量子最佳化控制演算法之新發展與時變密度泛函中之準確最佳有效場

Development of New Quantum Optimal Control Algorithms and Exact Optimized Effective Potential in Time-dependent Density Functional Theory

廖聖侖(Sheng-Lun Liao)

指導教授 ： 朱時宜

國立臺灣大學/理學院/物理學研究所/博士(2018年)

https://doi.org/10.6342/NTU201800385

摘要

現今科學的進展，不只在於觀測並解釋新穎的物理與化學現象，更試圖控制電子的即時超快運動。為此，針對量子系統最佳化控制的問題，我們發展了高效率的演算法；並且針對時變泛函理論中軌域相關泛函的計算， 推導出新的理論方程式 。

論文的第一部分，我們提出了快速起步演算法，用以尋找最佳化的雷射場以控制量子態之間的躍遷。對於起始條件比較差的狀況，此方法更凸顯其效率。此演算法是基於單調收斂疊代法(兩點邊界控制範例)，並額外加上特別設計的時變函數來加速搜尋最佳化控制場，且同時維持演算法的單調收斂性。我們數值模擬了量子系統的振動躍遷與超快電子穿隧現象，結果顯示，與原本的單調收斂疊代法相比，新方法的效率大幅增加（數十至數百倍）。即使我們限制了控制場的頻寬，數值上，新的方法依舊保持良好的單調收斂性。

我們更將此單調收斂疊代法推廣到混合態的量子最佳化控制。其中，我們探討了如何透過近單週期兆赫雷射來控制熱系綜下的分子轉動定向。我們對OSC 線性分子進行了大規模的計算，模擬了混合態中的動態轉動過程，並考慮了極廣的轉動角動量分布（轉動量子數J 的分佈從0 至100）。 我們理論分析了最大可致之轉動定向，並確認高度的轉動定向控制是可以透過最佳化的近單週期兆赫雷射來達成。

論文的第二部分闡述了我們最近的工作----在時變泛函理論架構下，針對軌域泛函，得到相對應之精準最佳有效場。如何精準地解最佳有效場積分方程是長久以來的一大難題。為此，我們推導了一個等價的時間局域性之斯徒姆-劉維微分方程式。透過此時間局域性方程式，以及時變KS 軌域、時變等效記憶軌域，我們得以即時地計算時變最佳有效場。透過數值計算，我們模擬了氫鏈中的多電子動態過程，且得到穩定收斂的時變偶極變化。更重要的是，我們確認新方程得到的解是滿足時變泛函理論中的不受力定理。

我們更近一步地利用此時間局域性方程，針對一維雙電子氦模型進行非絕熱動態過程的計算。透過比較偶極變化與時變機率密度，我們說明了最佳化有效場近似比常用的絕熱局部密度泛函近似或KLI近似更加準確。我們發現，新方程式中的若干項，彰顯了非絕熱與記憶效應，並精巧地改變了交換-關連性勢能，驅使系統得到準確的時變密度，其結果相當吻合於時變薛丁格方程式的準確解。對於未來時變密度泛函中記憶效應的研究，這些新發現將會是關鍵的步驟。

我們在量子最佳化控制方法上的發展，得以推廣到目前各類有趣的物理與化學的動態問題上。另一方面，我們在最佳有效場上的工作成果，是時變密度泛函中的一大進展，更是未來非絕熱動態模擬中重要的關鍵。在原子、分子、與凝態系統中，仍有著更多的多電子動態行為，特別是非絕熱過程的電子動力學，得以利用我們上述發展的方法去探討。

關鍵字

量子最佳化控制理論 ； 分子轉動控制 ； 時變密度泛函理論 ； 時變最佳有效場 ； 記憶效應 ； 非絕熱近似密度度泛函理論

並列摘要

Nowadays, advances in science not only aim at observing and uncovering novel physical and chemical phenomena but also attempt to control the ultrafast electronic dynamics. To this end, we develop efficient convergent algorithms for quantum optimal control problems and formulate solvable equations to implement orbital-dependent functionals in time-dependent density functional theory (TDDFT).

In the first major part of this thesis, a fast-kick-off search algorithm is presented for quickly finding optimal control fields in the state-to-state transition probability control problems, especially those with poorly chosen initial control fields. This new algorithm is based on the efficient monotonically convergent iteration algorithm, the two-point boundary-value quantum control paradigm (TBQCP) method, aided by the implementation of an instantaneous overlap function that monitors the search progress throughout. Our numerical control simulations for vibrational state-to-state transitions and for ultrafast electron tunneling have demonstrated that the new algorithm not only can greatly improve the search efficiency over its original one, but it also can attain good monotonic convergence quality in the case of the frequency constraints.

We also extend the TBQCP method to the mixed-states quantum optimal control problem, and study the maximum attainable field-free molecular orientation with optimally shaped linearly polarized near-single-cycle THz laser pulses of a thermal ensemble. Large-scale benchmark optimal control simulations are performed, including rotational energy levels with the rotational quantum numbers up to J = 100 for OCS linear molecules. As a result, it is shown that a very high degree of field-free orientation can be achieved by strong, optimally shaped near-single-cycle THz pulses.

The second major part of this thesis is devoted to our recent work on the exact time-dependent optimized effective potential (TDOEP) in TDDFT. In order to tackle the long-standing challenge in solving the exact TDOEP integral equation derived from orbital-dependent functionals, we formulate a completely equivalent Sturm-Liouville-type time-local TDOEP equation that admits a unique real-time solution in terms of time-dependent Kohn-Sham and effective memory orbitals. The time-local formulation is numerically implemented to study the many-electron dynamics of a one-dimensional hydrogen chain. It is shown that the long-time behavior of the electric dipole converges correctly and the zero-force theorem is fulfilled in the current implementation.

We further conduct the non-adiabatic TDDFT calculations for the one-dimensional two-electron helium model based on the time-local TDOEP equation. Through comparing the time-dependent dipole moment and probability density, we show that the TDOEP approach is more accurate than the adiabatic local spin density approximation (ALSDA) and the Krieger-Li-Iafrate (KLI) approximation. It is found that the non-adiabatic and memory-dependent terms in the time-local TDOEP equation elaborately refine the time-dependent structure of exchange-correlation potential and yield the resultant probability density evolution in consistent with the time-dependent Schrödinger equation solutions. These findings take a crucial step toward further studies on memory effects in TDDDFT.

Our new developments of the optimal control methods can be extended to the efficient and accurate investigation of a broad range of quantum optimal control problems in novel chemical and physical processes of current interest. And our new development of the time-local TDOEP equation represents a major breakthrough in the formulation of the non-adiabatic real-time TDDFT. Much remains to be explored for the many-electron non-adiabatic quantum dynamics of atomic, molecular, and condensed matter systems in the future.

並列關鍵字

quantum optimal control theory ； molecular orientation control ； time-dependent density functional theory ； TDOEP ； memory effects ； non-adiabatic TDDFT

參考文獻

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[5] A. Moulet et al., Science 357, 1134 (2017).