【作者】 张靖；

【导师】 李春文；

【作者基本信息】 清华大学 ， 控制科学与工程， 2006， 博士

【摘要】 量子相干性和量子纠缠是量子技术的基础。它们常常受到量子系统与环境间的相互作用引起的退相干效应的破坏,从而成为实现量子技术的最大障碍,因此对退相干进行抑制具有重要意义。本论文利用密度矩阵相干向量表示研究了退相干效应的抑制和相应的纠缠控制问题,论文主要工作包括:(1)在密度矩阵相干向量表示下,利用最优控制研究N能级Markov系统的退相干抑制问题。N能级Markov系统可以用主方程模型描述,在相干向量表示下具有标准的双线性形式。针对这个双线性系统,采用最优轨线跟踪的方法抑制环境噪声的影响。当系统满足完全退相干条件、可控性假设、收敛性假设时,最优控制给出了很好的结果。在渐近意义下,最优轨线是与系统自由演化轨线同步的等幅振荡过程,但振荡的振幅相比自由动态有所损失,最优控制律是在实验室中易于实现的正弦信号。针对最优控制失效的某些特定情形,引入反馈控制修正可以使得最优控制依然有效。(2)在相干向量表示下,引入su(N)李代数噪声解耦分解的概念,并用于相干向量部分状态变量与环境噪声的完全解耦问题。一方面,根据su(N)第一类噪声解耦分解,利用开环控制可以使得N能级系统部分状态变量与环境渐近解耦;另一方面,根据su(N)第二类噪声解耦分解,可以设计反馈控制使得部分状态变量与环境精确解耦,从而避免渐近解耦过渡过程所带来的影响。(3)对于非Markov系统的退相干控制进行了讨论。在这里利用了中间“库”的概念。在这种模型下,系统与库所组成的大系统可以用主方程模型描述。仿真表明,在渐近意义下,利用最优控制同样可以使得受控轨线是与目标轨线同步的等幅振荡过程,而控制律是正弦信号。(4)对量子纠缠这一典型量子现象进行了研究。首先提出了一种新的纠缠度量。相比于已有的度量,该度量适用于更广泛的一类量子态,并可以用相干向量解析表示。在这种纠缠度的基础上,结合前面提出的退相干控制方法,讨论了纠缠保持的问题。一般来讲,纠缠可以部分得到保持。不过,对于二比特系统,在渐近意义下可以实现纠缠的完全保持。论文最后对全文工作进行总结,并为进一步的研究指明了方向。更多还原

【Abstract】 Quantum coherence and quantum entanglement are the bases of the quantum tech-nologies. They will usually be destroyed by the decoherence induced by environmentalinteractions, which has become the main obstacle for the implementation of the quan-tum technologies. Therefore, it is of great importance to suppress the decoherence.The thesis discusses the decoherence suppression and the corresponding entanglementcontrol problems with the help of the coherence vector representation of the densitymatrix. The main contributions are as follows:(1) The decoherence suppression of N level Markovian systems is investigatedby the optimal control in the coherence vector representation. The N level Markoviansystem can be described by the master equation, and possesses a standard bilinear formin the coherence vector representation. For this bilinear system, the optimal trajectorytracking method is used to suppress the decoherence effects brought by the environ-mental noises. It is shown that, when the system satisfies the complete decoherenceconditions, the controllability and the convergence assumptions, the optimal controlstrategy gives good results. The optimal trajectories asymptotically synchronize withthe oscillating free-evolution trajectories with partial loss of the amplitudes, and theoptimal control laws are asymptotically sinusoidal signals that are easy to be realizedin the laboratory. For some special cases when the optimal control works badly, thefeedback control strategy is introduced to retrieve the effectiveness of the optimal con-trol.(2) In the coherence vector representation, the concept of the noise decoupling de-composition of the Lie algebra su(N) is introduced and applied to completely decouplea part of the variables in the coherence vector from the environmental noises. Basedon the first noise decoupling decomposition of su(N), it is shown that some variablesin the coherence vector of the N level systems can be asymptotically decoupled fromthe environments via the open-loop controls. On the other hand, based on the secondnoise decoupling decomposition of su(N), feedback control laws can be designed toexactly decouple some variables from the environments, and they are not affected bythe possible transitional processes.(3) The decoherence control of non-Markovian systems is explored. The conceptof the intermediate”bath”is introduced here. In this model, the total system of thesystem plus the bath can be described by a master equation. The simulation exhibitssimilar oscillating trajectories that asymptotically synchronize with the object trajecto-ries, and the control laws are asymptotically sinusoidal signals.(4) The quantum entanglement as a typical quantum phenomenon is studied.Firstly, a new entanglement measure is proposed. Compared with the existing entan-glement measures, this measure is applicable to a wider class of quantum states and canbe analytically expressed in terms of the coherence vector. Based on this measure, theproblem of the entanglement preservation is discussed combined with the former pro-posed decoherence suppression strategies. Generally, the entanglement can be partiallypreserved. Even better, for the two-qubit case, the entanglement can be completelypreserved in an asymptotic manner.The last part summarizes the results obtained in the thesis and the further researchdirections are pointed out.更多还原