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

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## 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

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**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.

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**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