Photon-momentum transfer in one- and two-photon ionization of atoms

The transfer of the linear photon momentum to the electron in the few-photon ionization of an atom is accurately computed through a numerical solution to the three-dimensional time-dependent Schrödinger equation (TDSE) beyond the dipole approximation. The characteristics of the transfer is studied in detail by comparing the TDSE results with those calculated by the perturbation theory (PT) using the exact scattering states. Our fully quantum-mechanical calculations show a discernible photoelectron momentum shift along the laser's propagation direction, which is caused by the nondipole effects. The momentum transfer rule for the single-photon ionization is shown to agree with results obtained from the perturbation theory. We identify some extraordinary “dips” in the photon-momentum transfer in the case of two-photon ionization, which depend on the laser ellipticity and are caused by the suppression of the nondipole transitions. In addition, the Coulomb screening effect on the photon-momentum transfer has also been investigated and we find that the Coulomb tail is negligible in the single-photon ionization case. However, the potential in the short range near the Coulomb center may influence the initial state, which will change the amount of the momentum that can be transferred.

Figure 1

(a) The 2D photoelectron momentum distribution in the $xz$ plane of a hydrogen atom by a linearly polarized laser along the $x$ axis. (b) The normalized angular distributions in the $xz$ plane for both the linear (red solid line) and the circular (dashed green line) polarization. (c) The normalized photoelectron momentum distribution $f\left(p_z\right)$ along the laser propagation direction. See the text for the laser parameters.

Figure 2

The expectation of photoelectron momentum $〈p_z〉$ along the laser propagation direction as a function of the laser frequency $\omega$ in the single-photon ionization regime. Red open circles (green solid regular triangles): $〈p_z〉$ for the LP (CP) light. Red open diamonds (green solid inverted triangles): the electron average kinetic energy $〈E_k〉$ divided by the vacuum light speed $c$ for the LP (CP) light. Black dashed line: the momentum transfer law given by the analytic first-order perturbation theory in Eq. (12). Black solid line: $(\omega -I_p)$ divided by the vacuum light speed $c$.

Figure 3

Same as those of Fig. 1, but for the two-photon ionization case at two different frequencies $\omega =0.4$ a.u. (first row) and $\omega =0.425$ a.u. (second row).

Figure 4

The weight of each partial wave. Black solid line with open triangles (blue solid line with open circles) in the first row: $W_{lm}$ in the CP field of the frequency $\omega =0.4$ a.u. ($\omega =0.425$ a.u.); red solid line with solid triangles (green solid line with solid circles) in the second row: $W_{lm}$ in the LP field of the frequency $\omega =0.4$ a.u. ($\omega =0.425$ a.u.). Inset: $W_{lm}$ in the log scale.

Figure 5

The expectation of photoelectron momentum $〈p_z〉$ along the laser propagation direction as a function of the laser frequency $\omega$ in the two-photon ionization regime. Red solid circles (blue solid line with circles): $〈p_z〉$ for the CP (LP) light. Red (blue) triangles: $〈E_k〉/c$ for the CP (LP) light. Black dashed line: $1.71(2\omega -I_{\mathrm{p}})/c$. Black solid line: $(2\omega -I_{\mathrm{p}})/c$.

Figure 6

The expectation of photoelectron momentum $〈p_z〉$ along the laser propagation direction as a function of $〈E_k〉/c$ in the one-photon ionization regime with different initial states: red solid line for the $1s$ (angular quantum number $l=0$ and magnetic quantum number $m=0$) state; blue dotted line with solid circles for the $2s$ ($l=0,m=0$) state; black dash-dotted line for the $2p$ ($l=1,m=0$) state; black dotted line with solid triangles for the $2p$ ($l=1,m=-1$) state; black dashed line for the $2p$ ($l=1,m=1$) state; green solid line with open inverted triangles for the $3d$ ($l=2,m=1$) state.

Figure 7

The expectation of photoelectron momentum $〈p_z〉$ along the laser propagation direction as a function of the laser ellipticity $\varepsilon$ at $\omega =0.4$ a.u.

Figure 8

The total weight of partial waves related to the nondipole transitions at different laser frequencies. Blue solid line with circles (red dashed line with triangles): $W^{\mathrm{nondipole}}$ for the LP (CP) pulse.

Figure 9

Top panels show different types of potential $V\left(r\right)$ in the regime of (a) $0<r<0.5$ a.u. and (b) $0<r<20$ a.u. The lower panel shows the expectation of photoelectron momentum $〈p_z〉$ along the laser propagation direction as a function of $〈E_k〉/c$ in the one-photon ionization regime with different potential types. Red solid line: results with the Coulomb potential given by $V\left(r\right)=-1/r$; thick green dashed line: results with the finite potential; thin blue dashed line: results with the Yukawa potential ($Z=1.905$, $\alpha =1$); black dash-dotted line: results with the Yukawa potential ($Z=1.5$, $\alpha =0.5365$).

Figure 10

The expectation of photoelectron momentum along the propagation direction $〈p_z〉$ as a function of $〈E_k〉/c$ in the one-photon ionization regime for three cases: hydrogen atom in the Coulomb potential (green solid line for PT and red solid circles for TDSE); hydrogen atom in the Yukawa potential ($Z=1.905$, $\alpha =1$) (yellow dashed line for PT and black solid triangles for TDSE); helium atom with a model potential (gray dash-dotted line for PT and blue solid squares for TDSE).