ac Stark effect in the electronic continuum and its impact on the photoionization of atoms by coherent intense short high-frequency laser pulses

We investigate the photoionization of an atom by a high-frequency strong coherent laser pulse, where the energy of a single photon is sufficient to ionize the system. Ab initio theory exemplified by numerical calculations of the photoionization of the hydrogen atom illustrates that the laser pulse induces a time-dependent ac (or dynamic) Stark shift of the binding energy of the system due to interaction with the electronic continuum. The energy shift follows the pulse envelope. Because of this shift, the energy of the emitted photoelectron also follows the intensity envelope of a short laser pulse. This results in a strong dynamic interference of the photoelectrons of the same kinetic energy emitted at different times, when the pulse arrives and expires, respectively. The dynamic interference leads to interesting modifications of the photoionization process and results in a multiple peak structure in the photoelectron spectrum. The present work provides a detailed theoretical and numerical study of the results reported in Demekhin and Cederbaum [Phys. Rev. Lett. 108, 253001 (2012)]. The working equations are derived explicitly and an extended analysis of the theory is provided. Additional results are presented. In particular, we demonstrate that the dynamic interference can be controlled by the intensity, duration, and shape of the pulse envelope.

Figure 1

Population of the neutral hydrogen atoms remaining in the ground electronic state after the laser pulse has expired as a function of the peak intensity of the pulse. Two Gaussian-shaped pulses with carrier frequency $\omega =53.6057$ eV have been considered: 30 fs (broken curve) and 3 fs (solid curve). For each pulse the interval of the peak intensities chosen for the computation of the photoelectron spectra in Fig. 2 is indicated separately as a shaded area.

Figure 2

Photoelectron spectra of the hydrogen atom exposed to Gaussian-shaped pulses of 30-fs (left panel) and 3-fs (right panel) durations with a carrier frequency of $\omega =53.6057$ eV computed numerically at different peak intensities. In part these results were published in our previous study [26]. In order to be consistent with Ref. [26], all computed spectra here and below are shown on the compressed scale of 1/20. Note that Fig. 1 provides information on the ionization probabilities and thus suffices to put all reported spectra on the absolute scale.

Figure 3

Photoelectron spectra of the hydrogen atom exposed to Gaussian-shaped pulses with carrier frequency $\omega =53.6057$ eV computed in different approximations. (Upper panel) Pulse with $\tau =$30 fs and $I_0=5\times {10}^{15}$ W/cm${}^2$. (Lower panel) Pulse with $\tau =$3 fs and $I_0=5\times {10}^{16}$ W/cm${}^2$. Shown are the numerical results (solid curves) and approximate results (broken curves) as discussed in the text.

Figure 4

Time evolution of photoelectron spectra computed numerically for hydrogen atom exposed to a Gaussian-shaped pulses of 30-fs duration, of carrier frequency $\omega =53.6057$ eV, and peak intensities of $5\times {10}^{15}$ W/cm${}^2$ (upper panel, taken from Ref. [26]) and $I_0=9\times {10}^{15}$ W/cm${}^2$ (lower panel).

Figure 5

Numerical calculations of the ground-state amplitude for a Gaussian-shaped pulse of 30-fs duration with carrier frequency $\omega =53.6057$ eV and peak intensities of $5\times {10}^{15}$ W/cm${}^2$ (solid curves) and of $9\times {10}^{15}$ W/cm${}^2$ (broken curves). (Upper panel) Population of the ground electronic state as a function of time. Time 0 fs corresponds to the pulse maximum. (Lower panel) Phase $f\left(t\right)$ of the amplitude for the population of the ground electronic state as a function of time. The tangent to each of the $f\left(t\right)$ functions at $t=0$ are also shown by straight lines. The derivatives $\dot{f}\left(t\right)$ at $t=0$ are equal to $-0.260$ eV and $-0.458$ eV for the weak and strong pulses, respectively. These derivatives can be used to determine the total ac Stark shift $\Delta$ in the continuum (see text).

Figure 6

(Upper panel) The photoelectron spectrum of the hydrogen atom exposed to a Gaussian-shaped pulse of 30-fs duration, carrier frequency $\omega =53.6057$ eV, and peak intensity of $5\times {10}^{15}$ W/cm${}^2$ computed numerically (open circles) compared to the spectrum computed in the local approximation (solid curve) using the quantities $\Gamma =0.044$ eV and $\Delta =0.26$ eV. (Lower panel) The spectrum obtained in the stationary phase approximation via Eq. (18) or (19) with the same parameters (solid curve). The two individual contributions to the spectrum arising from the negative and positive times $t_s=\pm t_1\left(\varepsilon \right)$ in Eq. (18) are shown by broken curves. These correspond to the contributions when the pulse arrives and expires, respectively. The figure is taken from our previous study [26].

Figure 7

The photoelectron spectrum of the hydrogen atom exposed to a Gaussian-shaped pulse of 30-fs duration, carrier frequency $\omega =53.6057$ eV, and peak intensity $9\times {10}^{15}$ W/cm${}^2$ computed numerically and in different approximations (see notations in Fig. 6).

Figure 8

The photoelectron spectrum of the hydrogen atom exposed to a single Gaussian-shaped pulse (upper panel, the same as in Fig. 6) and to a sequence of two well-separated Gaussian pulses (lower panel) computed in the local approximation (open circles) and by the stationary phase approximation (solid curves). See text in Sec. 5 for details.

Figure 9

Schematic picture of the emergence of dynamic interference. An intense laser pulse of sufficiently high carrier frequency ionizes the system by the absorption of a single photon and simultaneously leads to a decrease of the binding energy of the atom due to the ac Stark effect in the continuum. This situation can be viewed as if in the presence of the pulse the ground-state energy of the system acquires a time-dependent energy correction relative to the ionization threshold which may acquire such a shift itself. In the drawing we let for the sake of simplicity the ionization threshold coincide with the field-free threshold at all times, and thus it looks as if only the ground state shifts relative to the continuum. The energy shift of the binding energy follows the pulse intensity envelope $\Delta g^2\left(t\right)$, as illustrated at the bottom of the scheme. Because of this time-dependent shift, the photoelectrons emitted at different moments $t$ can have predominantly a specific kinetic energy $\varepsilon$. These are the times at which the energy of the ground state, dressed by the field (i.e., shifted upwards by the energy of absorbed photon $\omega$) and augmented by the time-depended shift $\Delta g^2\left(t\right)$, coincides with the energy of continuum state $\varepsilon$ (see upper part of the scheme). These times and electron energies are connected via the simple expression $\varepsilon =\omega -I_p+\Delta g^2\left(t\right)$. The pulse envelope first grows and then falls. Therefore, for a Gaussian pulse there are exactly two times when the emitted electrons have the same energy $\varepsilon$. These two electrons emitted at different times interfere, giving rise to the strongly modulated distribution of the photoelectron line shown in the scheme.