Strong-field control by reverse engineering

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.

Figure 1

Schematic representation of the strong-field two-photon control scenario discussed in this work. The shaped laser pulse $E\left(t\right)$, which is obtained by reverse engineering, interacts with the atomic target and induces its time evolution according to a prescribed quantum path. The amplitude and phase of $E\left(t\right)$ are engineered such that the relative dynamic Stark shift of interest, $3s$-$4s$ in the case of Na, is fully compensated and an efficient population transfer is achieved, which leads to the desired final quantum state superposition of the system. The bell-shaped dashed lines symbolize the dynamic Stark shifts of the strongly coupled states.

Figure 2

Control parameter dependence of the engineered laser pulse $E\left(t\right)$ [Eq. (27)] applied to atomic Na. (a) The temporal envelope profile ${\varepsilon }_{0}g\left(t\right)$ is presented for two different values of the final target population $p^f$. The larger the desired population change, the stronger and longer the required laser pulse. This is further illustrated in (b), where the FWHM pulse duration and the peak electric field strength ${\varepsilon }_0$ are shown as a function of transition rate. Fast transitions require strong and short pulses which might lead to the breakdown of $E\left(t\right)$ due to the application of AER (see the text for details). (c) Relative dynamic Stark shift that is responsible for the frequency chirping of $E\left(t\right)$ is presented for two transition rate values and population variations. (d) Pulse area of $E\left(t\right)$ calculated according to Eq. (40). The stars correspond to two-photon $\pi$ pulses that induce a total population inversion between the $3s$ and $4s$ states.

Figure 3

Strong-field parameters of the $3s\to 4s$ two-photon transition of atomic sodium. Peak values of the (a) $3s$ and $4s$ state dynamic Stark shifts are calculated according to Eq. (11), while the (b) two-photon Rabi frequency between the $3s$ and $4s$ states is obtained from Eq. (10). The solid lines correspond to full results, which are calculated in the space of 67 electronic states ($n<17,l<5$). Quantities denoted by dashed lines are calculated upon exclusion of the one-photon resonant $7p$ state from the description. Owing to the resonant $7p$ state, the $4s$ Stark shift becomes sensitive to small variations of the laser frequency near the two-photon resonance. The vertical dashed line indicates the bare two-photon resonance energy between the strongly coupled $3s$ and $4s$ states (${\omega }_{\text{res}}=1.5886001944$ eV).

Figure 4

Comparison of the time-dependent $3s$ and $4s$ state populations calculated with the TDEC and TDWP methods (see Sec. 4). For both (a) fast and (b) slow transitions, the two completely different methods give perfect agreement; furthermore, the simulated population curves follow the analytical target functions (dotted lines with open circles and open squares). (c) The slight increase in the ionization probability around $\alpha =0.001$ a.u. correlates with the maximum in the final $7p$ populations, indicating that a Stark shifted transient resonance between $4s$ and $7p$ opens up a resonance-enhanced ionization channel.

Figure 5

Application of the engineered laser pulse $E\left(t\right)$ [Eq. (27)] to control the population dynamics of the $3s\to 4s$ two-photon transition in atomic sodium. A wide range of the transition parameter $\alpha$ is considered (left column) to achieve complete control over the rate of transition and the final target $3s$ and $4s$ populations (dotted lines with open circles and squares). The different rows of panels correspond to increasing population modulations from bottom to top. (a)–(c) demonstrate a complete population inversion between the $3s$ and $4s$ states, while (d)–(f) depict an equal final population distribution of these states. Upon fast transitions we only approximately reach the target superpositions (middle column), while applying sufficiently slow transitions, any desired population dynamics is achieved with $E\left(t\right)$ (right column) (see the perfect matching of the control functions and the numerically obtained $3s$ and $4s$ populations). The full results are calculated in the space of 67 electronic states ($n<17,l<5$) with the TDEC method. Quantities denoted by dashed lines are calculated upon exclusion of the one-photon resonant $7p$ state from the description. Results are obtained by solving Eq. (4) with $E\left(t\right)$ in Eq. (27). The vertical dashed lines indicate the specific $\alpha$ values that are applied in the middle and right columns.

Figure 6

Energies of the one-photon nearly resonant $4s$ and $7p$ states of Na under the action of the engineered laser pulse given in Eq. (27). The Stark shifted $7p$ state energy (red dashed line) and the Stark shifted $4s$ state energy, dressed by the chirped photon energy $\omega \left(t_{\text{max}}\right)$ (black solid line) are calculated according to Eqs. (41a) and (41b) for different transition rate values (peak energies are considered at the maximum $t_{\text{max}}$ of the pulse). Upon increasing $\alpha$ (and hence increasing field strength and frequency modulation), these Stark shifted one-photon coupled states form an energy crossing that gives rise to an efficient population transfer from the $4s$ to the $7p$ state [see the shaded region and the final $7p$ populations shown by the green dotted line from Fig. 5].

Figure 7

Final $4s$ state populations of Na as a function of the laser parameter imperfections defined in Eqs. (42a)–(42c). Deviations from the optimal values of (a) the Rabi frequency and (b) the resonance laser frequency lead to a deterioration of the final $4s$ state population from its target value of 0.6. A proper modification of the frequency chirping of the laser pulse can however still maintain the target $4s$ population (see the dashed lines in both panels). The horizontal dash-dotted lines indicate transform-limited (unchirped) pulses, which can still drive the system close to the desired final target state in some cases. The vertical dotted line in (b) indicates the peak relative DSS per photon $\frac{1}{2}\delta S\left(t_{\text{max}}\right)$. The remaining fixed values of the applied laser pulses are $\alpha =0.0001$ a.u., $p^i=0.7$, and $p^f=0.4$. The presented results are obtained by the TDEC method solving Eq. (4) in the space of 67 electronic states, with $E\left(t\right)$ in Eq. (27).