# Hyperchaos in a Bose-Hubbard Chain with Rydberg-Dressed Interactions

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## Abstract

**:**

## 1. Introduction

## 2. Model and Method

#### 2.1. Extended Bose-Hubbard Model in the Semiclassical Limit

#### 2.2. Nonlinear Eigenenergies and Bogoliubov Spectra

#### 2.3. Poincaré Sections and Lyapunov Exponents

#### 2.4. Quenching Schemes

**Scheme$\mathbf{I}$**: First, we consider a linear quench of the potential bias [71]. The bias between two neighboring sites is given by the following function:

**Scheme$\mathbf{II}$**: Alternatively, we consider a hysteresis quench [74,75,76] where the system begins at ${\gamma}_{i}$ and then evolves to ${\gamma}_{f}$. At time $\tau ={\gamma}_{f}/\alpha $, the potential bias is quenched back towards ${\gamma}_{i}$. The function describing this scheme is as follows:

**Scheme$\mathbf{III}$**: In addition to quenching the level bias, we also change the two-body interaction strength through a linear ramp:

**Scheme$\mathbf{IV}$**: The hysteresis counterpart of the interaction quench is given by the following equation:

## 3. Stability of the Ground State

#### 3.1. Eigenenergies, Bogoliubov Spectra, and Lyapunov Exponents

#### 3.2. Quench Dynamics

## 4. Stability of the Localized State

#### 4.1. Bogoliubov Spectra and Lyapunov Exponents

#### 4.2. Quench Dynamics

## 5. Discussion

## 6. Conclusions and Outlook

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**(Color online)

**The extended Bose-Hubbard Chain and quenching schemes.**(

**a**) Nearest-neighbor (U) and next-nearest-neighbor (V) interactions between atoms in a one-dimensional optical lattice (lattice constant d). The tilting of the lattice is denoted by the parameter $\gamma $. We consider a linear quench in (

**b**) $\gamma $ and (

**c**) U towards a non-zero value (solid). When $\gamma $ (U) returns to the initial value (both solid and dashed), this is a hysteresis quench. The rate to quench $\gamma $ (U) is $\alpha $ ($\beta $). See text for details of the soft-core interaction and quenching protocols.

**Figure 2.**

**Eigenenergies, Bogoliubov spectra, and Lyapunov exponents when varying the tilt $\gamma $.**We show (

**a**) the nonlinear eigenenergy, the Bogoliubov spectra of (

**b**) the ground state and (

**c**) the first excited state, and (

**d**) the Lyapunov exponents of the ground state. The nonlinearity dominates when $\left|\gamma \right|$ is small, leading to loops in the eigenenergy. The Bogoliubov spectra are all real when the system is in the ground state (

**b**). The Bogoliubov spectra have complex components (red region) when the system is in the first-excited state. Positive Lyapunov exponents indicate the system exhibits chaos dynamically, which appear mostly in the loop region of the eigenenergy. Parameters are $L=3$ and $U=2V=5$.

**Figure 3.**

**Eigenenergies and Lyapunov exponents as a function of U.**We show eigenenergy for (

**a**) $L=3$ and (

**b**) $L=5$ when the trap is balanced ($\gamma =0$). Level crossings are found when the interaction is strong. Starting from the ground state, we calculate Lyapunov exponents for (

**c**) $L=3$ and (

**d**) $L=5$. For a given U, Lyapunov exponents of same value but opposite signs appear in pairs.

**Figure 4.**(Color online)

**Final population distribution of the ground state**. The population by quenching $\gamma $ with (

**a**) scheme $\mathbf{I}$ and (

**b**) scheme $\mathbf{II}$ is shown for $L=3$. In the numerical simulation, ${\gamma}_{i}=-{\gamma}_{f}=-10$ and the interaction strength is $U=5$. The interaction U is quenched with (

**c**) scheme $\mathbf{III}$ and (

**d**) scheme $\mathbf{IV}$, where ${U}_{i}=0$, ${U}_{f}=10$ and $\gamma =0$, respectively. In all the figures, the quench rates ($\alpha $ or $\beta $) are 1 (blue), $0.1$ (green), and $0.01$ (red). The total number of trajectories is $M=100$. The target final state is shown as the large black circle.

**Figure 5.**(Color online)

**Bogoliubov spectra and Lyapunov exponents of the localized state.**Dynamically unstable regions (dark red) for (

**a**) $L=3$ and (

**b**) L = 5 are shown as a function of U and $\gamma $. Panels (

**c**,

**d**) give the Lyapunov exponents as a function of U. Random perturbation to the initial state are examined for (

**e**) $L=3$ and (

**f**) $L=5$. The red lines show the maximal Lyapunov exponents in (

**c**,

**d**), correspondingly. Here, $\gamma =0$ in panels (

**c**,

**d**).

**Figure 6.**(Color online)

**Final population distribution of the localized state**. The first and second row show the linear and hysteresis quench of $\gamma $. Here, $U=5$, ${\gamma}_{i}=-{\gamma}_{f}=10$. The third and fourth row show the linear and hysteresis quench of U, with ${U}_{i}=0$, ${U}_{f}=10$, and $\gamma =0$. In (

**a**–

**d**), we consider three sites and the initial thermal state is $\overline{\Psi}=[0.1{e}^{i{\varphi}_{1}},\sqrt{0.98}{e}^{i{\varphi}_{2}},0.1{e}^{{\varphi}_{3}}]$, with ${\varphi}_{j}$ ($j=1,\phantom{\rule{0.166667em}{0ex}}2,\phantom{\rule{0.166667em}{0ex}}3$) being a random number in $[0,2\pi ]$. In (

**e**–

**h**), $L=5$ and the initial state is $\overline{\Psi}=[\sqrt{0.005}{e}^{i{\varphi}_{1}},\sqrt{0.98}{e}^{i{\varphi}_{2}},\sqrt{0.005}{e}^{i{\varphi}_{3}},\sqrt{0.005}{e}^{i{\varphi}_{4}},{\sqrt{0.005}}^{i{\varphi}_{5}}]$, with ${\varphi}_{j}$ ($j=1,\cdots ,5$) being randomly distributed in $[0,2\pi ]$. The small fraction in sites other than the localized state is used to trigger the hopping dynamics. The other parameters are the same as the ones in Figure 4.

**Figure 7.**(Color online)

**Lyapunov Exponents vs. System Size.**The maximum Lyapunov exponent and total number of positive Lyapunov exponents are shown in (

**a**,

**b**) for the ground state configuration. Panels (

**c**,

**d**) show the same quantity for the localized state. The larger the value of U is, the larger the maximal Lyapunov exponent. The maximal Lyapunov exponent decreases with increasing L. The number of Lyapunov exponents increases and then decreases with increasing L. For the localized state, $\eta $ increases almost linearly with increasing L. In each panel, $U=1$ (square), 3 (circle), and 5 (triangle).

**Figure 8.**(Color online)

**Poincaré Sections of the ground state and localized state on the ${\mathcal{U}}_{1}$-plane**. The Poincaré sections are shown for the ground state (

**a**–

**c**) and the localized state (

**d**–

**f**). Each point represents a numerical realization. We consider $L=5$ (

**a**,

**d**), $L=10$ (

**b**,

**e**), and $L=20$ (

**c**,

**f**). Other parameters are $U=3$ and $\gamma =0$.

**Figure 9.**(Color online)

**Areas of the Poincaré Sections**. We compare the fitted area (open shapes) of the Poincaré section with ${\lambda}_{m}$ (solid) for both the ground state (

**a**) and the localized state (

**b**), respectively. The blue circles are for $U=3$, and red triangles for $U=5$. In both situations, $\gamma =0$.

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**MDPI and ACS Style**

McCormack, G.; Nath, R.; Li, W.
Hyperchaos in a Bose-Hubbard Chain with Rydberg-Dressed Interactions. *Photonics* **2021**, *8*, 554.
https://doi.org/10.3390/photonics8120554

**AMA Style**

McCormack G, Nath R, Li W.
Hyperchaos in a Bose-Hubbard Chain with Rydberg-Dressed Interactions. *Photonics*. 2021; 8(12):554.
https://doi.org/10.3390/photonics8120554

**Chicago/Turabian Style**

McCormack, Gary, Rejish Nath, and Weibin Li.
2021. "Hyperchaos in a Bose-Hubbard Chain with Rydberg-Dressed Interactions" *Photonics* 8, no. 12: 554.
https://doi.org/10.3390/photonics8120554