High-Order Phase-Dependent Asymmetry in the Above-Threshold Ionization Plateau

Above-threshold ionization spectra from cesium are measured as a function of the carrier-envelope phase (CEP) using laser pulses centered at $3.1\mu \mathrm{m}$ wavelength. The directional asymmetry in the energy spectra of backscattered electrons oscillates three times, rather than once, as the CEP is changed from 0 to $2\pi$. Using the improved strong-field approximation, we show that the unusual behavior arises from the interference of few quantum orbits. We discuss the conditions for observing the high-order CEP dependence, and draw an analogy with time-domain holography with electron wave packets.

Figure 1

Experimental results for ATI of Cs by linearly polarized few-cycle mid-IR laser pulses (${\tau }_p=31\mathrm{fs}$, $\lambda =3100\mathrm{nm}$). (a) CEP-averaged photoelectron spectra measured for intensity values of approximately $0.7\mathrm{TW}/{\mathrm{cm}}^2$ (purple dotted line), $1.8\mathrm{TW}/{\mathrm{cm}}^2$ (blue dashed line), $3.2\mathrm{TW}/{\mathrm{cm}}^2$ (green dashed-dotted line), and $4.1\mathrm{TW}/{\mathrm{cm}}^2$ (yellow solid line), respectively. (b),(c) Measured CEP-dependent asymmetry parameter for 4.1 and $3.2\mathrm{TW}/{\mathrm{cm}}^2$, respectively. The black box in (b) marks the trifurcation of the asymmetry, which is absent in (c). (d) The average number of electrons detected per laser shot is compared to predictions based on the PPT tunneling rate (solid line). The error bars depict an estimated 15% uncertainty in the intensity determination. Panel (e) presents the CEP-dependent asymmetries at electron energies of $6U_p$, as marked by the black arrows in (a). The high-order $m=3$ asymmetry oscillation in the case of the highest intensity is clearly visible and distinct from the usual $m=1$ oscillations observed for the lower intensity values.

Figure 2

Calculated CEP-dependent asymmetry maps for (a) TDSE, (b) ISFA, (c) saddle-point calculations at intensity $I=4\mathrm{TW}/{\mathrm{cm}}^2$, averaged over the intensity distribution of a Gaussian focal volume. While the TDSE results contain contributions from both direct and recollision electrons, the ISFA and saddle-point results are obtained for recolliding electrons only.

Figure 3

Quantum orbit analysis. (a) The laser electric field of a coslike pulse ($\varphi =0$) is plotted along with the most relevant quantum orbits where the emission time $t_0$ is connected with the recollision time $t$. The solid (dashed) lines correspond to quantum orbits with electron emission in positive (negative) direction. Red integer (black half-integer) values correspond to long (short) orbits that lead to positive drift momenta. (b) For each quantum orbit, the electron yield is plotted against the electron kinetic energy. The sharp spikes at the cutoff energies are an artifact of the saddle-point method. For the intensity of $4\mathrm{TW}/{\mathrm{cm}}^2$, the resulting CEP-dependent asymmetry map is plotted for (c) long orbits only, (d) short orbits only, and (e) all orbits. The black boxes mark the trifurcation region.