Subradiance in Multiply Excited States of Dipole-Coupled V-Type Atoms

TL;DR

This paper extends the understanding of collective atomic subradiance to multilevel V-type atoms, revealing new dark states that enable long-term storage of multiple photonic excitations through dipole-dipole interactions.

Contribution

It introduces a theoretical model for multilevel atomic subradiance, demonstrating the existence of dark states that suppress decay in complex atomic ensembles.

Findings

Discovery of non-radiating dark states in multilevel atomic systems

Long-lived subradiance observed in geometric configurations like triangles and chains

Probabilistic methods for preparing subradiant states using phase-controlled laser pumping

Abstract

We generalize the theoretical modeling of collective atomic super- and subradiance to the multilevel case including spontaneous emission from several excited states towards a common ground state. We show that in a closely packed ensemble of $N$ atoms with $N-1$ distinct excited states each, one can find a new class of non-radiating dark states,, which allows for long-term storage of $N-1$ photonic excitations. Via dipole-dipole coupling only a single atom in the ground state is sufficient in order to suppress the decay of all $N-1$ other atoms. By means of some generic geometric configurations, like a triangle of V-type atoms or a chain of atoms with a $J=0 \to J=1$ transition, we study such subradiance including dipole-dipole interactions and show that even at finite distances long lifetimes can be observed. While generally hard to prepare deterministically, we identify various possibilities for a probabilistic preparation via a phase controlled laser pump and decay.

Tables

Table 1. Table 1: Feeding rates where the negative sign means that it gives the decay rate for the energetically higher lying state.

Feeding Rate from	$|e_1{e}_1{e}_1⟩$	$|e_1{e}_1{e}_2⟩$	$|e_1{e}_2{e}_2⟩$	$|e_2{e}_2{e}_2⟩$	$|e_1{e}_2{e}_1⟩$	$|e_2{e}_2{e}_1⟩$	$|e_2{e}_1{e}_1⟩$	$|e_2{e}_1{e}_2⟩$
to $|{\psi }_d^3⟩$	$0$	$-\frac{1}{3}\gamma +\frac{1}{3}{\Gamma }_1$	$-\frac{1}{3}\gamma +\frac{1}{3}{\Gamma }_2$	0	$-\frac{1}{3}\gamma +\frac{1}{3}{\Gamma }_1$	$-\frac{1}{3}\gamma +\frac{1}{3}{\Gamma }_2$	$-\frac{1}{3}\gamma +\frac{1}{3}{\Gamma }_1$	$-\frac{1}{3}\gamma +\frac{1}{3}{\Gamma }_2$
to $|{\psi }_{sr}^3⟩$	$0$	$-\frac{1}{3}\gamma -\frac{1}{3}{\Gamma }_1$	$-\frac{1}{3}\gamma -\frac{1}{3}{\Gamma }_2$	0	$-\frac{1}{3}\gamma -\frac{1}{3}{\Gamma }_1$	$-\frac{1}{3}\gamma -\frac{1}{3}{\Gamma }_2$	$-\frac{1}{3}\gamma -\frac{1}{3}{\Gamma }_1$	$-\frac{1}{3}\gamma -\frac{1}{3}{\Gamma }_2$

Table 2. Table 2: Decay rates, where the first column is the total decay rate of the two states and the other columns are the decay rates to the lower lying eigentstates. The last column is the ground state and the other six are the six lower lying eigenstates in the one-excitation manifold.

State	Total decay rate	1	2	3	4	5	6	$|g,g,g⟩$
$|{\psi }_d^3⟩$	$2\gamma -{\Gamma }_1-{\Gamma }_2$	$0$	$0$	$-\frac{1}{2}\left(\gamma -{\Gamma }_2\right)$	$-\frac{1}{2}\left(\gamma -{\Gamma }_1\right)$	$-\frac{1}{2}\left(\gamma -{\Gamma }_2\right)$	$-\frac{1}{2}\left(\gamma -{\Gamma }_1\right)$	$0$
$|{\psi }_{sr}^3⟩$	$2\gamma +{\Gamma }_1+{\Gamma }_2$	$-\frac{2}{3}\gamma -\frac{4}{3}{\Gamma }_1$	$-\frac{2}{3}\gamma -\frac{4}{3}{\Gamma }_2$	$-\frac{1}{6}\gamma +\frac{1}{6}{\Gamma }_2$	$-\frac{1}{6}\gamma +\frac{1}{6}{\Gamma }_1$	$-\frac{1}{6}\gamma +\frac{1}{6}{\Gamma }_2$	$-\frac{1}{6}\gamma +\frac{1}{6}{\Gamma }_1$	$0$

Equations

$$|{\psi }_{\mathrm{d}}^N⟩=\frac{1}{\sqrt{N!}}\sum _{\pi \in S_N}\mathrm{sgn}(\pi )\underset{i}{⨂}|s_{\pi (i)}⟩.$$

$$H=H_{\mathrm{A}}+H_{\mathrm{F}}+H_{\mathrm{int}}$$

$$H_{\mathrm{int}}=-\sum _{i=1}^N\sum _{j=1}^{N-1}\left(\overrightarrow{{\mu }_j^i}{\sigma }_j^{i+}\cdot \vec{E}\left({\vec{r}}_i\right)+h.c.\right),$$

$${\mu }_j^i\cdot {\mu }_{j^{\prime }}^i=0.$$

$$\dot{\rho }=i\left[\rho ,H\right]+\mathcal{L}\left[\rho \right]$$

$$H=\sum _{i=1}^N\sum _{j=1}^{N-1}{\omega }_j{\sigma }_j^{i+}{\sigma }_j^{i-}+\sum _{i\ne k}^N\sum _{j,j^{\prime }}^{N-1}{\Omega }_{j{j}^{\prime }}^{ik}{\sigma }_j^{i+}{\sigma }_{j^{\prime }}^{k-}$$

$$\mathcal{L}\left[\rho \right]=\sum _{i,k}^N\sum _{j,j^{\prime }}^{N-1}{\Gamma }_{j{j}^{\prime }}^{ik}\left(2{\sigma }_j^{i-}\rho {\sigma }_{j^{\prime }}^{k+}-{\sigma }_j^{i+}{\sigma }_{j^{\prime }}^{k-}\rho -\rho {\sigma }_j^{i+}{\sigma }_{j^{\prime }}^{k-}\right),$$

$$\begin{gathered} {\Omega }_{j{j}^{\prime }}^{ik}=\frac{3\sqrt{{\gamma }_j{\gamma }_{j^{\prime }}}}{2}[\left({\vec{\mu }}_j^i\cdot {\vec{\mu }}_{j^{\prime }}^k\right)P_R\left(k_0{r}_{ik}\right) \\ -\left({\vec{\mu }}_j^i\cdot {\vec{r}}_{ik}\right)\left({\vec{\mu }}_{j^{\prime }}^k\cdot {\vec{r}}_{ik}\right)Q_R\left(k_0{r}_{ik}\right)], \end{gathered}$$

$$\begin{gathered} {\Gamma }_{j{j}^{\prime }}^{ik}=\frac{3\sqrt{{\gamma }_j{\gamma }_{j^{\prime }}}}{2}[\left({\hat{\mu }}_j^i\cdot {\hat{\mu }}_{j^{\prime }}^k\right)P_I\left(k_0{r}_{ik}\right) \\ -\left({\hat{\mu }}_j^i\cdot {\hat{r}}_{ik}\right)\left({\hat{\mu }}_{j^{\prime }}^k\cdot {\hat{r}}_{ik}\right)Q_I\left(k_0{r}_{ik}\right)]. \end{gathered}$$

$$P_R(\xi )=$$

$$P_I(\xi )=$$

$$Q_R(\xi )=$$

$$Q_I(\xi )=$$

$$\begin{gathered} |{\Psi }_d^3⟩=\frac{1}{\sqrt{6}}\{|e_1{e}_{2}g⟩+|g{e}_1{e}_2⟩+|e_{2}g{e}_1⟩ \\ -|e_{1}g{e}_2⟩-|e_2{e}_{1}g⟩-|g{e}_2{e}_1⟩\}, \end{gathered}$$

$$⟨\psi |\mathcal{L}[|{\psi }_d^3⟩⟨{\psi }_d^3|]|\psi ⟩\text{and }⟨\psi |\mathcal{L}[|{\psi }_{sr}^3⟩⟨{\psi }_{sr}^3|]|\psi ⟩$$

$$⟨{\psi }_d^3|\mathcal{L}[|e_i{e}_j{e}_k⟩⟨e_i{e}_j{e}_k|]|{\psi }_d^3⟩\text{and }⟨{\psi }_{sr}^3|\mathcal{L}[|e_i{e}_j{e}_k⟩⟨e_i{e}_j{e}_k|]|{\psi }_{sr}^3⟩,$$

Full text

I Introduction

Quantum fluctuations in the electromagnetic vacuum field inevitably lead to energy dissipation from excited atomic states via the spontaneous emergence of photons Dirac (1927) known as spontaneous emission. In a quantum electrodynamics treatment the probability for this process and its corresponding decay rate $\Gamma=\omega_{0}^{3}\mu^{2}/(3\pi\epsilon_{0}\hbar c^{3})$ was first derived by Weiskopf and Wigner Weisskopf (1935). It is proportional to the third power of the transition energy between the excited and lower lying state as well as to the square of the transition dipole moment between those two states.

As there is only one electromagnetic vacuum, atoms in close proximity will experience correlated fluctuations inducing cooperative effects in their dissipative behavior. By means of constructive as well as destructive interference of the emerging photons the collective spontaneous emission rates are drastically modified as a function of distance Lehmberg (1970a, b); Ficek et al. (1987); Agarwal and Patnaik (2001). A strongly increased spontaneous emission is dubbed ’superradiance’ while a decreased rate is referred to as ’subradiance’ Gross and Haroche (1982).

Due to the quantum nature of atomic excitations, they can be delocalized and distributed over an entire atomic ensemble, exhibiting highly multi-partite entanglement Lukin et al. (2000); Chou et al. (2005); Plankensteiner et al. (2015). Well known examples are the single-excitation Bell states of two atoms Cabrillo et al. (1999); Raimond et al. (2001), the W-state Eibl et al. (2004); Zou et al. (2002) and many others.

Depending on the geometry of the atomic ensemble as well as on the local phase difference of the excitation amplitudes between the atoms, such delocalized excitation states can feature either super- or subradiance. For instance, for two closely spaced atoms ($d\ll\lambda_{0}=2\pi c/\omega_{0}$), the symmetric Bell state $\ket{+}=\left(\ket{eg}+\ket{ge}\right)/\sqrt{2}$ is superradiant, while its asymmetric analogue $\ket{-}=\left(\ket{eg}-\ket{ge}\right)/\sqrt{2}$ is strongly subradiant and decouples from the radiation field completely at distances close to zero Dicke (1954). This leads to the term ’dark state’. Because of the fact that their lifetime is often orders of magnitude longer than typical experimental cycles, those dark states are a valuable resource in quantum information storage and processing Fleischhauer and Lukin (2002); Chaneliere et al. (2005).

While subradiant states of dense atomic ensembles are easy to identify theoretically Temnov and Woggon (2005); Asenjo-Garcia et al. (2017), they have been quite elusive and hard to find in concrete experiments Guerin et al. (2016); Bromley et al. (2016), with directional emission patterns as one of the signatures of destructive interference leading to subradiance Bhatti et al. (2018). Besides the influence of motion and various dephasing mechanisms, it was recently pointed out, that the complex level structure of typical atoms beyond a two-level approximation will often prevent the appearance of perfectly dark states Hebenstreit et al. (2017). In particular, for excited atomic states, which can decay to different lower states via more than one decay channel, the observation of subradiance is much more challenging. It can be easily shown that for a system of two $\Lambda$-type atoms no dark state can be found, as both decay channels need to be blocked via interference, which cannot be achieved simultaneously. However, in earlier work Hebenstreit et al. (2017) we could show that an ensemble of $N$ $N$-level atoms with and $N-1$ independent decay channels from the excited state to $N-1$ different ground states, a unique perfectly dark state, can be identified. This completely anti-symmetric dark state has remarkable entanglement and symmetry properties making it a promising candidate for quantum information applications.

In this paper, we investigate a related system, namely the inverted energy level configuration involving $N$ atoms where one ground state is coupled to $N-1$ excited states $\ket{i}=\ket{s_{i}}=\ket{e_{i}}$. Each upper state can decay independently to a common ground state $\ket{0}=\ket{s_{0}}=\ket{g}$. Again the totally anti-symmetric state is a dark state of a similar form

[TABLE]

Here the sum runs over all permutations $\pi$ of $N$ elements. Using a spatially symmetric configuration of three atoms we will show below, that this $N$-level state of $N$ atoms is subradiant as well as an eigentstate of the Hamiltonian.

II Model

Let us consider a collection of $N$ identical V-level type atoms at fixed positions $\left\{\vec{r}_{i}\right\}_{i=1}^{N}$. Each atom features $N-1$ excited states $\left\{\ket{e_{j}}\right\}_{j=1}^{N-1}$ at energies $\omega_{j}$ with dipole coupling to a common ground state $\ket{g}$ via a transition dipole moment of $\vec{\mu}_{j}$.

The combined Hamiltonian of the atoms and the electromagnetic field is given by

[TABLE]

with the atomic part $H_{\mathrm{A}}=\sum_{i=1}^{N}\sum_{j=1}^{N-1}\omega_{j}\sigma_{j}^{i+}\sigma_{j}^{i-}$ and the field $H_{\mathrm{F}}=\sum_{\vec{k},\lambda}\omega_{k}{a}_{\vec{k},\lambda}^{\dagger}a_{\vec{k},\lambda}$.

The interaction between the atoms and the field in dipole approximation is then

[TABLE]

where $\vec{E}\left(\vec{r}_{i}\right)$ is the quantized electromagnetic field. When particularizing to $N=3$ below, we will consider a situation where the transition dipole matrix elements inside each atom are mutually orthogonal and real, that is

[TABLE]

After tracing out the electromagnetic field modes in a standard quantum optics fashion assuming the field in its vacuum state Gardiner et al. (2004); Ficek and Tanaś (2002); Agarwal and Patnaik (2001); Moy et al. (1999) the system dynamics can be described by the master equation

[TABLE]

with the effective Hamiltonian including dipole-dipole interaction

[TABLE]

and the Liouvillian in Lindblad form

[TABLE]

where $\sigma^{i\pm}_{j}$ denotes the rising (lowering) operator of the $j$-th transition in the $i$-th atom.

The coherent part of the dipole-dipole interaction induces energy shifts (see Fig. 1) due to the couplings

[TABLE]

while the incoherent collective dissipation is characterized by

[TABLE]

Furthermore, for brevity we have introduced the functions

[TABLE]

where $r_{ik}=\left|\vec{r}_{i}-\vec{r}_{k}\right|$ represents the interatomic distance between atom $i$ and atom $k$, and $k_{0}=\omega_{0}/c$ with $\omega_{0}=\left(\omega_{j}+\omega_{j^{\prime}}\right)/2$ and $\Gamma^{ii}_{jj^{\prime}}=\gamma_{j}=2\mu_{j}^{2}{\omega_{j}}^{3}/\left(3\epsilon_{0}c^{3}\right)$ is the spontaneous emission rate of a single atom on the $j$-th transition. The couplings for the energy shifts as well as the collective decays are plotted in Fig. 2 as a function of the interatomic distance, whereas varying the dipole moment orientations leads to oscillations of various amplitudes (see Fig. 3). The terms $\Gamma^{12}_{12}$ and $\Omega^{12}_{12}$ are dipole-dipole cross coupling coefficients, which couple dipoles even though they are orthogonal. Agarwal and Patnaik (2001).

III Equilateral Triangle: Analytical Treatment

For three 3-level atoms placed at the corners of an equilateral triangle with dipole orientations chosen such that the configuration features a $C_{3}$ symmetry (see Fig. 3), the states $\ket{\Psi_{d}^{3}}$ and $\ket{\Psi_{sr}^{3}}$ are both eigenstates of the Hamiltonian from eq. (6) whose energies can be calculated explicitly. For three V-type atoms $\ket{\Psi_{d}^{3}}$ is given by

[TABLE]

whereas in the superradiant analogue $\ket{\Psi_{sr}^{3}}$, which is comprised of the exact same bare states, all signs are positive.

Clearly, the dynamics of any eigenstate of the Hamiltonian is restricted to the decay towards other eigenstates $\ket{\psi_{\mathrm{eig}}}$ induced by the Liouvillian, i.e. $\dot{{\rho}}_{\mathrm{eig}}=\mathcal{L}\left[{\rho}_{\mathrm{eig}}\right]$ with $\rho_{\mathrm{eig}}=\ket{\psi_{\mathrm{eig}}}\bra{\psi_{\mathrm{eig}}}$. The corresponding rates can be found by calculating the overlap with all other states. The decay and feeding rates for a certain selection of states are shown in Fig. S1. Explicitly, the decay rate for the eigenstate $\ket{\psi_{\mathrm{eig}}}$ is given by $\Braket{\psi_{\mathrm{eig}}}{\mathcal{L}\left[{\rho}_{\mathrm{eig}}\right]}{\psi_{\mathrm{eig}}}$.

We find that the lowest lying energy state in the double excitation manifold corresponds to the antisymmetric dark state $\ket{\Psi_{d}^{3}}$, while the highest energy state is the superradiant state. With a more and more pronounced subradiance in $\ket{\Psi_{d}^{3}}$ at decreasing interatomic distances, also its feeding rate from higher lying states decreases, which culminates in a decoupling from all other states and the electromagnetic field. In particular, for the equilateral triangle configuration, the lower an eigenstate lies energetically, the smaller its decay rate, as can be seen for selected states in Fig. S1. A full account of all coupling and feeding rates is available in the supplementary information Holzinger R. (2019). Also note that all feeding and decay rates to and from a particular state sum up to zero.

IV Numerical Diagonalization for three and more Atoms

For the case of $N\geq 3$ atoms we analyze the scaling of the decay rates as a function of the interatomic distance for increasing atom numbers and different geometries.

In Fig. 5(a) the simple case of two two-level atoms is shown, where the sub- and superradiant decay rates oscillate around the independent decay rate $\Gamma$ with an amplitude decreasing with the interatomic distance, such that the super- and subradiant state switch their roles at each node. The black dashed curve corresponds to the lowest decay rate at any given distance and is generalized to more involved configurations in Fig. 5(b).

In Fig. 5 it can be seen, that the lowest collective decay rate for the $(N-1)$-excitation manifold for $N$ atoms goes to zero only if the interatomic distances approach zero, if all dipole transition moments are orthogonal to the plane of the atomic ensemble. For the equilateral triangle with symmetric dipole orientations and for $N\geq 4$ atoms with more than two transitions this is not possible anymore and the minimal decay rate is $\Gamma$.

V Dark State Preparation

In most geometric configurations apart from the equilateral triangle the anti-symmetric state $\ket{\psi_{d}^{3}}$ is not an exact eigenstate of the Hamiltonian from Eq. (6) Yet, its subradiant property will prevail as shown for a linear chain in Fig. 6. The state $\ket{\Psi_{\mathrm{unp}}}=\left(\ket{e_{1}e_{2}}+\ket{e_{2}e_{1}}\right)/\sqrt{2}\otimes\ket{g}$ denotes a product state with atoms $1$ and $2$ entangled and exhibits subradiance as well. Generally, subradiance becomes particularly apparent at small atomic distances, where the derivative of the incoherent coupling with respect to $r$ is almost zero.

At finite distances $\ket{\psi_{d}^{3}}$ can couple to to other states and will therefore decay as shown in 6. Naturally, this means that it can be populated via decay from a higher lying state, which in this case are all triply excited states. A typical case where the dark state becomes populated by photon emission for a three qutrit chain prepared in a totally inverted state, $\ket{e_{1},e_{2},e_{1}}$, is demonstrated in Fig. 8. Note that there are, in fact, eight different possibilities for triply excited states, i.e. $\ket{e_{i},e_{j},e_{k}}$ with $i,j,k\in\{1,2\}$, which lead to similar results. In Fig. 8 it can be seen, that the dark state can acquire a significant population, even via purely dissipative preparation, by choosing an appropriate geometric configuration. On the other hand, the feeding rate for the dark state becomes smaller with decreasing distances as it starts to decouple from the electromagnetic field.

As we have seen above, after an initial build-up of population in the dark state, the remainder of the population mostly ends up in the ground state. Hence, one can think of reusing the atoms in the ground state in order to further increase the occupation of the dark state. For this purpose , the preparation of the dark state $\ket{\Psi_{d}^{3}}$ or its superradiant analogue $\ket{\Psi_{sr}^{3}}$ can be facilitated by a continuous pump laser. It turns out that using different excitation phases for each atom can strongly improve the efficiency of this process, although this might be challenging to implement in practice.

We include a continuous pump in our model by adding the term ${H}_{\mathrm{pump}}=\sum_{i=1}^{3}\sum_{j=1}^{2}\eta_{i}({\sigma}^{i}_{j}+{\sigma}^{i\dagger}_{j})$ to the Hamiltonian with $\eta_{1}=\eta$, $\eta_{2}=\eta e^{i\varphi_{1}}$ and $\eta_{3}=\eta e^{i\varphi_{2}}$, assuming that all atoms are driven with the same strength $\eta$. In our example the atoms are initialized in the ground state, $\ket{ggg}$, and we look at the population of the dark state after a given laser illumination time.

In Fig. 9 the preparation probabilities for $\ket{\Psi_{d}^{3}}$ in a linear chain and for its superradiant analogue $\ket{\Psi_{sr}^{3}}$ in an equilateral triangle are shown as a function of the laser phase using a constant pump amplitude of $\eta=8.5\Gamma$ at an interatomic distance of $\lambda_{0}/50$ in both cases. For the linear chain it can be seen, that for instance if atom $2$ and $3$ are driven by phases $\varphi_{1}=\pi/2$ and $\varphi_{2}=\pi$ relative to atom $1$, the preparation probability for $\ket{\Psi_{d}^{3}}$ reaches $20\%$. The state will still decay, but with a small rate, as given in Fig. 6. In contrast, setting $\varphi_{1}=0$ and $\varphi_{2}=0$ in the equilateral triangle results in a preparation probability of $30\%$ for $\ket{\Psi_{sr}^{3}}$. We find a surprisingly high preparation probability after a time evolution of $\Gamma t=0.3$.

Now, we include different phases for different transitions by writing our pump Hamiltonian as ${H}_{\mathrm{pump}}=\sum_{i=1}^{3}\sum_{j=1}^{2}\eta^{i}_{j}({\sigma}^{i}_{j}+{\sigma}^{i\dagger}_{j})$. Figure 10 (a) shows the preparation probability for $\ket{\Psi_{d}^{3}}$ for a range of different phases, where for instance for $\varphi_{1}=\varphi_{2}=7/10\pi$ a maximum of $\approx 24\%$ is reached after $\Gamma t=0.3$. In Fig. 10 (b) we compare the time evolution of a pulsed laser with a continuous drive. Both cases lead to the same maximal value after $\Gamma t=0.3$, but, after turning off the laser the dissipative dynamics lead to larger preparation probabilities shortly after that. Only for times longer than $\Gamma t=1.5$ the laser driven system dominates the preparation probability. Specifically, for the case of pulsed lasing in Fig. 10 (b) the first peak corresponds to a preparation probability of $24\%$ and the second peak to $15\%$, both within an evolution time of $\Gamma t=1$.

VI Conclusions

We have generalized the concept of subradiance to multilevel emitters with several excited atomic levels decaying via independent decay channels towards a common ground state. In these systems the most subradiant states are completely anti-symmetric and maximally entangled. In contrast to ensembles of two-level emitters this multilevel type of dark states can hold several excitation quanta without decay. Hence detection could be facilitated by non-classical photon correlations at long time delays.

Entangled subradiant states have promising applications in quantum information processing and optical lattice clocks Krämer et al. (2016); Ostermann et al. (2014), amongst other key quantum technologies, where longer coherence times and a better understanding of energy level shifts induced via dipole-dipole interactions are crucial for improved accuracies.

States that do not decay as they decouple from the radiation field in turn are hard to access in order to prepare them directly. Yet, a probabilistic preparation can be achieved via spontaneous emission from higher lying states or in a much more efficient way by the application of laser pulses with spatial phase control.

Future work in this lines of studies will include coupling to a cavity field and analyzing the emission and absorption behaviour of multiple V-type emitters via an input/output formalism as in Plankensteiner et al. (2018). Another direction is the inclusion of vibrational degrees of freedom for each emitter as is demonstrated in Holstein (1959); Michael Reitz (2018) via a Holstein Hamiltonian for 2-level emitters.

Acknowledgements

Financial support for this publication has been provided by the European Research Commission through the Quantum Flagship project iqClock (R. H. and H. R.) as well as by the Austrian Science Fund FWF through project P29318-N27 (L. O.).

VII Decay Cascade For Three 3-level Emitters

By considering

[TABLE]

where $\ket{\psi}$ are all the lower lying eigenstates into which the superradiant and subradiant state decays, we obtain the decay rates into the respective states. For $\ket{\psi}=\ket{\psi^{3}_{d}}$ or $\ket{\psi}=\ket{\psi^{3}_{sr}}$ we obtain the total decay rate.

Whereas for

[TABLE]

where $i,j,k=1,2$ and $e_{i}e_{j}e_{k}$ are the 8 possible inverted states which can feed the Super- and Subradiant states, we obtain the feeding rates.

From the decay rate of the dark state we see that in the limiting case of infinitely close atoms, the decay could become even zero and the state would be indeed stationary under the Liouvillian superoperator. As is demonstrated in the main text, for the triangle it approaches $1\Gamma$ for infinitesimal distances and zero for the linear chain.