Subradiant Dimer Excitations of Emitter Chains Coupled to a 1D Waveguide

This Letter shows that chains of optical or microwave emitters coupled to a 1D waveguide support subradiant states with close pairs of excited emitters, which have longer lifetimes than even the most subradiant states with only a single excitation. Exact, analytical expressions for non-radiative excitation dimer states are obtained in the limit of infinite chains. To understand the mechanism underlying these states, we present a formal equivalence between subradiant dimers and single localized excitations around a chain defect (unoccupied site). Our analytical mapping permits extension to emitter chains coupled to the 3D free space vacuum field.

An exception is the so-called fermionic multi-excitation subradiant states in 1D systems [15][16][17]. While one might expect emitter saturation to play only a perturbative role in the few-excitation scenarios [30], it enforces an equivalence between the multi-excitation subradiant states and the Tonks-Girardeau gas of hard core bosons [18]. This suggests a class of subradiant states with state amplitudes which are anti-symmetric combinations of the oneexcitation subradiant states.
However, numerical analyses of emitter chains coupled to a 1D waveguide reveal the existence of another family of subradiant states with entirely different properties, see [18] and [31, Sec. A]. In this Letter, we first numerically demonstrate and assess the extraordinary properties of subradiant states with very close pairs of excited emitters, i.e., subradiant dimers. In particular, we find that for specific distances between the excited emitters their radiative lifetimes can be longer than the fermionic states and even than the most long-lived one-excitation states [32]. Then, we present an analytical treatment that explains the confinement mechanism that leads to the subradiant dimers by a mapping to the localized subradiant excitation near an unoccupied site (defect) in the chain. This confinement-localization mapping is valid under more general conditions and allows extension of our analysis, e.g., to emitter chains coupled to the 3D free space quantized field.
Spin Model. Consider a chain of N two-level emitters equally spaced by the distance d, as illustrated in Fig. 1(a). Each emitter has a ground state |g and an excited state |e with transition frequency ω 0 . The emitters are coupled to a 1D waveguide that supports light modes with a linear dispersion relation. Using the Born-Markov approximation, the waveguide modes can be eliminated to yield an effective theory for the emitters [33], which entails a non-Hermitian infinite-range spin-spin interac- Figure 1. (a) A chain of two-level emitters couple to a 1D waveguide with parameters introduced in the main text. In the limit of infinite chains, a dimer state has two excitations characterized by a short relative distance ∆ and the delocalized "center of mass" (corresponding to a well defined total wave number K). In this Letter we study two types of subra- The bare excitation energies of the individual emitters are not included, Γ 1D is the decay rate of an individual emitter coupled to the waveguide, k 1D is the wave number of the waveguide mode resonant with ω 0 , z m is the position coordinate of the m th emitter and σ † m = |e m g|. In a perfect experimental implementation of (1) Γ 1D should dominate all other decay processes, as in photonic crystal waveguides [37,38] and superconducting qubits coupled to transmission lines [39][40][41][42].
In the Monte Carlo wave function formalism [43], the state of the emitter chain evolves under Eq. (1), interrupted by stochastic quantum jumps representing spontaneous emission of a photon. The jump rate makes a system prepared in a right eigenstate of H eff maintain its excitation with a probability that decays with twice the negative imaginary part of the corresponding eigenvalue. In this work, we obtain these eigenstates by the exact diagonalization of H eff with use of the SLEPc (Scalable Library for Eigenvalue Problem Computations) [44].
Subradiant dimer excited states. We focus on the twoexcitation subspace of eigenstates of H eff , for lattices with 0 < k 1D d < π/2. This Hilbert space is spanned by states |n, m where the m th and n th emitters are excited. As we illustrate in Fig. 1, by numerical diagonalization of H eff we find subradiant dimer states with delocalized center of mass Z c = (z m + z n )/2 and well-defined distance ∆ = |z m − z n | between the excitations. To understand the appearance of these states and their properties, we introduce basis states, with center-of-excitation wave number K and spatial separation ∆ between the excitations. In a finite chain, K is not conserved but Z c distributions resembling standing waves appear due to the boundary conditions at the chain ends. The expansion of the identified type-I dimers on the basis states (2) with the dominant amplitude on ∆ = d, see Fig. 1 with the dominant amplitude on ∆ = 2d, see Fig. 1(c). These dimers are perfectly subradiant with vanishing decay rates on infinite chains, i.e., the corresponding eigenvalues of H eff are real. In [31, Sec. B], we derive the asymptotic eigenvalues of the two types of dimers, viz., ω I = 2Γ 1D cot(k 1D d) and ω II = 2Γ 1D cot(2k 1D d), and numerically obtain their corrections on a finite chain. Knowing these asymptotic values allows efficient search for the eigenstates for a finite chain with a large number of emitters by the Krylov-Schur algorithm [45] with the shift-and-invert method [46]. The dimers on finite chains have small but finite decay rates. We find that the minimal decay rates of the most subradiant type-I states scale asymptotically as nearly N −2 , see [31, Sec. C]. However, the type-II states show longer lifetimes and a more complex behaviour. The decay rate of the most subradiant type-II dimers is shown versus N and k 1D d in Fig. 2, where a narrow region between k 1D d = 0.16π and 0.17π is distinctively subradiant. Moreover, fringe textures are seen to the right hand side of that region. We compare the minimal decay rates of type-II dimers with those of the most subradiant fermionic states and one-excitation states, and obtain the critical parameters of k 1D d and N shown by the red and black boundary curves in Fig. 2. Specifically, Fig. 2 demonstrates that the type-II dimer can be more subradiant than the fermionic states, and even the one-excitation states. This observation disproves an unwritten orthodoxy that states with more excitations have shorter lifetimes, being true, for example, for the fermionic states that decay with the sum of their oneexcitation constituent decay rates [15][16][17]. Fig. 2 shows that a short chain with N = 48 emitters is sufficient to observe the even more subradiant dimer states, in the case of k 1D d = 0.1676π.
To study the extremely subradiant region in more detail, we plot decay rates of the most subradiant type-II dimers for a few values of N in Fig. 3(a). A sharp dip in decay rates appears around k 1D d = π/6 and a magnified view of the interval k 1D d/π ∈ [0.17, 0.18] shows the fringe textures observed in Fig. 2. The robustness of the results to position disorder is discussed in [31, Sec. D]. In Fig. 3(b), we observe different scalings of the decay rate with N ≤ 500 for different values of k 1D d. Around k 1D d = π/6, the decay rates thus fall off faster than N −3 and show oscillations breaking the conventional monotonicity with N . For k 1D d near 0.25π, the decay rate  When k 1D d = 0.25π, Eq. (4) vanishes so that amplitudes of ∆ > 2d are completely suppressed. On infinite chains this state is an eigenstate of both the centre-ofexcitation wave number, equivalent to a total momentum,p 1 +p 2 , and the relative position coordinate,x 1 −x 2 , i.e., it is an implementation of the Einstein-Podolsky-Rosen state [47]. A similar state is found for the type-I dimer for k 1D d = 0.5π.
Confinement-localization mapping. The Hamiltonian Eq. (1) does not provide any direct evidence for the emergence of the subradiant dimer states, and neither does our analysis based on the Holstein-Primakoff transformation [18]. Here we provide the physical mechanism leading to the long lived excitation dimers.
Applying H eff on the ansatz of Eq. (2) yields It separates H eff into contributions preserving K, i.e., the matrix H K defined by elements and remaining terms, denoted by "tails" that break the conservation of K, see [31, Sec. E]. The "tails" vanish when N → ∞. Thus the Hamiltonian H K acting on the relative position eigenstates is essential for the formation of dimers and must explain their vanishing decay rates on infinite chains.
Our key insight is that the eigenstates of H K can be uniquely mapped to the even-parity eigenstates of a Hamiltonian H K def with matrix elements where ∆/d, ∆ /d attain both positive and negative values {±1, ±2, · · · }. As shown in [31, Sec. F], for an eigenstate |ψ of H K (with eigenvalue λ), there is a corresponding even-parity eigenstate |ψ def (unnormalized) of H K def with the eigenvalue λ/2 that satisfies ∆|ψ def = ∆|ψ for indices ∆ > 0.
The above mapping implies that, the dimer state is equivalent to an eigenstate of the Hamiltonian H K def , describing a single excitation localized around a defect (unoccupied site) at ∆ = 0, as illustrated in Fig. 4(a). Actually for the case of K = 0, H K=0 def can be written as ∝ ∆ e iqI|∆| |∆ with q I d = −i ln cos(k 1D d). This is in agreement with the spatial factor Eq. (3) found for the type-I dimers. The eigenvalue of H K=0 def for the localized state is Γ 1D cot(k 1D d), exactly half of that of the type-I dimer ω I , as predicted by the confinement-localization mapping. The case of K = π/d (type-II dimers) is equivalent to the case of K = 0 [31, Sec. H], up to alternating sign flips and the replacement of d → 2d. This equivalence explains the resemblance between Eqs. (3) and (4), and between the eigenvalues ω I and ω II .
The excitation of an emitter blocks further excitation and the dimer state is stable because each excitation serves as a defect supporting the localization of the other one. For finite chains, as shown in Fig. 4(b) the localized state around a defect has a decay rate suppressed exponentially in the emitter number N , and if the defect is not at the chain center, the decay rate is determined by the length of the shorter subchain, see Fig. 4(c). This dependence is much faster than the N −3 scaling on chains free from defects [15][16][17][18] and it comes about because the localized state is a superposition of a left and a right excited subchain. Their destructive interference results in the extreme subradiance, seen also in Refs. [26][27][28].
The finite size effects shown in Figs. 2 and 3 are due to the "tails" of Eq. (5). Since the states |K, ∆ have finite width of ∆, the "tails" is restricted to a short section of length ∆ at each end, but their complicated expression [31, Sec. E] prevents analytical solution. We may, however, infer that the dimers become seriously affected at the chain ends, and hence an interplay between the length of the chain and the bond-length of the dimer may be responsible for the periodic oscillations seen in Fig. 3(b) and the fringe texture seen in Fig. 2. The strong dependence of the decay rates on the emitter separation d might be equivalent to the observation, see Ref. [21] of special emitter distances leading to extraordinary (single excitation) subradiant states.
Universality. The mapping between confinement and localization can be extended by linearity to Hamiltonians written as H eff = dµ(k 1D )H eff (k 1D ), where dµ(k 1D ) is an integral measure over the variable k 1D and H eff (k 1D ) refers to Eq. (1). This for example covers 1D emitter chains coupled with 3D free space modes where µ(k 1D ) is given in Ref. [18]. Also here, localized subradiant states will exist, and we show their excitation amplitude in Fig. 4(d,e) (see phase profiles in [31, Sec. I]), for emitters polarized both transverse and parallel to the chain. In contrast to the 1D waveguide, these localized states have finite decay rates in the infinite chain limit. Due to the mapping, we can conclude that subradiant excitation dimers exist and that they have intrinsic finite decay rates also in the limit of infinite chains.
Conclusion and discussion. In this Letter, we have introduced subradiant excited dimers of emitter chains coupled to a 1D waveguide. We showed that such (type-II) dimers can be more subradiant than even the longest lived one-excitation states of the system. Their decay rates show unusual dependence, including non-monotonic wiggles, as function of N when k 1D d is slightly larger than π/6. We identify the intrinsic emitter saturation and the resulting emergence of mutual defect modes as the cause of the long lived dimer states, and we use a rigorous mapping between confinement and localization to obtain their physical properties. This mapping is valid under general conditions and gives access to the properties of similar states of emitter chains decaying into 3D free space field modes.
We propose to verify the predictions experimentally, and to address single excited states around defects, e.g., by exciting emitters around a missing or suitably perturbed site, and waiting for excitation amplitudes on orthogonal excitation modes to decay. To verify the dimer subradiant states, one would excite a system uniformly, but one may exploit interactions to facilitate correlated excitation of dimers within certain distance ranges [48], and thus maximize the overlap with the long lived states identified in this Letter. Other efficient ways to couple the bound states or localized states may be mediated by ancillary emitters distributed off the 1D lattice sites [42,49]. Finally, we imagine that our method to formally map doubly excited states on localized states around defects may become a useful ingredient in other lattice models and contribute to the analysis of other quasiparticle confinement phenomena, cf., recent findings in Ising spin chains with long-range interactions [50].
This work was supported by the Villum Foundation, the European Unions Horizon 2020 research and innovation program (Grant No. 712721, NanOQTech), and the EuropeanUnion FETFLAG program (Grant No. 820391, SQUARE). The numerical results were obtained at the Center for Scientific Computing, Aarhus University.

A. Distributions of the eigenvalues in the two-excitation sector
The fermionic states and the dimers constitute all the subradiant states in the two-excitation sector. This is verified by Fig. 1, where we show all the eigenvalues of H eff for a chain of N = 50 emitters coupled to a 1D waveguide with k 1D d = 0.2π. In Fig. 1 five branches of subradiant states can be clearly identified. For the three branches of fermionic states, their asymptotic eigenvalues can be obtained from the supplemental material (Sec. A) of Ref. [18]. The asymptotic eigenvalues of the dimers are given in the main text and will be derived below in Sec. B.
where the coefficients are Suppose that q has a positive imaginary part and N is sufficiently large so that e iqN d 0. This implies that h 0 q = 0 and if we can find a value of q so that g 0 q = 0, the corresponding |q is an eigenstate of H 0 . For large N , this leads to the equation The solution is q = −i ln cos(k 1D d). Substituting this into the expression for ω q yields the real eigenvalue ω I = 2 cot(k 1D d). In Fig. 2, we plot the deviations of energy levels of the dimers from their N → ∞ asymptotic values. Counterparts of the type-II dimers can be obtained in the same way. One can also obtain them by using the equivalence between the two types of dimers as presented in Sec. H of this Supplemental Material.

C. Decay rates of the type-I dimers
The decay rates of the most subradiant type-I dimer states versus emitter number N and k 1D are plotted in Fig. 3. Compared with the results for the type-II states, shown in the main text, the curves are free from dips and wiggles.

D. Robustness against spatial disorder
In this section, we examine the influence of disorder of the spatial positions of the emitters. It is conceivable that for weak disorder, the eigenstates are only slightly perturbed, while for more significant disorder, the system may display new physics, such as Anderson localization effects.
We restrict ourselves here to the situation of weak disorder. Then the position of each emitter is assumed to be shifted randomly by a distance uniformly distributed in a small interval [−δ, δ]d. We show the decay rates of the most subradiant type-II dimer as a function of k 1D d in Fig. 4 for a number of random realizations of the disorder. Our simulations show that the dip near k 1D d = π/6 is robust against the disorder, and that disorder can even lead to further suppression of the decay rates. The omitted "tails" are expressed as where |G is the state where no emitters are excited, σ † p = m e ipzm σ † m , and the foot indices R and L restrict the summation over sites to the intervals (z N − ∆, z N ] and [z 1 , z 1 + ∆), respectively. For dimer states dominated by small values of ∆, the "tail" terms are well restricted to the ends of the chain.
The one-to-one correspondence between the eigenstates of H K and the even eigenstates of H K def can be understood from the following simple observation.
While these quantities are introduced for application to the case of ∆, ∆ > 0, they formally obey the symmetry, A + ∆,∆ = A − ∆,−∆ and, hence the action of H K on a general state obeys the following set of equations, In the last line we introduce the even states, |ψ def , defined for both positive and negative ∆, and satisfying ∆|ψ = ∆|ψ def for ∆ > 0, and we observe that the last expressions can be combined in a single sum G. Defect-induced localized subradiant states We denote the effective Hamiltonian of a chain with the m th emitter missing by H −m,def . This defect separates the chain into a left and a right subchain, where Bloch one-excitation states, |q L and |q R , are defined as Then we have where the coefficients are given as The expression of ω q is identical to that of Eq. (2a). We expect that the eigenvalues will be expressed by ω q for some specific values of q with contributions to the eigenstates from the degenerate states | ± q L and | ± q R . Indeed, one verifies by inspection that a superposition, c L |ψ q L + c R |ψ q R , of |ψ q L ∝ g L,−q |q L − g L,q | − q L and |ψ q R ∝ h R,−q |q R − h R,q | − q R , lead to cancellation of the |±k 1D;L and |±k 1D;R terms in Eq. (9) and that the coefficients c L and c R can be found if the determinant of the following matrix vanishes: This condition is further evaluated to be where It is expected that the solution for q has positive imaginary part, hence e iqN d ≈ 0. Then Eq. (12) can be evaluated to When the missing site is far from the chain ends so that the right hand side of the above equation can be ignored, we have with the solution q I = −i ln cos(k 1D d). Similarly for the case corresponding to the type-II states, we obtain q II = π−0.5i ln cos(2k 1D d). Substituting these expressions into ω q yields the asymptotic eigenvalues. Terms on the right hand side of Eq. (13) are suppressed exponentially in the length min N LR of the shorter subchain (left or right side of the missing site). The coupling at the end provides a correction, q = q I + δ, with Substituting this into the expression for ω q yield an exponentially suppressed decay rate of the localized state as function of min N LR . The localized states are exponentially suppressed at the chain ends and the eigenstates are approximately given in the form of |q I;R + | − q I;L as we present in the main text.
where the indices ∆ and ∆ have been transformed to dimensionless integers for convenience of notation. If ∆ and ∆ have opposite parity, i.e., one is even and the other is odd, |∆ − ∆ | will be odd and consequently e i π 2 |∆−∆ | + e −i π 2 |∆−∆ | = 0.
where the odd and even terms commute.
To summarize, the localization of H π/d def (k 1D d), which corresponds to the type-II dimers, is equivalent to that of H 0 def (2k 1D d) which belongs to the type-I dimers.
I. Phase profiles of the localized states For the 1D waveguide case, the phase of the localized state is uniform for 0 < k 1D d < π/2, and flips by π per site if π/2 < k 1D d < π. For coupling to the 3D free space vacuum field, the amplitude and phase profiles are not as regular as in the 1D waveguide case. In Fig. 5 we show the phase profiles of the localized states illustrated in Fig. 4(d,e) of the main text (the amplitude profiles are replicated for convenience).
In the figure, the emitter number is not specified, because profiles of the localized states are almost the same, as long as their widths are adequately shorter than the chain lengths.