Chin. Phys. B, 2020, Vol. 29(12): 120302    DOI: 10.1088/1674-1056/abc0dc
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# Chaotic dynamics of complex trajectory and its quantum signature

Wen-Lei Zhao(赵文垒)1,†, Pengkai Gong(巩膨恺)1, Jiaozi Wang(王骄子)2, and Qian Wang(王骞)3,
1 School of Science, Jiangxi University of Science and Technology, Ganzhou 341000, China; 2 Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China; 3 Department of Physics, Zhejiang Normal University, Jinhua 321004, China
Abstract  We investigate both the quantum and classical dynamics of a non-Hermitian system via a kicked rotor model with $\mathcalPT$ symmetry. For the quantum dynamics, both the mean momentum and mean square of momentum exhibit the staircase growth with time when the system parameter is in the neighborhood of the $\mathcalPT$ symmetry breaking point. If the system parameter is much larger than the $\mathcalPT$ symmetry breaking point, the accelerator mode results in the directed spreading of the wavepackets as well as the ballistic diffusion in momentum space. For the classical dynamics, the non-Hermitian kicking potential leads to the exponentially-fast increase of classical complex trajectories. As a consequence, the imaginary part of the trajectories exponentially diffuses with time, while the real part exhibits the normal diffusion. Our analytical prediction of the exponential diffusion of imaginary momentum and its breakdown time is in good agreement with numerical results. The quantum signature of the chaotic diffusion of the complex trajectories is reflected by the dynamics of the out-of-time-order correlators (OTOC). In the semiclassical regime, the rate of the exponential increase of the OTOC is equal to that of the exponential diffusion of the complex trajectories.
Keywords:  $\calPT$ symmetry      quantum-classical correspondence      quantum chaos
Received:  11 July 2020      Revised:  28 August 2020      Published:  13 November 2020
 PACS: 03.65.-w (Quantum mechanics) 03.65.Yz (Decoherence; open systems; quantum statistical methods) 05.45.-a (Nonlinear dynamics and chaos) 05.45.Mt (Quantum chaos; semiclassical methods)
Fund: Project partially supported by the National Natural Science Foundation of China (Grant Nos. 12065009, 11804130, and 11805165) and Zhejiang Provincial Nature Science Foundation, China (Grant No. LY20A050001).
Corresponding Authors:  Corresponding author. E-mail: wlzhao@jxust.edu.cn Corresponding author. E-mail: qwang@zjnu.edu.cn