周期驱动四次振子的功统计：经典动力学与量子动力学
Work statistics in the periodically driven quartic oscillator: classical versus quantum dynamics
摘要：
在纳米系统的热力学中，经典和量子力学描述之间的关系特别重要。为了仔细研究这种对应关系，我们研究了一个非谐振子驱动的周期性外力与缓慢变化的振幅经典和量子力学的框架内。驱动引起的振子能量变化与系统功的概率分布密切相关。随着驱动的幅度$λ(t)$从零增加到最大值$λ_{max}$，然后再次回到零，初始和最终哈密顿量一致。然后，主要的感兴趣的量是从初始能量$E_i$到最终能量$E_f$的跃迁的概率密度$P(E_f|E_i)$。在经典的情况下，非对角跃迁与$E_f\neq E_i$主要是由于分界线交叉的机制。我们表明，近似的分析结果内的钟摆近似是根据数值模拟。在量子的情况下，数值精确的结果补充分析参数采用Floquet理论。对于经典和量子情形，我们对$P(E_f|E_i)$随驱动的最大振幅$λ_{max}$的周期性变化提供了直观的解释.
英文摘要：
In the thermodynamics of nanoscopic systems the relation between classical and quantum mechanical description is of particular importance. To scrutinize this correspondence we study an anharmonic oscillator driven by a periodic external force with slowly varying amplitude both classically and within the framework of quantum mechanics. The energy change of the oscillator induced by the driving is closely related to the probability distribution of work for the system. With the amplitude $λ(t)$ of the drive increasing from zero to a maximum $λ_{max}$ and then going back to zero again initial and final Hamiltonian coincide. The main quantity of interest is then the probability density $P(E_f|E_i)$ for transitions from initial energy $E_i$ to final energy $E_f$. In the classical case non-diagonal transitions with $E_f\neq E_i$ mainly arise due to the mechanism of separatrix crossing. We show that approximate analytical results within the pendulum approximation are in accordance with numerical simulations. In the quantum case numerically exact results are complemented with analytical arguments employing Floquet theory. For both classical and quantum case we provide an intuitive explanation for the periodic variation of $P(E_f|E_i)$ with the maximal amplitude $λ_{max}$ of the driving.
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