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研究了半开放系统中粒子向开放空间的隧穿问题. 考虑由无限高的墙和多个
$\delta$ 函数势垒组成的半Dirac梳模型, 首先求解该模型的精确解析解, 其能量本征函数可以用递推关系以封闭解析的形式给出. 对单个势垒、多个势垒、无序势垒等不同情况, 利用傅里叶积分计算了任意时刻单粒子波函数的明确表示, 导出了由初态保真度定义的粒子生存几率闭合形式的表达式, 重点研究了粒子生存几率对势垒高度、无序强度等系统参数的依赖, 以及利用相关参数对衰减规律的操控及抑制. 发现多个势垒将大幅度提高粒子的生存几率, 无序的加入会极大地抑制其随时间的振荡.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 $\delta$ 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$\delta$ -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$\delta$ -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.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. Our calculation is expected to find applications in quantum control of particle escape problem in the disordered system. [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] -
图 1 . 初始位于宽度为a的无限深势阱中的粒子, 右势垒突然撤掉后向开放空间隧穿, 右方N个势垒与原来的系统组成了半Dirac梳. 图中显示了初始时刻及
$t=10 t_0$ 时在各阱的几率密度分布. 这里取$N=10$ -
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48]
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