## Abstract

We consider a periodic chain of oscillating dipoles, interacting via long-range dipole–dipole interactions, embedded inside a cuboid cavity waveguide. We show that the mixing between the dipolar excitations and cavity photons into polaritons can lead to the appearance of new states localized at the ends of the dipolar chain, which are reminiscent of Tamm surface states found in electronic systems. A crucial requirement for the formation of polaritonic Tamm states is that the cavity cross section is above a critical size. Above this threshold, the degree of localization of the Tamm states is highly dependent on the cavity size since their participation ratio scales linearly with the cavity cross-sectional area. Our findings may be important for quantum confinement effects in one-dimensional systems with strong light–matter coupling.

## 1 Introduction

In 1932, Tamm [1] showed the existence of surface states in a one-dimensional (1-D) crystal lattice, due to the abrupt termination of the periodic crystal at an interfacing surface, such as the vacuum [2], [3], [4]. This result highlighted a surprising failure of the theory of a periodic potential with cyclic boundary conditions at the elementary level of the electronic bandstructure, despite its great utility in explaining the bulk properties of solids [5], [6]. Tamm surface states have since been shown to have profound consequences for the rich field of surface science, including for photoluminescence in mesoscopic systems [7], photocurrents in superlattices [8], and the absorption spectra of molecular chains [9], [10], [11].

Over the last two decades, various theories of Tamm states in the latest condensed matter systems have been developed [12]. For example, with exciton–polaritons in multilayer dielectric structures [13], with plasmons at at the boundary between a metal and a dielectric Bragg mirror [14], and with phonons in graphene nanoribbons [15]. Pioneering experimental work has seen the observance of Tamm states in magnetophotonic structures [16], in organic dye-doped polymer layers [17], and latterly in photonic crystals [18], [19], [20].

Due to the rise of topological physics in photonics and plasmonics [21], [22], [23], [24], [25], [26], there is an ongoing interest in finding and classifying unconventional light–matter states. Indeed, the latest advances in topological matter have been made in photon-based systems, leading to the rapidly expanding subfield of topological nanophotonics [27], [28], [29], [30], [31], [32], [33]. It is therefore crucial to also classify and understand surface states of a nontopological origin, such as Tamm-like edge states, in systems with strong light–matter coupling. Indeed, there are recent experimental studies of polariton micropillars, where the localization of both topologically trivial and topologically nontrivial modes is examined in detail [34], [35].

In this work, we consider a nanophotonic system which exhibits Tamm-like edge states: a 1-D chain of regularly spaced nanoresonators, coupled via dipole–dipole interactions, which are housed inside a cavity waveguide (see Figure 1(a)). The linear dipolar chain is of some importance since it is a simple system where one may study the subwavelength transportation of energy and information [36], [37], [38], [39], [40]. In our theory, we place the dipolar chain inside a cuboid cavity in order to study the effect of controllable light–matter interactions. Modulating the cross-sectional area of the cavity allows one to tune both the light–matter coupling strength and the light–matter detuning. In the strong coupling regime, the dipolar excitations in the resonator chain hybridize with the cavity photons to form polaritonic excitations [41], [42]. The resulting polaritons, which display half-light and half-matter properties, can lead to the emergence of a highly localized edge state of a nontopological origin: a Tamm-like state. Notably, neither the dipolar chain nor the cavity photons display Tamm states when the light–matter interaction is switched off. We discuss the properties of the emergent polaritonic Tamm state, including how its creation requires the cavity cross section to be above a critical size and how its localization properties scale with the cavity cross-sectional area.

The presented theory of a chain of oscillating dipoles embedded inside a cuboid cavity may be realized in a wide range of dipolar systems, as alluded to in the sketches in Figure 1(b). At the subwavelength scale, exploiting the Mie resonances in a chain of dielectric nanoparticles is a promising option since the system does not suffer from high losses and is hence ideal for energy transportation [43], [44], [45]. Localized surface plasmons hosted by metallic nanoparticles are another accessible platform [46], [47], [48], and there are several recent experimental studies of plasmonic nanoparticles in cavity geometries [49], [50], [51], [52]. Exciting spin waves in magnetic microspheres is another appealing possibility [53] since cavity magnons have been well studied experimentally in recent years [54]. Finally, implementations with Rydberg [55], [56] and ultracold atoms [57], [58], as well as helical resonators [59], are also viable settings for the versatile theory presented here.

The rest of the manuscript is organized as follows: we describe our model in Section 2, we unveil the polaritonic Tamm states in Section 3, and we draw some conclusions in Section 4. The Supplementary material contains additional theoretical details.

## 2 Model

The Hamiltonian of a chain of oscillating dipoles embedded inside a cavity reads as follows [60], [61], [62]:

accounting for the dipolar dynamics, the cavity photons, and the light–matter coupling, respectively. Importantly, the couplings in *H*_{dp} go beyond the nearest-neighbor approximation [25], [40], which is essential for a proper treatment of the type of Tamm states discussed in this system.

We sketch in Figure 1(c) the model of our system: a 1-D array of dipoles, regularly spaced at the interval *d*, which is encased inside a cuboid cavity of dimensions *L*_{x} × *L*_{y} × *L*_{z} (see panel [a]). Tuning the size of the cavity cross-sectional area (*L*_{x} × *L*_{y}) modulates both the light–matter coupling strength and the light–matter detuning, such that polariton excitations may be formed by the mixing between the cavity photons and dipolar excitations (which are generally treated as harmonic oscillators, see Figure 1(b) for some physical realizations).

### 2.1 Dipolar Hamiltonian

The dipolar Hamiltonian (*H*_{dp} in Eq. (1)) describes a linear chain of

where the bosonic creation (annihilation) operator *q*, where *ω*_{0}, which is associated with the length scale *a* [63]. The weak dipolar coupling constant *d* is the center-to-center separation between the dipoles [63]. In Eq. (2), we have introduced the lattice sum _{s}(*z*) is the Clausen function of order *s*. Crucially, *f*_{q} takes into account long-range interactions between all of the resonators, which is known to be important in dipolar systems [40]. Significantly, going beyond the nearest-neighbor approximation also changes the constraints at the edge of the chain from standard hard-wall boundary conditions. Namely, the edge resonators 1 and *f*_{q} with the standard result

After ignoring the counter-rotating terms in Eq. (2) (see Ref. [63] for the full treatment), the eigenfrequencies

### 2.2 Polaritonic Hamiltonian

The photonic Hamiltonian (*H*_{ph} in Eq. (1)) describes the cavity photons inside the long cuboid cavity of dimensions *L*_{z} ≫* L*_{y} >* L*_{x} (see Figure 1[a]). In terms of the photonic creation (annihilation) operator

where the cavity photon dispersion

where *c* is the speed of light in vacuum. The cavity width is *L*_{y} and the cavity aspect ratio *L*_{y} >* L*_{x}, such that only the photonic band of Eq. (5) is relevant for the problem. The full light–matter coupling Hamiltonian (*H*_{dp−ph} in Eq. (1)) reads (see Ref. [63] for the derivation) as follows:

where, we have introduced the light–matter coupling constant as follows:

The diamagnetic term (on the second line of Eq. (6)) simply leads to a renormalization of the photon dispersion

a shift which can be safely disregarded throughout this work, since it only leads to small quantitative changes to the results presented here. The paramagnetic term (on the first line of Eq. (6)) is important and gives rise to the formation of polaritonic excitations.

Ignoring counter-rotating terms in the polaritonic Hamiltonian (formed by Eqs. (2), (4), and (6)), we may write the resulting rotating wave approximation polaritonic Hamiltonian as follows:

where the polaritonic Bloch Hamiltonian is

where the index *τ* = ± labels the upper and lower polariton bands. The polariton dispersion

where the average frequency of the uncoupled dispersions

The bosonic Bogoliubov operators

where the Bogoliubov coefficients are as follows:

in terms of the quantities defined in Eq. (12).

We plot in Figure 2 the polariton dispersion of Eq. (11) for the cavity heights *L*_{y} = 3*L*_{x} and the interdipole separation at *d* = 3*a*. With increasing cavity cross-sectional area in going from panel (a) to (b) to (c), the light–matter detuning Δ_{q} is reduced (cf. Eq. (12)), leading to increasingly noticeable deviations of the polariton bands (solid lines) from the uncoupled dispersions (dashed lines). The upper (lower) polariton band is given by the red (blue) lines. The photonic bandstructure is denoted by orange lines, while the dipolar bands are in cyan. Notably, in panel (a), only a single polaritonic band is visible on this scale since the (mostly photonic) upper polariton band lies significantly above the frequency scale of

The Bogoliubov operators of Eq. (13) imply the pair of polaritonic Bloch spinors *q*-dependent phase factor difference (like

where {*σ*_{x}, *σ*_{y}, *σ*_{z}} are the Pauli matrices, and *I*_{2} is the two-dimensional identity matrix. Despite this Dirac mapping, the associated spinors

Perhaps surprisingly, the mixing between the dipolar and photonic modes into polaritons also gives rise to the formation of Tamm-like edge states. These localized states are missing in Figure 2 since their emergence requires a finite system (which precludes the use of periodic boundary conditions).

## 3 Polaritonic Tamm states

In order to search for the edge states in our system of a chain of resonators inside a cavity, it is necessary to solve the eigenproblem of Eq. (1) in real space, thus removing the periodic boundary condition assumption in the Fourier space calculation of the previous section. This procedure leads to the eigenfrequencies *m*. Each eigenstate *m*), as defined as follows [73], [74]:

where the summations are over all of the dipole sites *n*. Extended states residing in the bulk part of the spectrum are characterized by a participation ratio scaling linearly with the system size and in the nearest-neighbor approximation

In Figure 3(a), we plot the polariton bandstructure from Eq. (11) with *L*_{x} = 10*a* (cf. Figure 2(b)), where all-neighbor coupling *τ* = −1). Clearly, the impact on the continuum bandstructure of going beyond the nearest-neighbor approximation is negligible, perhaps making the appearance of edge states in the corresponding finite system even more surprising.

Using Eq. (16), Figure 3(b) displays the participation ratio PR(*m*) for the equivalent problem in real space for a chain of ^{nn}(*m*) ≃ (2/3)1000 ≃ 667], while for the all-neighbor case, the participation ratio of the lower polariton band (blue circles) is markedly different, especially near to the top of the lower polariton band (TLB). In particular, the state at the very top of the lower polariton band in the nearest-neighbor approximation is associated with *m*_{Tamm}) has PR(*m*_{Tamm}) ≃ 41 (see the insert in panel (b) for a zoom in on the Tamm state). This last result suggests a highly localized state, induced by the different boundary conditions in the all-coupling case, as compared to the standard hard-wall conditions in the nearest-neighbor approximation.

We plot in Figure 3(c) the probability density *n*, for the polariton eigenstate at the top of the lower polariton band, where the nearest (all)-neighbor coupling is given by the thin green (thick red) line and is associated with the index *m*_{TLB} (*m*_{Tamm}). This panel clearly displays the emergence of the Tamm-like edge state in the all-coupling case, induced by (i) the strong light–matter coupling in the cavity and (ii) the all-coupling boundary conditions. This state is not associated with a topological invariant (see the discussion after Eq. (15)), and so we term it a polaritonic Tamm state. This is in direct analogy with the nontopological surface states studied in solid state physics, which also typically arise in 1-D tight-binding models.

In Figure 4(a) and (b), we show the dependence of the participation ratio *L*_{x} = 5.64*a* in panel (a) (the reason for this choice will become apparent in what follows) and *L*_{x} = 10*a* in panel (b). The results for the Tamm states are denoted by thin blue lines, while the results for the bulk states are represented by thick red lines, and the equation of the line is labeled nearby. These results confirm that the bulk states behave according to the standard formula *m*_{Tamm}) ≃ constant with increasing *L*_{x} in going from panel (a) to (b) in Figure 4 has led to an increased participation ratio of the Tamm states, suggesting weaker localization (explicitly a rise from PR(*m*_{Tamm}) ≃ 19.0 in panel [a] to PR(*m*_{Tamm}) ≃ 40.9 in panel (b)). This reveals the simple modulation of the cavity cross-sectional area as a tool to control the degree of localization of the edge state (supplementary plots for other cavity sizes are given in Ref. [63]).

We investigate the cavity size Tamm state relationship in Figure 4(c), which shows the minimum of the participation ratio min{PR(*m*)} as a function of the cavity height *L*_{x}, for chains of *L*_{Tamm} ≃ 5.30*a*, the cavity height above which the Tamm states first appear in the system. For subcritical cavities (*L*_{x} <* L*_{Tamm}), no edge states are present, as in a regular dipolar chain uncoupled to cavity photons, since the light–matter detuning is too great to significantly influence the dipolar modes. For supercritical cavities (*L*_{x} ≥* L*_{Tamm}), we observe the presence of Tamm-like edge states, which are characterized by a participation ratio which grows linearly with the cavity size for *L*_{x} ≳ 8a. Explicitly, the dependency here is PR(*m*_{Tamm}) ≃ 3.62(*L*_{x}/*a*) + 5.39, as shown by the dashed green line in panel (c) [75]. For smaller cavity sizes *L*_{Tamm} ≤ *L*_{x} ≲ 8a, there is an interesting nonmonotonous behavior, and a global minimum of PR(*m*_{Tamm}) ≃ 19.0 occurs at *L*_{x} = 5.64*a* (cf. the results of Figure 4[a]). Of course, these numbers depend on the chosen interdipole separation ratio *d*/*a*, dimensionless dipole strength *L*_{y}/*L*_{x}.

We have therefore demonstrated an unusual, nontopological (see the discussion after Eq. (15)) phase transition demarcating the absence and presence of Tamm-like edge states, which are induced by cavity interactions and boundary conditions in the chain beyond those used in the nearest-neighbor approximation. The observation of these proposed Tamm states requires careful sweeping in energy, due to their proximity to bulk states. Such careful measurements can be performed using the latest techniques in cathodoluminescence spectroscopy [76] and optical microscopy and spectroscopy [77]. The detection of such states in the strong coupling regime provides perspectives for the fundamental understanding of the interplay between edge states and light–matter coupling and for controlling the localization of polariton states in nanoscale waveguiding structures. Furthermore, while there is a great quest to find topological nanophotonic states [27], [28], [29], [30], [31], [32], [33], our findings highlight that after the experimental observation of an edge state, one should also consider possible nontopological mechanisms of generation.

## 4 Conclusion

We have presented a theory of polaritonic Tamm states, forged due to the mixing between the collective excitations in a dipolar chain and cavity photons. Importantly, the very existence of Tamm states requires the cavity cross-sectional area to be above a critical value. In this supercritical regime, the degree of localization of the Tamm states is highly dependent on the cavity size, with the participation ratio scaling linearly with the cavity cross-sectional area. We have also shown the crucial role played by dipole–dipole interactions beyond the celebrated nearest-neighbor approximation, without which the Tamm edge states do not form. The theory demonstrates the possibility of light trapping in nontopological 1-D structures, which may be important for waveguiding at the nanoscale. Our results also highlight how edge states may be generated via nontopological means, quite distinct from iconic topological models.

Our proposed model can be implemented in a host of systems based upon dipolar resonators, including dielectric and metallic nanoparticles [78]. Our theoretical proposal therefore offers the opportunity to finely control the propagation and localization of collective light–matter excitations at the subwavelength scale and provides perspectives for more complicated and higher dimensional nanophotonic systems [79], [80].

**Funding source: **Consejo Superior de Investigaciones Científicas

**Award Identifier / Grant number: **MAT2017-88358-C3-I-R

**Funding source: **Quantum Materials and Devices (Q-MAD)

**Funding source: **Ministerio de Economía y Competitividad (MINECO)

**Funding source: **Juan de la Cierva program

**Author contribution:**All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.**Research funding:**This work was supported by the Aragón government through the project Quantum Materials and Devices (Q-MAD) and Ministerio de Economía y Competitividad (MINECO) (Contract No. MAT2017-88358-C3-I-R). CAD acknowledges support from the Juan de la Cierva program (MINECO, Spain).**Conflict of interest statement:**The authors declare no conflicts of interest regarding this article.

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## Supplementary Material

The online version of this article offers supplementary material (https://doi.org/10.1515/nanoph-2020-0370).

**Received:**2020-07-03

**Accepted:**2020-08-22

**Published Online:**2020-09-14

© 2020 Charles A. Downing and Luis Martín-Moreno, published by De Gruyter, Berlin/Boston

This work is licensed under the Creative Commons Attribution 4.0 International License.