# Optical Properties of Concentric Nanorings of Quantum Emitters

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Theoretical Framework: Bloch Eigenmodes

#### 2.2. Bloch Eigenmodes in Rotationally Symmetric Ring Structures

## 3. Results

#### 3.1. Optical Properties of a Single Nano-Ring

#### 3.1.1. Dense and Large Ring Case (Quasi-Linear Chain Limit)

#### 3.1.2. Small Ring Case (Dicke Limit)

#### 3.2. Optical Properties of Two Coupled Nano-Rings

#### 3.2.1. Coupled Identical Non-Rotated Rings ($\delta =0$)

#### 3.2.2. Coupled Unequal Rings with Rotation ($\delta \ne 0$)

#### 3.3. B850 and B800 Bands in LH2

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Schematics of a single ring with lattice constant d. Each emitter features an optical dipole moment (indicated by the red solid arrow) with orientation $\widehat{\wp}=sin\theta cos\varphi \phantom{\rule{0.166667em}{0ex}}{\widehat{e}}_{\varphi}+sin\theta sin\varphi \phantom{\rule{0.166667em}{0ex}}{\widehat{e}}_{r}+cos\theta \phantom{\rule{0.166667em}{0ex}}{\widehat{e}}_{z}$, where $\theta $ and $\varphi $ are the polar and azimuth angle, respectively. The vertical, radial, and tangential unit vectors are indicated by ${\widehat{e}}_{z}$, ${\widehat{e}}_{r}$, and ${\widehat{e}}_{\theta}$, respectively. The red dashed arrow denotes the projection of the dipole onto the ring plane. (

**b**) Double-ring structure: two rings of radius ${R}_{1}$ and ${R}_{2}$ and lattice constants ${d}_{1}$ and ${d}_{2}$ are stacked concentrically and separated by the vertical distance z. The two rings are, in general, rotated by an angle $\delta $. The dashed-line rectangle encloses the two sites (one from each of the rings), forming a possible unit cell (see main text).

**Figure 2.**

**Single ring optical properties**. (

**a**,

**b**) Collective decay rates ${\mathsf{\Gamma}}_{m}$ and (

**c**,

**d**) frequency shifts ${\mathsf{\Omega}}_{m}$ versus angular momentum m, depending on polarization orientation (blue open, blue solid, and orange are for transverse, radial, and tangential polarization, respectively). Left panels correspond to a large ring with $d/\lambda =1/3$ and $N=100$. For comparison, solid lines show the result for an infinite linear chain with transverse (blue) and longitudinal (orange) polarization. Right panels are for $d/\lambda =0.05$ and $N=20$ (Dicke regime). In this case, there are only one (two) bright modes at $m=0$ ($m=\pm 1$) for transverse (tangential and radial) polarization. For tangential polarization, the bright (dark) modes are energetically low (high), whereas the opposite behavior is found for radial and transverse polarization.

**Figure 3.**

**Two coupled identical non-rotated nanorings ($\mathbf{\delta}=\mathbf{0}$)**. (

**a**–

**c**) Collective decay rates ${\mathsf{\Gamma}}_{m}$ and (

**d**–

**f**) frequency shifts ${\mathsf{\Omega}}_{m}$ versus angular momentum m, for two coupled rings of $N=9$ emitters each and $R/\lambda =0.05$. The blue (orange) dashed lines denote the symmetric (anti-symmetric) eigenmodes. For comparison, the single-ring solution for the same parameters is shown (grey solid line). The two rings are separated by the vertical distance $Z=0.5R$, and the emitters have transverse, radial, or tangential polarization (left, middle, or right panels, respectively). For transverse (radial and tangential) polarization, the symmetric band is lower (higher) in energy.

**Figure 4.**

**Two coupled identical non-rotated nanorings ($\mathbf{\delta}=\mathbf{0}$)**. Scaling of the most subradiant eigenmode decay rate for two coupled rings (blue) with $Z/\lambda =0.009$ versus the atom number N of each of the rings. For comparison, we overlay the most subradiant decay rate for a single ring of N atoms (

**a**) and a single ring of ${N}_{\mathrm{tot}}=2N$ atoms (

**b**) (green) with fixed inter-particle distance $d/\lambda =1/3$ and transverse polarization. Similar results are found in the case of tangential polarization.

**Figure 5.**

**Two coupled identical non-rotated nanorings ($\mathbf{\delta}=\mathbf{0}$)**. (

**a**) Most subradiant decay rate of two coupled rings with $N=9$ emitters and transverse polarization, as a function of ring constant $d/\lambda $ and inter-ring distance $z/\lambda $. Subradiant states can exist even beyond the threshold $d/\lambda <1/2$ and $z/\lambda <1/2$ due to destructive wave interference. (

**b**) Overlap of the most subradiant eigenmode with the Bloch waves corresponding to angular momentum $\left|m\right|$. The Bloch waves of each ring can form symmetric and antisymmetric superpositions, and it can be seen that, at various distances, the symmetric superposition of $m=0$ Bloch waves can be subradiant. The parameters are identical to (

**a**) and the overlap oscillates when varying the ring parameters as soon as $d,z\gtrsim \lambda /2$.

**Figure 6.**

**Two coupled identical non-rotated nanorings ($\mathbf{\delta}=\mathbf{0}$).**Field intensity pattern versus real space coordinates in units of the transition wavelength $\lambda $ (cut at $z=6R$) generated by the eigenmodes with $m=0,1,4$ as indicated in the panels. Middle and bottom rows correspond to the symmetric and anti-symmetric eigenmodes, respectively. Top panels are for the single ring, for comparison ($N=9$, $d/\lambda =0.1$, $Z/\lambda =0.2$, tangential polarization).

**Figure 7.**

**Two coupled identical non-rotated nanorings ($\mathbf{\delta}=\mathbf{0}$).**Field intensity pattern versus real space coordinates (cut at $y=6R$) generated by the eigenmodes with $m=0,1,4$ as indicated in the panels. Middle and bottom rows correspond to the symmetric and anti-symmetric eigenmodes, respectively. Top panels are for the single ring, for comparison ($N=9$, $d/\lambda =0.1$, $Z/\lambda =0.2$, tangential polarization).

**Figure 8.**

**Two coupled rotated nanorings**($\mathbf{\delta}\ne \mathbf{0}$). (

**Top panels)**Two identical nanorings ($\mathit{R}=\mathbf{0.05}\mathbf{\lambda}$) with transverse dipole orientation separated by a vertical distance $\mathit{Z}=\mathbf{0.1}\mathit{R}$, depending on the rotation angle $\mathbf{\delta}\in [\mathbf{0},\mathbf{2}\mathbf{\pi}/\mathbf{18}]$: (

**a**) decay rate and (

**b**) frequency shift of the two eigenmodes with $\mathit{m}=\lceil (\mathit{N}-\mathbf{1})/\mathbf{2}\rceil $. An avoided level crossing emerges at $\mathbf{\delta}\sim \mathbf{0.15}$, where the highest energy level changes from being subradiant to radiant, and from being antisymmetric to symmetric. (

**Bottom panels)**Two coplanar unequal nanorings ($\mathit{Z}=\mathbf{0}$) with radius ${\mathit{R}}_{\mathbf{1}}=\mathbf{0.05}\mathbf{\lambda}$ and ${\mathit{R}}_{\mathbf{2}}=\mathbf{0.9}{\mathit{R}}_{\mathbf{1}}$ and tangential dipole orientation, depending on the rotation angle $\mathbf{\delta}$: (

**c**) decay rate and (

**d**) frequency shift of the two eigenmodes with $\mathit{m}=\lceil (\mathit{N}-\mathbf{1})/\mathbf{2}\rceil $. Similarly, as before, an avoided level crossing (shown amplified in the inset) emerges at $\mathbf{\delta}\sim \mathbf{0.07}$, where the highest energy level changes from being radiant to subradiant, and from being symmetric to anti-symmetric.

**Figure 9.**

**LH2 dipole configuration**. (

**a**,

**b**) Collective decay rates and (

**c**,

**d**) frequency shifts as a function of angular momentum index m for the LH2 structure (B800 and B850 bands) parameterized according to [65]. Left and right panels correspond to uncoupled and coupled rings, respectively. The B850 band consists of a two-component unit cell ring with 9-fold symmetry (denoted by blue and orange), whereas the B800 band is a single-component ring with 9-fold symmetry (denoted by violet). The B800 ring is far in energy and thus only couples very weakly to the B850 rings. However, the two components of the B850 band are strongly coupled, due to the reduced inter-particle distance, which leads to a broad dispersion in the frequency shifts. Two bands emerge: a darker band that is higher in energy and close to the B800 band (denoted by cyan), and a brighter band (with two bright modes corresponding to $m=\pm 1$) that is lower in energy (denoted by green). This band structure is relevant for the excitation energy transfer occurring between the B800 and B850 bands.

**Figure 10.**

**LH2 dipole configuration**. Individual ring occupation probabilities for each of the three eigenmodes as a function of angular momentum index m. Blue and orange correspond to the two B850 rings (as indicated in Figure 9), whereas violet is the occupation of the B800 ring. Each panel is a different eigenmode, indicated with the same code color as in Figure 9.

**Figure 11.**

**LH2 dipole configuration**. (

**a**) Frequency shift and (

**b**) ring occupation probabilities for the third band as a function of the overall size of the molecule at the $m=1,4$ mode. The size of ring i is varied via ${R}_{\alpha ,i}=\alpha {R}_{i}$ around the actual size ($\alpha =1$). Solid (dashed) lines correspond to the $m=1$ ($m=4$) mode. The color code is equivalent to the one in Figure 9. Dependent on the mode m the second and third bands as well as the excited state populations cross at ${\alpha}_{\mathrm{c}}<1$. For systems with $\alpha <{\alpha}_{c}$, the third band is occupied by the B850 ring, whereas for $\alpha >{\alpha}_{c}$, it is occupied by the B800 ring.

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**MDPI and ACS Style**

Scheil, V.; Holzinger, R.; Moreno-Cardoner, M.; Ritsch, H.
Optical Properties of Concentric Nanorings of Quantum Emitters. *Nanomaterials* **2023**, *13*, 851.
https://doi.org/10.3390/nano13050851

**AMA Style**

Scheil V, Holzinger R, Moreno-Cardoner M, Ritsch H.
Optical Properties of Concentric Nanorings of Quantum Emitters. *Nanomaterials*. 2023; 13(5):851.
https://doi.org/10.3390/nano13050851

**Chicago/Turabian Style**

Scheil, Verena, Raphael Holzinger, Maria Moreno-Cardoner, and Helmut Ritsch.
2023. "Optical Properties of Concentric Nanorings of Quantum Emitters" *Nanomaterials* 13, no. 5: 851.
https://doi.org/10.3390/nano13050851