# A Versatile Quantum Simulator for Coupled Oscillators Using a 1D Chain of Atoms Trapped near an Optical Nanofiber

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Tailored Coupling of the Quantized Motion of a Trapped Atom Chain

#### 2.2. Model Assumptions and Limitations

## 3. Results

#### 3.1. Simulating Coulomb Interactions between Trapped Quantum Particles

#### 3.2. Bipartite Quantum Gates between Distant Particles

#### 3.2.1. Using the Two Lowest Oscillator States on a Qubit Basis

#### 3.2.2. Coherent States as a Computational Basis

#### 3.3. Entanglement Propagation via Controlled Long-Range Interaction

#### 3.4. State Read Out via the Outgoing Fiber Fields

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Data Values for Figure 4

**Table A1.**Data values for Figure 4.

Triangle | Triangle with Suppressed Interactions | |
---|---|---|

${\Omega}_{0}/\tilde{\Omega}$ | 251.5 | 251.4 |

${\Omega}_{1}/\tilde{\Omega}$ | 643 | 642.6 |

${\Omega}_{2}/\tilde{\Omega}$ | 580.5 | 580.1 |

${\Omega}_{3}/\tilde{\Omega}$ | 72 | 72 |

${\Omega}_{4}/\tilde{\Omega}$ | 0 | 0 |

${\Omega}_{5}/\tilde{\Omega}$ | 666.2 | 665.8 |

${\Omega}_{6}/\tilde{\Omega}$ | 1149.7 | 1149.1 |

${\Omega}_{7}/\tilde{\Omega}$ | 754.3 | 754.3 |

${\Omega}_{8}/\tilde{\Omega}$ | 104.7 | 104.8 |

${\Omega}_{9}/\tilde{\Omega}$ | 115.5 | 115.3 |

${\Omega}_{10}/\tilde{\Omega}$ | 591.3 | 590.8 |

${\Omega}_{11}/\tilde{\Omega}$ | 724.8 | 724.5 |

${\Omega}_{12}/\tilde{\Omega}$ | 392.4 | 392.4 |

${\Omega}_{13}/\tilde{\Omega}$ | 81.2 | 81.3 |

## Appendix B. Time Evolution of the Coherent States

## References

- Kaufman, A.M.; Lester, B.J.; Regal, C.A. Cooling a single atom in an optical tweezer to its quantum ground state. Phys. Rev. X
**2012**, 2, 041014. [Google Scholar] [CrossRef] [Green Version] - Sheremet, A.S.; Petrov, M.I.; Iorsh, I.V.; Poshakinskiy, A.V.; Poddubny, A.N. Waveguide quantum electrodynamics: Collective radiance and photon-photon correlations. arXiv
**2021**, arXiv:2103.06824v1. [Google Scholar] - Vetsch, E.; Reitz, D.; Sagué, G.; Schmidt, R.; Dawkins, S.; Rauschenbeutel, A. Optical interface created by laser-cooled atoms trapped in the evanescent field surrounding an optical nanofiber. Phys. Rev. Lett.
**2010**, 104, 203603. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Goban, A.; Choi, K.; Alton, D.; Ding, D.; Lacroûte, C.; Pototschnig, M.; Thiele, T.; Stern, N.; Kimble, H. Demonstration of a state-insensitive, compensated nanofiber trap. Phys. Rev. Lett.
**2012**, 109, 033603. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Béguin, J.B.; Müller, J.H.; Appel, J.; Polzik, E.S. Observation of quantum spin noise in a 1D light-atoms quantum interface. Phys. Rev. X
**2018**, 8, 031010. [Google Scholar] [CrossRef] [Green Version] - Meng, Y.; Liedl, C.; Pucher, S.; Rauschenbeutel, A.; Schneeweiss, P. Imaging and localizing individual atoms interfaced with a nanophotonic waveguide. Phy. Rev. Lett
**2020**, 125, 053603. [Google Scholar] [CrossRef] - Markussen, S.B.; Appel, J.; Østfeldt, C.; Béguin, J.B.S.; Polzik, E.S.; Müller, J.H. Measurement and simulation of atomic motion in nanoscale optical trapping potentials. Appl. Phys. B
**2020**, 126, 73. [Google Scholar] [CrossRef] - Jones, R.; Buonaiuto, G.; Lang, B.; Lesanovsky, I.; Olmos, B. Collectively enhanced chiral photon emission from an atomic array near a nanofiber. Phys. Rev. Lett.
**2020**, 124, 093601. [Google Scholar] [CrossRef] [Green Version] - Shomroni, I.; Rosenblum, S.; Lovsky, Y.; Bechler, O.; Guendelman, G.; Dayan, B. All-optical routing of single photons by a one-atom switch controlled by a single photon. Science
**2014**, 345, 903–906. [Google Scholar] [CrossRef] [Green Version] - Pivovarov, V.; Gerasimov, L.; Berroir, J.; Ray, T.; Laurat, J.; Urvoy, A.; Kupriyanov, D. Single collective excitation of an atomic array trapped along a waveguide: A study of cooperative emission for different atomic chain configurations. arXiv
**2021**, arXiv:2101.05398. [Google Scholar] - Holzmann, D.; Ritsch, H. Tailored long range forces on polarizable particles by collective scattering of broadband radiation. New J. Phys.
**2016**, 18, 103041. [Google Scholar] [CrossRef] [Green Version] - Prasad, A.S.; Hinney, J.; Mahmoodian, S.; Hammerer, K.; Rind, S.; Schneeweiss, P.; Sørensen, A.S.; Volz, J.; Rauschenbeutel, A. Correlating photons using the collective nonlinear response of atoms weakly coupled to an optical mode. Nat. Photonics
**2020**, 14, 719–722. [Google Scholar] [CrossRef] - Cirac, J.I. Atomic self-organization around tappered nanofibers. In Proceedings of the Laser Science 2012, Rochester, NY, USA, 14–18 October 2012; Optical Society of America: Washington, DC, USA, 2012; p. LW1J.6. [Google Scholar]
- Chang, D.; Jiang, L.; Gorshkov, A.; Kimble, H. Cavity QED with atomic mirrors. New J. Phys.
**2012**, 14, 063003. [Google Scholar] [CrossRef] [Green Version] - Metzger, N.K.; Wright, E.M.; Sibbett, W.; Dholakia, K. Visualization of optical binding of microparticles using a femtosecond fiber optical trap. Opt. Express
**2006**, 14, 3677–3687. [Google Scholar] [CrossRef] [PubMed] - Chang, D.E.; Cirac, J.I.; Kimble, H.J. Self-Organization of Atoms along a Nanophotonic Waveguide. Phys. Rev. Lett.
**2013**, 110, 113606. [Google Scholar] [CrossRef] [Green Version] - Buonaiuto, G.; Carollo, F.; Olmos, B.; Lesanovsky, I. Dynamical phases and quantum correlations in an emitter-waveguide system with feedback. arXiv
**2021**, arXiv:2102.02719. [Google Scholar] - Grießer, T.; Ritsch, H. Light-induced crystallization of cold atoms in a 1D optical trap. Phys. Rev. Lett.
**2013**, 111, 055702. [Google Scholar] [CrossRef] [Green Version] - Holzmann, D.; Sonnleitner, M.; Ritsch, H. Self-ordering and collective dynamics of transversely illuminated point-scatterers in a 1D trap. Eur. Phys. J. D
**2014**, 68, 352. [Google Scholar] [CrossRef] [Green Version] - Ostermann, S.; Sonnleitner, M.; Ritsch, H. Scattering approach to two-colour light forces and self-ordering of polarizable particles. New J. Phys.
**2014**, 16, 043017. [Google Scholar] [CrossRef] [Green Version] - Holzmann, D.; Sonnleitner, M.; Ritsch, H. Synthesizing variable particle interaction potentials via spectrally shaped spatially coherent illumination. New J. Phys.
**2018**, 20, 103009. [Google Scholar] [CrossRef] [Green Version] - Georgescu, I.M.; Ashhab, S.; Nori, F. Quantum simulation. Rev. Mod. Phys.
**2014**, 86, 153. [Google Scholar] [CrossRef] [Green Version] - Kim, E.; Zhang, X.; Ferreira, V.S.; Banker, J.; Iverson, J.K.; Sipahigil, A.; Bello, M.; González-Tudela, A.; Mirhosseini, M.; Painter, O. Quantum electrodynamics in a topological waveguide. Phys. Rev. X
**2021**, 11, 011015. [Google Scholar] - Feynman, R.P. Simulating physics with computers. Int. J. Theor. Phys.
**1982**, 21, 467–488. [Google Scholar] [CrossRef] - Hartmann, M.J. Quantum simulation with interacting photons. J. Opt.
**2016**, 18, 104005. [Google Scholar] [CrossRef] - Longhi, S. Optical realization of the two-site Bose–Hubbard model in waveguide lattices. J. Phys. B At. Mol. Opt. Phys.
**2011**, 44, 051001. [Google Scholar] [CrossRef] [Green Version] - Noh, C.; Angelakis, D.G. Quantum simulations and many-body physics with light. Rep. Prog. Phys.
**2016**, 80, 016401. [Google Scholar] [CrossRef] [Green Version] - Tashima, T.; Takashima, H.; Takeuchi, S. Direct optical excitation of an NV center via a nanofiber Bragg-cavity: A theoretical simulation. Opt. Express
**2019**, 27, 27009–27016. [Google Scholar] [CrossRef] [PubMed] - Huo, M.X.; Noh, C.; Rodríguez-Lara, B.; Angelakis, D.G. Quantum simulation of Cooper pairing with photons. Phys. Rev. A
**2012**, 86, 043840. [Google Scholar] [CrossRef] [Green Version] - Angelakis, D.G.; Huo, M.X.; Chang, D.; Kwek, L.C.; Korepin, V. Mimicking interacting relativistic theories with stationary pulses of light. Phys. Rev. Lett.
**2013**, 110, 100502. [Google Scholar] [CrossRef] [PubMed] - Davoudi, Z.; Hafezi, M.; Monroe, C.; Pagano, G.; Seif, A.; Shaw, A. Towards analog quantum simulations of lattice gauge theories with trapped ions. Phys. Rev. Res.
**2020**, 2, 023015. [Google Scholar] [CrossRef] [Green Version] - Cirac, J.I.; Zoller, P. Quantum computations with cold trapped ions. Phys. Rev. Lett.
**1995**, 74, 4091. [Google Scholar] [CrossRef] - Kewes, G.; Schoengen, M.; Neitzke, O.; Lombardi, P.; Schönfeld, R.S.; Mazzamuto, G.; Schell, A.W.; Probst, J.; Wolters, J.; Löchel, B.; et al. A realistic fabrication and design concept for quantum gates based on single emitters integrated in plasmonic-dielectric waveguide structures. Sci. Rep.
**2016**, 6, 28877. [Google Scholar] [CrossRef] [Green Version] - Paulisch, V.; Kimble, H.; González-Tudela, A. Universal quantum computation in waveguide QED using decoherence free subspaces. New J. Phys.
**2016**, 18, 043041. [Google Scholar] [CrossRef] - Leong, W.S.; Xin, M.; Chen, Z.; Chai, S.; Wang, Y.; Lan, S.Y. Large array of Schrödinger cat states facilitated by an optical waveguide. Nat. Commun.
**2020**, 11, 5295. [Google Scholar] [CrossRef] - Li, Y.; Aolita, L.; Chang, D.E.; Kwek, L.C. Robust-fidelity atom-photon entangling gates in the weak-coupling regime. Phys. Rev. Lett.
**2012**, 109, 160504. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Gonzalez-Tudela, A.; Martin-Cano, D.; Moreno, E.; Martin-Moreno, L.; Tejedor, C.; Garcia-Vidal, F.J. Entanglement of two qubits mediated by one-dimensional plasmonic waveguides. Phys. Rev. Lett.
**2011**, 106, 020501. [Google Scholar] [CrossRef] [PubMed] - Snyder, A. Optical Waveguide Theory; Springer: Boston, MA, USA, 1983. [Google Scholar]
- Le Kien, F.; Dutta Gupta, S.; Balykin, V.I.; Hakuta, K. Spontaneous emission of a cesium atom near a nanofiber: Efficient coupling of light to guided modes. Phys. Rev. A
**2005**, 72, 032509. [Google Scholar] [CrossRef] [Green Version] - Scarpelli, L.; Lang, B.; Masia, F.; Beggs, D.; Muljarov, E.; Young, A.; Oulton, R.; Kamp, M.; Höfling, S.; Schneider, C.; et al. 99% beta factor and directional coupling of quantum dots to fast light in photonic crystal waveguides determined by spectral imaging. Phys. Rev. B
**2019**, 100, 035311. [Google Scholar] [CrossRef] [Green Version] - Liu, F.; Brash, A.J.; O’Hara, J.; Martins, L.M.; Phillips, C.L.; Coles, R.J.; Royall, B.; Clarke, E.; Bentham, C.; Prtljaga, N.; et al. High Purcell factor generation of indistinguishable on-chip single photons. Nat. Nanotechnol.
**2018**, 13, 835–840. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Mirhosseini, M.; Kim, E.; Zhang, X.; Sipahigil, A.; Dieterle, P.B.; Keller, A.J.; Asenjo-Garcia, A.; Chang, D.E.; Painter, O. Cavity quantum electrodynamics with atom-like mirrors. Nature
**2019**, 569, 692–697. [Google Scholar] [CrossRef] - Wang, X.; Zhang, P.; Li, G.; Zhang, T. High-efficiency coupling of single quantum emitters into hole-tailored nanofibers. Opt. Express
**2021**, 29, 11158–11168. [Google Scholar] [CrossRef] [PubMed] - Bogdanov, Y.I.; Lukichev, V.F.; Orlikovsky, A.A.; Nuyanzin, S.A. Quantum Noise and the Quality Control of Hardward Components of Quantum Computers Based on Superconducting Phase Qubits. Russ. Microelectron.
**2012**, 41, 325–335. [Google Scholar] [CrossRef] - Kok, P.; Munro, W.J.; Nemoto, K.; Ralph, T.C.; Dowling, J.P.; Milburn, G.J. Linear optical quantum computing with photonic qubits. Rev. Mod. Phys.
**2007**, 79, 135. [Google Scholar] [CrossRef] [Green Version] - Jeong, H.; Kim, M.S. Efficient quantum computation using coherent states. Phys. Rev. A
**2002**, 65, 042305. [Google Scholar] [CrossRef] [Green Version] - Ralph, T.C.; Gilchrist, A.; Milburn, G.J.; Munro, W.J.; Glancy, S. Quantum computation with optical coherent states. Phys. Rev. A
**2003**, 68, 042319. [Google Scholar] [CrossRef] [Green Version] - Marek, P.; Fiurášek, J. Elementary gates for quantum information with superposed coherent states. Phys. Rev. A
**2010**, 82, 014304. [Google Scholar] [CrossRef] [Green Version] - Hümmer, D.; Schneeweiss, P.; Rauschenbeutel, A.; Romero-Isart, O. Heating in nanophotonic traps for cold atoms. Phys. Rev. X
**2019**, 9, 041034. [Google Scholar] [CrossRef] [Green Version] - Iversen, O.A.; Pohl, T. Strongly Correlated States of Light and Repulsive Photons in Chiral Chains of Three-Level Quantum Emitters. Phys. Rev. Lett.
**2021**, 126, 083605. [Google Scholar] [CrossRef] - Jen, H. Bound and Subradiant Multi-Atom Excitations in an Atomic Array with Nonreciprocal Couplings. arXiv
**2021**, arXiv:2102.03757. [Google Scholar] - Mahmoodian, S.; Calajó, G.; Chang, D.E.; Hammerer, K.; Sørensen, A.S. Dynamics of many-body photon bound states in chiral waveguide QED. Phys. Rev. X
**2020**, 10, 031011. [Google Scholar] [CrossRef] - Lodahl, P.; Mahmoodian, S.; Stobbe, S.; Rauschenbeutel, A.; Schneeweiss, P.; Volz, J.; Pichler, H.; Zoller, P. Chiral quantum optics. Nature
**2017**, 541, 473–480. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**Sketch of our system: N particles are confined in homogeneous traps next to a nanofiber. The particles are illuminated by multi-color transverse pump fields and scatter light into the fiber. Interference of the scattered fields in the fiber leads to effective forces between the particles.

**Figure 2.**Schematic mapping of three ions along a line, interacting via the Coulomb force separated at a distance D onto a system of particles along a nanofiber, with distances ${d}_{12}$ and ${d}_{23}$. This can be achieved by finding the right distances, frequencies and interactions strengths to solve Equations (16). The lower figures show the eigenenergies of these three harmonically trapped ions with Coulomb repulsion (blue) in comparison to particles trapped along the fiber with simulated interaction (red) as a function of the distance between the interacting particles. In the simulation, the particles are trapped along the fiber at fixed distances ${d}_{12}={d}_{23}=3/8\phantom{\rule{3.33333pt}{0ex}}{\lambda}_{0}$ and the pump laser parameters are adjusted to mimic Coulomb interaction at arbitrary distances between the ions. We used ${\Delta}_{k}=0.7\phantom{\rule{3.33333pt}{0ex}}{k}_{0}$ and ${\Omega}_{0}/\tilde{\Omega}=0.34+1.07\phantom{\rule{3.33333pt}{0ex}}{\tilde{k}}_{0}D$, ${\Omega}_{1}/\tilde{\Omega}=1.16+1.34\phantom{\rule{3.33333pt}{0ex}}{\tilde{k}}_{0}D$, ${\Omega}_{2}/\tilde{\Omega}=1.68+1.58\phantom{\rule{3.33333pt}{0ex}}{\tilde{k}}_{0}D$, ${\Omega}_{3}/\tilde{\Omega}=0.74+1.4\phantom{\rule{3.33333pt}{0ex}}{\tilde{k}}_{0}D$, with ${\tilde{k}}_{0}={k}_{0}{\delta}_{0}/{\delta}_{0}^{\prime}$. For this figure, we chose $\tilde{\Omega}/{\omega}_{T}$, such that every ${\Omega}_{l}/{\omega}_{T}\le 0.004$ is restricted as required by Equation (11), that is, $\tilde{\Omega}/{\omega}_{T}={\mathrm{max}}_{l}({\Omega}_{l}/\tilde{\Omega})/{(0.004{\tilde{k}}_{0}D)}^{3}$. The figure on the left side shows the energies corresponding to the first oscillator state and the right figure shows the energies corresponding to the second oscillator state.

**Figure 3.**We simulated the Coulomb interaction of three ions arranged in an equilateral triangle by a 1D particle chain along a nanofiber. The first three particles correspond to the interaction in the x direction, while the next three particles correspond to the interaction in the y direction. In Figure 4, we also show an example where the interaction between the ion numbers 1 and 3 is turned off.

**Figure 4.**Eigenenergies of three harmonically trapped ions with Coulomb repulsion ordered like a triangle (blue) in comparison to particles trapped along a fiber with light-induced interaction (red), as shown in Figure 3, as a function of the distance between the interacting ions. This can be achieved by finding the right distances, frequencies and interaction strengths to solve Equation (19). In the simulation, the particles are trapped along the fiber at fixed distances ${d}_{12}={d}_{23}=1/3\phantom{\rule{3.33333pt}{0ex}}{\lambda}_{0}$, ${d}_{34}={\lambda}_{0}$ and ${d}_{45}={d}_{56}=1/4\phantom{\rule{3.33333pt}{0ex}}{\lambda}_{0}$ and the pump laser parameters are adjusted to mimic Coulomb interactions at arbitrary distances between the ions with ${\Delta}_{k}=0.33\phantom{\rule{3.33333pt}{0ex}}{k}_{0}$. ${\tilde{k}}_{0}$ and $\tilde{\Omega}/{\omega}_{T}$ are defined as in Figure 2. The figures in the upper row show the eigenenergies when all particles are interacting, while in the lower figures the interaction between ion numbers 1 and 3 at the bottom of the triangle is suppressed. The figures on the left side show the energies corresponding to the first oscillator state and the right figures show the energies corresponding to the second oscillator state. Data values can be found in the Appendix A in Table A1.

**Figure 5.**Time evolution of the excited motional state occupation for three coupled particles. The blue line corresponds to the first particle, the orange line to the second particle and the green line to the third particle. We start from a state $|100\rangle $, where only the first particle is excited, and set the distances to ${d}_{12}=3/4\phantom{\rule{3.33333pt}{0ex}}{\lambda}_{1}$, ${d}_{23}=7/8\phantom{\rule{3.33333pt}{0ex}}{\lambda}_{1}$, ${k}_{2}=4/3{k}_{1}$ and ${\Omega}_{2}=0.82\phantom{\rule{3.33333pt}{0ex}}{\Omega}_{1}$. Choosing these parameters, the interaction between the third and the other two particles can be turned off, while the first two particles still interact with each other. Deactivating the interaction between specific particle pairs is necessary to implement bipartite quantum gates.

**Figure 6.**Entanglement propagation for three particles coupled via a single illumination beam as a function of time with ${d}_{12}={\lambda}_{0}/2$ and ${d}_{23}=\left(1/2+0.1\right){\lambda}_{0}$ for the initial state $|001\rangle $. The two curves show the entanglement entropy of the subsystem containing particles 2 and 3 (blue) and the subsystem containing particles 1 and 2 (yellow). So, the blue line describes the entanglement between the subsystem containing particle 1 and the subsystem containing particles 2 and 3, and the yellow line between the subsystem containing particle 3 and the subsystem containing particles 1 and 2. (▴) corresponds to the state $1/\sqrt{3}\left(|001\rangle -i|010\rangle +i|100\rangle \right)$, (×) to the state $1/\sqrt{2}\left(|01\rangle -|10\rangle \right)|0\rangle $, (+) to the state $\frac{1}{\sqrt{3}}\left(-|001\rangle +i|010\rangle -i|100\rangle \right)$, and (•) to the state $-|001\rangle $.

**Figure 7.**State-dependent light intensities emitted from the fiber to the left ${I}_{-}$ and to the right ${I}_{+}$ for a system with three particles. Here, the distance between the first two particles stays constant with ${d}_{12}={\lambda}_{0}$, while we vary the position of the third particle. Red lines correspond to the ground state $|000\rangle $, blue lines to the single excited state $|100\rangle $, green lines to a doubly excited state $|011\rangle $ and purple lines corresponds to the state $|111\rangle $. Note that in the left figure the green and blue lines overlap.

**Figure 8.**Average output power ${I}_{-}$ emitted on the left side of the fiber and ${I}_{+}$ on the right side for a system of two particles as a function of time. The initial condition is the same as in Figure 6. The blue line corresponds to the outgoing intensity to the left side $\langle {I}_{-}\rangle $ and the purple line to the (constant) outgoing intensity to right $\langle {I}_{+}\rangle $. (▴) corresponds to the state $\frac{1}{\sqrt{3}}\left(|001\rangle -i|010\rangle +i|100\rangle \right)$, (★) to the state $\frac{1}{\sqrt{2}}\left(|01\rangle -|10\rangle \right|0\rangle $, (+) to the state $\frac{1}{\sqrt{3}}\left(-|001\rangle +i|010\rangle -i|100\rangle \right)$ and (·) to $-|001\rangle $. The light leaving the system thus contains information about the entangled motional states of the particles.

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

© 2021 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**

Holzmann, D.; Sonnleitner, M.; Ritsch, H.
A Versatile Quantum Simulator for Coupled Oscillators Using a 1D Chain of Atoms Trapped near an Optical Nanofiber. *Photonics* **2021**, *8*, 228.
https://doi.org/10.3390/photonics8060228

**AMA Style**

Holzmann D, Sonnleitner M, Ritsch H.
A Versatile Quantum Simulator for Coupled Oscillators Using a 1D Chain of Atoms Trapped near an Optical Nanofiber. *Photonics*. 2021; 8(6):228.
https://doi.org/10.3390/photonics8060228

**Chicago/Turabian Style**

Holzmann, Daniela, Matthias Sonnleitner, and Helmut Ritsch.
2021. "A Versatile Quantum Simulator for Coupled Oscillators Using a 1D Chain of Atoms Trapped near an Optical Nanofiber" *Photonics* 8, no. 6: 228.
https://doi.org/10.3390/photonics8060228