Field ionization of Rydberg atoms in a single-cycle pulse

We study the ionization of Rydberg atoms in a single-cycle-pulse electric field based on both classical and quantum calculations. The ionization-probability curve exhibits a “ripple” structure as a function of the pulse duration and the field amplitude. These ripple structures are found to be dependent on the angular distribution of the initial state. A large electron-emission asymmetry is observed, and the ionized electron is almost completely emitted to one side of the atom except when the pulse length is roughly one Rydberg period. In both the long-pulse and the short-pulse regimes, larger electron energy can be expected from the ionization of lower-lying Rydberg states, matching the observation in a recent experiment [S. Li and R. R. Jones, Phys. Rev. Lett. 112, 143006 (2014)]. This trend is closely related to the electron-emission asymmetry associated with the field-direction change in a single-cycle pulse. The possible implications of the different energy transfer in a single-cycle pulse from that in a multicycle pulse are also discussed briefly for the strong-field ionization of the ground atomic state.

Figure 1

Ionization-probability curves as a function of the field parameters. (a)–(c) The ionization-probability curves as a function of the field amplitude for the hydrogen $15s, 15p$, and $15d$ states, respectively, with $t_w=0.5$ ps (approximately one Rydberg period). (d)–(f) The ionization-probability curves as a function of the pulse duration setting $F_m=20$ kV/cm. The results from the CTMC simulation and solving the TDS equation are shown by the blue solid lines and the red solid circles, respectively, in each figure. The blue dashed and dot-dashed lines in (a)–(c) give the ionization probabilities from the CTMC simulation for the ionized electron emitted to the right side and that to the left side, respectively.

Figure 2

Landscape for the dependence of the ionization probability on the field parameters obtained from CTMC simulations. (a)–(c) Calculations for the hydrogen $15s, 15p$, and $15d$ states, respectively. (d),(e) Results for the hydrogen $15p$ and $15d$ states, respectively, but using the same angular distribution as that for an $s$ state. The values of ionization probabilities are indicated by a color bar on the left bottom. The green solid curve in each figure indicates the field-ionization threshold given by Eq. (6) in Ref. [2].

Figure 3

Dependence of the ionization probability on the field parameters obtained from CTMC simulations for individual Rydberg orbits from the hydrogen $15s$ state with different angles. (a) $\theta =0^{\circ }$, (b) $\theta ={90}^{\circ }$, (c) $\theta ={180}^{\circ }$, and (d) $\theta ={270}^{\circ }$ correspond to orbits with their outer turning points located, respectively, on the positive $z$ axis, positive $x$ axis, negative $z$ axis, and negative $x$ axis. The initial direction of the electron moving around the nucleus is always kept to be anticlockwise, and all of the orbits are in the $xz$ plane. The color bar on the right shows the corresponding ionization-probability values, and the green solid curve in each figure indicates the ionization threshold given by Eq. (6) in Ref. [2].

Figure 4

Dependence of the electron-emission asymmetry on the single-cycle pulse duration. (a)–(c) The hydrogen $15s, 15p$, and $15d$ states, respectively. The thin black solid line shows the total ionization probability. The blue dashed and dot-dashed lines present the ionization probabilities for the electron emitted to the right side and to the left side, respectively. The asymmetry parameter ${\alpha }_A$ is given by the thick red solid line.

Figure 5

Electron-emission asymmetry for the hydrogen $15d$ state ionized by a stronger pulse field than that in Fig. 4. The field amplitude used in the CTMC simulation is indicated in each figure. The arrangement of different curves is the same as that in Fig. 4.

Figure 6

Electron energy spectra and left-right asymmetry for the hydrogen $15s$ state. The total ionization spectrum and the left-side and right-side ionization spectra are, respectively, displayed in order as (a)–(c) in the top panel ($F_m=$ 50 kV/cm, $t_w=0.5$ ps) and (d)–(f) in the bottom panel ($F_m=50$ kV/cm, $t_w=0.1$ ps). The thin gray curves present the CTMC-simulation results, while the thick red curves are calculated by solving the TDS equation. There are no adjustable parameters between the classical and the quantum results. The definition of the energy distribution $dP/dE$ is the probability to find an electron per unit energy interval [20].

Figure 7

Comparison between the energy spectra from Figs. 6 and 6 on a logarithmic scale. The thin blue and thick red solid lines display the quantum spectra from Fig. 6 ($t_w=0.5$ ps) and Fig. 6 ($t_w=0.1$ ps), respectively. The corresponding spectra from CTMC calculations are shown by the thin gray curves.

Figure 8

Comparison between the ionization spectra for sodium atoms in different bound states. A single-cycle pulse in Eq. (2) is used with $F_m=430$ kV/cm and $t_w=0.5$ ps (about one Rydberg period). The same model potential as that in Ref. [2] is adopted. (a) The final energy spectra after interacting with the full single-cycle pulse; (b) the ionization spectra at the end of the first half cycle. In both (a) and (b), the thick red curve, the blue curve, and the thin gray curve correspond to the CTMC-simulation spectra for the sodium $15d, 9d$, and $6d$ states, respectively. The connection between the energy spectra in (a) and (b) is demonstrated in (c) by taking $n=9$ as an example. The thick green curve in (c) is the full energy spectrum reproduced from (a). The evolution of electron trajectories with $E\ge 25$ and $0\le E<25$ eV in (b) is tracked, and their final energy distributions after the full single-cycle pulse are displayed, respectively, by the red dashed curve and the thin blue solid curve in (c).

Figure 9

Comparison between the ionization spectra for hydrogen atoms in different bound states. The same field parameters as those in Fig. 8 are used. The thick red curve and the thin blue curve correspond to the CTMC-simulation spectra for the hydrogen $15d$ and $9d$ states, respectively. No ionization signal is obtained for the hydrogen $6d$ state because the ionization threshold is higher than the field amplitude used here (430 kV/cm).

Figure 10

Same as Fig. 9 but in the short-pulse regime. A single-cycle pulse in Eq. (2) is used with $t_w=2.5$ fs and a peak intensity of $2.7\times {10}^{15} {\mathrm{W}/\mathrm{cm}}^2$. The thick red curve, the blue curve, and the thin gray curve correspond to the CTMC-simulation spectra for the hydrogen $15d, 9d$, and $6d$ states, respectively.

Figure 11

Comparison of a single-cycle pulse (thick red lines) with an approximate three-cycle pulse (thin blue lines). The field profile and the energy transfer from the field to an electron are compared in (a) and (b), respectively. The single-cycle pulse and the three-cycle pulse are given by Eqs. (7) and (8), respectively, with $t_w=30$ a.u. and $F_m=0.075$ a.u.

Figure 12

Strong-field ionization of the hydrogen ground state. The detailed field profile, the energy transfer from the field to an electron, the time-dependent ionization probability, and the final energy spectrum, respectively, for (a)–(d) a single-cycle pulse and (e)–(h) an approximate three-cycle pulse. The correspondences between different quantities are indicated by the dotted and the dashed lines for the central one cycle. The single-cycle pulse and the three-cycle pulse used in this figure are the same as those compared in Fig. 11.