Dynamic Interference of Photoelectrons Produced by High-Frequency Laser Pulses

The ionization of an atom by a high-frequency intense laser pulse, where the energy of a single photon is sufficient to ionize the system, is investigated from first principles. It is shown that as a consequence of an ac Stark effect in the continuum, the energy of the photoelectron follows the envelope of the laser pulse. This is demonstrated to result in strong dynamic interference of the photoelectrons of the same kinetic energy emitted at different times. Numerically exact computations on the hydrogen atom demonstrate that the dynamic interference spectacularly modifies the photoionization process and is prominently manifested in the photoelectron spectrum by the appearance of a distinct multipeak pattern. The general theory is well approximated by explicit analytical expressions that allow for a transparent understanding of the discovered phenomena and for making predictions on the dependence of the measured spectrum on the pulse.

Figure 1

Photoelectron spectra of the hydrogen atom exposed to Gaussian-shaped pulses of 30 fs (left panel) and 3 fs (right panel) durations with carrier frequency $\omega =53.6057\mathrm{eV}$ computed exactly for different peak intensities. In weak-field single-photon ionization, the photoelectron line appears as a Gaussian peak centered around the energy of ${\epsilon }_0=\omega -IP=40\mathrm{eV}$ (indicated by the vertical broken line) and a width provided by the pulse: FWHM of about 52 meV for the 30 fs pulse, and 0.52 eV for the 3 fs pulse. The presently computed strong pulse ionization spectra are, however, much broader (by several times), shifted to higher energies from the central electron energy of ${\epsilon }_0=40\mathrm{eV}$, and posses distinct modulations of the electron intensity. It will be shown that these modulations are due to dynamic interference in the continuum.

Figure 2

Time evolution of the photoelectron spectrum computed exactly for the hydrogen atom exposed to a Gaussian-shaped pulse of 30 fs duration, of carrier frequency $\omega =53.6057\mathrm{eV}$, and peak intensity of $5\times {10}^{15}\mathrm{W}/{\mathrm{cm}}^2$. At very early times, the spectrum starts to develop symmetrically around the central electron energy of ${\epsilon }_0=40\mathrm{eV}$. At later times, but before the maximum of the pulse arrives, the maximum of the spectrum moves to higher electron energies (see the spectrum computed at $-10\mathrm{fs}$). When the pulse arrives (the time 0 fs corresponds to the pulse maximum), the spectrum is not any more symmetric but still does not possess intensity modulations. The first modulations start to show up in the spectrum only after the pulse maximum (see the spectrum computed at $+10\mathrm{fs}$) has arrived.

Figure 3

Upper panel: The photoelectron spectrum of the hydrogen atom exposed to a Gaussian-shaped pulse of 30 fs duration, carrier frequency $\omega =53.6057\mathrm{eV}$, and peak intensity $5\times {10}^{15}\mathrm{W}/{\mathrm{cm}}^2$ computed exactly (open circles) compared with the spectrum computed in the local approximation (solid curve) with the quantities $\Gamma =0.044\mathrm{eV}$ and $\Delta =0.26\mathrm{eV}$. Lower panel: The spectrum obtained in the stationary phase approximation via Eq. (8) with the same parameters (solid curve). The two individual contributions to the spectrum from the negative and positive times $t_s=\pm t_1(\epsilon )=\pm \tau \sqrt{\ln [\Delta /(\epsilon -{\epsilon }_0)]/2}$ describing in Eq. (8) the contributions when the pulse arrives and expires, respectively, are shown by broken curves. The energy-dependent interference term in Eq. (8) is proportional to $\cos \{-2(\epsilon -{\epsilon }_0)t_1-\Delta [J(-t_1)-J(+t_1)]-\pi /2\}$. In the energy interval $\epsilon -{\epsilon }_0\in [0,\Delta ]$ for which a stationary phase can be found, its argument accumulates a phase of $\Delta J(+\infty )$, which for a Gaussian pulse is equal to $\Delta \tau \sqrt{\pi /2}$. For the present parameters, the interference term accumulates a phase of $14.85\approx 2.36\times 2\pi \mathrm{rad}$, and, thus, the spectrum exhibits around two and a half oscillations of the intensity profile as seen in the figure.

Figure 4

The intense laser pulse of sufficient high frequency to ionize the system by one-photon absorption induces a time-dependent ac Stark shift of the ground state of the system. Because of this dynamic shift, the photoelectron emitted along the pulse envelope has at every moment $t$ predominantly a specific kinetic energy $\epsilon$ which is determined by that $t$ (at a given time $t$, the phase contributing to the integrand in Eq. (7) is stationary at $\epsilon ={\epsilon }_0+g^2(t)\Delta$, for details see text). Owing to the fact that the pulse envelope first grows and then falls, there are for a Gaussian pulse exactly two times (indicated as $t^{-}$ and $t^{+}$ in the figure) at which the emitted electron wave has the same energy $\epsilon$. These two waves emitted with a time delay with respect to each other interfere. This dynamic interference gives rise to the strongly modulated intensity distribution of the photoelectron spectrum seen in Figs. 1, 2, 3. The dynamic interference can be controlled by the intensity, duration, carrier frequency, and shape of the pulse. A pulse envelope with more than one maximum leads to more than two waves that interfere and higher laser intensity to larger ac Stark shift $\Delta$ and to more oscillations of the spectrum.