量子临界动力学调控的理论研究

In recent years, the control of quantum critical dynamics has attracted a lot of attention because of its high relevance to topological computation and quantum simulation. There exists a generic class of quantum critical systems that can be mapped into free-fermion models. These systems are not only relevant to a variety of important practical systems, but also serve as a test-bed for investigating quantum critical non-equilibrium dynamics, owning to the fact that they can generally be exactly soluble. In this project, we will study in detail the optimal control of quantum critical dynamics and non-equilibrium quench processes in these types of systems. We will focus on the following three subjects: 1) Using the optimal control algorithm recently developed by the applicant that can be applied to a class of many-body systems, we will study the optimal preparation of macroscopic entanglement states in several critical spin systems; 2) Develop a gradient-based multi-variable optimal control algorithm that combines the transitionless quantum driving method；3) Study quench dynamics in systems that can support topological phases. In particular, we will study the formation or decay of Majorana zero modes under the quench of boundary parameters in a Kitaev one-dimensional p-wave superconductor chain with twisted boundary conditions, which was recently proposed by the applicant and coauthors.

近几年来，量子临界动力学的调控因其与拓扑量子计算和量子模拟的密切关系而受到了广泛关注。常见的量子临界系统中，存在一类可以化归为自由费米子模型的系统。 它们不仅与多种重要的实际体系相关，更因其能够精确求解而成为研究量子临界非平衡动力学的良好试验体系。本项目拟对该类体系中量子临界动力学的最优化控制和非平衡淬火过程展开深入研究。具体研究内容包括：1) 利用申请人最近发展的可用于一类多体问题的最优化算法，研究几种临界自旋系统中宏观纠缠态的最优化制备；2) 发展与无跃迁量子驱动方法相结合的多变量最优化梯度算法；3) 研究拓扑相变体系中的淬火动力学。特别地，在申请人及其合作者最近提出的扭曲边界Kitaev一维p-波超导链中，研究边界参数淬火下Majorana零模的形成或破坏过程。

本项目旨在理论研究量子临界动力学的动力学调控，为实验上实现量子模拟、拓扑量子相等提供理论指导。虽然对临界系统的讨论大多集中在热力学及线下，然而真实的系统总是有限的。鉴于此，本项目在有限尺寸低维自旋系统（主要是具有量子相变的系统）中若干精确结果及其应用方面做了深入探究，并得到了以下研究成果：1. 得到了有限尺度非均匀量子XX链中自旋算符矩阵元的行列式表达式；2. 利用“算符方法”得到了高自旋非均匀XXZ中心自旋模型的可解性条件，并得到了相应的Bethe ansatz方程；亦证明了高自旋XX中心自旋模型的部分可解性；3. 找到了有限尺度非均匀一维横场compass模型中精确Majorana零模的个数的解析表达及其波函数；5. 揭示了有限尺度量子Ising模型中铁磁态与动量空间基态之间的解析关系，并提出了可以计算大尺度系统中破坏费米子数宇称的纵向磁矩动力学的Pfaffian方法。.以上精确结果在中心自旋动力学、量子电池的储能、有机分子晶体与光场的强耦合、量子计量学、拓扑量子计算、离散时间晶体等研究中具有重要应用。