量子链中信息的完美传输和系统布居完全转移研究
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摘要
本论文的主要工作是研究和讨论量子链中信息的完美传输和在脉冲驱动下系统布居的完全转移。对于量子信息传输,主要研究了如何构造系统的耦合强度以及外加控制场参数,从而使量子信息在相邻耦合的自旋链中实现完美传输,并给出了求解该问题解析解的方法。在一大类N量子比特自旋链中,与量子信息传输对应的2N维希尔伯特空间可以简化为一个N+1维子空间。系统的哈密顿量也因此可以化为一个三对角矩阵。可以证明只有当量子比特间的耦合强度和外加控制场的强度满足特定条件时,才能使得量子信息在自旋链中实现完美传输。进一步的,我们可以通过量子信息完美传输的条件确定系统哈密顿量的本征值谱的取值,从而建立自旋链参数与哈密顿量本征值之间的函数关系。求解完美信息传输问题即转化为了已知本征值求解控制参数的逆问题。通过求解多项式方程,我们可以得到自旋链信息完美传输的解析解,该结果可以用来分析自旋链解的结构和寻找有优化性质的特解。
对于系统布居转移,我们研究了相邻耦合的有限维多能级系统在脉冲驱动下的量子动力学行为。我们把拉比振荡的概念从二能级系统推广到了多能级系统,并且求解了布居完全转移时的含时量子动力学方程。通过用Groebner基分析的方法,我们得到了多能级系统布居完全转移的解析解,给出了从二能级到九能级系统解析解的表达式,以上这些结果可以用来设计一般意义上的有限维量子系统的最优控制策略。
In this paper, we explore the perfect information transfer in spin chains and the perfect population transfer in pulse-driven quantum chains.We study how to engineer parameters for perfect information transfer in neighbor-coupled spin chains. The 2N-dimension Hilbert space associated with quantum information transfer over the spin chain can be projected into an N+1-dimension subspace, so the Hamiltonian of the system will be reduced to a tridiagonal matrix in standard basis. The functional relation between the parameters of the spin chain and the eigenvalue spectrum of the Hamilto-nian, which can be determined by the perfect transfer conditions, are established. The task of finding all solutions to the parameters of perfect information transfer is accom-plished by solving polynomial equations.The results could be used to analyze structures of the chain or find particular chains with optimal properties.
The quantum dynamics of a pulse-driven finite-dimensional quantum chain with only nearest-neighbor coupling is studied. We extend the concept of Rabi oscilla-tions from two-level quantum systems to the multi-level quantum chains. The time-dependent quantum dynamics and solutions producing perfect population transfer are obtained for up to five-level quantum chains. The Groebner basis analysis technique is used to generalize o the results and get all analytical solutions for perfect population transfer. The explicit formulas for the solutions up to nine levels are presented. These results could be used to design control strategies for general finite-dimensional quantum systems.
对于系统布居转移,我们研究了相邻耦合的有限维多能级系统在脉冲驱动下的量子动力学行为。我们把拉比振荡的概念从二能级系统推广到了多能级系统,并且求解了布居完全转移时的含时量子动力学方程。通过用Groebner基分析的方法,我们得到了多能级系统布居完全转移的解析解,给出了从二能级到九能级系统解析解的表达式,以上这些结果可以用来设计一般意义上的有限维量子系统的最优控制策略。
In this paper, we explore the perfect information transfer in spin chains and the perfect population transfer in pulse-driven quantum chains.We study how to engineer parameters for perfect information transfer in neighbor-coupled spin chains. The 2N-dimension Hilbert space associated with quantum information transfer over the spin chain can be projected into an N+1-dimension subspace, so the Hamiltonian of the system will be reduced to a tridiagonal matrix in standard basis. The functional relation between the parameters of the spin chain and the eigenvalue spectrum of the Hamilto-nian, which can be determined by the perfect transfer conditions, are established. The task of finding all solutions to the parameters of perfect information transfer is accom-plished by solving polynomial equations.The results could be used to analyze structures of the chain or find particular chains with optimal properties.
The quantum dynamics of a pulse-driven finite-dimensional quantum chain with only nearest-neighbor coupling is studied. We extend the concept of Rabi oscilla-tions from two-level quantum systems to the multi-level quantum chains. The time-dependent quantum dynamics and solutions producing perfect population transfer are obtained for up to five-level quantum chains. The Groebner basis analysis technique is used to generalize o the results and get all analytical solutions for perfect population transfer. The explicit formulas for the solutions up to nine levels are presented. These results could be used to design control strategies for general finite-dimensional quantum systems.
引文
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