Dynamic interference in below-threshold ionization

We study the temporal interference of photoelectrons emitted during the rising and falling edges of intense femtosecond laser pulses, that can ionize atoms via near-resonant transitions. Due to the near-resonant coupling with the laser pulse, the emerging atomic dressed states repel each other, giving rise to the Rabi splitting that is primarily controlled by the laser intensity and detuning. Our numerical and analytical analysis reveals that dynamic interference is observed when the Rabi splitting of the dressed states is maximized while the ionization of the atom remains small. We demonstrate that these two conditions are fulfilled when the atom is driven off resonantly within the one-Rabi-cycle regime at perturbative laser intensities. As a result, the single peak of the photoelectron spectrum found in the weak-field limit is replaced by a pronounced multipeak pattern.

Figure 1

Sequential two-photon ionization of hydrogen by Gaussian pulses of $T=30$ fs duration set to near resonance with the $1s\text{−}2p$ transition (${\omega }_{\text{res}}=3/8\text{a.u.}\approx 10.204\text{eV}$). Shown are the populations of the $2p$ state (a) when the pulse has expired and (b) when the pulse is maximal. The white dashed lines indicate the boundary between the laser parameter regions where the atom completes a single or multiple Rabi cycles. (c) Ratio of the $1s\text{−}2p$ Rabi splitting ($W$) and the ionization rate of the $2p$ state ($\mathrm{\Gamma }$). The necessary condition for dynamic interference [Eq. (6)] is well satisfied in the whole parameter range considered ($W\gg \mathrm{\Gamma }$). The open circles indicate the laser parameters at which the spectra are calculated in Figs. 3 and 3.

Figure 2

Conditions for dynamic interference in the one-Rabi-cycle regime for different laser intensities and detunings. (a) Detuning dependence of condition 1 [Eq. (4)] : Resonant pulses do not and off-resonant pulses do satisfy condition 1. (b) Detuning dependence of condition 2 [Eq. (5)]: Although condition 2 is well satisfied at any detuning value, far off-resonant driving leads to marginal excitation and ionization probabilities. (c), (d) Intensity dependence of conditions 1 and 2, respectively, for a fixed detuning value (${\mathrm{\Delta }}_{\text{rel}}=-5\%$): (c) For increasing intensity, condition 1 is always satisfied; (d) condition 2 might be violated at very strong couplings. The threshold for condition 2 is underestimated by the simple analytical formula of Eq. (5) (solid blue line with circles, $\sim 1\times {10}^{14}{\text{W/cm}}^2$) in contrast to the numerical prediction $P_{\text{ion}}(t=0)<1/2$ (solid red line, $>1\times {10}^{14}{\text{W/cm}}^2$).

Figure 3

Photoelectron spectra of hydrogen after sequential two-photon ionization, induced by (a), (b) resonant and (c), (d) off-resonant pulses in the one-Rabi-cycle regime. (a), (b) Resonant pulses ($\omega =0.375$ a.u.) do not satisfy condition 1 [Eq. (4)] and thus the spectra possess no intensity modulations. Here, the laser parameters are chosen to keep the pulse area constant ($\theta =2\pi$). (c), (d) For off-resonant pulses (${\mathrm{\Delta }}_{\text{rel}}=-5\%$), the atom still completes no more than one Rabi cycle (see Fig. 1), and furthermore, conditions 1 and 2 are simultaneously fulfilled (see Fig. 2), allowing dynamic interference to come into play. This is manifested in the pronounced intensity modulations of the spectra for increasing coupling strength (both in the $s$ and $d$ channels). Here, the laser intensities are chosen according to $I_0\left(k\right)={10}^{k/5}\times {10}^{13}{\text{W/cm}}^2$ ($k=0,1,...,5$). For better visualization the spectra are shifted apart from each other, and the spectra in the $s$ channel are multiplied by a factor of $|{\mu }_{2p{\varepsilon }_{0}d}{|}^2/{\left|{\mu }_{2p{\varepsilon }_{0}s}\right|}^2$ to have identical peak heights in the perturbative limit. The vertical dashed lines indicate the nominal positions of the spectra ${\omega }_{{\varepsilon }_0}={\omega }_{1s}+2\omega$ expected in the weak-field limit. The solid lines represent the full solution of the TDSE in the length gauge (multilevel method [30]) while the model solutions [Eqs. (3a)–(3c)] are shown by the dashed curves.

Figure 4

(a) Time evolution of the electronic state populations and (b) of the corresponding total photoelectron spectrum of hydrogen, induced by a single Gaussian laser pulse of $T=30$ fs duration, $\omega \approx 9.694$ eV carrier frequency (${\mathrm{\Delta }}_{\text{rel}}=-5\%$), and $I_0=1\times {10}^{14}$ W/${\mathrm{cm}}^2$ peak intensity [see the uppermost spectra in Figs. 3 and 3(d)]. On the rising edge of the pulse but before its maximum ($t<0$) the spectrum is nearly symmetric, and its maximum is gradually shifted away from the nominal spectrum position (${\omega }_{{\varepsilon }_0}={\omega }_{1s}+2\omega$, white dashed line). On the falling edge of the pulse after its maximum ($t>0$), pronounced intensity modulations start to develop in the spectrum, similarly to that reported previously for one-photon ionization [1]. The presented results are obtained by the multilevel method [30].