Effect of nonresonant states in near-resonant two-photon ionization of hydrogen

By numerically solving the three-dimensional time-dependent Schrödinger equation, two-photon ionization of hydrogen is investigated at the near-resonant frequencies of the $1s\text{−}2p$ transition. Due to the Rabi oscillations between the $1s$ and $2p$ states, the photoelectron energy spectra exhibit the Autler-Townes (AT) doublets and we focus on the energy spacing and the asymmetry of the doublets. Our results show that the laser frequency for the minimum energy spacing of AT doublets locates at the resonant frequency of the ac-stark-shifted states, while the symmetry of the AT doublets is affected both by the ac-stark shift and the nonresonant ionization pathway. Developing the minimal three-state model including all of the nonresonant (nonessential) states, the effects of the ac-stark shift and the nonresonant ionization pathway on the AT doublets due to the nonresonant states are identified. Furthermore, due to the nonresonant ionization pathway, the photoelectron angular distributions are distinctly different for the lower- and higher-energy peaks in the AT doublets and these angular distributions sensitively depend on the laser intensity and pulse duration. These results are well reproduced by our minimal three-state model.

Figure 1

The PES as a function of the laser frequency obtained from the 3D-TDSE. The laser intensity is $I=5\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2$ and the pulse durations are $N=60$, 300 for (a), (b), respectively. In each case, the dashed white line corresponds to the photon energy for the minimum energy spacing of the AT doublet.

Figure 2

(a), (b) The energy spacing of the doublets ($W$) as a function of the laser frequency. (c), (d) The asymmetry of the doublets ($A$) as a function of the laser frequency. The laser intensities are $5\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2$ in (a) and (c), and $1\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$ in (b), (d). The blue dotted, green dashed, and red solid lines represent the results from TDSE with the pulse durations $N=60$, 120, and 300, respectively.

Figure 3

The PES at the laser frequency $\omega =0.375$ a.u. obtained from the minimal three-state model (red dashed lines) and TDSE (blue solid lines). The laser intensity is $5\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2$, and the pulse durations are $N=60$ and 300 in (a), (b), respectively. The black dashed lines denote the expected kinetic energy (0.25 a.u.) of a photoelectron absorbing two resonant photons.

Figure 4

(a), (b) The energy spacing of the doublets ($W$) as a function of the laser frequency obtained from the minimal three-state model. (c), (d) The asymmetry of the doublets ($A$) as a function of the laser frequency obtained from the minimal three-state model. The laser intensities are $5\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2$ and $1\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$ in (a), (c) and (b), (d), respectively. The solid and dotted lines represent the results calculated from the minimal three-state model with ${\tilde{M}}_{I\varepsilon }=-7.5$ a.u. and ${\tilde{M}}_{I\varepsilon }=0$ a.u., respectively. In each case, the gray dashed line corresponds to the frequency ${\omega }_{\mathrm{m}}^{\mathrm{A}}$ or ${\omega }_{\mathrm{m}}^{\mathrm{W}}$ obtained from TDSE (as shown in Fig. 2).

Figure 5

(a) The PEMD in the polarization plane with the laser frequency $\omega =0.375$ a.u. The laser intensity is $1\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$ and the pulse duration is $N=60$. (b) The asymmetry parameters ${\beta }_n$ of PEMDs as a function of the laser frequency. Dashed lines represent the asymmetry parameters ${\beta }_n$ of the inner ring in the PEMDs and solid lines represent ${\beta }_n$ of the outer ring. The asymmetry parameters ${\beta }_0$ of the inner and outer rings overlap.

Figure 6

The percentages of the $S$ (red $S$ lines) and $D$ (green $D$ lines) partial waves as a function of laser frequency. The solid, dashed, and dotted lines represent the results obtained from TDSE, Eq. (18), and Eq. (21), respectively. (a)–(d) The results for the outer ring in PEMDs and (e,f) the results for the inner ring. The pulse durations for the left and right columns are $N=60$ and 300, respectively. The laser intensities are $5\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2$ in (a), (b), (e), and (f) and $1\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$ in (c), (d).

Figure 7

The sketch of the energy shift and the splitting in the near-resonant laser field. The ac-stark shifts $S_R\left(t\right), S_I\left(t\right), S_{\varepsilon }\left(t\right)$ as well as the photon energy detuning $\mathrm{\Delta }\omega$ affect the splitting and shift of the energy levels.