations on the effective interaction between two atomic rings. We observe that there is a critical point where the effective interaction strength changes dramatically owing to the cooperative effect of the subwavelength atomic array. We believe that our results could be important for quantum information processing based on atomic arrays.

By utilizing a closed configuration with no open ends, this shortcoming can be effectively overcome.The subradiant modes of the circular ring configuration can be interpreted as 'whispering gallery modes', in which excitations only radiate via the finite radius of curvature [34].
For ring atomic arrays, the efficiency of ring-to-ring energy transfer is enhanced by the physical properties of long-lived subradiance [45,46], which features the shielding effect from electromagnetic environments with an almost zero decay rate.The coupling efficiency of subradiant spin waves increases as we reduce the separation between atomic rings.However, the near-field effects of closed-by atom pairs are introduced as the separation between atomic arrays decreases [47][48][49].As tunable parameters of atomic arrays, the atomic transition dipole moments can lead to scattering of the spin wave eigenmodes when they differ between atoms on the two rings.Thus, the configuration with optimal spin wave exchange is worth exploring.
In this paper, we explore the exchange dynamics of subradiant spin waves between two equally-sized atomic rings in the single-excitation regime.We first introduce the theoretical formalism known as spin model, which describes the full quantum interaction between atoms and radiation filed by utilizing electromagnetic Green's function.By numerically diagonalizing effective non-Hermitian Hamiltonian, we obtain the eigenmodes of an individual ring and the effective coherent or dissipative interactions between two rings.Then, we investigate the dynamics for one of the atomic rings at the most subradiant state and another at its ground state with different separations between two rings.We observe different phenomena induced by subradiant feature and near-field interaction, i.e. subradiant shielding effect, optimal spin wave exchange, and the dephasing mechanism of close-by atom pairs, respectively.Furthermore, we explored the case of an increased number of atoms on the ring, and observed that the strengthening subradiant shielding effect requires spin wave transfer at closer separation between two rings.Additionally, in the case of imbalance atom numbers, mode mismatch makes it challenging for excitation to transfer to the second ring.Next, we considered the scenario of rotating one of the rings, where specific rotation angles lead to destructive interference.Moreover, the effective coherent coupling between the two rings exhibits a sharp variation behavior as the dipole orientation of one of the rings is rotated.We demonstrate that this sharp variation is due to the cooperative effect of collective dipole-dipole interactions by providing the examples of two atoms coupling.
This paper is organized as follows: in section 2, we introduce the model of two atomic ring arrays and the theoretical formalism for all-to-all atomic interactions known as spin model.Next, we investigate the spin wave exchange dynamics between two rings with different configurations in section 3. The experimental feasibility of our scheme is discussed in section 4. Conclusion and further discussion are given in section 5.

Model
We start with consideration of two coplanar ring structure system consisting of N i (i = 1, 2) identical two-level atoms with an excited state |e⟩ and a ground state |g⟩ (see figure 1), separated in a narrow optical resonance frequency by ω 0 = k 0 c = 2π c/λ 0 , and the spontaneous emission of the excited state with a rate Γ 0 = |µ 0 | 2 ω 3 0 /(3πhc 3 ).Each atom is fixed at positions r j with given atomic transition dipole moment μj (j = 1, 2, . . ., N).Here, we assume that the numbers of atoms on the left ring and right ring are N 1 and N 2 , respectively.The inherent coupling between each pair of individual atoms in free space arises from effective dipole-dipole interactions, which are induced by exchanging virtual photons [34].
Under the Born-Markov approximation, the dipole dynamics is obtained from a master equation where ρ is the atomic density matrix and the effective Hamiltonian (h = 1) reads [50] where σ − j = |g j ⟩⟨e j | is the atomic lowering operator of the jth atom.The Lindblad operator accounting for the collective decay is given as the coherent (J ij ) and dissipative (Γ ij ) interaction strengths between atoms i and j are given by The minimum spacing between the two rings is denoted as a.Each atom has a transition dipole moment μj , whose polar angle is θ j and azimuthal angle is ϕ j .We assume that the dipole orientation of the atoms in each ring is identical.
where G 0 (r i − r j , ω 0 ) is the free-space Green's tensor [51], which is physical interpreted as the propagator of the electromagnetic field between r i and r j [34].Since the atom is assumed as a point-like radiant source, the form of Green's tensor in free-space can be written as where r = |r| denotes the distance between atoms.For an individual atom, the spontaneous emission is evaluated by equation ( 2) with Γ ii = Γ 0 .

Exchange of subradiant spin wave
In the single-excitation regime, the delocalized eigenmode of a collection of N atoms is the 'timed Dicke state' [31,32], also known as spin wave, in which unique cooperative effects are critical to quantum optics and quantum information processing.In the subwavelength atomic array, photon storage and retrieval is achieved by switching spin wave modes between subradiant and selected-radiant modes [34].Moreover, the spin wave acts as a steady propagating mode with minimal radiative loss and has been studied in 1D [52,53] or ring-like [41] atomic array.For an atomic ring, due to its symmetry configuration and the limitation of the single-excitation manifold, collective eigenmodes are obtained as perfect spin waves by analytically diagonalizing its effective Hamiltonian [45,46].From the discussion in appendix, it is apparent that the most subradiant spin waves lead to the greatest transfer efficiency [45,46].Indeed, the feature of subradiant physics substantially inhibits the dissipation of excitation into free space, and the excitation transfer is possible due to the coupling of the collective spin waves in the subwavelength regime.In this section, we focus on the exchange of subradiant spin waves between the two rings with different configurations.

Spin wave exchange with different separations and spatial disorders
Physically, the subradiant ring-to-ring spin wave transfer is due to the local near-field dipole-dipole interactions.In addition, the subradiant state is naturally decoupled from the vacuum's electromagnetic modes, it is intuitively that the two rings will decouple as the separation increases.Similarly, in the case where the two rings are extremely close to each other, we investigate whether near-field effects affect the transfer of spin waves.
In this subsection, we consider two equally sized (N 1 = N 2 = 20, R 1 = R 2 = 0.3λ 0 ) atomic rings separated by the variable distances a, as is shown in figure 1, and the polarization of all atoms is μ = ẑ orientated.Without loss of generality, we assume that one of the rings is initially in the most subradiant state and another is in the ground state: The absolute value of coupling strength Jm 11 ,m21 (logarithmic scale) as a function of the separation a between two rings (see also in [45,46]).The insets show the evolution of population for a separation with a = 0.05λ0 and a = 0.25λ0, respectively.
where |e j ⟩ is the excited state of the jth atom and |G⟩ represents the second ring's ground state, and the symbol ⊗ represents the tensor product.In figure 2(a), we show the evolution of the population for each ring with a fixed separate distance a = 0.15λ 0 .Here, the excitation's loss of system is defined as We observe the Rabi-like oscillation of excitation's population, which means the spin wave is completely and repeatedly transferred between two rings.By utilizing optical control techniques [54,55] to rapidly separate or bring together two atomic arrays, an on/off switch is realized for the transfer of spin waves.Intuitively, the spin wave will scatter if it is not an eigenmode of an atomic array.In this configuration, the subradiant spin wave exchange between two rings without loss because their eigenmodes are identical.
In general, the effective transfer of excitation between two systems depends on the coupling strength, detuning, and decay rate.The coherent and dissipative coupling between two rings is defined by the complex eigenvalues λ m 1i ,m 2j [45,46], where J m 1i ,m 2j = Re{λ m 1i ,m 2j } and Γ m 1i ,m 2j = −2Im{λ m 1i ,m 2j } represent the coherent and dissipative coupling between the m i th eigenmode of the first ring and the m j th eigenmode of the second ring (see appendix), respectively.The m j eigenmode with j = 1, 2, • • • , N is sorted in ascending order according to the decay rates of each individual ring.Throughout this paper, we choose the global phase between two eigenstates m 1i and m 2j to be zero, i.e. the phase difference between the coefficient wavefunctions c i,m11 and c j,m21 is zero.To quantify how efficiently an excitation can be transfered between the two rings, the ratio is defined as [45,46] where The effective interaction between the equally sized rings with a subradiant initial state, the decay Γ and detuning ∆ can be neglected in equation (6).Thus, the exchange of spin wave in this case is quantified by the strength of coherent coupling J m11,m21 (see appendix), where the subscript m ij represents the ith ring's jth eigenmode.In figure 2(b), we plot the absolute value of coupling J m11,m21 as a function of the separation a.A significant reduction in the coupling strength is observed as the separation between the two rings increases, which is due to the feature of subradiant state with exponential suppression of single ring's decay rate [34].
The insets in figure 2(b) show the evolution of population of the considered setup, for the separation a = 0.25λ 0 .There is no excitation transfer due to the subradiant shielding effect, i.e. once the emitter system exceeds the effective coupling distance, it cannot interact with the subradiant emitter system.A smaller separation leads to a larger coupling strength J m11,m21 .However, we observe rapid oscillations in the exchange of spin wave for a = 0.05λ 0 , and an increasing loss of excitation P loss , which is attributed to the strong near-field interaction that scales with 1/r 3 .(See equation ( 4)).The close-by neighbors of atoms exhibit the induced inhomogeneous broadening [47][48][49] (i.e.undesirable dephasing mechanism) which leads to part of spin wave excitation being lost.To achieve optimal spin wave exchange, the appropriate physical spacing between the two rings is required, which ensures the coupling strength is large enough and no dephasing mechanism is introduced.
The atomic positions would fluctuate in any subwavelength array experimental realizations.Here, we consider the same configuration in figure 2(a) and introduce the disorder by displacing atoms randomly from their positions.In figure 3, we present the excitation population of the two rings as a function of time, considering various levels of disorder with a maximum deviation of δl.For the weak disorder with δl = 0.01d, as shown in figure 3(a), the exchange of spin wave is almost unaffected compared with the perfect lattice situation.However, we observed a slight enhancement in the Rabi frequency due to the disorder.In figure 3(b), with the disorder of δl = 0.02d, the spin wave exchange becomes incomplete.As shown in figure 3(c), with an increase in the maximum deviation of disorder to δl = 0.05d, the excitation of the exchange between the two rings weakens, but the obvious exchange can still be observed.For the strong disorder with δl = 0.10d, as illustrated in figure 3(d), the exchange dynamics of spin waves are disrupted by the disorder.The excitation leakage is induced by disorder, which leads the system to an imperfect dark state.

Number variation of atoms and a rotated ring
In the single-excitation regime, the decay rate Γ m1 of the most subradiant state in an individual atomic rings is exponential suppressed versus the atom number N [34], i.e.Γ m1 ∼ exp(−N).Under a fixed spatial size, as the number of atoms increases, the subradiant shielding effect gradually isolates the atomic system from electromagnetic modes in the near-field region.Thus, the optimal separation for spin wave exchange between two rings decreases as the number of atoms increases.In this subsection, we explore the influence of other geometric parameters of the two rings on the exchange of the most subradiant spin wave.
To begin with, we investigate the exchange of the most subradiant spin wave as a function of the atom number in two rings by assuming N 1 = N 2 .In figure 4(a), we plot the efficiency of the most subradiant spin wave exchange as functions of the number of atoms and the separations between two rings.The efficiency η m11,m21 is defined by equation (6).It is evident that, for any number of atoms, increasing the separation between the two rings reduces the efficiency of effective excitation transfer.Furthermore, the subradiant system will decouple from the external system when the separation between the two systems exceeds the effective coupling distance, i.e. the long range interaction is effectively shielded.We observe that the dark blue region in figure 4(a) expands towards smaller separations a as the atom number N increases.This trend indicates a decrease in the effective coupling distance.In figure 4(b), at a constant separation of a = 0.15λ 0 , the efficiency of excitation transfer tends to decrease after it surpassing the maximum efficiency at N = 24.In figure 4(c), we present the population dynamics of the two rings and the total system loss for the scenario where N 1 = N 2 = 50 and a = 0.15λ 0 .In this case, the lossless exchange of the most subradiant spin wave exhibits a smaller Rabi frequency compared to the scenario depicted in figure 2(a).Hence, an increase in the number of atoms in the ring necessitates a smaller separation to observe a more rapid Rabi oscillation of the spin wave exchange.Our discussion has focused on scenarios where the number of atoms in both rings is equal, and we will consider an imbalance case in the following discussion.
Moreover, the spin wave can smoothly transfer between identical atomic rings; however, the situation with the imbalanced atom numbers between two rings still not be considered in our discussion.Here, we consider the efficiency of excitation transfer with the different atom number between two rings.In figure 5, we depict the efficiency as it varies with the atom numbers N 2 , while the other ring maintains a fixed atom number of N 1 = 20.However, the efficiency is exceedingly low except when the atom numbers are equal, i.e.N 1 = N 2 = 20.Thus, achieving subradiant spin wave exchange requires an equal number of atoms in the two atomic rings.
In addition, we examine the scenario where one of the rings is rotated by a small angle around the central axis of its circle.The angle between two adjacent atoms is defined as ϕ N max = 2π/N, as illustrated in figure 6(a), and we denote the rotation angle as ϕ N rot .In the case of an even-numbered atomic ring, the most subradiant mode corresponds to a checkerboard pattern, where any two neighboring atoms exhibit a phase difference of π, forming a subradiant state involving two atoms.However, in the case of an odd-numbered atomic ring, the most subradiant state does not exhibit a checkerboard pattern.This lack of a checkerboard pattern results in the subradiant state on the ring lacking rotational symmetry.Therefore, we will exclusively focus on the rotation of even-numbered atomic rings in the following discussion.In figure 6(b), we plot the absolute value of the coherent coupling strength |J m11,m ′ 21 | as a function of the rotation angle ϕ N rot for various atom numbers.For each even-numbered ring scenario, the minimum coupling value occurs at ϕ N rot = ϕ N max /2; this symmetry configuration introduces a form of destructive interference that acts as an obstacle to the transfer of excitation between the two rings.In figures 6(c) and (d), we investigate the dynamics of spin wave transfer for the configuration in figure 2(a), where one of the rings is rotated.We use the notation 'rp' to represent the rotated ratio ϕ N rot /ϕ N max .It can be observed that spin waves can still exchange without loss when a ring is rotated by a small angle.In this scenario, the line connecting the two nearest atoms of the rotated ring does not pass through the center of the ring.This leads to a partial breaking of translational symmetry, resulting in a reduction of the Rabi frequency of spin wave exchange.However, at the rotated angle ϕ N rot /ϕ N max = 1/2, as shown in figure 6(d), strong destructive interference prevents the excitation from transferring to the second ring.
For atomic arrays, in addition to their customizable geometric configurations, the intrinsic transition dipole moments of atoms can also be tuned.The orientation of atomic transition dipole moments also influences the coupling strength between atoms.The subsequent discussion revolves around the impact of different transition dipole moments on spin wave exchange.

Critical point of transition dipole orientation
In this subsection, we consider the configuration in which the first atomic ring has a specific dipole orientation, while the second ring has a fixed transition dipole moment in the μ = ẑ direction.To illustrate the orientation of atomic transition dipole, we introduce spherical coordinates as is shown in figure 1.For an individual atom the transition dipole vector is parametrized as μj = {θ j , ϕ j }, where the polar angle θ j and the azimuthal angle ϕ j represent the angle between the dipole vector and the ring plane, and the angle between vector êx and projection vector in atomic ring plane, respectively.For simplicity, the dipole moment vector is denoted as μ = {θ, ϕ} since we assume that all of the dipole moments are identical on the same atomic ring.Generally, the eigenstate of a single atom will not change drastically since the transition dipole orientation is varied adiabatically (i.e.slowly and continuously).However, this continuously changing eigenstate is not widely satisfied in the case of many-body quantum emitters.Similar to the previously considered case, we assume that the excited ring is initially in its most subradiant state, while the other ring is in its ground state.Note that the strength of the coherent interaction between two rings is expressed as J m11,m ′ 21 .We start with rotating the dipole moment of the first ring from the θ = π/2 direction to the μ = {0, π/2} direction, while the second ring maintains a fixed dipole orientation of θ = π/2.In figure 7(a), we plot the coupling J m11,m ′ 21 as a function of the angle θ.It can be seen that a discontinuity acts like a step function at critical point θ c ≈ 0.228π.Here, the critical point is at the same polar angle θ c when the azimuthal angle is ϕ = 0.In the following results, we show that this a sharp variation behavior is due to the cooperative effect of collective dipole-dipole interaction.
To illustrate the discontinuity behavior of the coupling strength J m11,m ′ 21 is induced by cooperative effect of collective dipole-dipole interaction, we consider the simplest model of two atoms with different dipole orientations as shown in figure 7(b).By referring to the size of two rings in our model, we select the separation between two atoms from x = 0.1λ 0 to x = 0.4λ 0 .Intuitively, one can imagine that as the dipole orientation of one atom gradually turns towards the other one, the dipole-dipole interaction between the two atoms would change drastically at a critical point, leading to sharp variation behavior.However, we observed that the strength of the interaction between two atoms varies continuously with dipole orientation for different atomic distances.Thus, the sharp variation behavior of coupling strength is attributed to the collective dipole-dipole interaction between distinct eigenmodes of the rings.
Taking the dipole moment θ = π/2 as a comparison, a slight rotation of the polar angle θ leads to a greater coupling strength of the two rings.Generally, a larger coupling strength results in a more intense and sufficient energy exchange between systems, but it is not entirely true in our study.In figure 7(c), we demonstrate the dynamics of the spin wave excitation of the system for rotated dipole moments of the first The coherent dipole-dipole interaction strength between two atoms with different transition dipole moments.One of the transition dipole moment orientations µ j is perpendicular and another µ i is rotating from parallel to perpendicular.The separation between two atoms is denoted as x.(c) The evolution of populations for the dipole moment of the first ring has a fixed azimuthal angle ϕ = π/2, for different polar angles θ.The initial state of the first ring is at its most subradiant state and the second ring is at the ground state.ring, while other parameters remain the same as those in figure 2(a).For the polar angle is θ = 0.45π, we observe that most of the spin wave excitation is transferred to the second ring and then completely transferred back to the first ring, which has a larger Rabi frequency than the perpendicular dipole situation.However, when the polar angle is θ = 0.40π, the dynamics exhibit small-amplitude fast oscillation, even though the coherent interaction is larger compared with that in θ = 0.45π situation.The difference in configuration between two rings results in a mismatch of their eigenmodes, making it challenging to transfer the subradiant spin wave to the second ring.For θ = 0.20π, the interaction strength is almost negligible due to the strong subradiant cooperative effect, resulting in the two rings being in their steady states.
The Rabi frequency of spin wave exchange increases when the first ring's dipole moments are slightly tilted, but at the expense of a small portion of the spin wave excitation not being transferred to the second ring.As we continue to rotate the dipole polar angle θ, the spin wave exchange disappears due to the mismatch of eigenmodes, although the energy exchange between the rings persists.When the polar angle θ of the dipole moment is smaller than the critical point θ c , the interaction between the rings becomes decoupled due to the cooperative effect of dipole-dipole interactions.

Experimental feasibility
The subwavelength atomic array is an experimental reality, which overcomes the challenge of diffraction limit by utilizing two different atomic transitions.One of the atomic transitions is used for trapping the atom and another is used for driving the optical excitation.The experimental setup mainly consists of neutral atoms trapped in the optical lattice [8,14,54,[56][57][58][59][60] and optical tweezers [61][62][63][64][65][66][67][68].Generally, the optical lattice can only create gridlike arrays.However, by using the technique of spatial light modulators or holographic metasurfaces, the optical tweezers allow positioning atoms arbitrarily.To prepare the atomic array experimentally, the first step is precooling dilute atomic vapor released by chamber.The 87 Rb atoms are captured by a six-arm 3D magneto-optical trap, where the beam is detuned by −18 MHz from the 5S 1/2 (F = 2) → 5P 3/2 (F ′ = 3) hyperfine transition [61].Each optical tweezer has a waist radius of approximately ∼ 1 µm, and via the light-assisted collision process [69] to ensures that only a single atom is trapped.It should be noted that the atoms need to be trapped as tightly as possible to avoid spatial disorder.Besides the tweezers, with the aid of the blue-detuned optical lattices could confine the atoms more tightly and precisely [70].The hyperfine structure of the atom increases the complexity of the experimental reality, and it can be solved by utilizing the bosonic species that lack the hyperfine structure [71,72].
To efficiently prepare the initial spin wave state of an atomic ring, a laser pulse with imprinted phase and suitable temporal duration is required.The phase-mismatched spin wave state, i.e. the subradiant spin wave with the wave vector beyond the light cone, satisfying |k| > ω 0 /c, is obtained by cyclically driving an auxiliary transition with subnanosecond shaped laser pulses [40].Similar to the mechanism of spin wave exchange in our discussion, by focusing the external light into an atomic array in the regime of near-field, the laser pulse can effectively be coupled with it.In addition, the most subradiant modes of ẑ−polarized even number atomic ring is the checkerboard pattern, i.e. the phase difference between nearest-neighbors is π.By controlling the Zeeman level shifts of atoms in the 2D array, subradiance-protected excitation is achieved through the coherently spreading of single-photon excitation into the atomic array [44].Moreover, the spatial light modulators or the waveguide can be employed to focusing of light that enhances the coupling with atomic array.The spin wave state can be extracted in reverse and propagated through the waveguide, and the optical properties of atomic arrays can be controlled dynamically via external dress field.
The spin wave excitation may also be observed in other solid-state qubits platforms [73], such as color centers [74], rare-earth ions [75], and quantum dots [76,77].However, compared with the atomic array, these solid-state quantum platforms have some issues such as spatial disorder induced by fabrication, inhomogeneous broadening, and nonradiative decay.

Conclusion and discussion
In conclusion, we have demonstrated that the subwavelength ring atomic array can support subradiant or superradiant states, where the eigenmodes of the array exhibit degenerate spin waves propagating in opposite directions.The ring-to-ring energy transfer efficiency is significantly enhanced when the rings are loaded with two same (opposite) propagating spin waves.One of the atomic rings is initially at the most subradiant state, and another is at the ground state.At an appropriate separation, we observe that optimal spin wave exchange is due to the near-field dipole-dipole interactions and the mode-matching spin wave.However, in the regime of close distances, the dephasing mechanism of close-by atoms pair will be introduced, which leads to energy dissipation.On the contrary, there is almost zero interaction strength between two rings due to strong subradiant physics when two rings are separated by a large distance.Increasing the number of atoms on the ring enhances the subradiant shielding effect, resulting in a shorter effective coupling distance and requiring a smaller separation between the two rings for spin wave transfer.The imbalance in the number of atoms makes the transfer of excitation between the rings challenging.For the even-numbered atomic rings, rotating one of the rings to break the translational symmetry of the system leads to a reduction in the Rabi frequency of spin wave exchange.Furthermore, when the rotation angle satisfies ϕ N rot = ϕ N max /2, the atoms neighboring the two rings form a destructive interference pattern, preventing the excitation from transferring to the second ring.Moreover, by tuning the orientation of dipole moments, the effective interaction between two rings can change dramatically, exhibiting a sharp variation behavior.We further demonstrate that this sharp variation behavior is a cooperative effect by referring to the dipole-dipole interaction between two atoms.In summary, when the configurations of the two rings satisfy optimal translational symmetry and are positioned at an appropriate coupling distance, lossless spin wave exchange can occur between the two rings.
where J ij and Γ ij are given in equation ( 3).The complex eigenvalues λ m = J m − iΓ m /2 is obtained through diagonalizing the effective non-Hermitian Hamiltonian with m = 1, 2, . . ., N, where J m and Γ m are the frequency shift and the decay rate for the mth mode; the corresponding eigenstate is , where σ+ m is the collective raising operator and c j,m is the coefficient wave function.For simplicity, we sort the eigenvalues and eigenstates from minimum to maximum based on the magnitude of Γ m , i.e. with the index m = 1 corresponding to the most subradiant state and the index m = N corresponding to the most superradiant state.Here, we consider a subwavelength atomic ring array with radius R = 0.3λ 0 and the number of atoms N = 20.Figure 8 shows the single-excitation decay rate Γ m and the frequency shift |J m | for each collective spin wave mode, which are normalized by the single-atom spontaneous emission rate Γ 0 .Some of the modes are significantly suppressed (Γ m /Γ 0 ≪ 1), which corresponds to the phase-mismatch spin wave far beyond the light cone [10,34]; mode inside the light cone is the radiant mode corresponding to the phase-match spin wave.Additionally, for this symmetric ring structure, the inherent paired degeneracy modes correspond to the two spin waves propagating in opposite directions.
In the case of two coplanar atomic rings, the ring-to-ring coupling strength is quantified by the effective Hamiltonian in the angular momentum basis [45,46].Here, we rewrite the effective Hamiltonian describing the coupling between two rings under spin wave eigenstates |ψ m ⟩ as follows: where J m 1i ,m 2j = Re{λ m 1i ,m 2j } and Γ m 1i ,m 2j = −2Im{λ m 1i ,m 2j } represent the coherent and dissipative coupling between the m i th eigenmode of the first ring and the m j th eigenmode of the second ring, respectively.The complex eigenvalue is given as where two sets of indices correspond to sites of the first ring R 1 = {1, 2, . . ., N 1 }] and the second ring R 2 = {N 1 + 1, . . ., N 1 + N 2 }, respectively.The primary goal to achieve efficient energy transfer is to prevent the excitation of the rings from escaping into free space.We consider the configuration shown in figure 1 with atom number N 1 = N 2 = 20, the radius of atomic rings R 1 = R 2 = 0.3λ 0 , a fixed separation between two rings a = 0.15λ 0 and all of the atoms are polarized along the μ = ẑ direction.In figure 9(a), we plot the strength of dissipative couplings as functions of the spin wave state of the two rings m 1 and m 2 , where the indices are sorted from subradiance to superradiance by the magnitude of Γ m .We find that the dissipation is suppressed when the system is in the subradiant eigenmode of two rings, which is the result of the two subradiant spin wave modes being decoupled from free space and not scattered between each other.However, the radiant mode leads to leakage of excitation and therefore is not suitable as an appropriate initial state for achieving high energy transfer.Furthermore, in order to quantify how efficiently an excitation can be transferred between the two rings, the ratio is defined as [45,46] where ∆ m 1i ,m 2j = |J m 1i − J m 2j |.For the equally sized rings, the transfer efficiency is generally higher when two modes are the spin waves with same (or opposite) propagating directions (see figure 9(b)).In particular, for the degenerate spin waves, the transfer efficiency increases as its decay rate decreases.

Figure 1 .
Figure1.Schematic of the model.Two atomic rings lie in the x − y plane.Each two-level atom has an identical frequency ω0 and a spontaneous emission rate Γ0.The two ring's radiuses are R1 and R2, while the atom numbers of each ring are N1 and N2.The minimum spacing between the two rings is denoted as a.Each atom has a transition dipole moment μj , whose polar angle is θ j and azimuthal angle is ϕ j .We assume that the dipole orientation of the atoms in each ring is identical.

Figure 2 .
Figure 2. Two ẑ-polarization rings with N1 = N2 = 20 and R1 = R2 = 0.3λ0.(a)The dynamics of two rings' population and total loss of system.One of the ring is initially in the most subradiant spin wave state and another is in the ground state.The lattice constants of two ring arrays are d1 = d2 ≈ 0.0938(6)λ0.(b) The absolute value of coupling strength Jm 11 ,m21 (logarithmic scale) as a function of the separation a between two rings (see also in[45,46]).The insets show the evolution of population for a separation with a = 0.05λ0 and a = 0.25λ0, respectively.

Figure 3 .
Figure 3.The evolution of population for two atomic rings with spatial disorder.Besides the atomic positions fluctuating at each site with a maximal deviation of l, the other parameters remain the same as in figure 2(a).We consider the position fluctuations in three dimensions with (a) δl = 0.01d, (b) δl = 0.02d, (c) δl = 0.05d and (d) δl = 0.10d, where d is the lattice constant.The results are averaged over 100 individual realizations of spatial disorders.

Figure 4 .
Figure 4.The efficiency of the most subradiant spin wave exchange as a function of the number of atoms in two rings.(a) The efficiency ηm 11 ,m21 is plotted against the separation of two rings a and the number of atoms in both rings, where N1 = N2 ≡ N and the radiuses R1 = R2 = 0.3λ0.(b) The efficiency η m11,m ′ 21 as a function of the atoms number N with the separation a = 0.15λ0.The results in (a) and (b) are shown on the logarithmic scales.(c) The population dynamic of two ring with the atoms number N1 = N2 = 50 and the separation between two ring is a = 0.15λ0.The transition dipoles of atoms are polarized along the z-direction, and the lattice constants are d1 = d2 ≈ 0.0376(7)λ0.

Figure 5 .
Figure 5.The transfer efficiency η m11,m ′ 21 of the most subradiant spin wave as a function of the number of second atoms N2 with a fixed atoms number N1 = 20.The separation between two ring is a = 0.15λ0.The result is shown on the logarithmic scale.

Figure 6 .
Figure 6.Rotating one of the rings around the central axis of the circle.(a) The schematic diagram of the rotating operation for the example of N1 = N2 ≡ N = 12.The angle between two adjacent atoms is ϕ N max = 2π/N.One of the rings is rotated by an angle of ϕ N rot .(b) The absolute value of coupling strength J m11,m ′ 21 (logarithmic scale) as a function of the rotated angle ϕ N rot , normalized by ϕ N max .The ratio ϕ N rot /ϕ N max is represented as 'rp' in the subsequent results.(c) The population dynamics of the two rings when one of the rings is rotated by an angle of ϕ N rot with rp = 0.25, rp = 0.35, in (d) rp = 0.45, rp = 0.48, and rp = 0.50, respectively.All other parameters remain consistent with those shown in figure 2(a).Notice that the time scales of the two subgraphs (c) and (d) are distinct.

Figure 7 .
Figure 7. Tilted transition dipole moments.(a) The coupling strength J m11,m ′ 21 as a function of the polar angle θ with ϕ = π/2 and ϕ = 0, respectively.The critical point is at θc ≈ 0.228π.These results are shown on logarithmic scale.(b)The coherent dipole-dipole interaction strength between two atoms with different transition dipole moments.One of the transition dipole moment orientations µ j is perpendicular and another µ i is rotating from parallel to perpendicular.The separation between two atoms is denoted as x.(c) The evolution of populations for the dipole moment of the first ring has a fixed azimuthal angle ϕ = π/2, for different polar angles θ.The initial state of the first ring is at its most subradiant state and the second ring is at the ground state.

Figure 8 .
Figure 8.The real and imaginary parts of the complex eigen-energy for an individual ẑ−polarization atomic ring with the radius R = 0.3λ0, the number of atoms N = 20, and the lattice constance d ≈ 0.0938(6)λ0.The dash line corresponds to the value of a single-atom spontaneous emission rate Γ0, and the gray shadow region denotes the light cone.