损益平衡振荡器中$\mathcal{PT}$对称性对随机性的恢复力
Resilience of $\mathcal{PT}$ symmetry against stochasticity in a gain-loss balanced oscillator
摘要：
我们调查的二分法噪声的影响，添加到一个经典的谐振子的形式随机的时间依赖的增益和损失的状态，其持续时间从两个不同的指数等待时间分布采样。尽管随机性，稳定性标准可以制定时，平均在许多实现的渐近时间限制和服务，以确定边界线在参数空间中的分离区域的增长幅度从衰减的。此外，$\mathcal{PT}$对称性的概念仍然适用于这样的随机振荡器，我们用它来区分欠阻尼对称相位和过阻尼非对称相位。在前一种情况下，稳定性的极限由增益和损耗状态的相同平均持续时间来标记，而在后一种情况下，需要更长的损耗状态持续时间来保持系统稳定。过阻尼相具有施加位置-速度比锁定的有序结构，并且被视为从欠阻尼相的相变，欠阻尼相相反地显示出宽的和更无序的，但是仍然是$\mathcal{PT}$对称结构。我们还解决了短的时间限制和动态的时刻的位置和速度，目的是揭示这个显然是相当简单的机械系统提供了非常丰富的动态。迄今为止建立的概念可以扩展和应用在超材料和光纤中的光传播的稳定与随机分布区域的非对称有源和无源介质。
英文摘要：
We investigate the effects of dichotomous noise added to a classical harmonic oscillator in the form of stochastic time-dependent gain and loss states, whose durations are sampled from two distinct exponential waiting time distributions. Despite the stochasticity, stability criteria can be formulated when averaging over many realizations in the asymptotic time limit and serve to determine the boundary line in parameter space that separates regions of growing amplitudes from those of decaying ones. Furthermore, the concept of $\mathcal{PT}$ symmetry remains applicable for such a stochastic oscillator and we use it to distinguish between an underdamped symmetric phase and an overdamped asymmetric phase. In the former case, the limit of stability is marked by the same average duration for the gain and loss states, whilst in the the latter case, a higher duration of the loss state is necessary to keep the system stable. The overdamped phase has an ordered structure imposing a position-velocity ratio locking and is viewed as a phase transition from the underdamped phase, which instead displays a broad and more disordered, but nevertheless, $\mathcal{PT}$ symmetric structure. We also address the short time limit and the dynamics of the moments of the position and the velocity with the aim of revealing the extremely rich dynamics offered by this apparently quite simple mechanical system. The notions established so far may be extended and applied in the stabilization of light propagation in metamaterials and optical fibres with randomly distributed regions of asymmetric active and passive media.
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