# Signatures of Quantum Chaos of Rydberg-Dressed Bosons in a Triple-Well Potential

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## Abstract

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## 1. Introduction

## 2. Model

## 3. Results

#### 3.1. Level Statistics

#### 3.2. Entanglement Entropy

#### 3.3. Survival Probability and Variance of Populations

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Saturation Values of ${\mathit{S}}_{\mathit{E}\mathit{E}}$

## References

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**Figure 1.**(

**a**) Three-well trapping potential and soft-core-shaped interaction between Rydberg-dressed atoms. The soft-core radius ${R}_{c}$ is larger than the distance between neighboring sites. By tuning ${R}_{C}$, both the nearest-neighbor and next-nearest-neighbor interaction can be made to be strong. (

**b**,

**c**) Lyapunov exponents in the semiclassical regime. Parameters are $\gamma /J=2.5$ and $U/J=7.0$ in (

**b**) and (

**c**), respectively. See text for details.

**Figure 2.**(

**a**,

**b**) Eigenspectrum as a function of $U/J$. When U is large, avoided crosses are found. In the calculation, $\gamma /J=2.5$ and $N=12$. (

**c**) Chaos indicator $\beta $ as a function of $U/J$. Maximal $\beta $ is found at around $U/J=7$, where the level spacing approaches the WD distribution. (

**d**–

**f**) show examples of level spacing distributions. In (

**d**–

**f**), $U/J$ is fixed to be 3, 7 and 13, respectively. The dashed blue and solid red lines in (

**d**–

**f**) are the Poissonian and WD distribution, respectively. When $U/J=7$, the level is close to the WD distribution. Other parameters are $\gamma /J=2.5$ and $N=120$ in (

**d**–

**f**).

**Figure 3.**(

**a**,

**b**) Eigenspectra at different intervals of $\gamma /J$ with $U/J=7$ and $N=12$. Direct and avoided level crossings are found in (

**a**). In (

**b**), only avoided level crossings are found. (

**c**) Chaos indicator $\beta $ as a function of $\gamma /J$. Around $\gamma /J=2.5$, $\beta $ reaches the maximal value, where the level spacing approaches the WD distribution. (

**d**–

**f**) show examples of level spacing distributions. In (

**d**–

**f**), $\gamma /J$ is fixed to be 0, 2.5 and 7, respectively. The dashed blue and solid red lines in (

**d**–

**f**) represent the Poissonian and WD distribution, respectively. When $\gamma /J=2.5$, the levels approach the WD distribution. Other parameters are $U/J=7$ and $N=120$ in (

**d**–

**f**).

**Figure 4.**(

**a**) Dynamical evolution of entanglement entropy for $U/J=3.0$, $7.0$ and $13.0$. (

**b**) Normalized average entropy ${\overline{S}}_{EE}$ as a function of $U/J$ for different N. (

**c**) Dynamical evolution of ${S}_{EE}$ for $\gamma /J=1.0$, $4.0$ and $7.0$. (

**d**) ${\overline{S}}_{EE}$ with respect to $\gamma /J$ for different N. In (

**a**,

**b**) panels, $\gamma /J=2.5$. In (

**c**,

**d**) panels, $U/J$ is fixed to be 7.0.

**Figure 5.**Moving averages of survival probabilities. We set the temporal window size to be $0.04/J$ in numerical calculations. $U/J$ was fixed to be 7.0 in (

**a**). $\gamma /J$ was fixed be 2.5 in (

**b**). The dashed blue lines in both panels denote the saturation value of survival probability of GOE matrices. The insets of both panels show the moving averages of survival probabilities beyond ${\mathrm{log}}_{10}\left(t\right)=2$. In both panels, $N=90$.

**Figure 6.**(

**a**) Population variances with respect to $\gamma $. Evolution of population when $\gamma /J=0$ (

**b**), $\gamma /J=2$ (

**c**) and $\gamma /J=7$ (

**d**). In these panels, blue, green and red lines denote the expectation values in the leftmost, middle and rightmost site, respectively. In all panels, we choose $U/J=7.0$ and $N=90$.

**Figure 7.**(

**a**) Population variances with respect to U. Evolution of populations when $U=3$ (

**b**), $U=7$ (

**c**) and $U=13$ (

**d**). In each panel, blue, green and red lines denote the population in the leftmost, middle and rightmost site, respectively. In the calculation, $\gamma =2.5$ and $N=90$.

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**MDPI and ACS Style**

Yan, T.; Collins, M.; Nath, R.; Li, W.
Signatures of Quantum Chaos of Rydberg-Dressed Bosons in a Triple-Well Potential. *Atoms* **2023**, *11*, 89.
https://doi.org/10.3390/atoms11060089

**AMA Style**

Yan T, Collins M, Nath R, Li W.
Signatures of Quantum Chaos of Rydberg-Dressed Bosons in a Triple-Well Potential. *Atoms*. 2023; 11(6):89.
https://doi.org/10.3390/atoms11060089

**Chicago/Turabian Style**

Yan, Tianyi, Matthew Collins, Rejish Nath, and Weibin Li.
2023. "Signatures of Quantum Chaos of Rydberg-Dressed Bosons in a Triple-Well Potential" *Atoms* 11, no. 6: 89.
https://doi.org/10.3390/atoms11060089