# Dynamical Stability in a Non-Hermitian Kicked Rotor Model

^{*}

## 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

- Haake, F. Quantum Signatures of Chaos, 3rd ed.; Springer Series in Synergetics; Springer: Berlin, Germany, 2010. [Google Scholar]
- Wang, W.G.; He, L.W.; Gong, J. Preferred States of Decoherence under Intermediate System-Environment Coupling. Phys. Rev. Lett.
**2012**, 108, 070403. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Wang, W.G.; Li, B.W. Uniform semiclassical approach to fidelity decay: From weak to strong perturbation. Phys. Rev. E
**2005**, 71, 066203. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Cao, A.; Sajjad, R.; Mas, H.; Simmons, E.Q.; Tanlimco, J.L.; Nolasco-Martinez, E.; Shimasaki, T.; Kondakci, H.E.; Galitski, V.; Weld, D.M. Interaction-driven breakdown of dynamical localization in a kicked quantum gas. Nat. Phys.
**2022**, 18, 1302. [Google Scholar] [CrossRef] - Martinez, M.; Larré, P.; Delande, D.; Cherroret, N. Low-energy prethermal phase and crossover to thermalization in nonlinear kicked rotors. Phys. Rev. A
**2022**, 106, 043304. [Google Scholar] [CrossRef] - Peres, A. Stability of quantum motion in chaotic and regular systems. Phys. Rev. A
**1984**, 30, 1610. [Google Scholar] [CrossRef] - Casati, G.; Chirikov, B.V.; Izrailev, F.M.; Ford, J. Lecture Notes in Physics; Springer: Berlin, Germany, 1979; Volume 90, p. 334. [Google Scholar]
- Fishman, S.; Grempel, D.R.; Prange, R.E. Chaos, Quantum Recurrences, and Anderson Localization. Phys. Rev. Lett.
**1982**, 49, 509. [Google Scholar] [CrossRef] - Santhanam, M.S.; Paul, S.; Kannan, J.B. Quantum kicked rotor and its variants: Chaos, localization and beyond. Phys. Rep.
**2022**, 956, 1. [Google Scholar] [CrossRef] - Wang, W.G.; Casati, G.; Li, B.W. Stability of quantum motion: Beyond Fermi-golden-rule and Lyapunov decay. Phys. Rev. E
**2004**, 69, 025201(R). [Google Scholar] [CrossRef] [Green Version] - Hainaut, C.; Fang, P.; Rançon, A.; Clément, J.F.; Szriftgiser, P.; Garreau, J.C.; Tian, C.; Chicireanu, R. Experimental Observation of a Time-Driven Phase Transition in Quantum Chaos. Phys. Rev. Lett.
**2018**, 121, 134101. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Zhou, L.; Gu, Y.; Gong, J. Dual topological characterization of non-Hermitian Floquet phases. Phys. Rev. B
**2021**, 103, L041404. [Google Scholar] [CrossRef] - Gong, J.; Wang, Q. Time-dependent $\mathcal{PT}$-symmetric quantum mechanics. J. Phys. A Math. Theor.
**2013**, 46, 485302. [Google Scholar] [CrossRef] [Green Version] - Zhang, Z.; Zhang, Y.; Sheng, J.; Yang, L.; Miri, M.A.; Christodoulides, D.N.; He, B.; Zhang, Y.; Xiao, M. Observation of Parity-Time Symmetry in Optically Induced Atomic Lattices. Phys. Rev. Lett.
**2016**, 117, 123601. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Li, J.; Harter, A.K.; Liu, J.; Melo, L.; Joglekar, Y.N.; Luo, L. Observation of parity-time symmetry breaking transitions in a dissipative Floquet system of ultracold atoms. Nat. Commun.
**2019**, 10, 855. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Zhao, X.M.; Guo, C.X.; Yang, M.L.; Liu, W.M.; Kou, S.P. Anomalous non-Abelian statistics for non-Hermitian generalization of Majorana zero modes. Phys. Rev. B
**2021**, 104, 214502. [Google Scholar] [CrossRef] - Yu, Z.F.; Xue, J.K.; Zhuang, L.; Zhao, J.; Liu, W.M. Non-Hermitian spectrum and multistability in exciton-polariton condensates. Phys. Rev. B
**2021**, 104, 235408. [Google Scholar] [CrossRef] - Zhao, X.M.; Guo, C.X.; Kou, S.P.; Zhuang, L.; Liu, W.M. Defective Majorana zero modes in a non-Hermitian Kitaev chain. Phys. Rev. B
**2021**, 104, 205131. [Google Scholar] [CrossRef] - Hu, H.; Sun, S.; Chen, S. Knot topology of exceptional point and non-Hermitian no-go theorem. Phys. Rev. Res.
**2022**, 4, L022064. [Google Scholar] [CrossRef] - Gopalakrishnan, S.; Gullans, M.J. Entanglement and Purification Transitions in Non-Hermitian Quantum Mechanics. Phys. Rev. Lett.
**2021**, 126, 170503. [Google Scholar] [CrossRef] - Pires, D.P.; Macrì, T. Probing phase transitions in non-Hermitian systems with multiple quantum coherences. Phys. Rev. B
**2021**, 104, 155141. [Google Scholar] [CrossRef] - Bergholtz, E.J.; Budich, J.C.; Kunst, F.K. Exceptional topology of non-Hermitian systems. Rev. Mod. Phys.
**2021**, 93, 015005. [Google Scholar] [CrossRef] - Zou, D.; Chen, T.; He, W.; Bao, J.; Lee, C.H.; Sun, H.; Zhang, X. Observation of hybrid higher-order skin-topological effect in non-Hermitian topolectrical circuits. Nat. Commun.
**2021**, 12, 7201. [Google Scholar] [CrossRef] [PubMed] - Xue, Y.; Hang, C.; He, Y.; Bai, Z.; Jiao, Y.; Huang, G.; Zhao, J.; Jia, S. Experimental observation of partial parity-time symmetry and its phase transition with a laser-driven cesium atomic gas. Phys. Rev. A
**2022**, 105, 053516. [Google Scholar] [CrossRef] - Budich, C.; Bergholtz, E.J. Non-Hermitian Topological Sensors. Phys. Rev. Lett.
**2020**, 125, 180403. [Google Scholar] [CrossRef] [PubMed] - McDonald, A.; Clerk, A.A. Exponentially-enhanced quantum sensing with non-Hermitian lattice dynamics. Nat. Commun.
**2020**, 11, 5382. [Google Scholar] [CrossRef] - Dai, C.; Shi, Z.; Yi, X. Floquet theorem with open systems and its applications. Phys. Rev. A
**2016**, 93, 032121. [Google Scholar] [CrossRef] [Green Version] - Xiao, L.; Deng, T.; Wang, K.; Wang, K.; Zhu, G.; Wang, Z.; Yi, W.; Xue, P. Non-Hermitian bulk–boundary correspondence in quantum dynamics. Nat. Phys.
**2020**, 16, 761–766. [Google Scholar] [CrossRef] [Green Version] - West, C.T.; Kottos, T.; Prosen, T. $\mathcal{PT}$-Symmetric Wave Chaos. Phys. Rev. Lett.
**2010**, 104, 054102. [Google Scholar] [CrossRef] [Green Version] - Zhao, W.L.; Wang, J.; Wang, X.; Tong, P. Directed momentum current induced by the $\mathcal{PT}$-symmetric driving. Phys. Rev. E
**2019**, 99, 042201. [Google Scholar] [CrossRef] [Green Version] - Zhao, W.L. Quantization of out-of-time-ordered correlators in non-Hermitian chaotic systems. Phys. Rev. Res.
**2022**, 4, 023004. [Google Scholar] [CrossRef] - Zhao, W.L.; Zhou, L.W.; Liu, J.; Tong, P.; Huang, K.Q. Superexponential diffusion in nonlinear non-Hermitian systems. Phys. Rev. A
**2020**, 102, 062213. [Google Scholar] [CrossRef] - Haug, F.; Bienert, M.; Schleich, W.P.; Seligman, T.H.; Raizen, M.G. Motional stability of the quantum kicked rotor: A fidelity approach. Phys. Rev. A
**2015**, 71, 043803. [Google Scholar] [CrossRef] [Green Version] - Shrestha, R.K.; Wimberger, S.; Ni, J.; Lam, W.K.; Summy, G.S. Fidelity of the quantum δ-kicked accelerator. Phys. Rev. E
**2013**, 87, 020902. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Probst, B.; Dubertrand, R.; Wimberger, S. Fidelity of the near-resonant quantum kicked rotor. J. Phys. A Math. Theor.
**2011**, 44, 335101. [Google Scholar] [CrossRef] - Chirikov, B.V. A universal instability of many-dimensional oscillator systems. Phys. Rep.
**1979**, 52, 263. [Google Scholar] [CrossRef] - Fishman, S.; Prange, R.E.; Griniasty, M. Scaling theory for the localization length of the kicked rotor. Phys. Rev. A
**1989**, 39, 1628. [Google Scholar] [CrossRef] [PubMed] - Satpathi, U.; Sinha, S.; Sorkin, R.D. A quantum diffusion law. J. Stat. Mech.
**2017**, 2017, 123105. [Google Scholar] [CrossRef] [Green Version] - Shepelyansky, D.L. Some statistical properties of simple classically stochastic quantum systems. Phys. D
**1983**, 8, 208. [Google Scholar] [CrossRef] - Shirley, J.H. Solution of the Schrödinger Equation with a Hamiltonian Periodic in Time. Phys. Rev.
**1965**, 138, B979. [Google Scholar] [CrossRef] - Sambe, H. Steady States and Quasienergies of a Quantum-Mechanical System in an Oscillating Field. Phys. Rev. A
**1973**, 7, 2203. [Google Scholar] [CrossRef] - Longhi, S. Localization, quantum resonances, and ratchet acceleration in a periodically kicked $\mathcal{PT}$-symmetric quantum rotator. Phys. Rev. A
**2017**, 95, 012125. [Google Scholar] [CrossRef] - Edwards, J.T.; Thouless, D.J. Numerical studies of localization in disordered systems. J. Phys. C Solid State Phys.
**1972**, 5, 807. [Google Scholar] [CrossRef] [Green Version] - Izrailev, F. Simple models of quantum chaos: Spectrum and eigenfunctions. Phys. Rep.
**1990**, 196, 299. [Google Scholar] [CrossRef] - Keser, A.C.; Ganeshan, S.; Refael, G.; Galitski, V. Dynamical many-body localization in an integrable model. Phys. Rev. B
**2016**, 94, 085120. [Google Scholar] [CrossRef] [Green Version] - Rozenbaum, E.B.; Galitski, V. Dynamical localization of coupled relativistic kicked rotors. Phys. Rev. B
**2017**, 95, 064303. [Google Scholar] [CrossRef] [Green Version] - Čadež, T.; Mondaini, R.; Sacramento, P.D. Dynamical localization and the effects of aperiodicity in Floquet systems. Phys. Rev. B
**2017**, 96, 144301. [Google Scholar] [CrossRef] [Green Version] - Harper, F.; Roy, R.; Rudner, M.S.; Sondhi, S.L. Topology and Broken Symmetry in Floquet Systems. Annu. Rev. Condens. Matter Phys.
**2020**, 11, 345–368. [Google Scholar] [CrossRef] [Green Version] - Liang, H.; Li, L. Topological properties of non-Hermitian Creutz ladders. Chin. Phys. B
**2022**, 31, 010310. [Google Scholar] [CrossRef] - Zhou, L.; Gu, Y. Topological delocalization transitions and mobility edges in the nonreciprocal Maryland model. J. Phys. Condens. Matter
**2022**, 34, 115402. [Google Scholar] [CrossRef] - Fleckenstein, C.; Bukov, M. Thermalization and prethermalization in periodically kicked quantum spin chains. Phys. Rev. B
**2021**, 103, 144307. [Google Scholar] [CrossRef] - Xu, J.; Zhong, C.; Han, X.; Jin, D.; Jiang, L.; Zhang, X. Floquet Cavity Electromagnonics. Phys. Rev. Lett.
**2020**, 125, 237201. [Google Scholar] [CrossRef] - Kumar, U.; Banerjee, S.; Lin, S. Floquet engineering of Kitaev quantum magnets. Commun. Phys.
**2022**, 5, 157. [Google Scholar] [CrossRef] - Quito, V.L.; Flint, R. Floquet Engineering Correlated Materials with Unpolarized Light. Phys. Rev. Lett.
**2021**, 126, 177201. [Google Scholar] [CrossRef] [PubMed] - Topp, G.E.; Jotzu, G.; McIver, J.W.; Xian, L.; Rubio, A.; Sentef, M.A. Topological Floquet engineering of twisted bilayer graphene. Phys. Rev. Res.
**2019**, 1, 023031. [Google Scholar] [CrossRef] [Green Version] - Menu, R.; Roscilde, T. Anomalous Diffusion and Localization in a Positionally Disordered Quantum Spin Array. Phys. Rev. Lett.
**2020**, 124, 130604. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Bitter, M.; Milner, V. Control of quantum localization and classical diffusion in laser-kicked molecular rotors. Phys. Rev. A
**2017**, 95, 013401. [Google Scholar] [CrossRef] [Green Version] - Paul, N.; Amir, A. Quantum diffusion in the strong tunneling regime. Phys. Rev. B
**2019**, 100, 024110. [Google Scholar] [CrossRef]

**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 ${\hslash}_{\mathrm{eff}}=0.25$.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2022 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**

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