Particle escape problem in semi-open system
Authors: Zhang Mao-Fang , You Hui-Min , Yin Xiang-Guo , Zhang Yun-Bo ,   State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Theoretical Physics, Shanxi University, Taiyuan 030006, China,   Key Laboratory of Optical Field Manipulation of Zhejiang Province, Physics Department, Zhejiang Sci-Tech University, Hangzhou 310018, China
Published: May 2022

<sec>We investigated the tunneling properties of a particle in a semi-open system. Starting initially from the eigenstate of the particle in the one-dimensional infinite well, we quench the infinitely high barrier on the right into a series of <inline-formula><tex-math id="M2">\begin{document}$\delta$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220450_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220450_M2.png"/></alternatives></inline-formula> barriers to observe the survival probability which is defined as the fidelity to the initial state. This constitutes a semi-Dirac comb model consisting of an infinitely high wall and multiple equally spaced <inline-formula><tex-math id="M3">\begin{document}$\delta$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220450_M3.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220450_M3.png"/></alternatives></inline-formula>-potential barriers. We first solve the exact analytical solution of this model and obtain the closed analytic form of the eigen function expressed by a recursive relation. For a single barrier, multiple potential barriers, the disordered potential barriers, the closed-form expression of the survival probability i.e., the initial state fidelity, is given for any evolution time and it is used to reveal the mechanism of the particle escape process. The dependence of survival probability on the strength of barrier, number of barriers, and disorder strength is calculated numerically based on fast Fourier transform method. The relevant parameters are used to control and suppress the particle escape problem. We found that for a single <inline-formula><tex-math id="M4">\begin{document}$\delta$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220450_M4.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220450_M4.png"/></alternatives></inline-formula>-potential barrier, the survival probability of the particle follows different trends in different decay time ranges. The particle in the ground state or excited states decays exponentially in a short time. After some time, the decay of the excited state will proceed with the same decay constant as that of the ground state. Finally, the survival probability follows a long-time inverse power law. The curve changes abruptly at different decay time intervals and is accompanied by significant oscillations. These oscillations in the transition region are caused by the interference of the exponential rate and the inverse power-law term, while the long-time non-exponential decay is due to the fact that the system energy spectrum has a lower bound. Increasing the barrier strength will greatly increase the probability of particles remaining in the well.</sec><sec>For multiple potential barriers, the reflection and transmission of particles between the potential barriers interfere with each other. When the strength of the potential barrier is small, the particle still decays exponentially. For a larger potential barrier strength, the probability of particle reflection increases, and the particles that tunnel out may be bounced back. The survival probability oscillates sharply, reaching higher fidelity at certain moments. The oscillatory maximum of the survival probability decreases linearly with the number of barriers, while the moment corresponding to the oscillatory maximum shows a parabolic increase with the number of barriers. The introduction of a series disordered barriers can significantly improve the survival probability and greatly suppress its oscillations over time.</sec><sec>Our calculation is expected to find applications in quantum control of particle escape problem in the disordered system.</sec>