Based on the idea of reverse engineering, we design an optimal laser pulse to control strong-field multiphoton atomic transitions. Starting from the time-dependent Schrödinger equation of the full system, we adiabatically eliminate the nonessential states and apply the rotating-wave approximation to arrive at an effective two-state representation that involves dynamic Stark shifts and multiphoton coupling. Solving this equation inversely for the field, we obtain an analytical laser pulse shape that is expected to induce the full system's evolution according to user-defined quantum pathways. In our procedure, the amplitude and phase of the laser pulse are engineered such that the dynamically shifted electronic states are resonantly coupled during the action of the pulse at each moment of time. As a result, the driven system evolves from an arbitrary initial population distribution to any desired final quantum state superposition at a predefined rate. The proposed scheme is demonstrated using the example of the $3s\to 4s$ two-photon transition of atomic sodium. By solving the time-dependent Schrödinger equation of the single-active electron with two different methods, either propagating time-dependent coefficients of many field-free states or directly propagating the three-dimensional electronic wave packet on a grid, we demonstrate the robustness as well as the limitations of the presented reverse engineering scheme.

中文翻译：

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逆向工程强场控制

基于逆向工程的思想，我们设计了一个最优的激光脉冲来控制强场多光子原子跃迁。从整个系统的时间相关薛定谔方程开始，我们绝热地消除非本质状态并应用旋转波近似来获得有效的两态表示，其中涉及动态斯塔克位移和多光子耦合。对场逆求解该方程，我们获得了一个分析激光脉冲形状，预计会根据用户定义的量子路径诱导整个系统的演化。在我们的程序中，激光脉冲的幅度和相位经过设计，使得动态位移的电子状态在每个时刻的脉冲作用期间共振耦合。其结果，驱动系统以预定义的速率从任意初始种群分布演变为任何所需的最终量子态叠加。使用以下示例演示了所提出的方案$3秒\to 4秒$原子钠的双光子跃迁。通过用两种不同的方法求解单活性电子的瞬态薛定谔方程，要么传播许多无场状态的瞬态系数，要么直接在网格上传播三维电子波包，我们证明了鲁棒性为以及所提出的逆向工程方案的局限性。