AI 中文总结
该研究提出带均匀记忆的量子随机重置协议，推导任意时不变哈密顿量下密度矩阵元的精确演化，区分有隙与无隙量子系统的不同动力学行为，并以具体系统为例验证结果。
AI 中文摘要
我们提出了一种具有均匀记忆的量子随机重置协议，其中每次重置事件都会将系统返回到其整个历史中随机均匀选择的某个时刻所访问过的状态。由此产生的动力学是非幺正的、非马尔可夫的，是经典优先重定位模型的直接量子推广。我们在能量本征基下工作，对任意不随时间变化的哈密顿量，推导了所有密度矩阵元的精确演化，结果表明哈密顿量仅通过对应的玻尔频率参与动力学过程，这自然将量子系统分为两类：有隙系统和无隙系统。对于有隙系统（具有离散能谱的系统），其对角元保持不变，而能量本征基下密度矩阵的非对角元以连续变化的指数代数衰减，振幅随log t周期性振荡，因此系统会趋近于一个与重置速率无关的稳态，且保留对初始状态的强记忆。对于无隙系统（具有连续能谱的系统），任意小的玻尔频率会阻止稳态的出现，取而代之的是，位置分布在通用的超慢尺度log(rt)/r上扩散，且与初始状态和哈密顿量的细节无关。我们以二能级系统、谐振子和自由量子粒子为例说明了这些结果，并将其与经典对应物进行了对比。
英文摘要
We introduce a quantum stochastic resetting protocol with uniform memory, in which each resetting event returns the system to a state visited at a time chosen uniformly from its entire history. The resulting dynamics is nonunitary, non-Markovian and a direct quantum generalization of the classical preferential relocation model. Working in the energy eigenbasis, we derive the exact evolution of every density-matrix element for an arbitrary time-independent Hamiltonian and show that the Hamiltonian enters the dynamics only through the corresponding Bohr frequencies. This leads to a natural distinction between two classes of quantum systems: gapped and gapless. In \emph{gapped systems} (systems with a discrete energy spectrum), while the diagonal elements remain unchanged, the off-diagonal elements of the density matrix in the energy eigenbasis decay algebraically with a continuously varying exponent and with an amplitude that oscillates periodically in $\log t$. The system therefore approaches a stationary state that is independent of the resetting rate and retains a strong memory of the initial state. In \emph{gapless systems} (systems with a continuous energy spectrum), arbitrarily small Bohr frequencies prevent stationarity. Instead, the position distribution spreads on the universal (ultra-slow) scale $\log(rt)/r$, independently of the initial state and of the details of the Hamiltonian. We illustrate these results with a two-level system, a harmonic oscillator, and a free quantum particle, and contrast them with their classical counterparts.
