# Dynamical Stability in a Non-Hermitian Kicked Rotor Model

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## Abstract

**:**

## 1. Introduction

## 2. Dynamical Stability Induced by Non-Hermitian Driven Potential

## 3. Enhancement of Dynamical Localization by Non-Hermitian Driven Potential

## 4. Mechanism of the Enhancement of Dynamical Localization by Non-Hermiticity

## 5. Conclusions and Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The $\overline{\mathcal{L}}$ versus time with $\u03f5={10}^{-3}$ for $\lambda =0$ (squares), $\pm {10}^{-4}$ (circles), $\pm 2\times {10}^{-4}$ (triangles), and $\pm {10}^{-2}$ (diamonds). Solid (empty) symbols represent positive (negative) $\lambda $. Note that the $\overline{\mathcal{L}}$ for $\lambda =\pm {10}^{-2}$ almost completely overlaps with each other. Red line indicates the exponential decay $\overline{\mathcal{L}}\propto {e}^{-\gamma t}$ with the Lyapunov exponent $\gamma =ln(K/2)$. The parameters are $K=5$ and ${\hslash}_{\mathrm{eff}}=3\times {10}^{-5}$.

**Figure 2.**(

**a**) The $\langle {p}^{2}\rangle $ versus time with $\lambda $ = −0.003 (empty triangles), −0.002 (empty circles), 0 (squares), 0.002 (solid circles), and 0.003 (solid triangles). Arrow marks the threshold time ${t}^{*}$. (

**b**) Momentum distributions at the time ${t}_{n}=1000$ for $\lambda =0$ (squares) and −0.003 (circles). Solid lines indicates the exponential function ${\left|\psi \left(p\right)\right|}^{2}\propto {e}^{-\left|p\right|/\xi}$ with $\xi \approx 23$ and 15 for $\lambda =0$ and −0.003, respectively. Other parameters are $K=5$ and ${\hslash}_{\mathrm{eff}}=0.25$.

**Figure 3.**(

**a**) The time-averaged value of mean energy $\langle {\overline{p}}^{2}\rangle $ in the parameter space $(K,\lambda )$ with ${\hslash}_{\mathrm{eff}}=0.25$. (

**b**) The $\langle {\overline{p}}^{2}\rangle $ versus $\lambda $ with $K=7$. (

**c**) The $\langle {\overline{p}}^{2}\rangle $ versus K with $\lambda =0.004$.

**Figure 4.**Left panels: Dependence of $\mathcal{F}$ at the time ${t}_{n}=1000$ on the imaginary part of the quasienergy ${\epsilon}_{i}$ with $\lambda =0.003$ (

**a**) and −0.003 (

**c**). Right panels: Comparison of the probability density distributions between the state $\left|\psi \right({t}_{n}=1000)\rangle $ (circles) and the quasieigenstate $|{\phi}_{\epsilon}\rangle $ (squares) of the maximum value of $\mathcal{F}$ (red diamonds) with $\lambda =0.003$ (

**b**) and −0.003 (

**d**). (

**d**) Red lines indicate the exponentially-localized shape ${\left|\psi \left(p\right)\right|}^{2}\propto {e}^{-\left|p\right|/\xi}$ with $\xi \approx 19$ (

**b**) and 15 (

**d**). Other parameters are the same as in Figure 2a.

**Figure 5.**Top two panels: $\langle \mathcal{I}\rangle $ versus K (

**a**) and $\lambda $ (

**b**) with $\hslash =0.1$ (squares), 0.25 (triangles), and 0.4 (circles). In (

**a**): Red lines indicate the function $\langle \mathcal{I}\rangle \propto \eta {K}^{2}$ with $\eta \approx 11$, 4.1, and 2.6 for ${\hslash}_{\mathrm{eff}}=0.1$, 0.25, and 0.4, respectively. The parameter is $\lambda =0.003$. In (

**b**): Dash-dotted line (in cyan) indicates the function $\langle \mathcal{I}\rangle \propto -ln\left(\lambda \right)$. Solid lines in red denote $\langle \mathcal{I}\rangle \propto -\alpha \lambda $ with $\alpha \approx 74$ and 45 for ${\hslash}_{\mathrm{eff}}=0.25$ and 0.4, respectively. The parameter is $K=5$. (

**c**) The $\langle \mathcal{I}\rangle $ in the parameter space $(K,\lambda )$ with ${\hslash}_{\mathrm{eff}}=0.25$.

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**MDPI and ACS Style**

Zhao, W.; Zhang, H.
Dynamical Stability in a Non-Hermitian Kicked Rotor Model. *Symmetry* **2023**, *15*, 113.
https://doi.org/10.3390/sym15010113

**AMA Style**

Zhao W, Zhang H.
Dynamical Stability in a Non-Hermitian Kicked Rotor Model. *Symmetry*. 2023; 15(1):113.
https://doi.org/10.3390/sym15010113

**Chicago/Turabian Style**

Zhao, Wenlei, and Huiqian Zhang.
2023. "Dynamical Stability in a Non-Hermitian Kicked Rotor Model" *Symmetry* 15, no. 1: 113.
https://doi.org/10.3390/sym15010113