Circularly polarized RABBITT applied to a Rabi-cycling atom

We utilize the reconstruction of attosecond beating by interference of two-photon transitions (RABBITT) technique to study the phase of a Rabi-cycling atom using circularly polarized extreme ultraviolet and infrared fields, where the infrared field induces Rabi oscillations between the $2s$ and $2p$ states of lithium. Autler-Townes splittings are observed in sidebands of the photoelectron spectra and the relative phases of outgoing electron wave packets are retrieved from the azimuthal angle. In this RABBITT scheme, more ionization pathways beyond the usual two-photon pathways are required. Our results show that the polar-angle-integrated and polar-angle-resolved RABBITT phases have different behaviors when the extreme ultraviolet and infrared fields have co- and counter-rotating circular polarizations, which are traced back to the different ionization channels according to the selection rules in these two cases and their competition relying on the propensity rule in laser-assisted photoionization.

Figure 1

Photoionization cross sections from the $2s$ and $2p$ ($m=0$) states of the lithium atom as the functions of the photoelectron energy.

Figure 2

The schematic of ionization pathways for ${\mathrm{P}}^{\mathrm{h}}$ with the photoelectron energy of $2q\omega -I_p^{2s}+{\omega }_{\mathrm{R}}$ (${\omega }_{\mathrm{R}}\approx {\mathrm{\Omega }}_{\mathrm{R}}^0/2$). The purple and red arrows indicate the transition by exchanging the XUV and IR photons, respectively. The essential pathways are drawn in thick-line arrows. The less-contributed pathways are represented by thin-line arrows.

Figure 3

(a) The schematic of the essential states of the lithium atom coupled to the IR field. The double-side (single-side) red arrows denote the strong (weak) coupling. The thick horizontal solid (dashed) lines indicate the states that are allowed (forbidden) to be populated according to the dipole selection rules. The detuning of the $2p-3d$ transition is $\mathrm{\Delta }≔\omega -E_{3d}+E_{2p}$, with $E_{2p}$ and $E_{3d}$ the energies of the $2p$ and $3d$ states of the lithium atom, respectively. (b)–(d) The populations of the $2s$ (red-solid lines), $2p$ (orange-dashed lines), $3d$ states of lithium (bluish-gray-dotted lines) and their summation (thin-purple-solid lines) as a function of the atom-field interaction time (the beginning time is set as zero). The intensities of the left-hand circularly polarized IR field are (b) $1\times {10}^{10}$ (corresponding to the energy spacing of 0.0367 eV), (c) $3\times {10}^{10}$ (corresponding to the energy spacing of 0.0642 eV), and (d) $1\times {10}^{11}\mathrm{W}/{\mathrm{cm}}^2$ (corresponding to the energy spacing of 0.1154 eV).

Figure 4

From top to bottom: (a1), (b1) The RABBITT phases extracted from the photoelectron spectra within the polarization plane of the left-hand circularly polarized XUV and IR fields for ${\mathrm{P}}^{\mathrm{l}}$ as a function of the photoelectron energy. (a2), (b2) The same as panels (a1), (b1), but for ${\mathrm{P}}^{\mathrm{h}}$. (a3), (b3) The relative RABBITT phases between ${\mathrm{P}}^{\mathrm{h}}$ and ${\mathrm{P}}^{\mathrm{l}}$, as a function of the photoelectron energy. The different columns correspond to the results obtained (a1), (a2), (a3) by solving the TDSE and (b1), (b2), (b3) by perturbation theory including all pathways, respectively. The purple circles, blue rhombuses and orange squares correspond to the IR intensities of $1\times {10}^{10}$, $3\times {10}^{10}$, and $1\times {10}^{11}\mathrm{W}/{\mathrm{cm}}^2$, respectively.

Figure 5

(a) The schematic of the ionization channels for pathways 1 and 3 when using the left-hand circularly polarized (denoted ${\sigma }_{+}$) XUV and IR fields. (b) The same as panel (a), but for pathways 6 and 9. The purple and red arrows indicate the transition via exchanging the XUV and IR photons, respectively. The solid (dashed) arrows denote the relatively more (less) probable transition according to the propensity rule in laser-assisted photoionization when comparing each step of absorption and emission from the same state. The partial waves of each ionization channel are illustrated as the real spherical harmonics represented on polar plots. Pathway 5, having the same partial waves as pathway 1, is not shown. From top to bottom of the two rightmost columns: (c1), (d1) The RABBITT phases extracted from the photoelectron spectra within the polarization plane of the circularly polarized fields for ${\mathrm{P}}^{\mathrm{l}}$ as a function of the photoelectron energy. (c2), (d2) The same as panels (c1), (d1), but for ${\mathrm{P}}^{\mathrm{h}}$. (c3), (d3) The relative RABBITT phases between ${\mathrm{P}}^{\mathrm{h}}$ and ${\mathrm{P}}^{\mathrm{l}}$, as a function of the photoelectron energy. The different columns correspond to the results obtained by perturbation theory only including (c1), (c2), (c3) pathways 1, 3, 5, 6, 9 and (d1), (d2), (d3) pathways 1, 3, 5, respectively. The purple circles, blue rhombuses, and orange squares correspond to the IR intensities of $1\times {10}^{10}$, $3\times {10}^{10}$, and $1\times {10}^{11}\mathrm{W}/{\mathrm{cm}}^2$, respectively.

Figure 6

From top to bottom: (a1), (b1), (c1) The relative polar-angle-resolved RABBITT phases for ${\mathrm{P}}^{\mathrm{l}}$ as a function of polar emission angle of photoelectrons, with the intensities of the left-hand circularly polarized XUV and IR fields $1\times {10}^{13}$ and $1\times {10}^{10}\mathrm{W}/{\mathrm{cm}}^2$, respectively. (a2), (b2), (c2) The same as (a1), (b1), (c1), but for ${\mathrm{P}}^{\mathrm{h}}$. (a3), (b3), (c3) The relative phases between ${\mathrm{P}}^{\mathrm{h}}$ and ${\mathrm{P}}^{\mathrm{l}}$, as a function of the polar emission angle of photoelectrons. The different columns respectively correspond to the results obtained by perturbation theory including (a1), (a2), (a3) all pathways; (b1), (b2), (b3) only pathways 1, 3, 5, 6, and 9; and (c1), (c2), (c3) only pathways 1, 3, and 5. The circles, pluses, squares, rhombuses, crosses and triangles correspond to SBs 10, 12, 14, 16, 18 and 20, respectively.

Figure 7

(a) The schematic of the ionization channels for pathways 1 and 3 when using the left-hand circularly polarized (denoted ${\sigma }_{+}$) XUV field and the right-hand circularly polarized (denoted ${\sigma }_{-}$) IR field. (b) The same as panel (a), but for pathways 6 and 9. The purple and red arrows indicate the transition via exchanging the XUV and IR photons, respectively. The solid (dashed) arrows denote the relatively more (less) probable transition according to Fano's propensity rule and the propensity rule in laser-assisted photoionization when comparing each step of absorption and emission from the same state. The partial waves of each ionization channel are illustrated as the real spherical harmonics represented on polar plots. Pathway 5, having the same partial waves as pathway 1, is not shown. From top to bottom of the two rightmost columns: (c1), (d1) The polar-angle-resolved RABBITT phases for ${\mathrm{P}}^{\mathrm{l}}$ as a function of polar emission angle of photoelectrons, with the intensities of the XUV and the IR fields $1\times {10}^{13}$ and $1\times {10}^{10}\mathrm{W}/{\mathrm{cm}}^2$, respectively. (c2), (d2) The same as (c1), (d1), but for ${\mathrm{P}}^{\mathrm{h}}$. (c3), (d3) The relative RABBITT phases between ${\mathrm{P}}^{\mathrm{h}}$ and ${\mathrm{P}}^{\mathrm{l}}$, as a function of the polar emission angle of photoelectrons. The different columns corres- pond to the results obtained by perturbation theory including (c1), (c2), (c3) all pathways and (d1), (d2), (d3) only pathways 1, 3, 5, respectively. The circles, pluses, squares, rhombuses, crosses and triangles correspond to SBs 10, 12, 14, 16, 18 and 20, respectively.

Figure 8

From top to bottom: (a1), (b1) The polar-angle-integrated RABBITT phases for ${\mathrm{P}}^{\mathrm{l}}$ as a function of the photoelectron energy. (a2), (b2) The same as panels (a1), (b1), but for ${\mathrm{P}}^{\mathrm{h}}$. (a3), (b3) The relative RABBITT phases between ${\mathrm{P}}^{\mathrm{h}}$ and ${\mathrm{P}}^{\mathrm{l}}$, as a function of the photoelectron energy. The different columns correspond to the results obtained by perturbation theory including (a1), (a2), (a3) all pathways and (b1), (b2), (b3) only pathways 1, 3, and 5, respectively. The purple circles, blue rhombuses, and orange squares correspond to the IR intensities of $1\times {10}^{10}$, $3\times {10}^{10}$, and $1\times {10}^{11}\mathrm{W}/{\mathrm{cm}}^2$, respectively.