包含连续阈值效应的两个合并本征值附近的特征动力学
Characteristic dynamics near two coalescing eigenvalues incorporating continuum threshold effects
摘要：
在文献中已经报道，在两个本征态合并的例外点附近的生存概率$P(t)$通常应该表现出$P(t) \sim t^2 e^{-Γt}$的演化，其中$Γ$是合并本征态的衰减率;这已经在微波台球实验中得到验证[B. Dietz等人，Phys.Rev.E75，027201（2007）]。然而，启发式的有效哈密顿，通常用来获得这一结果忽略了可能的影响的连续阈值上的动力学。相比之下，在这项工作中，我们采用了一种分析方法，从微观哈密顿代表两个简单的模型，以表明连续阈值有很强的影响，在各种情况下的异常点附近的动态。为了报告我们的结果，我们将厄米开放量子系统中的例外点分为两种情况：在EP 2A中，两个虚束缚态在形成具有复共轭本征值的共振、反共振对之前合并，而在EP 2B中，两个共振在形成两个不同的共振之前合并。对于EP 2B，这是在微波台球实验中研究的情况下，我们验证的生存概率表现出先前报道的修改后的指数衰减中间的时间尺度，但这是取代了逆幂律非常长的时间尺度。同时，对于EP 2A，连续阈值的影响是如此之强，以至于在所有时间尺度上的进化都是非指数的，启发式方法完全失败。当EP 2A出现非常接近的阈值，我们得到了新的发展$P(t) \sim 1 - C_1 \sqrt{t} + D_1 t$的中间时间尺度上，而进一步远离抛物线衰减（芝诺动力学）的短时间尺度上的增强。我们还在附录中讨论了EP 2A的参数包围。
英文摘要：
It has been reported in the literature that the survival probability $P(t)$ near an exceptional point where two eigenstates coalesce should generally exhibit an evolution $P(t) \sim t^2 e^{-Γt}$, in which $Γ$ is the decay rate of the coalesced eigenstate; this has been verified in a microwave billiard experiment [B. Dietz, et al, Phys. Rev. E 75, 027201 (2007)]. However, the heuristic effective Hamiltonian that is usually employed to obtain this result ignores the possible influence of the continuum threshold on the dynamics. By contrast, in this work we employ an analytical approach starting from the microscopic Hamiltonian representing two simple models in order to show that the continuum threshold has a strong influence on the dynamics near exceptional points in a variety of circumstances. To report our results, we divide the exceptional points in Hermitian open quantum systems into two cases: at an EP2A two virtual bound states coalesce before forming a resonance, anti-resonance pair with complex conjugate eigenvalues, while at an EP2B two resonances coalesce before forming two different resonances. For the EP2B, which is the case studied in the microwave billiard experiment, we verify the survival probability exhibits the previously reported modified exponential decay on intermediate timescales, but this is replaced with an inverse power law on very long timescales. Meanwhile, for the EP2A the influence from the continuum threshold is so strong that the evolution is non-exponential on all timescales and the heuristic approach fails completely. When the EP2A appears very near the threshold we obtain the novel evolution $P(t) \sim 1 - C_1 \sqrt{t} + D_1 t$ on intermediate timescales, while further away the parabolic decay (Zeno dynamics) on short timescales is enhanced. We also discuss the parametric encirclement of the EP2A in an appendix.
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