哈密顿量诱导的量子演化速度

在研究量子态的演化问题时,量子演化速率往往定义为量子初态与其演化态之间的态距离随时间的变化率.本文将量子演化的基本理论与线性代数方法相结合,通过量子态演化的路径距离来研究量子系统的演化.量子幺正演化系统中,量子演化算符包含了量子演化的路径信息,路径距离的大小则决定于演化算符本征值的幅角主值分布.由量子态演化的路径距离随时间的变化率而得到的量子瞬时演化速率则正比于系统哈密顿量的最大与最小本征值之差.作为应用之一,利用量子演化的路径距离及哈密顿量诱导的瞬时演化速率,可以给出量子演化新的时间下限.此时间下限只与系统的演化算符及哈密顿量有关,而与量子初态的具体形式无关,这与量子系统真实演化时间所具有的性质一致.严格的理论证明以及两个演化实例的数值结果均表明,在[0,π/(2ω_H)]时间范围内,本文给出的演化时间下限与真实演化时间重合,是真实演化时间的准确预测.通过量子演化的路径距离及相应演化速率来研究量子系统的演化,为相关问题的解答提供了新的思路和方法.     

Quantum speed limit for mixed states in a unitary system

Since the evolution of a mixed state in a unitary system is equivalent to the joint evolution of the eigenvectors contained in it, we could use the tool of instantaneous angular velocity for pure states to study the quantum speed limit (QSL) of a mixed state. We derive a lower bound for the evolution time of a mixed state to a target state in a unitary system, which automatically reduces to the quantum speed limit induced by the Fubini-Study metric for pure states. The computation of the QSL of a degenerate mixed state is more complicated than that of a non-degenerate mixed state, where we have to make a singular value decomposition (SVD) on the inner product between the two eigenvector matrices of the initial and target states. By combing these results, a lower bound for the evolution time of a general mixed state is presented. In order to compare the tightness among the lower bound proposed here and lower bounds reported in the references, two examples in a single-qubit system and in a single-qutrit system are studied analytically and numerically, respectively. All conclusions derived in this work are independent of the eigenvalues of the mixed state, which is in accord with the evolution properties of a quantum unitary system.     

两类辛算法的稳定性比较

主要对一阶隐式Euler法、二阶隐式Euler中点公式分别嵌入到一阶显辛Euler法和二阶leapfrog显辛积分器所得算法M1和M2进行了稳定性比较。对于坐标与速度耦合的线性哈密顿系统,理论分析寻找稳定区比较困难,依靠数值方法定性地比较了他们的稳定性。结果表明M1算法一般优于M2算法。     

一类3维反转系统中的异维环分支

研究了一类3维反转系统中包含2个鞍点的对称异维环分支问题, 且仅限于研究系统的线性对合R的不变集维数为1的情形. 给出了R-对称异宿环与R-对称周期轨线存在和共存的条件, 同时也得到了R-对称的重周期轨线存在性. 其 次, 给出了异宿环、 同宿轨线、 重同宿轨线和单参数族周期轨线的存在性、 唯一性和共存性等结论, 并且发现不可数无穷条周期轨线聚集在某一同宿轨线的小邻域内. 最后给出了相应的分支图.