Breakdown of the isotropic asymptotic approximation in two-colour photoionisation

Abstract

The Wigner delay is defined as the energy derivative of the scattering phase of a particle in a given potential, unveiling the time taken (or gained) due to the interaction. The characterisation of this delay plays a central role in attosecond science, where the time resolution allows to gain information on the time interval required for a photoelectron to be emitted into the continuum after the absorption of a single photon. Attosecond interferometric techniques, based on two-colour (extreme ultraviolet and near-infrared) photoionisation schemes, cannot provide a direct measurement of the Wigner delay, because the low-frequency photon contributes with an additional delay, which is imprinted on the outgoing photoelectron. The isolation of the Wigner delay is usually achieved by appealing to the asymptotic approximation, which assumes that the two-photon delay is separable into a Wigner and a near-infrared-induced phase and provides a universal analytical expression for the latter. In this study, we introduce a self-referencing approach based on the implementation of non-consecutive extreme ultraviolet harmonics, in order to test the validity of the asymptotic approximation. We demonstrate its breakdown by observing a deviation of a few tens of milliradians (corresponding to a few attoseconds) between its predictions and the experimentally measured phases of the sideband oscillations generated in our scheme, in agreement with full-dimensional simulations.

Figures (9)

• Figure 1: Energy scheme and self-refencing approach for the investigation of the breakdown of the isotropic asymptotic approximation in two-colour photoionization. a) Schematic view of the energy levels of a two-colour photoionisation scheme leading to the population of the two energy levels $a$ and $b$ through the exchange (absorption and emission) of a single NIR photon. $I_p$ indicates the ionisation potential of the target atom. b) Schematic view of the energy levels of a self-referencing two-colour photoionisation scheme based on the measurement of two sidebands $S^{(+)}_{q,q+1}$ and $S^{(-)}_{q,q+1}$ between the main photoelectron peaks generated by the absorption of one XUV photon from the harmonics $q$ and $q+1$ of a driving field with angular frequency $3\omega$. c) XUV photon spectra (upper line) consisting of the four non-consecutive harmonics $H_6$, $H_7$, $H_9$ and $H_{10}$. Photoelectron spectra (left hand side) generated in helium by single-photon ionisation using the XUV harmonics with (red spectrum) and without (blue spectrum) the NIR field. The energy levels reached in the single and two-photon ionisation processes are schematically presented. d) Photoionisation pathways leading to the population of the sidebands $S^{(\pm)}_{9,10}$ between the consecutive harmonics $H_9$ and $H_{10}$. The red arrows with solid lines indicate the exchange of NIR photons that differs in the population between the sidebands $S^{(-)}_{9,10}$ and $S^{(+)}_{9,10}$. e) Photoionisation pathways leading to the population of the sidebands $S^{(\pm)}_{7,9}$ between the nonconsecutive harmonics $H_7$ and $H_{9}$. Similarly to panel d, the red arrows with solid lines indicate the exchange of two NIR photons that differs in the population between the sidebands $S^{(+)}_{7,9}$ and $S^{(-)}_{7,9}$.
• Figure 2: Sideband oscillations and determination of the phase differences $\Delta\chi_{3\omega}^{\mathrm{exp}}$, $\Delta\chi_{6\omega}^{\mathrm{exp}}$, and $\Delta\Psi_{6\omega}^{\mathrm{exp}}$. Experimental oscillations of the sidebands $S^{(\pm)}_{6,7}$ (a), $S^{(\pm)}_{9,10}$ (b), $S^{(-,0)}_{7,9}$ (c), $S^{(0,+)}_{7,9}$ (d), and $S^{(-,+)}_{7,9}$ (e) and corresponding fits according to Eq. \ref{['Eq1']}. Component oscillating at $3\omega$ of the fit for the sidebands $S^{(\pm)}_{6,7}$ (f), and $S^{(\pm)}_{9,10}$ (g). Component oscillating at $6\omega$ of the fit for the sidebands $S^{(-,0)}_{7,9}$ (h), $S^{(0,+)}_{7,9}$ (i), and $S^{(-,+)}_{7,9}$ (j). The intensity of the NIR field was $I_2=2.2\times10^{12}$ W/cm$^2$. The second-order phase difference was $\Delta\varphi_{6,7,9,10}=\Delta\varphi_A$. The error bars in the plots were calculated as the standard deviation of the sideband amplitudes for the corresponding phase value $\Phi$.
• Figure 3: Comparison between simulations and experimental results for the phase differences $\Delta\chi_{3\omega}$, $\Delta\chi^{(+0,0-)}_{6\omega}$, and $\Delta\Psi_{6\omega}$ in helium. Comparison of the theoretical simulations (diamonds) obtained using the TDSE and the experimental data (coloured symbols) for the phase differences $\Delta\chi_{3\omega}$ (a,d), $\Delta\chi^{(+0,0-)}_{6\omega}$ (b,e), and $\Delta\Psi_{6\omega}$ (c,f) for three different NIR intensities: $I_1=1.2\times10^{12}$ W/cm$^2$ (green circles), $I_2=2.2\times10^{12}$ W/cm$^2$ (red squares), and $I_3=4.0\times10^{12}$ W/cm$^2$ (blue triangles). The second-order phase difference $\Delta\varphi_{6,7,9,10}$ for the measurements presented in panels (a-c) and (d-f) were $\Delta\varphi_A$ and $\Delta\varphi_B$, respectively. The experimental data were corrected for the contribution of the harmonic $H_8$. The error bars are derived by propagation of the uncertainties in the fitting procedure of the sideband oscillations. The open (full) diamonds in panels (b,e) indicate the phase difference $\Delta\chi_{6\omega}^{(+0)}$ ($\Delta\chi_{6\omega}^{(0-)}$).
• Figure 4: Origin of the breakdown of the isotropic asymptotic approximation. a) Scheme employed in b) and c) to obtain the continuum-continuum phases. The Wigner phase has already been subtracted in b) and c). b) Continuum-continuum phase $\delta_l$ obtained by solution of the TDSE for the hydrogen atom neglecting the centrifugal potential (solid lines) and with the centrifugal potential for final angular momentum $l=0$ (dashed lines) and $l=2$ (dotted lines), for absorption (red lines), emission (blue lines), and the resulting sum of absorption and emission lines (black lines) in the two-colour photoionisation process of hydrogen. c) Sum of absorption and emission phases for $l=0$ (dashed lines) and $l=2$ (dotted lines) for the hydrogen atom (black lines) and two helium SAE model potentials: "Tong" (green lines) and "He$^+_{1s}$" (orange), see main text. The result for hydrogen atom without the centrifugal potential (solid black line) is also shown. d) Scheme employed in e) and f) where the phases correspond to those of two NIR photon transitions, the Wigner phase has been subtracted as for b) and c). e) Two-photon continuum-continuum phase without centrifugal potential (solid lines) and for $l=1$ (dashed lines) and $l=3$ (dotted lines) for the same potentials as c. f) same as c), but for the two-photon transitions. The intensity of the XUV and IR radiation used in the simulation was $10^{11}$ W/cm$^2$ and $10^{10}$ W/cm$^2$, respectively. The pulses have a $\sin^8$ envelope with a total duration of 40 fs.
• Figure Extended Data Fig. 1: Schematic representation of the sideband spectrum created by absorption or emission of NIR photons from the mainband created by harmonic $H_i$. The phases corresponding to each NIR photon emission and absorption event are indicated below, showing that sideband $\alpha_{i+n}$ obtains complex phase $e^{-i n \varphi_\mathrm{NIR}}$.