Phase delays in $\omega -2\omega$ above-threshold ionization

Relative phases of atomic above-threshold ionization wave packets were investigated in an experiment [L. J. Zipp, A. Natan, and P. H. Bucksbaum, Optica 1, 361 (2014)] exploiting interferences between different pathways in a weak probe field at half the frequency of the strong ionization pulse. In this work we explore theoretically the extraction of phase delays and time delays of attosecond wave packets formed in strong-field ionization. We perform simulations solving the time-dependent Schrödinger equation and compare these results with the strong-field and Coulomb-Volkov approximations. In order to disentangle short- from long-range effects of the atomic potential, we also perform simulations for atomic model potentials featuring a Yukawa-type short-range potential. We find significant deviations of the ab initio phase delays between different photoelectron pathways from the predictions by the strong-field approximation even at energies well above the ionization threshold. We identify similarities to but also profound differences from the well-known interferometric extraction of phase and time delays in one-photon ionization.

Figure 1

Comparison between (a) multiphoton strong-field interference (MPSFI) and (b) the standard RABBIT protocol for two interfering pathways from the initial bound state $|i\rangle$ to final states $|f\rangle$ in the continuum. While RABBIT applies to two pathways involving ionization by one photon with energies $(2m-1)\omega$ and $(2m+1)\omega$ generated by HHG, MPSFI involves (at least) two ATI peaks generated by absorbing $N$ or $N+1$ photons from the strong pump field with frequency $2\omega$. The final state $|f\rangle$ is reached in either case by the absorption ($\omega$) or emission ($-\omega$) of one photon of the weak probe field. Each arrow denotes a one-photon transition.

Figure 2

(a) Total photoelectron spectrum (logarithmic scale), (b) forward emission spectrum integrated over the $+z$ hemisphere (logarithmic scale), and (c) asymmetry parameter $A$ (linear scale) for argon as a function of the relative phase $\varphi$ and the electron energy (in eV) calculated within the TDSE. The laser intensities are $I_{2\omega }=8\times {10}^{13}$ W/${\mathrm{cm}}^2$ and $I_{\omega }=4\times {10}^{11}$ W/${\mathrm{cm}}^2$ for the respective frequencies $2\omega$ and $\omega =0.057$ a.u. with pulse duration $\tau =881.85$ a.u., corresponding to eight full cycles of the latter. (d)–(f) Close-up in linear scale corresponding to (a)–(c), respectively.

Figure 3

Continuum phase shifts $\delta \left(E\right)$ extracted from the asymmetry $A(E,\varphi )$ as a function of the emission energy from the TDSE (thick black solid line), SFA (blue dotted line), and CVA (thin red solid line) results. Solid vertical gray lines denote ATI peak energies and dashed vertical yellow lines sideband energies according to Eq. (7). The horizontal dashed line corresponds to the strong-field limit for ATI phase shifts [$\delta \left(E\right)=\pi$].

Figure 4

Examples of pairs of quantum paths reaching (a) the sideband energy $E_n=E_{15}$ or (b) the ATI (or main) peak energy $E_n=E_{14}$ for argon. In (a) the set $P_i$ ($i=1,...$) features pairs, each with absorption [left (L)] or emission [right (R)] of one weak probe photon $\omega$, and the set $P_i^{\prime }$ features pairs with one additional absorption and emission of the strong pump photon (R) compared to the direct path (L) while both absorbing one weak probe photon $\omega$. The absorption of the probe photons may occur in the continuum ($P_1, P_2$, and $P_1^{\prime }$) or in virtual intermediate bound states ($P_3$ and $P_2^{\prime }$). In (b) the direct ATI process $P_0$ can, to lowest order, interfere with path pairs $P_1^{\prime \prime }$ involving absorption or emission of two $\omega$ photons; $P_2^{\prime \prime }$ are examples of a process involving four $\omega$ photons. The $P_0^{\prime }$ represents one contribution to the dressing of the ATI electron by the probe field.

Figure 5

Energy spectrum and angular momentum distribution after strong-field ionization of argon by the one-color $2\omega$ field with the same parameters as in Fig. 2: (a) photoelectron spectrum, (b) electron distribution as a function of the energy and angular momentum on a logarithmic scale covering three orders of magnitude, and (c) normalized $p_{\ell }$ (integrated over energy) for the first three ATI peaks from threshold.

Figure 6

(a) Electron spectra for a Yukawa potential [Eq. (3)] with $a=4$ and $b=0.629$ calculated for one-color $2\omega$ (black line) and two-color $\omega \text{−}2\omega$ (red line) laser fields with $\varphi =0$. (b) Phase delays $\delta \left(E\right)$ in units of $\pi$ calculated from the asymmetry $A(E,\varphi )$ integrated over hemispheres [see Eq. (9)]. For reference we also convert the phase delay into a time delay [Eq. (11)] (right side axis). The laser intensities are $I_{2\omega }={10}^{11}$ W/${\mathrm{cm}}^2$ and $I_{\omega }=5\times {10}^8$ W/${\mathrm{cm}}^2$. The other laser parameters are the same as in Fig. 2.

Figure 7

TDSE phase delays $\delta \left(E\right)$ calculated as a function of the emission energy for (a) ATI peaks and (b) sidebands for the same pulse parameters as in Fig. 2. Phase shifts were extracted from data for forward half spheres $S_{+}(E,\varphi )$ (squares) and asymmetry $A(E,\varphi )$ (circles) with integration over the energy window around each peak energy (closed symbols) and at the energy peak only (open symbols). Closed green circles correspond to experimental data by Zipp et al. [40] normalized to the TDSE result at the highest sideband energy ($\sim 15\mathrm{eV}$).

Figure 8

Interference phase delay $\delta \left(E\right)$ as a function of the probe laser intensity $I_{\omega }$ extracted from asymmetry parameter $A$ integrated over hemispheres for (a) three ATI peaks and (b) three sidebands with energies as indicated. All other laser parameters are the same as in Fig. 2. The horizontal dashed line corresponds to the strong-field limit for ATI phase shifts [$\delta \left(E\right)=\pi$]; the SFA limit for the sidebands is $\delta \left(E\right)=0$ (not shown).