# Perfect Photon Indistinguishability from a Set of Dissipative Quantum Emitters

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}molecules was developed in 2013 [7]; (Quantum Metrology) A IQP platform based on defects in diamond was used for extremely efficient detectors of magnetic fields with unprecedent sensitivity [8]. In contrast to other QT platforms, IQP leverage commercially available systems from the integrated photonics industry, which provide reliable devices for on-chip integration [9] and metamaterial systems for broadband operation [10,11]. In this context, IQP shows a new leading candidate for the future q-bit in QT: the indistinguishable single photon.

^{(2)}(0) = 7 $\times $ 10

^{−3}and I = 0.96 was reported with InAs/GaAs QDs embedded in a micropillar cavity at 4.3 K [17]; g

^{(2)}(0) = 1.2 $\times $ 10

^{−2}and I = 0.97 with InAs/GaAs QDs integrated in a DBR microcavity at 4.2 K [18]; g

^{(2)}(0) = 2.8 $\times $ 10

^{−3}and I = 0.99 with InGaAs/GaAs QDs inside DBR micropillars at 4 K [19]. However, for T above the cryogenic regime, QDs are subject to pure dephasing mechanisms which reduce the coherence of the emission [20,21,22]: g

^{(2)}(0) = 0.47 with InGaAs/GaAs QDs at 120 K [23]; g

^{(2)}(0) = 0.34 with InAs/InP QDs at 80 K [24]; g

^{(2)}(0) = 0.48 with GaAs/GaAsP QDs at 160 K [25]. For T > 200 K the best reported value is g

^{(2)}(0) = 0.34 [26]. As a consequence, I is reduced to non-practical values for quantum information tasks: I > 0.79 for most quantum information processing schemes and I > 0.5 for QKD protocols [27]. In this regard, QDs for SPS operation are restricted to low T. In an attempt to overcome this limitation, a variety of cavity-engineering approaches have been conducted [28,29]. However, several theoretical works [30,31,32] indicate that cavity quality factors (Q) above 4 $\times $ 10

^{7}are required for QDs to function at room T, while, to date, the highest reported Q coupled to a quantum emitter is about Q = 55,000 [33]. In this regard, the theoretical exploration over new strategies for enhancing I in the presence of dephasing processes is especially relevant.

## 2. Materials and Methods

#### 2.1. Dipole-Dipole Coupling Model

#### 2.2. Larger Systems

#### 2.3. Machine Learning Scheme

^{5}evaluations of the fitness function. If we directly use QRT for each evaluation, the optimization would require excessive computational times. Instead, in our approach we first generate a data set ($\mathit{\omega}$, I) with the results obtained from 2000 iterations. With these data, we train a deep NN which learns to estimate the outcome of $I$ for any possible set of random positions $\overrightarrow{\omega}$. Now, each time the GA creates a random vector $\mathit{\omega}$, the evaluation of the fitness function obtains I from the estimation of the NN. This way, each evaluation takes just a few seconds. Through the iteration of cross-over and mutation, the GA finds the optimal configuration for maximizing I after a certain number of generations. Therefore, with our NN-GA scheme we reduce the number of actual numerical simulations for the dataset by two orders of magnitude.

^{−3}, giving enough accuracy for the estimation of I and the optimization model. The genetic algorithm uses decimal representation for the genes, one-point crossover and uniform mutation. The total initial population was set to 5000, the number of parents mattings = 2500, number of weights = 1000. Using these values, we needed over 216 generations to find each optimal geometry.

## 3. Results

#### 3.1. Indistinguishability of Dipole Coupled Emitters

#### 3.2. Larger Systems

_{eff}, and Q).

_{ij}) that can be tuned by setting the relative distances between the QEs, so we have enough parameters to perform a sufficiently complex optimization. Figure 2d shows a layout of the system where each Bloch-sphere represents the time evolution of each QE

_{i}, and each arrow represents the specific transfer rate between the QE

_{i}and QE

_{j}. Our aim now is to find the geometrical configuration of the QEs that provides the optimal set of R

_{ij}that keep high I for high $\kappa $ values. This goal involves an optimization task with 10 degrees of freedom, which is a highly non-trivial problem and computationally very time-consuming. Nevertheless, similar optimization problems have been recently solved using machine-learning methods [28,47,48,49,50]. Employing a similar approach, we developed a machine-learning scheme based on a hybrid NN-GA algorithm which is able to solve the optimization problem in very short computational times providing the best geometrical configuration for the emitters.

#### 3.3. Machine Learning Optimization

_{ij}(I, j = 1, …, 5) between QEs leads to a dipolar interaction strength ${\mathsf{\Omega}}_{ij}$ and modified decay rate ${\gamma}_{ij}$. Since this scheme requires solving a system of 144 coupled differential equations, we are not able to derive an analytic expression for I such as in the two-QE case. Instead, we numerically solve the Lindblad equation of the system and compute I via QRT. At each iteration we generate a vector $\mathit{\omega}$ with five random positions for the QEs and we calculate I via QRT for a fixed g and $\kappa $. The data set ($\mathit{\omega}$, I) is then used to train the NN-GA algorithm which finds the optimal positions for maximum I for that g and $\kappa $. In Figure 3a–e we report the obtained optimal geometries for g = $\gamma $ and $\kappa $ = 10 $\gamma $, 50 $\gamma $, 100 $\gamma $, 500 $\gamma $ and 1000 $\gamma $, respectively. All these geometries provide perfect I (I = 1) with minimum distances d

_{ij}$~$0.1 $\lambda $, a value compatible with experimental realizations [34,35,36,37,38,39,40,41]. Each geometry leads to the right transfer rates R

_{ij}between the subsystems for keeping the stability at the specific rates g and $\kappa $. For a fixed geometry, small changes in g and $\kappa $ drastically reduce I. This is displayed in Figure 3f, which shows I versus normalized g/$\gamma $ and $\kappa /\gamma $ for the optimal geometry obtained for g = $\gamma $, $\kappa $ = 10 $\gamma $. The plot shows a small “bubble” of high I at the (g/$\gamma $, $\kappa /\gamma $) = (1,10) point, while in the neighbor regions of the bubble I reduces to 0. Figure 3a–e also shows the positioning tolerances for each QE for obtaining I > 0.9. The tolerances for the accuracy in the position depend on the specific QE and the (g, $\kappa $) values.

_{eff}and Q were obtained from the field profile (see Figure 3g) and frequency analysis of the resonance. For a QE with ($\gamma ,$ ${\gamma}^{*}$, $\mathit{\omega}$) = (160 MHz, 400 GHz and 400 THz) such as color centers in diamond [52] we obtained (g, $\kappa $) ≈ (1, 100). The radius and distances between the holes of the PCc were set to 120 nm and 50 nm, respectively, which is compatible with most fabrication techniques [53,54,55]. To highlight the benefits of our strategy we have contrasted the obtained performance with standard single-emitter-cavity systems [30] for different QEs at high T. Diamond color centers, InGaAs QDs, GaAs QDs and single molecules at 300 K has a pure dephasing of 1000 $\gamma $, 600 $\gamma $, 1450 $\gamma $ and 10

^{4}γ, respectively [20,21,52,56]. Considering the same standard PCc with (g, $\kappa $) ≈ (1, 100), a single-emitter-cavity system leads to I$~$0.01 for all these emitters, whereas the five-QEs optimized platform provides I = 1. For these emitters, obtaining I = 1 with a single-emitter-cavity at room T would require at least a cavity with Q above 4 $\times $ 10

^{7}, which is beyond the state of the art for most current fabrication technologies.

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Monroe, C.; Meekhof, D.M.; King, B.E.; Itano, W.M.; Wineland, D.J. Demonstration of a fundamental quantum logic gate. Phys. Rev. Lett.
**1995**, 75, 4714. [Google Scholar] [CrossRef] [PubMed] - Makhlin, Y.; Scöhn, G.; Shnirman, A. Josephson-junction qubits with controlled couplings. Nature
**1999**, 398, 305–307. [Google Scholar] [CrossRef] - Politi, A.; Cryan, M.J.; Rarity, J.G.; Yu, S.; O’brien, J.L. Silica-on-silicon waveguide quantum circuits. Science
**2008**, 320, 646–649. [Google Scholar] [CrossRef] [PubMed] - Bunandar, D.; Lentine, A.; Lee, C.; Cai, H.; Long, C.M.; Boynton, N.; Martinez, N.; DeRose, C.; Chen, C.; Grein, M. Metropolitan quantum key distribution with silicon photonics. Phys. Rev. X
**2018**, 8, 021009. [Google Scholar] [CrossRef] - Lago-Rivera, D.; Grandi, S.; Rakonjac, J.V.; Seri, A.; de Riedmatten, H. Telecom-heralded entanglement between multimode solid-state quantum memories. Nature
**2021**, 594, 37–40. [Google Scholar] [CrossRef] [PubMed] - Zhong, H.S.; Wang, H.; Deng, Y.H.; Chen, M.C.; Peng, L.C.; Luo, Y.H.; Qin, J.; Wu, D.; Ding, X.; Hu, Y. Quantum computational advantage using photons. Science
**2020**, 370, 1460–1463. [Google Scholar] [CrossRef] [PubMed] - Peruzzo, A.; McClean, J.; Shadbolt, P.; Yung, M.H.; Zhou, X.Q.; Love, P.J.; Aspuru-Guzik, A.; O’brien, J.L. A variational eigenvalue solver on a photonic quantum processor. Nat. Commun.
**2014**, 5, 4213. [Google Scholar] [CrossRef] [PubMed] - Acosta, V.M.; Bauch, E.; Ledbetter, M.P.; Santori, C.; Fu, K.M.; Barclay, P.E.; Beausoleil, R.G.; Linget, H.; Roch, J.F.; Treussart, F. Diamonds with a high density of nitrogen-vacancy centers for magnetometry applications. Phys. Rev. B
**2009**, 80, 115202. [Google Scholar] [CrossRef] - Pelucchi, E.; Fagas, G.; Aharonovich, I.; Englund, D.; Figueroa, E.; Gong, Q.; Hannes, H.; Liu, J.; Lu, C.Y.; Matsuda, N. The potential and global outlook of integrated photonics for quantum technologies. Nat. Rev. Phys.
**2022**, 4, 194–208. [Google Scholar] [CrossRef] - Zhang, H.; Zhang, H.F.; Liu, G.B.; Li, H.M. Ultra-broadband multilayer absorber with the lumped resistors and solid-state plasma. Results Phys.
**2019**, 12, 917–924. [Google Scholar] [CrossRef] - Cheben, P.; Halir, R.; Schmid, J.H.; Atwater, H.A.; Smith, D.R. Subwavelength integrated photonics. Nature
**2018**, 560, 565–572. [Google Scholar] [CrossRef] [PubMed] - Zhang, Y.; Wang, Z.; Su, Y.; Zheng, Y.; Tang, W.; Yang, C.; Tang, H.; Qu, L.; Li, Y.; Zhao, Y. Simple vanilla derivatives for long-lived room-temperature polymer phosphorescence as invisible security inks. Research
**2021**, 2021, 8096263. [Google Scholar] [CrossRef] [PubMed] - Wu, S.; Xia, H.; Xu, J.; Sun, X.; Liu, X. Manipulating luminescence of light emitters by photonic crystals. Adv. Mater.
**2018**, 30, 1803362. [Google Scholar] [CrossRef] [PubMed] - Juska, G.; Dimastrodonato, V.; Mereni, L.O.; Gocalinska, A.; Pelucchi, E. Towards quantum-dot arrays of entangled photon emitters. Nat. Photonics
**2013**, 7, 527–531. [Google Scholar] [CrossRef] - Gérard, J.M.; Sermage, B.; Gayral, B.; Legrand, B.; Costard, E.; Thierry-Mieg, V. Enhanced spontaneous emission by quantum boxes in a monolithic optical microcavity. Phys. Rev. Lett.
**1998**, 81, 1110. [Google Scholar] [CrossRef] - Hennessy, K.; Badolato, A.; Winger, M.; Gerace, D.; Atatüre, M.; Gulde, S.; Fält, S.; Hu, E.L.; Imamoğlu, A. Quantum nature of a strongly coupled single quantum dot–cavity system. Nature
**2007**, 445, 896–899. [Google Scholar] [CrossRef] - Wang, H.; Duan, Z.C.; Li, Y.H.; Chen, S.; Li, J.P.; He, Y.M.; Chen, M.C.; He, Y.; Ding, X.; Peng, C.Z.; et al. Near-transform-limited single photons from an efficient solid-state quantum emitter. Phys. Rev. Lett.
**2016**, 116, 213601. [Google Scholar] [CrossRef] - He, Y.M.; He, Y.; Wei, Y.J.; Wu, D.; Atatüre, M.; Schneider, C.; Höfling, S.; Kamp, M.; Lu, C.Y.; Pan, J.W. On-demand semiconductor single-photon source with near-unity indistinguishability. Nat. Nanotechnol.
**2013**, 8, 213–217. [Google Scholar] [CrossRef] - Somaschi, N.; Giesz, V.; De Santis, L.; Loredo, J.C.; Almeida, M.P.; Hornecker, G.; Portalupi, S.L.; Grange, T.; Anton, C.; Demory, J.; et al. Near-optimal single-photon sources in the solid state. Nat. Photonics
**2016**, 10, 340–345. [Google Scholar] [CrossRef] - Borri, P.; Langbein, W.; Schneider, S.; Woggon, U.; Sellin, R.L.; Ouyang, D.; Bimberg, D. Ultralong dephasing time in InGaAs quantum dots. Phys. Rev. Lett.
**2001**, 87, 157401. [Google Scholar] [CrossRef] - Bayer, M.; Forchel, A. Temperature dependence of the exciton homogeneous linewidth in In 0.60 Ga 0.40 As/GaAs self-assembled quantum dots. Phys. Rev. B
**2002**, 65, 041308. [Google Scholar] [CrossRef] - Berthelot, A.; Favero, I.; Cassabois, G.; Voisin, C.; Delalande, C.; Roussignol, P.; Ferreira, R.; Gérard, J.M. Unconventional motional narrowing in the optical spectrum of a semiconductor quantum dot. Nat. Phys.
**2006**, 2, 759–764. [Google Scholar] [CrossRef] - Mirin, R.P. Photon antibunching at high temperature from a single InGaAs/GaAs quantum dot. Appl. Phys. Lett.
**2004**, 84, 1260–1262. [Google Scholar] [CrossRef] - Dusanowski, Ł.; Syperek, M.; Misiewicz, J.; Somers, A.; Hoefling, S.; Kamp, M.; Reithmaier, J.P.; Sęk, G. Single-photon emission of InAs/InP quantum dashes at 1.55 μm and temperatures up to 80 K. Appl. Phys. Lett.
**2016**, 108, 163108. [Google Scholar] [CrossRef] - Yu, P.; Li, Z.; Wu, T.; Wang, Y.T.; Tong, X.; Li, C.F.; Wang, Z.; Wei, S.H.; Zhang, Y.; Liu, H.; et al. Nanowire quantum dot surface engineering for high temperature single photon emission. ACS Nano
**2019**, 13, 13492–13500. [Google Scholar] [CrossRef] [PubMed] - Arakawa, Y.; Holmes, M.J. Progress in quantum-dot single photon sources for quantum information technologies: A broad spectrum overview. Appl. Phys. Rev.
**2020**, 7, 021309. [Google Scholar] [CrossRef] - Bylander, J.; Robert-Philip, I.; Abram, I. Interference and correlation of two independent photons. Eur. Phys. J. D-At. Mol. Opt. Plasma Phys.
**2003**, 22, 295–301. [Google Scholar] [CrossRef] - Guimbao, J.; Sanchis, L.; Weituschat, L.; Manuel Llorens, J.; Song, M.; Cardenas, J.; Aitor Postigo, P. Numerical Optimization of a Nanophotonic Cavity by Machine Learning for Near-Unity Photon Indistinguishability at Room Temperature. ACS Photonics
**2022**, 9, 1926–1935. [Google Scholar] [CrossRef] - Guimbao, J.; Weituschat, L.M.; Montolio, J.L.; Postigo, P.A. Enhancement of the indistinguishability of single photon emitters coupled to photonic waveguides. Opt. Express
**2021**, 29, 21160–21173. [Google Scholar] [CrossRef] - Grange, T.; Hornecker, G.; Hunger, D.; Poizat, J.P.; Gérard, J.M.; Senellart, P.; Auffèves, A. Cavity-funneled generation of indistinguishable single photons from strongly dissipative quantum emitters. Phys. Rev. Lett.
**2015**, 114, 193601. [Google Scholar] [CrossRef] - Choi, H.; Zhu, D.; Yoon, Y.; Englund, D. Cascaded cavities boost the indistinguishability of imperfect quantum emitters. Phys. Rev. Lett.
**2019**, 122, 183602. [Google Scholar] [CrossRef] [PubMed] - Saxena, A.; Chen, Y.; Ryou, A.; Sevilla, C.G.; Xu, P.; Majumdar, A. Improving indistinguishability of single photons from colloidal quantum dots using nanocavities. ACS Photonics
**2019**, 6, 3166–3173. [Google Scholar] [CrossRef] - Ota, Y.; Iwamoto, S.; Kumagai, N.; Arakawa, Y. Spontaneous two-photon emission from a single quantum dot. Phys. Rev. Lett.
**2011**, 107, 233602. [Google Scholar] [CrossRef] [PubMed] - Shlesinger, I.; Senellart, P.; Lanco, L.; Greffet, J.J. Time-frequency encoded single-photon generation and broadband single-photon storage with a tunable subradiant state. Optica
**2021**, 8, 95–105. [Google Scholar] [CrossRef] - Shlesinger, I.; Senellart, P.; Lanco, L.; Greffet, J.J. Tunable bandwidth and nonlinearities in an atom-photon interface with subradiant states. Phys. Rev. A
**2018**, 98, 013813. [Google Scholar] [CrossRef] - Schilder, N.J.; Sauvan, C.; Sortais, Y.R.P.; Browaeys, A.; Greffet, J.J. Near-resonant light scattering by a subwavelength ensemble of identical atoms. Phys. Rev. Lett.
**2020**, 124, 073403. [Google Scholar] [CrossRef] [PubMed] - Ficek, Z.; Tanas, R.; Kielich, S. Cooperative effects in the spontaneous emission from two non-identical atoms. Opt. Acta
**1986**, 33, 1149–1160. [Google Scholar] - Lehmberg, R.H. Radiation from an N-atom system. I. General formalism. Phys. Rev. A
**1970**, 2, 883. [Google Scholar] [CrossRef] - Hettich, C.; Schmitt, C.; Zitzmann, J.; Kühn, S.; Gerhardt, I.; Sandoghdar, V. Nanometer resolution and coherent optical dipole coupling of two individual molecules. Science
**2002**, 298, 385–389. [Google Scholar] [CrossRef] - Vogl, T.; Campbell, G.; Buchler, B.C.; Lu, Y.; Lam, P.K. Fabrication and deterministic transfer of high-quality quantum emitters in hexagonal boron nitride. ACS Photonics
**2018**, 5, 2305–2312. [Google Scholar] [CrossRef] - Schröder, T.; Trusheim, M.E.; Walsh, M.; Li, L.; Zheng, J.; Schukraft, M.; Sipahigil, A.; Evans, R.E.; Sukachev, D.D.; Nguyen, C.T.; et al. Scalable focused ion beam creation of nearly lifetime-limited single quantum emitters in diamond nanostructures. Nat. Commun.
**2017**, 8, 15376. [Google Scholar] [CrossRef] - Hail, C.U.; Höller, C.; Matsuzaki, K.; Rohner, P.; Renger, J.; Sandoghdar, V.; Poulikakos, D.; Eghlidi, H. Nanoprinting organic molecules at the quantum level. Nat. Commun.
**2019**, 10, 1880. [Google Scholar] [CrossRef] [PubMed] - Bayer, M.; Hawrylak, P.; Hinzer, K.; Fafard, S.; Korkusinski, M.; Wasilewski, Z.R.; Stern, O.; Forchel, A. Coupling and entangling of quantum states in quantum dot molecules. Science
**2001**, 291, 451–453. [Google Scholar] [CrossRef] - Kim, H.; Kyhm, K.; Taylor, R.A.; Kim, J.S.; Song, J.D.; Park, S. Optical shaping of the polarization anisotropy in a laterally coupled quantum dot dimer. Light Sci. Appl.
**2020**, 9, 100. [Google Scholar] [CrossRef] - Takagahara, T. Excitonic Structures and Optical Properties of Quantum Dots. In Semiconductor Quantum Dots; Springer: Berlin/Heidelberg, Germany, 2002; pp. 59–114. [Google Scholar]
- Senellart, P.; Solomon, G.; White, A. High-performance semiconductor quantum-dot single-photon sources. Nat. Nanotechnol.
**2017**, 12, 1026–1039. [Google Scholar] [CrossRef] [PubMed] - Sanchis, L.; Cryan, M.J.; Pozo, J.; Craddock, I.J.; Rarity, J.G. Ultrahigh Purcell factor in photonic crystal slab microcavities. Phys. Rev. B
**2007**, 76, 045118. [Google Scholar] [CrossRef] - Sanchis, L.; Håkansson, A.; López-Zanón, D.; Bravo-Abad, J.; Sánchez-Dehesa, J. Integrated optical devices design by genetic algorithm. Appl. Phys. Lett.
**2004**, 84, 4460–4462. [Google Scholar] [CrossRef] - Morgado-León, A.; Escuín, A.; Guerrero, E.; Yáñez, A.; Galindo, P.L.; Sanchis, L. Genetic Algorithms Applied to the Design of 3D Photonic Crystals. In Proceedings of the International Work-Conference on Artificial Neural Networks, Torremolinos, Spain, 8–10 June 2011; Springer: Berlin/Heidelberg, Germany, 2011; pp. 291–298. [Google Scholar]
- Marques-Hueso, J.; Sanchis, L.; Cluzel, B.; de Fornel, F.; Martínez-Pastor, J.P. Genetic algorithm designed silicon integrated photonic lens operating at 1550 nm. Appl. Phys. Lett.
**2010**, 97, 071115. [Google Scholar] [CrossRef] - 3D Electromagnetic Simulator; Lumerical Inc.: Vancouver, BC, Canada, 2022.
- Neu, E.; Hepp, C.; Hauschild, M.; Gsell, S.; Fischer, M.; Sternschulte, H.; Steinmüller-Nethl, D.; Schreck, M.; Becher, C. Low-temperature investigations of single silicon vacancy colour centres in diamond. New J. Phys.
**2013**, 15, 043005. [Google Scholar] [CrossRef] - Englund, D.; Faraon, A.; Zhang, B.; Yamamoto, Y.; Vučković, J. Generation and transfer of single photons on a photonic crystal chip. Opt. Express
**2007**, 15, 5550–5558. [Google Scholar] [CrossRef] [PubMed] - Chang, W.H.; Chen, W.Y.; Chang, H.S.; Hsieh, T.P.; Chyi, J.I.; Hsu, T.M. Efficient single-photon sources based on low-density quantum dots in photonic-crystal nanocavities. Phys. Rev. Lett.
**2006**, 96, 117401. [Google Scholar] [CrossRef] [PubMed] - Clark, A.S.; Husko, C.; Collins, M.J.; Lehoucq, G.; Xavier, S.; De Rossi, A.; Combrié, S.; Xiong, C.; Eggleton, B.J. Heralded single-photon source in a III–V photonic crystal. Opt. Lett.
**2013**, 38, 649–651. [Google Scholar] [CrossRef] [PubMed] - Huang, L.; Krasnok, A.; Alu, A.; Yu, Y.; Neshev, D.; Miroshnichenko, A. Enhanced Light-Matter Interaction in Two Dimensional Transition Metal Dichalcogenides. Rep. Prog. Phys.
**2022**, 85, 046401. [Google Scholar] [CrossRef] - Liu, S.; Srinivasan, K.; Liu, J. Nanoscale Positioning Approaches for Integrating Single Solid-State Quantum Emitters with Photonic Nanostructures. Laser Photonics Rev.
**2021**, 15, 2100223. [Google Scholar] [CrossRef] - Thon, S.M.; Rakher, M.T.; Kim, H.; Gudat, J.; Irvine, W.T.; Petroff, P.M.; Bouwmeester, D. Strong coupling through optical positioning of a quantum dot in a photonic crystal cavity. Appl. Phys. Lett.
**2009**, 94, 111115. [Google Scholar] [CrossRef] - Liu, J.; Davanço, M.I.; Sapienza, L.; Konthasinghe, K.; De Miranda Cardoso, J.V.; Song, J.D.; Badolato, A.; Srinivasan, K. Cryogenic photoluminescence imaging system for nanoscale positioning of single quantum emitters. Rev. Sci. Instrum.
**2017**, 88, 023116. [Google Scholar] [CrossRef] [PubMed] - Gschrey, M.; Schmidt, R.; Schulze, J.H.; Strittmatter, A.; Rodt, S.; Reitzenstein, S. Resolution and alignment accuracy of lowtemperature in situ electron beam lithography for nanophotonic device fabrication. J. Vac. Sci. Technol. B
**2015**, 33, 021603. [Google Scholar] [CrossRef] - Elshaari, A.W.; Pernice, W.; Srinivasan, K.; Benson, O.; Zwiller, V. Hybrid integrated quantum photonic circuits. Nat. Photonics
**2020**, 14, 285–298. [Google Scholar] [CrossRef] - Wan, N.H.; Lu, T.J.; Chen, K.C.; Walsh, M.P.; Trusheim, M.E.; De Santis, L.; Bersin, E.A.; Harris, I.B.; Mouradian, S.L.; Christen, I.R.; et al. Large-scale integration of near-indistinguishable artificial atoms in hybrid photonic circuits. arXiv
**2019**. Available online: https://arxiv.org/abs/1911.05265 (accessed on 16 July 2022).

**Figure 1.**(

**a**) The two interacting QEs with $\gamma $ coupled to the cavity field with g are equivalent to a single QE with $2\gamma $ coupled to the cavity with $\sqrt{2}$ g, each sphere represents a single two-level-system. Indistinguishability of the effective QE versus the normalized $\kappa $ and g in the (

**b**) incoherent regime and (

**c**) coherent regime. Contour map of regions with I > 0.9 for different distances between the emitters from $d=6.9\times {10}^{-2}\mathsf{\lambda}$ to $d=8.5\times {10}^{-2}\mathsf{\lambda}$ (

**d**) incoherent regime and (

**e**) coherent regime. (

**f**) Indistinguishability versus normalized g for $d=7.2\times {10}^{-2}\mathsf{\lambda}$ (yellow), $d=7\times {10}^{-2}\mathsf{\lambda}$ (green) and $d=6.9\times {10}^{-2}\mathsf{\lambda}$ (blue); solid lines calculated using Equation (1); colored dots obtained from numerical integration of the Lindblad equation with two QEs.

**Figure 2.**(

**a**) $\kappa $-parameter space of the stability of rate equations of a single QE system coupled to a cavity. Black dots correspond to bounded points while the gradient colors represent the degree of stability. (

**b**) Characteristic equation of (2) (blue line); tangent line with slope ${P}^{\prime}\left(0\right)$. The cut of the tangent line with the x-axis is given by $\frac{P\left(0\right)}{{P}^{\prime}\left(0\right)}$. The arrows indicate consecutive ${\lambda}_{n}$ values of the iteration process. (

**c**) Indistinguishability versus normalized ${\kappa}_{2}$ for ${\mathrm{g}}_{1}$ = (green), 2$\gamma $ (red) and 3$\gamma $ (yellow). (

**d**) Bloch-spheres of the five-QE system with population rate transfers R

_{ij}between each subsystem.

**Figure 3.**(

**a**) Optimal configuration of the 5-QEs system in a 2D plane for (

**a**) $\kappa =10\gamma $, (

**b**) $\kappa =50\gamma $, (

**c**) $\kappa =100\gamma $, (

**d**) $\kappa =500\gamma $ and (

**e**) $\kappa =1000\gamma $. The circles around each QE position corresponds to the positioning tolerance for having I > 0.9. (

**f**) Indistinguishability versus normalized $\kappa $ and g for the optimized system shown in (

**a**). (

**g**) Field profile ${\left|E\right|}^{2}$ of the hexagonal PC-cavity-mode with a point source placed at the antinode.

**Figure 4.**(

**a**) Probability distributions with standard deviation ${\sigma}_{n}$ for n = 1 …20 for the normalized detuning values $\Delta /\gamma $. At each iteration we set a random $\Delta $ value for each QD according to the corresponding distribution. (

**b**) Average value of the indistinguishability obtained for each of the 20 probability distributions.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Guimbao, J.; Sanchis, L.; Weituschat, L.M.; Llorens, J.M.; Postigo, P.A.
Perfect Photon Indistinguishability from a Set of Dissipative Quantum Emitters. *Nanomaterials* **2022**, *12*, 2800.
https://doi.org/10.3390/nano12162800

**AMA Style**

Guimbao J, Sanchis L, Weituschat LM, Llorens JM, Postigo PA.
Perfect Photon Indistinguishability from a Set of Dissipative Quantum Emitters. *Nanomaterials*. 2022; 12(16):2800.
https://doi.org/10.3390/nano12162800

**Chicago/Turabian Style**

Guimbao, Joaquin, Lorenzo Sanchis, Lukas M. Weituschat, Jose M. Llorens, and Pablo A. Postigo.
2022. "Perfect Photon Indistinguishability from a Set of Dissipative Quantum Emitters" *Nanomaterials* 12, no. 16: 2800.
https://doi.org/10.3390/nano12162800