Many fundamental processes in nature such as the formation and dissociation of molecules are governed by multi-electron dynamics (MED). Understanding MED is one of the grand contemporary challenges of quantum physics. While the ground state properties of multi-electron systems are described with reasonable accuracy, the modelling of their dynamics is still in its infancy. Even for rather simple systems involving only a few interacting electrons such as low-Z atoms beyond helium (Z being the nuclear charge), the exact theoretical treatment of MED is hardly practicable. The identification of suitable approximations that capture the essential features of MED relies on the development of experiments aimed at resolving the electron dynamics on the characteristic attosecond timescale of their motion. An established technique for resolving one-electron dynamics on sub-femtosecond scales involves the use of a strong few-cycle infrared laser field as a clock^{1, 2, 3}. The well-known concept of attosecond streaking spectroscopy has permitted the observation of single-electron dynamics with attosecond time resolution⁴. Despite this progress, however, the ultrafast dynamics of multiple electrons has resisted direct observation, even though far-reaching conclusions were drawn from kinematically complete strong-field double^{5, 6} and multiple ionization⁷ experiments.

Among countless phenomena involving the motion of multiple electrons, non-sequential double ionization (NSDI) of atoms is a pure and, meanwhile, classic example for MED in the presence of strong fields. As reported for rare gases almost 30 years ago⁸, it manifests itself as an unexpected, orders-of-magnitude enhancement in the strong-field double ionization yield. This finding cannot be explained within the framework of the single active electron approximation, where both electrons are assumed to be ionized sequentially and independently of each other. A mechanism correlating the emission of the two electrons must be included to adequately describe this effect and different models have been proposed^{5, 9}.

Since the pioneering works, NSDI has been the subject of numerous experimental and theoretical studies (for a recent review, see ref. 6 and references therein). Recording the momentum spectra of doubly^{10, 11} and multiply⁷ charged ions provided clear evidence that the emission of the second electron is triggered by the laser-driven recollision of the first electron with its parent ion. Even more insight into the process was gained in kinematically complete studies where the correlated two-electron momentum distributions were measured¹¹ and investigated for different laser parameters^{12, 13, 14, 15}. Measurements of the dynamics of NSDI in near-single cycle laser pulses have been restricted to recoil-ion spectra, which did not allow for the direct retrieval of the correlated electron-emission dynamics^{16, 17}. The sub-cycle dependence of electron correlation spectra has been explored by various theoretical studies, where the double-ionization dynamics were confined to a single cycle of a laser pulse^{12, 18}. A direct test of the validity of these theoretical models, however, has been impeded so far by the lack of kinematically complete experimental data obtained with ultrashort pulses and precise control or knowledge of the electric field.

Here we present measurements of correlated two-electron momentum spectra for the strong-field NSDI of argon atoms in the single-cycle limit. The measured spectrum exhibits a cross-shaped structure that qualitatively differs from spectra recorded in all previous experiments using many-cycle pulses. Confining the ionization dynamics to a single laser cycle by choosing an adequate carrier envelope phase (CEP) allows us to perform a clean experiment with only one single recollision event contributing to NSDI, thus excluding the contribution of multiple recollisions that hamper the interpretation of experimental results. From the CEP-resolved spectra the correlated emission of two electrons is traced on sub-femtosecond timescales. In particular, we can determine at which time the second electron is released within the laser pulse.

Experimental set-up

A detailed sketch of our experimental set-up is given in the Methods. Briefly, the experiments were performed using linearly polarized laser pulses with a central wavelength of 750 nm and a full-width at half-maximum (FWHM) of the temporal intensity envelope of 4 fs, which were generated at a repetition rate of 3 kHz¹⁹. The electric field of the laser pulse can be approximated as E(t)=E₀ cos²(t/τ) cos(ωt+φ), where E₀ is the field amplitude, ω is the carrier frequency, τ is the pulse duration, and φ is the CEP. Combining a reaction microscope (REMI)²⁰ with a newly developed phase-tagging technique²¹, we obtained the CEP of the laser pulse for each and every ionization event²² recorded with the REMI. The laser was focussed to a maximum instantaneous intensity of into a cold supersonic gas jet of neutral argon atoms located in the centre of the REMI. The momenta of electrons and ions created in the laser focus were measured in coincidence in the REMI. For the conditions of our experiment, the ratio of the measured Ar²⁺ to Ar⁺ yields was about 0.1%.

Two-electron momentum distributions

Figure 1 shows experimental correlated two-electron momentum distributions, where the number of recorded double-ionization events is plotted as a function of the momenta p₁ and p₂ along the laser polarization axis of electron 1 and electron 2, respectively. The spectra are symmetric with respect to the p₂=p₁ diagonal because the two electrons, the recolliding electron and the one liberated as a result of this recollision, cannot be distinguished in the experiment. The CEP-averaged correlation spectrum is shown in Fig. 1a. The cross-shaped structure indicates that the momentum of one electron is always close to zero whereas the momentum of the other electron varies between −1.5 to 1.5 atomic units. Similar structures were found in a recent theoretical study on NSDI of argon employing S-matrix theory¹⁸. So far, however, such a cross-shaped two-electron momentum distribution has not been observed experimentally. This strongly suggests that the physical mechanisms that govern NSDI in the near single-cycle regime differ from the many-cycle regime, where previous experiments were performed. The CEP-resolved correlation spectra presented in Fig. 1b–e exhibit a strong CEP dependence. For CEP values of 155° (Fig. 1c) and 335° (Fig. 1e), the signal is predominantly (with more than 80% of the counts) located below and above the red diagonal line (p₂=−p₁), respectively. For intermediate CEP values of 65° (Fig. 1b) and 245° (Fig. 1d), the signal is distributed on both sides of the red diagonal.

The semiclassical model

To interpret the experimental results, we investigate the recollision-induced ionization dynamics using a semiclassical model (for details, see Methods). In the present intensity regime, essentially two mechanisms may contribute: the first is the (e, 2e)-mechanism, in which the recolliding electron directly promotes a second electron into the continuum via electron impact ionization²³. The second is recollision-induced excitation with subsequent ionization (RESI)^{24, 25}, where the recolliding electron excites the parent ion, which is subsequently ionized by the laser field as depicted in Fig. 2a. A key property of the RESI process is the delayed emission of the second electron with respect to the recollision event, the latter occurring near the zero crossing of the electric field of the laser. As shown below, our experimental results identify RESI as the dominant process, and, more importantly, enable the determination of the emission delay.

In our semiclassical model, the probability to liberate the first electron at any moment in time is calculated using the instantaneous Ammosov–Delone–Krainov (ADK) tunnelling rate including a correction factor to account for over-the-barrier ionization²⁶. The propagation of the electron after the ionization in the electric field of the laser pulse is treated classically, neglecting the Coulomb interaction with the ionic core and with the other electron. For the RESI process, the calculation is performed as follows: if the first electron recollides with the parent ion with sufficient kinetic energy, the lowest excited state of Ar⁺ is populated and the second electron is liberated by subsequent field ionization. For the (e, 2e) process, we consider only recolliding electrons having sufficient energy to overcome the second ionization potential of argon. In this case, the second electron is assumed to be liberated with zero-initial momentum via electron impact ionization. In both the RESI and (e, 2e) processes, the magnitude of the first electron momentum, just after the recollision, is determined by energy conservation. The direction of this momentum vector, described by the scattering angle β (where β is the angle between the momentum vectors just before and just after the recollision), is undetermined in our model and constitutes the only free parameter. The dependence of the simulated correlation spectra on the scattering angle is discussed in the Methods.

Comparison of measured and calculated spectra

The results of our calculations are displayed in Fig. 2b–d, for the instantaneous peak intensity of I₀=2.9×10¹⁴ Wcm⁻² (the cycle-averaged peak intensity equals I₀/2) and a pulse duration of 3.8 fs (FWHM in intensity). In Fig. 2c, the measured CEP-averaged momentum distributions of Ar⁺ and Ar²⁺ are compared with their simulated counterparts for the (e, 2e) and RESI mechanisms. The calculated width of the Ar⁺ spectrum is in good agreement with the measurement. The double hump structure of the Ar²⁺ spectrum is also correctly reproduced by the simulation for the RESI mechanism, where the best agreement was reached with β=20°. We note that small contributions of higher excited Ar⁺ states or the (e, 2e) mechanism would lead to a slightly broader momentum distribution of the Ar²⁺ ions in the calculations. The prediction for the (e, 2e) mechanism alone (red line in Fig. 2c), however, does not reproduce the experimental Ar²⁺ spectrum even if we use β=0°, where the best agreement was reached for this mechanism. Interestingly, the double hump structure, which was previously believed to originate from an (e, 2e) mechanism²⁷ can be reproduced considering the RESI mechanism only. Figure 2b shows the calculated CEP-averaged correlation spectrum for the RESI process, which exhibits the characteristic cross-shaped structure observed in the experiment. The corresponding two-electron momentum distribution for the (e, 2e) process is shown for comparison in the Methods.

Comparison of measured and calculated asymmetries

In earlier work, the (e, 2e) mechanism was invoked to explain the CEP-dependent Ar²⁺ recoil-ion spectra at about twice the intensity^{16, 28}. Our results clearly indicate that, under the conditions of the present experiment, electron impact excitation to the lowest excited Ar⁺ state and subsequent field ionization (RESI) constitutes the main contribution to NSDI of argon. To discuss the CEP dependence of NSDI, it is instructive to introduce the asymmetry parameter A=(N₊−N₋)/(N₊+N₋), where N₊ and N₋ denote the number of ions with positive and negative momentum along the polarization axis, respectively. In Fig. 2d, the measured asymmetries of the Ar⁺ and Ar²⁺ yields as a function of CEP are compared with the results of our calculations (using the parameters given above). The experimental CEP is only determined up to a constant offset (see Methods). We use the theoretical Ar⁺ asymmetry as a reference to set the value of the offset such that the maximum of the measured Ar⁺ asymmetry curve coincides with the maximum of the calculated one. Remarkably, both the amplitude and the shape of the simulated Ar⁺ asymmetry curve exactly match the measurement. Whereas the amplitudes of the calculated and measured Ar²⁺ asymmetry curves are in good agreement, the calculated phase shifts between the Ar⁺ and the Ar²⁺ asymmetry curves (154° for RESI and 177° for (e, 2e)) do not match the measured phase shift of 114°. We would like to emphasize that such asymmetry data obtained simultaneously for single and double ionization is only accessible in the few and near single-cycle regime and was not available before. The measured phase shift represents a useful observable to test the validity of theoretical models. In spite of the small discrepancies between the experimental data and the simulation results, the overall agreement is impressive, considering the simplicity of the theoretical model.

In light of the good agreement between simulation and experiment, the electron dynamics can now be analysed in more detail. In Fig. 3, experimental (Fig. 3a,b) and simulated (Fig. 3c,d) correlation spectra are compared for CEP values that correspond to maximum (Fig. 3a,c) and zero (Fig. 3b,d) Ar²⁺ asymmetry, respectively. Figures 3e and 3f show the calculated ionization rates of the first (green area) and second (red area) electron together with the temporal profile of the laser electric field for these two CEP values. Due to depletion of the excited Ar⁺ population, the second electron is emitted before the maximum of the laser electric field. Our analysis thus reveals that under our experimental conditions, it is the momentum of the first electron that is close to zero while the momentum of the second electron is non-zero and varies as a function of the CEP. Averaged over the CEP, this results in the cross-shaped structures seen in Figs 1a and 2b. This finding is in contrast to the common assumption that the second electron in the RESI mechanism is released at the peak of the electric field and acquires a small momentum distributed around zero⁷.

Multiple recollision events, which occur in the several-to-multi-cycle regime, severely hamper the interpretation of measured correlation spectra. Figure 3 illustrates how the short pulses used in our study allow for a clean experiment by confining NSDI to essentially one single recollision event for an appropriate choice of the CEP yielding the maximum Ar²⁺ asymmetry. For a CEP corresponding to a zero Ar²⁺ asymmetry (Figs 3b and d), precisely two recollision events from consecutive half-cycles contribute to NSDI. Here the second electron, liberated as a result of the first recollision, acquires a higher momentum with positive sign whereas the second recollision results in a lower momentum with negative sign. Even in this case, the change in sign of the slope of the electric field allows for distinguishing the contributions from the two consecutive recollision events.

With the analysis presented in Fig. 3, we can also unambiguously determine whether excitation of the singly charged ion occurs without recollision, that is, via laser excitation alone. Recollision-induced excitation takes place about three quarters of a cycle after the emission of the first electron. In contrast, a laser-excited ion would be efficiently ionized in the half-cycle following ionization of the first electron, leading to a signal on either side of the red diagonal in Fig. 3a, which is not observed. Disentangling laser excitation from recollisional excitation is possible only because the double ionization process has been confined to a single cycle.

Using the infrared field as an inherent clock and neglecting the effect of the Coulomb potential as commonly assumed in the attosecond streaking scenario^{1, 29}, we can retrieve the electron's most likely emission time, which is directly mapped onto its drift momentum. Specifically, Δt=arccos(p₀/p_max)/ω, where p₀ and p_max are the most probable and the maximum momentum of electron 2, respectively. From the experimental data, we find the maximum emission probability of the second electron at 210±40 attoseconds before the peak of the half-cycle following the recollisional excitation. This is in good agreement with our simulations predicting the emission leading the maximum of the field by 230 attoseconds.

In conclusion, using a near-single-cycle laser pulse with appropriate CEP, the double ionization dynamics of argon could be confined to an isolated recollision event, allowing us to investigate the elementary process responsible for NSDI. The measured correlated two-electron momentum spectrum has a cross-shaped structure, which has not been observed in previous experiments, all of them using longer laser pulses. This suggests that the transition from the multi-cycle to the near-single-cycle regime substantially modifies the dynamics of NSDI. Resolving the CEP-dependence of the correlated two-electron spectra has provided new insights into the double ionization dynamics. In particular, we have shown that, owing to depletion of the excited Ar⁺ ions, the second electron is emitted before the maximum of the laser electric field. The mean delay between the zero crossing of the electric field and the emission of the second electron could be determined unambiguously. More generally, the highly differential data obtained in our experiment provide strong constraints to multi-electron theories and may greatly stimulate their development. Whereas our semiclassical model captures the essential features of the data, electron correlations enter the model only indirectly via the ionic excitation and the scattering angle. More sophisticated models accounting for Coulomb and exchange interaction are certainly needed to fully resolve the correlated electron motion and predict the correct value for the phase shift observed between the Ar⁺ and Ar²⁺ asymmetry curves. Extension of our approach to the study of molecular dynamics may give a more detailed insight into the breaking of chemical bonds.

Experimental technique

The experiment was performed at the AS-1 attosecond beamline at the Max Planck Institute of Quantum Optics¹⁹. Linearly polarized laser pulses with a central wavelength of 750 nm, an energy of 400 μJ and a FWHM of the temporal intensity envelope of 4 fs are generated at a repetition rate of 3 kHz in an amplified laser system. Active stabilization of the CEP is difficult to conciliate with the long data-acquisition times (32 h for the present experiment) imposed by coincident particle-detection schemes. As a significant advance over previous efforts to measure the sub-femtosecond dynamics of the NSDI of argon, we recorded the CEP of each laser pulse by sending a fraction of the laser beam (25 μJ) into a single-shot stereographic above-threshold ionization (Stereo-ATI) phase-meter²¹ (see Fig. 4).

A detailed description of the CEP measurement technique using ATI and the Stereo-ATI phase meter can be found in refs 17,21,22,30,31. Briefly, the time-of-flight spectra of electrons emitted along both directions of the laser polarization axis are recorded on either side of the apparatus (left and right detector in Fig. 4). For each laser shot, the CEP is inferred from the asymmetry between the left and the right TOF spectra. The CEP is measured up to a constant offset in the phase meter.

A fraction (11 μJ) of the remaining part of the laser beam is focussed with a spherical mirror (f=25 cm) into a well-localized (1 mm width) and cold supersonic gas jet of neutral argon atoms in the centre of a REMI²⁰. Ions and electrons created in the laser focus (~40 μm diameter) are extracted by a weak electric field towards microchannel plate (MCP) detectors equipped with delay line anodes for position readout at the left and right side of the instrument, respectively. A pair of Helmholtz coils generates a homogenous magnetic field to guide the electrons towards their detector reaching a 4π acceptance angle for all electrons of interest (Fig. 4). The 3D momentum vectors of ions and electrons are reconstructed from their measured flight times and impact positions on the corresponding detectors. The low density of the jet allows for detection of on average less than one ionization event per laser shot, an important constraint to minimize the number of false coincidences, when measuring ions and electrons in coincidence. In the present experiment, 0.25 ions per laser shot were detected on average.

Data processing and false coincidences

To generate the correlated two-electron momentum distributions shown in Fig. 1, we select laser shots where at least one electron and one Ar²⁺ ion are detected. As most of the time only one of the two electrons generated together with Ar²⁺ is detected (because of the single-particle detection efficiency of ~50%), the momentum of the missing second electron is calculated using momentum conservation. Electrons emerging from single ionization of argon introduce a false coincidence background in the electron correlation spectra of less than 15%. The contribution of false Ar²⁺ events (<0.5% of the total Ar²⁺ signal) is negligible.

Laser intensity and CEP offset measurement

The analysis of the Ar⁺ spectra, recorded in the same measurement, provides important information allowing for in situ determination of the peak intensity of the laser pulse in the interaction region, as shown in Fig. 5. Furthermore, the comparison of the measured and calculated Ar⁺ asymmetry curves allows for the determination of the absolute CEP of the laser pulses in the interaction region in the REMI.

Simulation procedure

The yield Y(t) of electrons emitted in a time interval [t, t+dt] is calculated using the instantaneous ADK tunnelling rate w_ADK (refs 32,33) summed over all occupied degenerate states of the outer shell. This rate is further multiplied with the correction factor from (ref. 26) to account for above the barrier ionization:

Here I_p is the ionization potential and α is a numerical factor adjusted such that the rate, w(t), fits the result obtained by solving the Schrödinger equation numerically. The value of parameter α is 9 for Ar and 8 for Ar⁺ (ref. 26).

The ionization yield is calculated as

where N(t_i) is the initial population of the electronic state under consideration. After ionization, at time t₀, the electron enters the continuum with zero momentum at a distance x_t of the parent ion, where x_t is the exit of the tunnel. The electron is propagated classically in the electric field of the laser pulse, neglecting the interaction with the parent ion. The electron may return at a later time t₁, to the position at x₀, with sufficient energy to either excite the parent ion in the case of RESI or to directly ionize the parent ion in the case of the (e, 2e)-mechanism.

For the RESI process, the calculation is performed as follows: if the kinetic energy, E_kin(t₁) of the returning electron is higher than the excitation energy, E_exc=13.5 eV, needed to reach the lowest excited state of Ar⁺, this state, (3s 3p⁶) ²S_1/2, is populated with the probability Y(t₀) and the propagation of the re-scattered electron is continued with an initial kinetic energy E′_kin(t₁)=E_kin(t₁)−E_exc. For the ionization of the second electron from the excited ion, we proceed in the same way as for the ionization of the first electron, using the parameters of the excited state for the ionization rate.

To simulate the (e, 2e) process, a recolliding electron with sufficient energy to overcome the second ionization potential of argon is assumed to produce a second electron with zero-initial momentum. The magnitude of the first electron momentum just after the recollision is determined by energy conservation. As indicated in the main text, the experimental data cannot be described by our model assuming the (e, 2e) process. In addition to the predicted recoil-ion momentum spectrum and asymmetry curve presented in Fig. 2, the two-electron correlation spectrum calculated assuming the (e, 2e) mechanism is shown in Fig. 6.

For both the (e, 2e) and RESI processes, the final drift momentum of the first electron depends on the scattering angle β (the angle between the momentum vectors just before and just after the recollision). The sensitivity of the simulation on the scattering angle is shown in Fig. 7 for the RESI process. We find that the cross-shaped structure is significantly altered if β is varied by±10 degrees. Focal volume averaging is taken into account by assuming a Gaussian intensity profile, neglecting the intensity variation across the argon gas jet of 1 mm diameter.

How to cite this article: Bergues, B. et al. Attosecond tracing of correlated electron-emission in non-sequential double ionization. Nat. Commun. 3:813 doi: 10.1038/ncomms1807 (2012).

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We acknowledge fruitful discussions with M. Lezius and H. Schröder and experimental support by C.-D. Schröter and N. Kurz. We are grateful for support by the Max Planck Society, the Chemical sciences, Geosciences, and Biosciences Division, Office of Basic Energy Sciences, Office of Science, U.S. Department of Energy under DE-FG02-86ER13491 and DE-FG02-00ER15053, the National Science Foundation under CHE-0822646 and the DFG via the Emmy-Noether program, the International Collaboration in Chemistry program and the Cluster of Excellence: Munich Center for Advanced Photonics (MAP). We acknowledge support by the EU via the ITN ATTOFEL and LaserLab Europe.

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