## Watching as helium goes topsy-turvy

Theorists have long pondered the underpinnings of the Fano resonance, a spectral feature that resembles adjacent rightside-up and upside-down peaks. An especially well-studied instance of this feature appears in the electronic spectrum of helium as a transient state undergoes delayed ionization. Two studies have now traced the dynamics of this state in real time. Gruson *et al.* used photoelectron spectroscopy to extract the amplitude and phase of the electron wave packet after inducing its interference with reference wave packets tuned into resonance at variable delays. Kaldun *et al.* used extreme ultraviolet absorption spectroscopy to probe the transient state while variably forcing ionization with a strong near-infrared field.

## Abstract

The dynamics of quantum systems are encoded in the amplitude and phase of wave packets. However, the rapidity of electron dynamics on the attosecond scale has precluded the complete characterization of electron wave packets in the time domain. Using spectrally resolved electron interferometry, we were able to measure the amplitude and phase of a photoelectron wave packet created through a Fano autoionizing resonance in helium. In our setup, replicas obtained by two-photon transitions interfere with reference wave packets that are formed through smooth continua, allowing the full temporal reconstruction, purely from experimental data, of the resonant wave packet released in the continuum. In turn, this resolves the buildup of the autoionizing resonance on an attosecond time scale. Our results, in excellent agreement with ab initio time-dependent calculations, raise prospects for detailed investigations of ultrafast photoemission dynamics governed by electron correlation, as well as coherent control over structured electron wave packets.

Tracking electronic dynamics on the attosecond time scale and angstrom length scale is a key to understanding and controlling the quantum mechanical underpinnings of physical and chemical transformations (*1*). One of the most fundamental electronic processes in this context is photoionization, the dynamics of which are fully encoded in the released electron wave packet (EWP) and the final ionic state. The development of broadband coherent sources of attosecond pulses has opened the possibility of investigating these processes with attosecond resolution. On such a short time scale, few techniques (*2*–*5*) are able to provide access to both spectral amplitude and phase. The spectral derivative of the phase, the group delay, is a practical quantity for describing general wave packet properties reflecting the ionization dynamics. Recently, photoemission delays have been measured in a variety of systems: noble gas atoms (*6*–*8*), molecules (*9*), and solids (*10*). In the gas phase, these attosecond delays give insight into the scattering of the electron in the ionic potential; in the solid state, they provide information on the transport dynamics toward the surface. However, the physical relevance of group delays is restricted to fairly unstructured wave packets.

The necessity to go beyond simple delays arises for more complex ionization dynamics when the broadband excitation encompasses continuum structures associated with, for example, autoionizing states, shape resonances, and Cooper minima (*11*–*13*). These structures induce strong spectral variations of the amplitude and phase of the EWP corresponding to different time scales, ranging from the attosecond to the femtosecond domains. In general, the long-term evolution of the EWP amplitude [e.g., the lifetime of Fano autoionizing resonances (*14*)] can be characterized directly in the time domain (*15*), or in the spectral domain with the use of conventional spectroscopic techniques (*16*). However, the EWP phase is required for reconstruction of the full ionization dynamics. In particular, the short-term response associated with broadband excitation remains unexplored (*17*). It is mainly determined by the spectral phase variation over the resonance bandwidth, which has so far not been measured. An additional difficulty is that the characterization techniques often involve strong infrared probe fields that (i) strongly perturb the resonant structures (*18*–*20*) so that the field-free intrinsic dynamics cannot be accessed, and (ii) require elaborate theoretical input for decoding the electron spectrograms (*21*).

Here, we extend attosecond photoionization spectroscopy to the full reconstruction of the time-dependent EWPs produced by coherent broadband excitation through resonant structures. To this end, we have developed a perturbative interferometric scheme enabling the direct measurement of the spectral amplitude and phase of the unperturbed resonant EWP. Interferences between the latter and a reference nonresonant EWP are achieved through two-photon replicas obtained by photoionizing the target with an extreme ultraviolet (XUV) harmonic comb combined with the mid-infrared (MIR) fundamental field. This spectrally resolved technique is easy to implement and offers straightforward access to the EWP characteristics without complex analysis or theoretical input. We apply it to the investigation of the test case of the doubly excited 2s2p autoionizing resonance of helium, for which ab initio time-dependent calculations can be performed (*22*, *23*), thereby providing a benchmark for our experimental study.

Autoionization occurs when a system is excited in structured spectral regions where resonant states are embedded into a continuum. The system can then either directly ionize or transiently remain in the resonant bound state before ionizing. Coupling between the resonant state and continuum states of the same energy through configuration interaction leads to the well-known Fano spectral line shapes (*14*). Of particular interest is the autoionization decay from doubly excited states (*16*) that is a direct consequence of the electron-electron repulsion. Using our spectrally resolved technique, we directly access the complete ionization dynamics (including interferences at birth time) and monitor the resonance buildup on a subfemtosecond time scale—an endeavor of attosecond science (*17*, *24*).

The concept of the method is shown in Fig. 1A. We photoionize helium with a comb of mutually coherent odd harmonics derived from an optical parametric amplifier (OPA) MIR source. The harmonic of order 63 (H_{63}) is driven into the 2s2p resonance, at 60.15 eV from the ground state, by tuning the OPA central wavelength λ_{OPA} to 1295 nm. Because the harmonic width (400 meV) is much larger than the resonance width (Γ = 37 meV), a broad resonant EWP with complex spectral amplitude* A*_{R}(*E*) is produced. Simultaneously, nonresonant EWPs are created by the neighboring harmonics H_{61} and H_{65} in smooth regions of the continuum; each of these can serve as a reference, denoted* A*_{NR}(*E*), to probe the resonant EWP.

To induce interference, we use two-photon transitions to create replicas that spectrally overlap with each other. A weak fraction of the fundamental MIR pulse, of angular frequency ω_{0} = 2π*c*/λ_{OPA}, is superimposed on the harmonic comb with a delay τ. Its intensity (~2 × 10^{11} W/cm^{2}) is sufficiently high to induce perturbative two-photon XUV-MIR transitions but is low enough to avoid transitions involving more than one MIR photon [e.g., depletion of the doubly excited state by multiphoton ionization (*15*), or distortion of the resonance line shape (*19*)]. Most important, the MIR spectral width (26 meV) is smaller than both the harmonic and resonance widths, ensuring that each EWP produced in the two-photon process is a faithful, spectrally shifted, replica of the unperturbed EWP produced in the XUV one-photon process. Because of the frequency relation between the odd-harmonic XUV comb and the fundamental MIR laser, the resonant EWP upshifted by absorption of a MIR photon,* A*_{R+1}(τ, *E* + *ħ*ω_{0}) ∝ *A*_{R}(*E*) exp(*i*ω_{0}τ), and the reference EWP downshifted by stimulated emission of a MIR photon,* A*_{NR–1}(τ, *E* + *ħ*ω_{0}) ∝ *A*_{NR}(*E* + 2*ħ*ω_{0}) exp(–*i*ω_{0}τ), coherently add up in the single sideband (SB_{64}) that lies between the lines associated with H_{63} and H_{65}. Similarly, the resonant EWP downshifted by emission of a MIR photon interferes in sideband SB_{62} with the EWP upshifted by absorption of a MIR photon from H_{61}. We designate *E* the photoelectron energy in the resonant EWP, and = *E* ± *ħ*ω_{0} the photoelectron energy of the resonant EWP replicas in SB_{64} and SB_{62}, respectively.

The spectrum of these sidebands is thus modulated by the interference between the resonant and nonresonant replicas, depending on the XUV-MIR delay τ (*25*). For SB_{64}, the spectrum is given by (1)where the two contributions to the replicas’ relative phase are (i) 2ω_{0}τ, the phase introduced by the absorption or simulated emission of the MIR photon, and (ii) the relative phase between the initial one-photon EWPs. The latter is split into , the phase difference between the two ionizing harmonics, and , the difference between the nonresonant and resonant scattering phases of the two intermediate states. In our conditions, the variations over the sideband width of both and are negligible in comparison with that of the resonant scattering phase (*25*). The latter contains information about the scattering of the photoelectron by the remaining core, including strongly correlated scattering by the other electron close to the resonance. This phase is the measurable quantity addressed by our study.

Using a high-resolution (~1.9%) magnetic-bottle spectrometer with a length of 2 m, we have access to the photoelectron spectrogram—electron yield as a function of energy *E *and delay τ—spectrally resolved within the harmonics and sideband widths (Fig. 1B). As a result of its large bandwidth, H_{63} produces a photoelectron spectrum exhibiting a double structure with a Fano-type resonant peak and a smoother peak. This shape is replicated on each of the closest resonant sidebands (SB_{62} and SB_{64}). Strikingly, the components of the double structure oscillate with different phases when τ is varied, in both SB_{62} and SB_{64}.

These phase variations are further evidenced by a spectrally resolved analysis: For each sampled energy within the sideband width, we perform a Fourier transform of* S*_{63±1}(τ, *E* + *ħ*ω_{0}) with respect to τ to extract the amplitude and phase of the component oscillating at 2ω_{0} (see Eq. 1 and Fig. 2). The SB_{62} phase shows a strong increase of ~1 rad within the resonant peak, followed by a sudden drop at the amplitude minimum ( ~ 34.75 eV), and a rather flat behavior under the smooth peak. The SB_{64} phase has a very similar shape and magnitude but with an opposite sign due to opposite configuration of the resonant and reference EWPs in the interferometer. This correspondence confirms the direct imprint of the intermediate resonance on the neighboring sidebands.

The 2ω_{0} component of the resonant sidebands thus provides a good measure of the |*A*_{R}(*E*)| exp[*i*η_{scat}(*E*)] EWP that would result from one-photon Fourier-limited excitation. This allows a detailed study of the temporal characteristics of resonant photoemission, in particular of the electron flux into the continuum, through the direct reconstruction of this EWP in the time domain: (2)The temporal profile obtained from SB_{64} is shown in Fig. 3A. It presents a strong peak at the origin—given by the maximum of the Fourier transform of the nonresonant SB_{66} (*25*)—followed by a deep minimum around 4 fs and then a revival with a decay within ~10 fs. The presence of a fast phase jump (~2 rad within ~2 fs) at the position of the minimum indicates that it results from a destructive interference between two wave packet components, the origin of which is detailed below.

To benchmark the measured data, we theoretically investigated the multicolor XUV + MIR ionization of He in the vicinity of the 2s2p resonance. Fully correlated ab initio time-dependent calculations (*22*) were used to validate an analytical model of the two-photon transitions accounting for the actual pulses’ bandwidths (*23*). The simulated photoelectron spectrogram, taking into account the spectrometer resolution, remarkably reproduces the structured shape of the resonant sidebands as well as the dephasing between their two components (Fig. 1C). The analysis of the 2ω_{0} oscillations of SB_{62} and SB_{64} gives spectral phase variations in excellent agreement with the experimental data (Fig. 2). The temporal profile obtained by Fourier transform (Fig. 3) is also well reproduced, with a smaller revival but a similar decay time of ~10 fs. This reduced effective lifetime is a direct consequence of the finite spectrometer resolution. When the latter is assumed infinite, the time profile has the same behavior at short times but a longer decay, corresponding to the 17-fs lifetime of the resonance. Analytical calculations show that in our conditions, the reconstructed EWP does mirror the one-photon resonant EWP (*25*). These findings confirm that, except for a faster decay of the long-term tail due to our current electron spectrometer resolution, the essential physics of the early time frame of EWP creation is directly accessed from purely experimental data.

To further highlight the insight provided by this experimental technique, we undertook an in-depth analysis of the measured EWP characteristics in terms of Fano’s formalism for autoionization (*14*). Resonant ionization can be described as the interference between two distinct paths: (i) the direct transition to the continuum, and (ii) the resonant transition through the doubly excited state that eventually decays in the continuum through configuration interaction within the resonance lifetime (Fig. 3B). The normalized total transition amplitude can then be written as the coherent sum of two contributions, a constant background term and a Breit-Wigner amplitude for the resonance: (3)where ε = 2(*E* – *E*_{R})/Γ is the reduced energy detuning from the resonance at energy* E*_{R}, in units of its half width Γ/2. The Fano parameter *q* [–2.77 for the He(2s2p) resonance (*16*)] measures the relative weight of the two paths. Their interference leads to the well-known asymmetric Fano line shape |*R*(*E*)|^{2} and to the resonant scattering phase: η_{scat}(*E*) = arg *R*(*E*) = atan(ε) + π/2 – πΘ(ε + *q*), where Θ is the Heaviside function. This phase is experimentally accessed here (Fig. 2).

The spectral amplitude of an EWP created by Gaussian harmonic excitation *H*(*E*) is given by *R*(*E*)*H*(*E*). Its temporal counterpart is , where and are Fourier transforms of the spectral amplitudes, in particular (4)(*24*). The temporal profile thus decomposes into a Gaussian nonresonant term and a resonant contribution, like our experimental data (Fig. 3A). The destructive temporal interference between the two terms leads to the amplitude minimum and phase jump identified around *t* = 4 fs.

To illustrate how the interference between the two paths governs the formation of the resonance line shape, Wickenhauser *et al*. (*17*) introduced a time-frequency analysis based on the limited inverse Fourier transform: (5)which shows how the spectrum builds up until accumulation time* t*_{acc}. The result of this transform applied to the experimental EWP in Fig. 3A is shown in Fig. 3, C and D. The chronology of the resonance formation can be nicely interpreted within Fano’s formalism. In a first stage until ~3 fs, a close-to-Gaussian spectrum reflecting the ionizing harmonic spectral shape emerges: The direct path to the continuum dominates. Then the resonant path starts contributing as the populated doubly excited state decays in the continuum: Interferences coherently build up until ~20 fs, consistent with the temporal profile in Fig. 3A, to eventually converge toward the asymmetric measured spectrum. The resonance growth can thus be decomposed in two nearly consecutive steps governed by fairly different time scales.

The buildup of the resonant profile reveals the presence of a notable point around *E* = 35.6 eV where, as soon as the direct ionization is completed, the spectrum barely changes with *t*_{acc} any longer. This can be explained by splitting the |*R*(*E*)|^{2} spectrum from Eq. 3 into three terms: (6)(*26*). At this isosbestic-like point—that is, for ε = [(1/*q*) – *q*]/2—the bound (second term) and coupling (third term) contributions ultimately cancel each other, leaving only the direct continuum contribution (first term). This point thus gives a useful landmark in the resonant line shape (e.g., for cross section calibration or reference purposes).

Spectrally resolved electron interferometry thus provides insight into the ultrafast strongly correlated multielectron dynamics underlying autoionization decay. Given the generality and wide applicability of the Fano formalism [see, e.g., (*26*)], we anticipate that our approach, combined with progress in attosecond pulse production and particle detection (e.g., access to photoelectron angular distributions), will open prospects for studies of complex photoemission dynamics close to resonances and, more generally, structured EWP dynamics in a variety of systems, from molecules (*27*–*29*) and nanostructures (*26*) to surfaces (*10*). Furthermore, the well-defined amplitude and phase distortions induced by the resonance offer a means for shaping the broadband EWP, bringing opportunities for coherent control in the attosecond regime.

## Supplementary Materials

## References and Notes

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**Acknowledgments:**We thank S. Weber for crucial contributions to the PLFA attosecond beamline; D. Cubaynes, M. Meyer, F. Penent, and J. Palaudoux for setup and testing of the electron spectrometer; and O. Smirnova for fruitful discussions. Supported by European Union grant H2020-MSCA-ITN-MEDEA-641789, Agence Nationale de la Recherche grants ANR-15-CE30-0001-01-CIMBAAD, and ANR11-EQPX0005-ATTOLAB, European Research Council advanced grant XCHEM 290853, European COST Action grant XLIC CM1204, and MINECO Project grant FIS2013-42002-R. We acknowledge allocation of computer time from CCC-UAM and Mare Nostrum BSC.