We study the effect of the initial-state energy variance to the short-time behavior of the Loschmidt echo (LE) in a purely dephasing model. We find that the short-time LE behaves as a Gaussian function with the width determined by the initial-state energy variance of the interaction Hamiltonian, while it is a quartic decaying function with the width determined by the initial-state energy variance of the commutator between the interaction Hamiltonian and the environmental Hamiltonian when the initial state is an eigenstate of the interaction Hamiltonian. Furthermore, the Gaussian envelope in the temporal evolution of LE in strong coupling regime is determined by the inband variance. We will also verify the above conclusion in the XY spin model (as environment).

中文翻译：

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退相干中的能量方差*

我们研究了纯相位模型中初始状态能量方差对Loschmidt回波（LE）短时行为的影响。我们发现短时LE表现为高斯函数，其宽度由相互作用哈密顿量的初始能方差确定，而它是四次衰减函数，其宽度由换向器的初始能方差确定当初始状态是相互作用哈密顿量的本征态时，相互作用哈密顿量和环境哈密顿量之间的关系。此外，在强耦合状态下LE的时间演化中的高斯包络由带内方差确定。我们还将在XY自旋模型（作为环境）中验证以上结论。