Role of dynamic Stark shifts in strong-field excitation and subsequent ionization

The energy levels of atoms and molecules exposed to intense, short, high-frequency laser pulses undergo time-dependent distortion known as dynamic Stark shift (DSS). These level shifts are induced by the coupling with nonresonant (nonessential) states and follow the temporal intensity profile of the laser pulse. Owing to the different DSSs of the individual atomic levels, transient resonance suppressions and enhancements significantly modify the multiphoton ionization pathway, leaving clear fingerprints (asymmetry, splitting, and shifting) in the energy spectrum of emitted photoelectrons. Here we investigate this phenomenon by solving the time-dependent Schrödinger equation of the valence electron of Na and Li in increasing levels of complexity: (i) developing minimal models for two-photon transition and ($2+1$)-photon ionization, (ii) applying a spectral method without continuum-continuum couplings, and (iii) propagating the electron wave packet on a large spatial grid accounting for all possible couplings. We show that appropriately detuned transform limited pulses provide a high level of control in strong-field excitation, when the atomic levels are dynamically shifted. Furthermore, we demonstrate the role of DSSs in the multipeak structure of the Autler-Townes doublet, found when several Rabi oscillations are induced during the strong-field resonant ionization.

Figure 1

Energy-level scheme of (a) sodium and (b) lithium, showing the two-photon transition and the ($2+1$)-photon resonance ionization processes discussed in this work. The bell-shaped dashed lines represent the dynamic Stark shifts of the strongly coupled atomic levels.

Figure 2

(a) Final $4s$ populations of Li and Na computed in the TLA (5) (solid lines) and in the MLA (3) (dotted lines) for detuned $\pi /2$ pulses of 100-fs FWHM duration. The average relative DSS of the strongly coupled levels is compensated with negatively (Li) or positively (Na) detuned pulses that satisfy the $\mathrm{\Delta }=0.76\times \delta S_0$ condition (vertical dashed lines) and as a result population is efficiently transferred to the $4s$ excited levels. This is reflected by the $\left|\kappa \right|$ curves too [Eq. (9), dashed lines], which peak around the same region of the two-photon detuning. (b) Atom-field phase for Li [Eq. (8)], computed for several detuning values around the optimal one, shown in (a). (c) Same as (b) but for Na. The bell-shaped dashed lines represent the scaled intensity profile of the applied Gaussian laser pulses. The peak laser intensities were determined according to the $\pi /2$-pulse condition (10). The definition of $\mathrm{\Delta }$ is given around Eq. (6).

Figure 3

Final populations of (a) Li and (b) Na computed in the multilevel approximation for detuned $\pi /2$ pulses of increasing duration. The two-photon detuning is chosen according to ${\mathrm{\Delta }}_{\mathrm{opt}}=0.76\times \delta S_0$ [see Fig. 2] to induce a complete population inversion between the ground and the $4s$ levels. For short pulses, owing to the large bandwidth and the high peak intensity, several states are populated by the laser. For long enough pulses, state selectivity is fulfilled and as a result the atoms are efficiently driven to the target $4s$ state (horizontal dashed lines). The presented results were obtained by solving Eq. (3) in the space of 55 bound states for Li and 67 bound states for Na, considering the atoms initially in their ground states. For each pulse duration value, the peak laser intensity was determined according to the $\pi /2$-pulse condition (10).

Figure 4

Resonance-enhanced ($2+1$)-photon ionization of Li computed for two-photon resonant ($\mathrm{\Delta }=0\to \omega \approx 2.169\text{eV}$) Gaussian laser pulses of 2-ps FWHM duration. Shown are the populations of the $2s$ and $4s$ states after the pulse has expired. Upon absorption of a third photon from the same pulse, the two-photon Rabi dynamics between the resonantly coupled $2s$ and $4s$ states is damped, leading to enhanced ionization for increasing coupling strength. The vertical dashed lines indicate the laser intensities at which the system completes an integer number of Rabi cycles (for the corresponding photoelectron spectra, see Fig. 5). Note that the results obtained from the minimal three-level model [Eqs. (15a)–(15c)] are well supported by the multilevel solutions [Eq. (3)].

Figure 5

Resonance-enhanced ($2+1$)-photon ionization of Li computed for two-photon resonant ($\mathrm{\Delta }=0\to \omega \approx 2.169\text{eV}$) Gaussian laser pulses of 2-ps FWHM duration. Shown are the photoelectron spectra computed for laser intensities at which the atom completes an integer number of Rabi cycles between the resonantly coupled $2s$ and $4s$ states (see the vertical dashed lines in Fig. 4). The spectral features given by the minimal three-level model [Eqs. (15a)–(15c)] are fully supported by the multilevel solutions [Eq. (3)]. The vertical dashed line indicates the nominal position of the spectra (${\omega }_{{\varepsilon }_0}={\omega }_{2s}+3\omega$). The green circles represent the energies (at $t=0$) given by the simple analytical model of decoupled resonances [Eq. (29)]: ${\omega }_{{\varepsilon }_0}+\text{Re}\left[E_{\pm }\left(0\right)\right]$. For transparency, the individual spectra are vertically shifted apart from each other by factors proportional to the actual peak laser intensity.

Figure 6

(a) Asymmetry of the photoelectron spectra of Li, computed for different trapezoidal envelope functions defined in Eq. (30). When the DSSs of the resonantly coupled states are negligible, the spectrum is symmetric irrespective of the shape of the envelope (green dashed line). In contrast, when the DSSs are significant, the spectrum exhibits strong asymmetry which is very sensitive to the shape of the envelope (red solid line). (b) Photoelectron spectra computed for a rectangular pulse ($\mathrm{\Delta }T=0$, open circle in (a)). The observed asymmetry is opposite that found for the sine-squared $[\mathrm{\Delta }T=0.5$, X symbol in (a)] or Gaussian envelope shapes (Fig. 5). The slight deviations between the model and multilevel spectra are attributed to the $2p$ state. The vertical dashed line indicates the nominal position of the spectra (${\omega }_{{\varepsilon }_0}={\omega }_{2s}+3\omega$). The laser parameters are $T=2$ ps and $\omega \approx 2.169\text{eV}$ ($\mathrm{\Delta }=0$). The green circles indicate the analytical energies [Eq. (29)] ${\omega }_{{\varepsilon }_0}+\text{Re}\left[E_{\pm }\left(0\right)\right]$. For each envelope shape, $I_0$ is chosen to keep the pulse area constant so as to remain consistent with the uppermost spectrum in Fig. 5 ($\mathrm{\Theta }=5.545\times \pi /2$).

Figure 7

Resonance-enhanced multiphoton ionization of Li computed by the 3D TDSE method for a two-photon resonant ($\mathrm{\Delta }=0\to \omega \approx 2.169\text{eV}$) Gaussian laser pulse of 100-fs FWHM duration and $I_0=3.08\times {10}^{12}{\text{W/cm}}^2$ peak intensity. (a) Three Rabi cycles are completed by the valence electron between the resonantly coupled $2s$ and $4s$ states, while the intermediate $2p$ state and the one-photon resonant $4d$ state also participate in the dynamics. (b) The corresponding photoelectron spectrum exhibits pronounced intensity modulations with three distinct peaks, which are the result of two competing ionization pathways: (i) the $2s\to 4s\to \text{continuum}$ pathway and (ii) the $2s\to 2p\to 4d\to \text{continuum}$ pathway. (c) The emitted $p$ and $f$ electrons contribute with different weights to the three spectrum peaks, which is also reflected by the angular distributions computed for the three peak energies. Here each curve is normalized to its maximal value.

Figure 8

Comparison between the photoelectron spectra of Li computed with (a) the 3D TDSE method and (b) the multilevel approximation for the laser parameters applied in Fig. 7. In the 3D TDSE method, the ponderomotive motion of the emitted electron is properly described. In contrast, the multilevel solution neglects continuum-continuum couplings and hence cannot account for the dynamic Stark shift of the continuum states. Thus, on the one hand, the time-dependent ponderomotive energy of the continuum states $\left[U_p\left(t\right)\right]$ causes shifting of the spectrum towards lower energies (ranging between approximately 68% and 88% of its peak value $U_{p0}$). On the other hand, $U_p\left(t\right)$ modifies the shape of the spectrum by even changing its symmetry (see the $p$ channel). The rightmost peak of the $p$ channel spectrum remains the highest even when the bound $d$ states are omitted from the 3D TDSE simulations, confirming that it is indeed not the result of ionization via the $2s\to 2p\to 4d\to p\text{−}\text{continuum}$ pathway.