驱动量子动力学：它会混合吗？
Driven quantum dynamics: will it blend?
摘要：
随机性在密码学、黑洞物理学、随机矩阵理论和蒙特卡罗抽样等现代科学的许多学科中都是必不可少的工具。在量子系统中，由于所谓的q-设计，随机操作可以通过随机电路获得，并且在黑洞的快速置乱猜想中发挥着核心作用。在这里，我们考虑一个更物理动机的方式产生随机演化，利用多体动力学的量子系统驱动的随机外部脉冲。我们结合量子控制，开放量子系统和精确可解模型（通过Bethe-Answer）的技术来产生驱动多体系统中的Haar均匀随机操作。我们证明了任何完全可控的系统在长时间极限下都收敛于幺正q-设计。此外，我们还研究了一个驱动自旋链的收敛时间，通过将其随机演化映射到一个具有可积Liouville的半群上，并找到它的间隙。值得注意的是，我们发现通过Bethe-Anglands技术，差距是独立的q。我们使用平均场技术认为，这一属性可能是典型的其他可控系统，虽然我们明确地构建反例通过对称性破缺参数表明，这并不总是如此。我们的发现开辟了新的物理方法，通过量子多体动力学和随机驱动的结合，将经典随机性转化为量子随机性。
英文摘要：
Randomness is an essential tool in many disciplines of modern sciences, such as cryptography, black hole physics, random matrix theory and Monte Carlo sampling. In quantum systems, random operations can be obtained via random circuits thanks to so-called q-designs, and play a central role in the fast scrambling conjecture for black holes. Here we consider a more physically motivated way of generating random evolutions by exploiting the many-body dynamics of a quantum system driven with stochastic external pulses. We combine techniques from quantum control, open quantum systems and exactly solvable models (via the Bethe-Ansatz) to generate Haar-uniform random operations in driven many-body systems. We show that any fully controllable system converges to a unitary q-design in the long-time limit. Moreover, we study the convergence time of a driven spin chain by mapping its random evolution into a semigroup with an integrable Liouvillean and finding its gap. Remarkably, we find via Bethe-Ansatz techniques that the gap is independent of q. We use mean-field techniques to argue that this property may be typical for other controllable systems, although we explicitly construct counter-examples via symmetry breaking arguments to show that this is not always the case. Our findings open up new physical methods to transform classical randomness into quantum randomness, via a combination of quantum many-body dynamics and random driving.
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