Physics Department, Holon Institute of Technology, 58102 Holon, Israel.
Phys Rev E. 2017 Jan;95(1-1):012119. doi: 10.1103/PhysRevE.95.012119. Epub 2017 Jan 12.
Decay of the metastable state is analyzed within the quantum Kramers model in the weak-to-intermediate dissipation regime. The decay kinetics in this regime is determined by energy exchange between the unstable mode and the stable modes of thermal bath. In our previous paper [Phys. Rev. A 42, 4427 (1990)PLRAAN1050-294710.1103/PhysRevA.42.4427], Grabert's perturbative approach to well dynamics in the case of the discrete bath [Phys. Rev. Lett. 61, 1683 (1988)PRLTAO0031-900710.1103/PhysRevLett.61.1683] has been extended to account for the second order terms in the classical equations of motion (EOM) for the stable modes. Account of the secular terms reduces EOM for the stable modes to those of the forced oscillator with the time-dependent frequency (TDF oscillator). Analytic expression for the characteristic function of energy loss of the unstable mode has been derived in terms of the generating function of the transition probabilities for the quantum forced TDF oscillator. In this paper, the approach is further developed and applied to the case of the continuous frequency spectrum of the bath. The spectral density functions of the bath of stable modes are expressed in terms of the dissipative properties (the friction function) of the original bath. They simplify considerably for the one-dimensional systems, when the density of phonon states is constant. Explicit expressions for the fourth order corrections to the linear response theory result for the characteristic function of the energy loss and its cumulants are obtained for the particular case of the cubic potential with Ohmic (Markovian) dissipation. The range of validity of the perturbative approach in this case is determined (γ/ω_{b}<0.26), which includes the turnover region. The dominant correction to the linear response theory result is associated with the "work function" and leads to reduction of the average energy loss and its dispersion. This reduction increases with the increasing dissipation strength (up to ∼10%) within the range of validity of the approach. We have also calculated corrections to the depopulation factor and the escape rate for the quantum and for the classical Kramers models. Results for the classical escape rate are in very good agreement with the numerical simulations for high barriers. The results can serve as an additional proof of the robustness and accuracy of the linear response theory.
在弱到中等耗散范围内，通过量子 Kramers 模型分析亚稳态的衰减。在此范围内的衰减动力学由不稳定模式与热浴稳定模式之间的能量交换决定。在我们之前的论文中[Phys. Rev. A 42, 4427 (1990)PLRAAN1050-294710.1103/PhysRevA.42.4427]，我们扩展了 Grabert 对离散浴中势阱动力学的微扰方法[Phys. Rev. Lett. 61, 1683 (1988)PRLTAO0031-900710.1103/PhysRevLett.61.1683]，以考虑稳定模式的经典运动方程（EOM）中的二阶项。考虑到非周期性项，稳定模式的 EOM 可以简化为具有时变频率（TDF 振荡器）的受迫振荡器。不稳定模式的能量损耗特征函数的解析表达式是根据量子受迫 TDF 振荡器的跃迁概率生成函数推导出来的。在本文中，该方法得到了进一步发展，并应用于浴的连续频谱的情况。稳定模式的浴的谱密度函数表示为原始浴的耗散特性（摩擦函数）。对于一维系统，当声子态密度恒定时，它们会大大简化。对于立方势与欧姆（马尔可夫）耗散的特殊情况，得到了线性响应理论结果的能量损耗特征函数及其累积量的四阶修正的显式表达式。在这种情况下，微扰方法的有效范围（γ/ω_{b}<0.26）确定了，包括转折点区域。与线性响应理论结果的主要修正相关的是“功函数”，它导致平均能量损耗及其分散的减少。这种减少随着方法的有效范围内耗散强度的增加而增加（高达约 10%）。我们还计算了量子和经典 Kramers 模型的排空因子和逃逸率的修正。对于高势垒，经典逃逸率的结果与数值模拟非常吻合。这些结果可以作为线性响应理论的稳健性和准确性的额外证明。
