Analysis of the asymmetry of Autler–Townes doublets in the energy spectra of photoelectrons produced at two-photon ionization of atoms by strong laser pulses

Abstract

We analyzed the asymmetry of Autler–Townes doublets in the calculated photoelectron energy spectra for the two-photon ionization of the hydrogen atom by an intense short laser pulse that resonantly couples its ground state with the excited 2p state. The spectra are calculated applying the method of time-dependent amplitudes using two approaches of different level of approximation. The first approach involves solving a complete set of equations for amplitudes, which, in addition to amplitudes of coupled (1s, 2p) and continuous states, also includes the amplitudes of other discrete (nonessential) states. The second approach is the three-level model that includes only the amplitudes of the two coupled states and those of continuum states. By comparing the spectra obtained using these two approaches it is confirmed that the shift of the Autler–Townes doublets, which exists only in the spectra obtained by solving the full set of equations, can be attributed to the AC Stark shift, which is a consequence of the coupling with nonessential states. Finally, it was found that the asymmetry in the intensity of the Autler–Townes doublet components, which appears in the spectra obtained using both computational approaches, is primarily due to the decrease in the transition probability between the 2p and continuum states with increasing photoelectron energy.

Graphical Abstract

Appendix A: Matrix elements for dipole transitions

The matrix elements that occur in Eqs. (4), (5) and (8) are conveniently calculated using the coordinate representation for the discrete and continuum energy eigenstates of the bare atom

$$\begin{aligned} |nl\rangle \rightarrow \psi _{nl}({\mathbf {r}})= & {} \mathrm {R}_{nl}(r) \mathrm {Y}_{l0}(\varOmega ), \end{aligned}$$

(15)

$$\begin{aligned} |\varepsilon l\rangle \rightarrow \psi _{\varepsilon l}({\mathbf {r}})= & {} \sqrt{\frac{2}{\pi k}}\, i^l e^{-i\sigma _l(k)} \frac{\mathrm {F}_l(\eta ,kr)}{r} \mathrm {Y}_{l0}(\varOmega ),\nonumber \\ \end{aligned}$$

(16)

where $\mathrm {R}_{nl}(r)$ and $\mathrm {Y}_{l0}(\varOmega )$ are the radial wave functions of hydrogen and the spherical harmonics with $m = 0$, respectively, whereas $\sigma _l = \arg \Gamma (l+1+i\eta )$ and $\mathrm {F}_l(\eta ,kr)$ are the Coulomb phase shift and the regular Coulomb functions [15], where $\eta = -1/k$ and $k = \sqrt{2\varepsilon }$ (in atomic units). Then

$$\begin{aligned} \langle nl|z|n'l'\rangle= & {} \int \psi _{nl}^*({\mathbf {r}})\, r\cos \vartheta \, \psi _{n'l'}({\mathbf {r}})\, \mathrm {d}^3{\mathbf {r}} = I^\mathrm {(dis)}_{nl,n'l'} J_{ll'},\nonumber \\ \end{aligned}$$

(17)

$$\begin{aligned} \langle nl|z|\varepsilon l'\rangle= & {} \int \psi _{nl}^*({\mathbf {r}})\, r\cos \vartheta \, \psi _{\varepsilon l'}({\mathbf {r}})\, \mathrm {d}^3{\mathbf {r}}\nonumber \\= & {} \sqrt{\frac{2}{\pi k}}\, i^{l'} e^{-i\sigma _{l'}(k)} I^\mathrm {(cont)}_{nl,\varepsilon l'} J_{ll'}, \end{aligned}$$

(18)

where

$$\begin{aligned} I^\mathrm {(dis)}_{nl,n'l'}= & {} \int _0^\infty \!\! \mathrm {R}_{nl}(r)\, \mathrm {R}_{n'l'}(r)\, r^3 \mathrm {d}r, \end{aligned}$$

(19)

$$\begin{aligned} I^\mathrm {(cont)}_{nl,\varepsilon l'}= & {} \int _0^\infty \!\! \mathrm {R}_{nl}(r)\, \mathrm {F}_{l'}(\eta ,kr)\, r^2 \mathrm {d}r, \end{aligned}$$

(20)

$$\begin{aligned} J_{ll'}= & {} \int _\varOmega \mathrm {Y}_{l0}(\varOmega )\, \mathrm {Y}_{l'0}(\varOmega ) \cos \vartheta \, \mathrm {d}\varOmega . \end{aligned}$$

(21)