Asymmetry of above-threshold ionization of metal clusters in two-color laser fields: A time-dependent density-functional study

Above threshold ionization (ATI) spectra of small metal clusters (e.g., ${\mathrm{Na}}_4$ and $\mathrm{Na}_4{}^{+}$) are calculated numerically using a spherical jellium model and time-dependent density functional theory for two-color (1064 and $532\mathrm{nm}$) ultrashort $\left(25\mathrm{fs}\right)$ laser pulses as a function of phase difference between the two fields. ATI spectra and ionized electron fluxes are obtained in the two opposite directions of the linearly polarized laser fields. The asymmetry, defined as the difference in electron yield, is shown to depend strongly on the carrier-envelope phase of the second-harmonic $\left(2\omega \right)$ field. The ATI spectra allow one to identify the range of kinetic energies of the ionized electrons where the asymmetry mainly occurs. Comparisons are made between calculations with and without self-interaction correction and also with previous exact numerical solutions of the one-electron systems $\mathrm{H}$ and $\mathrm{H}_2{}^{+}$ [A. D. Bandrauk and S. Chelkowski, Phys. Rev. Lett. 84, 3562 (2000)] where such asymmetry effects had first been observed. We find that ATI spectra in the clusters generally have much longer energy plateaus than in previously studied one-electron systems, with cutoffs up to $30–40$ times the ponderomotive energy $U_p$. In high-harmonic generation spectra, on the other hand, no extended plateaus are observed.

Figure 1

(a) Electric fields $E_1\left(t\right)$ and $E_2\left(t\right)$ associated with a two-color $25\text{−}\mathrm{fs}$ laser pulse: ${\lambda }_1=1064\mathrm{nm}$, ${\lambda }_2=532\mathrm{nm}$, $I_1=6\times {10}^{12}\mathrm{W}∕{\mathrm{cm}}^2$, $I_2=I_1∕4$, and relative phase $\varphi =0$. (b)–(d) Resulting net field $E\left(t\right)$ [see Eq. (1)] for $\varphi =0$, $\varphi =\pi ∕4$, and $\varphi =\pi ∕2$, respectively.

Figure 2

ATI spectra of a ${\mathrm{Na}}_4$ jellium cluster, calculated with KLI-SIC. (a) Single-color excitation; (b)–(f) two-color excitations with varying relative phase $\varphi$, as indicated.

Figure 3

Same as Fig. 2, for $\mathrm{Na}_4{}^{+}$.

Figure 4

Left and right probability fluxes $P_{\mathrm{b}}\left(t\right)$ [Eq. (13)] for ${\mathrm{Na}}_4$ and $\mathrm{Na}_4{}^{+}$, calculated with KLI-SIC.

Figure 5

Same as Fig. 4, calculated with LSD.

Figure 6

HHG spectra for ${\mathrm{Na}}_4$ and $\mathrm{Na}_4{}^{+}$, calculated with LSD, for one- and two-color (phase $\varphi =\pi ∕2$) excitations. The arrows indicate the positions of the Mie plasmon energies.