Angular dependence of the Wigner time delay upon tunnel ionization of H 2
Daniel Trabert, Simon Brennecke, Kilian Fehre, Nils Anders, Angelina Geyer, Sven Grundmann, Markus S. Schöffler, Lothar Ph. H. Schmidt, Till Jahnke, Reinhard Dörner, Maksim Kunitski, Sebastian Eckart
Angular dependence of the Wigner time delay upon tunnel ionization of (cid:1) (cid:2)
D. Trabert*, K. Fehre, N. Anders, A. Geyer, S. Grundmann, M. S. Schöffler, L. Ph. H. Schmidt, T. Jahnke, R. Dörner, M. Kunitski, and S. Eckart*
Institut für Kernphysik, Goethe-Universität Frankfurt am Main, Max-von-Laue-Straße 1, 60438 Frankfurt am Main, Germany. *Corresponding authors: e-mail: (D.T.) [email protected]; (S.E.) [email protected] ore than 100 years after its discovery and its explanation in the energy domain , the duration of the photoelectric effect is still heavily studied. The emission time of a photoelectron can be quantified by the Wigner time delay . Experiments addressing this time delay for single-photon ionization became feasible during the last 10 years . A missing piece, which has not been studied, so far, is the Wigner time delay for strong-field ionization of molecules. Here we show experimental data on the Wigner time delay for tunnel ionization of (cid:1) (cid:2) molecules and demonstrate its dependence on the emission direction of the electron with respect to the molecular axis. We find, that the observed changes in the Wigner time delay can be quantitatively explained by elongated/shortened travel paths of the electrons that are due to spatial shifts of the electron’s birth position after tunneling. This introduces an intuitive perspective towards the Wigner time delay in strong-field ionization. Main Text:
How long does it take an electron to leave an atom or molecule in photoionization and which parameters affect the duration of this process? These questions can be addressed with the concept of the Wigner time delay, τ (cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10) , which was originally introduced to describe scattering processes . τ (cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10) is defined as the derivative of the phase of the electron’s wave function ψ with respect to the electron’s energy (cid:12) : τ (cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10) = ℏ (cid:15) (cid:16)(cid:17)(cid:18)(cid:19)(cid:20)(cid:21)(cid:15)(cid:22) (1) In the context of a scattering process, the origin of a phase shift is the potential (by which the electron is scattered) as it modulates the electron’s wavelength upon passage. In an ionization processes, however, the electron resides initially inside the potential of its parent ion and finally escapes from it. In such a “half-scattering” scenario further parameters influence the electron’s final phase shift, for example, details of the interaction process that launches the electron wave and the exact location from which the wave emerges . While the absolute phase of a quantum mechanical wave function is experimentally not accessible, relative phases can be measured via interference. Many experimental techniques that measure changes of the Wigner time delay, Δτ (cid:24)(cid:25)(cid:18)(cid:26)(cid:27)(cid:17) , employ two or more interfering pathways. For single photon ionization one such scheme is RABBITT (reconstruction of attosecond harmonic beating by interference of two-photon transitions) . In RABBITT, there are two pathways that lead to the same final electron energy. On each pathway two photons from two different laser pulses are absorbed, and the time delay between the two pulses is varied (see Ref. 10 for a proposed generalization of RABBITT to the multiphoton regime). The main challenge for corresponding studies with respect to strong-field tunnel ionization, is that many photons are absorbed during the ionization process. This leads to a plethora of possible pathways in the energy domain, that must be considered in order to understand the observed interference . To meet that challenge, Eckart has recently suggested an alternative interferometric approach towards τ (cid:24)(cid:25)(cid:18)(cid:26)(cid:27)(cid:17) in momentum space termed “holographic angular streaking of electrons” (HASE) . This scheme exploits semi-classical trajectories to model the interference occurring in a tunnel ionization process triggered by a co-rotating two color (CoRTC) laser field. According to that model, a change of the Wigner time delay manifests as a macroscopic rotation of a characteristic interference pattern in the lectron momentum distribution. Here we present a first experiment exploiting this perspective in order to retrieve the angular dependence of the Wigner time delay from the measured electron momentum distributions. The tailored laser electric field (cid:12)(cid:28)⃗(cid:19)(cid:30)(cid:21) , that is used in our experiment, is shown in Fig. 1(a). (cid:12)(cid:28)⃗(cid:19)(cid:30)(cid:21) is a CoRTC field that is generated by superimposing two femtosecond laser pulses: one pulse with high intensity and a central wavelength of 390 nm and one low intensity pulse at a central wavelength of 780 nm. Both single-color fields are circularly polarized and have the same helicity. The electric field is strong enough to bend the binding potential of the atom, giving rise to a rotating barrier through which an electron can tunnel from its bound state. For an electron that tunnels at time (cid:30) (cid:31) with an initial momentum ⃗ (cid:25)(cid:26)(cid:25)! , the final electron momentum ⃗ (cid:27)"(cid:27) , which is gained by the electron until the end of the laser pulse, is given by ⃗ (cid:27)"(cid:27) = ⃗ (cid:25)(cid:26)(cid:25)! − %⃗(cid:19)(cid:30) (cid:31) (cid:21) (neglecting the Coulomb interaction after tunneling). %⃗ is the laser’s vector potential and ⃗ (cid:25)(cid:26)(cid:25)! is assumed to be perpendicular to the electric field at the instant of tunneling (i.e. p(cid:28)⃗ (cid:6)(cid:8)(cid:6)’ ⋅ E(cid:28)(cid:28)⃗(cid:19)t (cid:31) (cid:21) = 0 for every trajectory) . The two half-cycles (labeled as “c1” and “c2” in Fig. 1(a)) differ in their electric field, (cid:12)(cid:28)⃗(cid:19)(cid:30)(cid:21) , as well as in their negative vector potential, – %⃗(cid:19)(cid:30)(cid:21) . Consequently, there are two different combinations of %⃗(cid:19)(cid:30) (cid:31) (cid:21) and ⃗ (cid:25)(cid:26)(cid:25)! within one full cycle of the laser field that lead to the same final electron momentum. These give rise to interference and allow for the retrieval of changes in the Wigner time delay (see Methods or Ref. 12 for further details). Noteworthy, the changes of the Wigner time delay can be measured with attosecond precision, although laser pulses with durations of several dozens of femtoseconds are used. In order to measure Δτ (cid:24)(cid:25)(cid:18)(cid:26)(cid:27)(cid:17) as a function of the emission direction of the electron with respect to the molecular axis, the electron momentum vector as well as the spatial orientation of the molecule have to be measured for each ionization event. To that end, we focus the laser pulses onto a cold molecular beam of (cid:1) (cid:2) and detect the electron and the proton momentum for ionization and subsequent dissociation into (cid:1) (cid:31) and (cid:1) - using a Cold Target Recoil Ion Momentum Spectroscopy (COLTRIMS) reaction microscope (see Methods). The dissociation is much faster than the rotation of the intermediately formed (cid:1) (cid:2)- molecule and thus the directions of the fragments’ momenta coincide with the molecular axis at the instant of ionization . Fig. 1(b) shows the electron momentum distribution in the laser polarization plane which exhibits an alternating half-ring pattern. This pattern in momentum space can be divided into half-rings that belong to above-threshold ionization (ATI) peaks in the electron energy spectrum and half-rings that are related to sideband (SB) peaks . This definition is chosen such that the sidebands vanish if the pulse with a central wavelength of 780 nm is switched off (the remaining ATI half-rings turn into full rings in this case). As indicated in Fig. 1(b), for each half-ring, there exists a most probable electron emission angle α . In a next step, the changes of α are investigated as a function of / . Here / is the electron’s emission angle with respect to the molecular axis, as illustrated in Fig. 1(a). Fig. 1(c) shows the difference
Δα(cid:19)/ ) = α(cid:19)/(cid:21) − α , where α is the most probable electron emission angle integrated over all values of β (see Methods and Fig. S1). For each ATI/SB half-ring α and Δα(cid:19)/ ) are determined independently. It is evident that Δα(cid:19)/ ) varies on the order of a few degrees and that the overall shape of the curves is similar for ATI peaks and SB peaks. A decrease in the modulation amplitude for higher electron energies is bserved. The curves showing Δα as a function of β (depicted in Fig. 1(c)) contain all information that is needed to calculate the changes of the Wigner time delay, Δ5 (cid:24)(cid:25)(cid:18)(cid:26)(cid:27)(cid:17) , as a function of the electron’s emission direction in the molecular frame and the electron’s energy (see Methods and Fig. S2). The experimentally obtained values for Δ5 (cid:24)(cid:25)(cid:18)(cid:26)(cid:27)(cid:17) = (cid:24)(cid:25)(cid:18)(cid:26)(cid:27)(cid:17) (cid:19)/(cid:21) − 5 (cid:24)(cid:25)(cid:18)(cid:26)(cid:27)(cid:17),0(cid:27)(cid:16)(cid:26) are presented in Fig. 2(a) and vary from -50 attoseconds (as) to + 50 as. (cid:24)(cid:25)(cid:18)(cid:26)(cid:27)(cid:17),0(cid:27)(cid:16)(cid:26) is the mean Wigner delay integrated over all electron emission directions. For Fig. 2(a) the molecular axis is aligned as indicated and used as a reference which allows for the visualization of Δ5 (cid:24)(cid:25)(cid:18)(cid:26)(cid:27)(cid:17) in the molecular frame. If the emission angle of the electron is parallel or perpendicular to the molecular axis ( β = 0° or β = ±90°(cid:21) then Δ5 (cid:24)(cid:25)(cid:18)(cid:26)(cid:27)(cid:17) is close to zero. If the electron is emitted at an angle of β = −45° ( β = +45° ) then negative (positive) values for Δ5 (cid:24)(cid:25)(cid:18)(cid:26)(cid:27)(cid:17) are measured. The magnitude of Δ5 (cid:24)(cid:25)(cid:18)(cid:26)(cid:27)(cid:17) decreases for increasing electron energy. Fig. 2(b) shows the theoretically calculated values for Δ5 (cid:24)(cid:25)(cid:18)(cid:26)(cid:27)(cid:17) using a theoretical model that is completely independent of the experimental data and has no free parameters. The theoretical results are in good agreement with the experiment. Deviations are seen for low energies, where the calculation yields smaller magnitudes for Δ5 (cid:24)(cid:25)(cid:18)(cid:26)(cid:27)(cid:17) compared to the experiment (also see Fig. S3). In the following we present an intuitive model, which quantitatively reproduces the experimental results on Δ5 (cid:24)(cid:25)(cid:18)(cid:26)(cid:27)(cid:17) and reveals its microscopic origin. For simplicity we employ the LCAO (linear combination of atomic orbitals) approach to obtain the single electron wave function of each electron of (cid:1) (cid:2) as the sum of two orbitals, that are separated by the hydrogen internuclear distance of 0.74Å . The resulting electron density in the plane of polarization |ψ(cid:19)x, y(cid:21)| (cid:2) is shown in Fig. 3(a). The tunneling direction rotates in the polarization plane as the laser electric field evolves with time. Tunneling projects the bound electronic wave function to the direction s E that is perpendicular to the tunneling direction . This is illustrated in Fig. 3(a), showing different orientations of the tunnel (orange arrows) relative to the molecular axis. In more detail, the electron density of (cid:1) (cid:2) that is evaluated along s E at the tunneling distance of 15.75 a.u. is shown in Fig. 3(b) for various field directions relative to the molecular axis . Due to the shape of the bound electronic wave function of (cid:1) (cid:2) , the peak position of these one-dimensional slices through the electron density shifts as a function of β . This position offset is referred to as Δs E . Neglecting Coulomb interaction after tunneling, an offset in position space of Δs E just shifts the whole trajectory in position space . Accordingly, the Wigner time delay of the electron is directly connected to Δs E . The electron’s final momentum, after the end of the laser pulse, is perpendicular to the laser electric field at the instant of tunneling. Thus, the final electron momentum ⃗ (cid:27)"(cid:27) is parallel or antiparallel to s E . Eventually, this allows one to map any offsets in position space Δ@ ⟂ directly to a change in the Wigner time delay as illustrated in Fig. 3(c): Δ5 (cid:24)(cid:25)(cid:18)(cid:26)(cid:27)(cid:17) = GHI J ∙ 0 L |M⃗ LNLO | (2) Equation 2 links the obtained attosecond time delays to microscopic position offsets of the initial electronic wave packet upon tunneling. The amplitude of Δ@ ⟂ is up to 0.13 Å (for β = ±45 °, see Fig. 3(b)). The vanishing time delays for β = 0° and β = ±90° can be explained by vanishing position offsets according to the symmetry of (cid:1) (cid:2) . The results of this simple model are in very good greement with our experimental data as shown in Fig. 2. The absolute magnitude of Δ5 (cid:24)(cid:25)(cid:18)(cid:26)(cid:27)(cid:17) (note that the color scale in Fig. 2(a) and 2(b) is identical), the change of sign between the quadrants and the decrease of time delay with electron energy are faithfully reproduced. Most importantly, the model illuminates the physics behind the Wigner time delay in strong-field ionization, which is the spatial displacement of the tunneling wave packet that gives the electron a head start, or a longer way to travel. The obtained attosecond time delays are explained as microscopic position offsets of the initial electronic wave function upon tunneling, which are a fingerprint of the spatial dimensions of hydrogen. The simple orbital shape of the hydrogen molecule allows for an intuitive interpretation of the experimental results. The quantitative agreement of experiment and theory is an important benchmark, showing that spatial information regarding the molecular orbital is accessible in strong-field ionization. Our findings pave the way towards a new class of experiments that can measure sub-Ångstrom position offsets and related changes of the Wigner time delay for electrons that tunnel from atoms and molecules. ETHODS
Laser Setup and gas target preparation.
The CoRTC pulses are generated in an interferometric setup based on a 200-µm / -barium borate crystal to double the frequency of laser pulses with a central wavelength of 780 nm (KMLabs Dragon, 40-fs FWHM, 8 kHz). The optical setup is the same as in Refs. 28,29. The light is focused by a spherical mirror (f=80 mm) onto a cold supersonic jet of (cid:1) (cid:2) . Intensity calibration is done as in Ref. 30 and yields intensities of P QR(cid:31)(cid:26)0 = 9.4 ⋅10 TQ U/WX (cid:2) , P YZ(cid:31)(cid:26)0 = 5.5 ⋅ 10 TT U/WX (cid:2) (corresponding to peak electric fields of (cid:12)
QR(cid:31)(cid:26)0 = (cid:12) YZ(cid:31)(cid:26)0 = Δα or Δ5 (cid:24)(cid:25)(cid:18)(cid:26)(cid:27)(cid:17) . The supersonic gas jet was created by expanding hydrogen gas at a pressure of 2.5 bar through a 30 µ m diameter nozzle into vacuum. Particle detection and analysis.
We use a COLTRIMS reaction microscope with an electron acceleration length of 390 mm and an ion acceleration length of 66 mm. An electric field of 10.9 V/cm and a magnetic field of 8.1 Gauss are applied to guide the charged particles to the respective detectors. The detectors are each comprised of a double-stack of micro-channel plates (diameter electron detector: 120 mm; ion detector: 80 mm) followed by position- and time-sensitive hexagonal delay-line anodes . Position and time-of-flight information is used to calculate the three-dimensional momentum vectors of all charged particles in coincidence. Only events corresponding to the main dissociation channel (via the [ (cid:18)(cid:27)(cid:17)(cid:16)\(cid:27) state) have been selected by gating on a kinetic energy release between 0.8 and 2.2 eV . The measured photoelectron momentum distribution in Fig. 1(b) can be considered as the product of an interference pattern and an intensity envelope. The envelope is defined by the tunneling probability which is a function of initial momentum and the time-dependent laser electric field . As shown in Fig. S1(b), the envelope can be extracted from the full electron momentum spectrum presented in Fig. S1(a) by filtering higher Fourier components along the radial direction while maintaining the integral for each angle (see Ref. 34). Division of the full distribution by the extracted envelope leads to the normalized spectrum shown in Fig. S1(c). This procedure is conducted independently for every value of β . For each peak in the radial direction in Fig. S1(c), the most probable electron emission angle α was determined from the one-fold symmetric angular distribution (as described in Ref. 12). The values for α(cid:19)/(cid:21) are depicted in Fig. S1(d), (e), (f). For each energy peak a reference value ] (mean of all values of α for this energy peak) has been calculated (indicated in Fig. S1(e), (f) by horizontal lines). These reference values have been subtracted for each energy peak independently in order to obtain the angles Δ](cid:19)/(cid:21) = ](cid:19)/(cid:21) − ] that are depicted in Fig. 1(c). The value of / is between -90° and +90° which is assured in the analysis by taking the two-fold symmetry of (cid:1) (cid:2) into account and subtracting or adding 180° to / , if necessary (see Fig. S1 for the unsymmetrized data). The situation depicted in Fig. 1(a) corresponds to a positive value of β . Error estimation has been done throughout this work by dividing the entire data set into three subsets which were analyzed eparately. The error was calculated as the standard deviations from the three independent results. Thus, error bars show statistical errors only. Extraction of changes of the Wigner time delay from the experimentally accessible quantity ^_ . The changes of the Wigner time delay (Fig. 2(a)) are calculated from the data that is shown in Fig. 1(c). The theoretical framework that is used for this purpose is described in detail in Ref. 12 and summarized in the following. In Ref. 12 a semi-classical, trajectory-based model is introduced that reproduces the alternating half-ring (AHR) pattern for CoRTC fields. Neglecting Coulomb interaction after tunneling, the final momentum ⃗ (cid:27)"(cid:27) of an electron that tunnels at time (cid:30) (cid:31) can be expressed by: ⃗ (cid:27)"(cid:27) =– %⃗(cid:19)(cid:30) (cid:31) (cid:21) + ⃗ (cid:25)(cid:26)(cid:25)! (3) Where %⃗(cid:19)(cid:30) (cid:31) (cid:21) is the vector potential at the time (cid:30) (cid:31) at which the electron tunnels. Here, we use the laser electric field and the vector potential that is shown in Fig. 1(a). The initial momentum ⃗ (cid:25)(cid:26)(cid:25)! is perpendicular to the laser electric field at the instant of tunneling ( p(cid:28)⃗ (cid:6)(cid:8)(cid:6)’ ⋅ E(cid:28)(cid:28)⃗(cid:19)t (cid:31) (cid:21) = 0 ). Since the momenta in the theoretical model are restricted to the polarization plane, the value for ⃗ (cid:25)(cid:26)(cid:25)! can be unambiguously expressed by the scalar value of (cid:31) : (cid:31) = ‘ | ⃗ (cid:25)(cid:26)(cid:25)! |, ab | ⃗ (cid:27)"(cid:27) | > |%⃗(cid:19)(cid:30) (cid:31) (cid:21)|−| ⃗ (cid:25)(cid:26)(cid:25)! |, ab | ⃗ (cid:27)"(cid:27) | < |%⃗(cid:19)(cid:30) (cid:31) (cid:21)| (4) Within one optical cycle of the pulse at a central wavelength of 780 nm, there exist two possible combinations of (cid:31) and (cid:30) (cid:31) , that lead to the same ⃗ (cid:27)"(cid:27) which we find numerically. Those two combinations are T , (cid:30) T and (cid:2) , (cid:30) (cid:2) . To be able to model not only sub-cycle interference but also inter-cycle interference, the release times in a subsequent laser cycle are also considered. They are (cid:30) Q = (cid:30) T + e YZ(cid:31) and (cid:30) f = (cid:30) (cid:2) + e YZ(cid:31) with Q = T and f = (cid:2) . Here, e YZ(cid:31) is the duration of one optical cycle of light at a wavelength of 780 nm. Thus, one set of initial conditions for a given final electron momentum ⃗ (cid:27)"(cid:27) is fully defined by the time (cid:30) (cid:26) at which the electron tunnels and the initial momentum (cid:26) (as defined in Eq. 4) where g is the trajectory number. For every final electron momentum ⃗ (cid:27)"(cid:27) four trajectories are calculated ( g ∈ {1,2,3,4} ). The phase accumulated on the corresponding semi-classical trajectory is given by : m (cid:26) (cid:19) ⃗ (cid:27)"(cid:27) (cid:21) = Tħ oP M (cid:30) (cid:26) − p M q,rs (cid:19)!(cid:21)-M t,rs (cid:19)!(cid:21)(cid:2)0 L ! u ! v w(cid:30)x + m yzz (cid:19) (cid:26) (cid:21) (5) Here, ℏ is the reduced Planck constant, X (cid:27) is the electron’s mass, P M is the ionization potential, (cid:30) z is the final time at which all phases are evaluated. (cid:30) z is set to (cid:30) z = (cid:30) f without loss of generality (see Ref. 12). {,! (cid:19)(cid:30)(cid:21) and |,! (cid:19)(cid:30)(cid:21) are the time-dependent electron momentum components that are equal to the initial momentum ⃗ (cid:25)(cid:26)(cid:25)! for (cid:30) = (cid:30) (cid:26) for each trajectory. m yzz (cid:19) (cid:31) (cid:21) is an offset phase that depends on the initial momentum (cid:31) and which is external to the model. Using this procedure one can calculate the expected intensity modulation }(cid:19) ⃗ (cid:27)"(cid:27) (cid:21) that is due to sub-cycle and inter-cycle interference. }(cid:19) ⃗ (cid:27)"(cid:27) (cid:21) is defined as the absolute square of the semi-classically modeled wave function : }(cid:19) ⃗ (cid:27)"(cid:27) (cid:21) = ~ ∑ (cid:128)(cid:129) (cid:130)a m (cid:26) (cid:19) ⃗ (cid:27)"(cid:27) (cid:21)(cid:131) f(cid:26)(cid:132)T ~ (cid:2) (6) he calculated distribution }(cid:19) ⃗ (cid:27)"(cid:27) (cid:21) shows the alternating half-ring pattern. Analysis of the angle Δ] – in analogy to the experimental data – for each energy peak shows that the values for Δ] depend on the phase gradient m (cid:133)yzz (cid:19) (cid:31) (cid:21) = \(cid:134) (cid:135)uu (cid:19)M (cid:136) (cid:21)\M (cid:136) . Analyzing how an assumed value of m (cid:133)yzz leads to changes of ] allows one to create a look-up-table that provides Δm (cid:133)yzz for a measured combination of electron energy, (cid:12) (cid:27)"(cid:27) = |M⃗ (cid:137)(cid:138)(cid:137)(cid:139) | s (cid:2)0 L , and Δα for the laser electric field that is used in the experiment. The result is shown in Fig. S2(b). For a given electron energy and small values of ∆] there is an almost linear relation between Δα and (cid:141)m′ yzz . Since by the Fourier transform a linear phase gradient in momentum space is equivalent to a shift of amplitudes in position space, the values of (cid:141)m′ yzz can be used to calculate the position offset, Δ@ E , that is parallel to the final electron momentum, ⃗ (cid:9)(cid:143)(cid:9)(cid:144) . Using the intuitive explanation from Fig. 3, this allows one to calculate the change of the Wigner time delay directly from (cid:141)m′ yzz (see Eq. 11 from Ref. 12): Δ5 (cid:24)(cid:25)(cid:18)(cid:26)(cid:27)(cid:17) = ℏ (cid:145) (cid:137) |M⃗ (cid:137)(cid:138)(cid:137)(cid:139) | (cid:141)m (cid:133)yzz (7) For the experiment, this allows to calculate ∆5 (cid:24)(cid:25)(cid:18)(cid:26)(cid:27)(cid:17) for all measured combinations of (cid:12) (cid:27)"(cid:27) and ∆] . The result is shown in Fig. 2(a). We note that our experiment gives access to the phase gradient m (cid:133)yzz (cid:19) (cid:31) (cid:21) = \(cid:134) (cid:135)uu (cid:19)M (cid:136) (cid:21)\M (cid:136) using a CoRTC field. As described above, in CoRTC fields there are two values of (cid:31) for each final electron momentum. The values for Δ5 (cid:24)(cid:25)(cid:18)(cid:26)(cid:27)(cid:17) as a function of energy are calculated using the approximation that the final electron energy (cid:12) (cid:27)"(cid:27) is unambiguously linked to the initial momentum, (cid:31) , by (cid:12) (cid:27)"(cid:27) = |M (cid:136) -(cid:146) (cid:147)(cid:148)(cid:136) | s (cid:2)0 L , where % QR(cid:31) is the absolute value of the vector potential of the laser pulse with a central wavelength of 390 nm. This approximation is exact for a vanishing intensity of the light pulse at a central wavelength of 780 nm. Within our model, the measured rotational offsets in final electron momentum space (measured as Δα for every energy peak and every β ) can be translated to: (i) changes of the phase gradient ( Δm (cid:133)yzz as a function of (cid:12) (cid:27)"(cid:27) and β ), (ii) changes of the Wigner time delay ( Δ5 (cid:24)(cid:25)(cid:18)(cid:26)(cid:27)(cid:17) as a function of (cid:12) (cid:27)"(cid:27) and β ) and (iii) position offsets that are parallel to the final electron momentum ( Δ@ E as a function of β if m yzz is linear in (cid:31) ). All three quantities can be viewed as different ways to model the same physical reality. Modeling of the spatial displacements perpendicular to the tunnel exit.
We use the LCAO single electron wave function for an internuclear distance R=0.74 Å. We calculate the tunnel exit to be at a radius of (cid:149) (cid:27) = (cid:150) (cid:151) (cid:152) (cid:153)L(cid:154)v = 15.75 a.u. from the origin of the coordinate system, for the peak electric field in the experiment and the ionization potential of (cid:1) (cid:2) ( P M =15.43 eV). Here, (cid:12) =0.036 a.u. is the average of the absolute value of the laser electric field for one cycle of the CoRTC field that is used in our experiment. The coherent superposition of the two 1s orbitals is represented by the wave function in position space: ψ(cid:19)x, y(cid:21)~exp(cid:130)−(cid:158)(cid:19)(cid:129) − (cid:159)/2(cid:21) (cid:2) + (cid:160) (cid:2) (cid:131) + exp(cid:130)−(cid:158)(cid:19)(cid:129) + (cid:159)/2(cid:21) (cid:2) + (cid:160) (cid:2) (cid:131) (8) his expression is evaluated at the tunnel exit’s position along a straight line @ E that is a tangent to the circle with the radius (cid:149) (cid:27) and is perpendicular to the laser electric field at the instant of tunneling (see Fig. 3(a)). This results in a one-dimensional subset in position space ψ E (cid:19)@ E (cid:21) that models the wave function in position space perpendicular to the direction of the laser electric field at the instant at tunneling: ψ E (cid:19)@ E (cid:21)~ψ(cid:19)x(cid:19)@ E (cid:21), y(cid:19)@ E (cid:21)(cid:21) (9) Here, x(cid:19)@ E (cid:21) = (cid:149) (cid:31) (cid:19)@ E (cid:21) ⋅ cos (cid:19)¢(cid:19)@ E (cid:21)(cid:21) and y(cid:19)@ E (cid:21) = (cid:149) (cid:31) (cid:19)@ E (cid:21) ⋅ sin (cid:19)¢(cid:19)@ E (cid:21)(cid:21) with (cid:149) (cid:31) (cid:19)@ E (cid:21) = (cid:158)(cid:149) (cid:27)(cid:2) + @ E(cid:2) and ¢(cid:19)@ E (cid:21) = ⁄ − tan GT o ¥ J (cid:17) L x, ϴ denotes the relative angle of the laser electric field at the instant of tunneling with respect to the molecular axis. In the following, we assume that the final electron emission angle is perpendicular to the laser electric field at the instant of tunneling ( ⁄ = / + 90° ). Thus, we conclude that ψ E (cid:19)@ E (cid:21) can be calculated using Eq. 9 and: (cid:129)(cid:19)@ E (cid:21) = −(cid:158)(cid:149) (cid:27)(cid:2) + @ E(cid:2) ⋅ sin o/ − tan GT o ¥ J (cid:17) L xx y(cid:19)@ E (cid:21) = (cid:158)(cid:149) (cid:27)(cid:2) + @ E(cid:2) ⋅ cos o/ − tan GT o ¥ J (cid:17) L xx (10) Exemplarily, Fig. 3(b) shows | ψ E (cid:19)@ E (cid:21)| (cid:2) for four different values of β . Fourier transformation of ψ E (cid:19)@ E (cid:21) results in the corresponding complex valued, initial momentum distribution perpendicular to the tunneling direction . Also note that the tunnel exit radius in Fig. 3(a) is chosen to be 1.6 a.u. for illustrational purposes only. In order to calculate the results in Fig. 3(b) the realistic value for the tunnel exit radius of (cid:149) (cid:27) = Calculation of the changes of the Wigner time delay as a function of ƒ and the electron energy using the theoretical model. In order to calculate the Wigner time delay independent of the experiment (results are shown in Fig. 2(b) and Fig. S3(b)) we take the complex valued initial momentum distribution upon tunnel ionization of (cid:1) (cid:2) (Eq. 17 from Ref. 23) and evaluate the phase gradient at the most probable initial momentum ( p (cid:31) = 0 a.u., Ref. 35). This phase gradient is termed m (cid:133)yzz (in analogy to above). The entire procedure is carried out for each β separately using that Eq. 17 from Ref. 23 depends on the molecular orientation. We make the approximation that the electric field at the instant of tunneling is perpendicular to the final electron momentum. Then, for a given β the value of m (cid:133)yzz can be directly translated into a change of (cid:24)(cid:25)(cid:18)(cid:26)(cid:27)(cid:17) using Δ5 (cid:24)(cid:25)(cid:18)(cid:26)(cid:27)(cid:17) = ℏ (cid:145) (cid:137) |M⃗ (cid:137)(cid:138)(cid:137)(cid:139) | m (cid:133)yzz (see Eq. 11 from Ref. 12). Here we know that m′ yzz = (cid:141)m′ yzz which was not assumed above to be as general as possible. We use that | ⃗ (cid:9)(cid:143)(cid:9)(cid:144) | = (cid:158)2 X (cid:27) (cid:12) (cid:27)"(cid:27) where (cid:12) (cid:27)"(cid:27) is the final electron energy. The assumption that m (cid:133)yzz does not depend on (cid:12) (cid:27)"(cid:27) is a good approximation if m yzz is linear in (cid:31) . It should be noted that a linear phase gradient in momentum space corresponds to a position offset. For typical tunneling geometries the phase of the initial momentum distribution is almost linear. Thus, the value of m (cid:133)yzz is very similar for all initial momentum components that have non-vanishing amplitudes (see Ref. 23 for examples). EFERENCES
1. Einstein, A. Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt.
Ann. Phys. , 132–148 (1905). 2. Wigner, E. P. Lower limit for the energy derivative of the scattering phase shift.
Phys. Rev. , 145–147 (1955). 3. Schultze, M. et al. Delay in photoemission. Science , 1658–1662 (2010). 4. Klünder, K. et al. Probing single-photon ionization on the attosecond time scale.
Phys. Rev. Lett. , 143002 (2011). 5. Huppert, M., Jordan, I., Baykusheva, D., von Conta, A. & Wörner, H. J. Attosecond delays in molecular photoionization.
Phys. Rev. Lett. , 93001 (2016). 6. Vos, J. et al. Orientation-dependent stereo wigner time delay and electron localization in a small molecule.
Science , 1326 (2018). 7. Grundmann, S. et al. Zeptosecond birth time delay in molecular photoionization.
Submitt.
8. Muller, H. G. Reconstruction of attosecond harmonic beating by interference of two-photon transitions.
Appl. Phys. B , s17 (2002). 9. Paul, P. M. et al. Observation of a train of attosecond pulses from high harmonic generation. Science , 1689–1692 (2001). 10. Zipp, L. J., Natan, A. & Bucksbaum, P. H. Probing electron delays in above-threshold ionization.
Optica , 361 (2014). 11. Kerbstadt, S. et al. Control of photoelectron momentum distributions by bichromatic polarization-shaped laser fields. New J. Phys. , 103017 (2017). 12. Eckart, S. Holographic angular streaking of electrons and the wigner-time delay. Preprint at https://arxiv.org/abs/2003.07249 (2020). 13. Shvetsov-Shilovski, N. I. et al. Semiclassical two-step model for strong-field ionization. Phys. Rev. A , 013415 (2016). 14. Feng, Y. et al. Semiclassical analysis of photoelectron interference in a synthesized two-color laser pulse. Phys. Rev. A , 63411 (2019). 15. Eckart, S. et al. Subcycle interference upon tunnel ionization by counter-rotating two-color fields.
Phys. Rev. A , 041402 (2018). 16. Eckart, S. et al. Sidebands in electron energy upon multi-photon ionization depend on the relative helicity in ( ω , 2 ω ) fields. Preprint at https://arxiv.org/abs/2005.04148 (2020). 17. Shvetsov-Shilovski, N. I. & Lein, M. Semiclassical two-step model with quantum input: Quantum-classical approach to strong-field ionization. Phys. Rev. A , 053411 (2019). 18. Ivanov, M. Y., Spanner, M. & Smirnova, O. Anatomy of strong field ionization.
J. Mod. Opt. , 165–184 (2005). 19. Ullrich, J. et al. Recoil-ion and electron momentum spectroscopy: reaction-microscopes. Reports Prog. Phys. , 1463–1545 (2003). 20. Wood, R. M., Zheng, Q., Edwards, A. K. & Mangan, M. A. Limitations of the axial recoil approximation in measurements of molecular dissociation. Rev. Sci. Instrum. , 1382 1997). 21. Freeman, R. R. et al. Above-threshold ionization with subpicosecond laser pulses. Phys. Rev. Lett. , 1092 (1987). 22. Han, M., Ge, P., Shao, Y., Gong, Q. & Liu, Y. Attoclock photoelectron interferometry with two-color corotating circular fields to probe the phase and the amplitude of emitting wave packets. Phys. Rev. Lett. , 73202 (2018). 23. Liu, M. & Liu, Y. Application of the partial-Fourier-transform approach for tunnel ionization of molecules.
Phys. Rev. A , 43426 (2016). 24. Murray, R., Liu, W.-K. & Ivanov, M. Y. Partial fourier-transform approach to tunnel ionization: Atomic systems. Phys. Rev. A , 23413 (2010). 25. Liu, M.-M. & Liu, Y. Semiclassical models for strong-field tunneling ionization of molecules. J. Phys. B At. Mol. Opt. Phys. , 105602 (2017). 26. Eckle, P. et al. Attosecond ionization and tunneling delay time measurements in helium. Science , 1525–1529 (2008). 27. Bray, A. W., Eckart, S. & Kheifets, A. S. Keldysh-rutherford model for the attoclock.
Phys. Rev. Lett. , 123201 (2018). 28. Eckart, S. et al. Nonsequential double ionization by counterrotating circularly polarized two-color laser fields.
Phys. Rev. Lett. , 133202. 29. Eckart, S. et al. Ultrafast preparation and detection of ring currents in single atoms.
Nat. Phys. , 701-704 (2018). 30. Eckart, S. et al. Direct experimental access to the nonadiabatic initial momentum offset upon tunnel ionization. Phys. Rev. Lett. , 163202 (2018). 31. Jagutzki, O. et al. Multiple hit readout of a microchannel plate detector with a three-layer delay-line anode.
IEEE Trans. Nucl. Sci. , 2477–2483 (2002). 32. Wu, J. et al. Electron-nuclear energy sharing in above-threshold multiphoton dissociative ionization of H (cid:2) . Phys. Rev. Lett. , 23002 (2013). 33. Delone, N. B. & Krainov, V. P. Energy and angular electron spectra for the tunnel ionization of atoms by strong low-frequency radiation.
J. Opt. Soc. Am. B , 1207 (1991). 34. Ge, P., Han, M., Deng, Y., Gong, Q. & Liu, Y. Universal description of the attoclock with two-color corotating circular fields. Phys. Rev. Lett. , 13201 (2019). 35. Arissian, L. et al. Direct test of laser tunneling with electron momentum imaging.
Phys. Rev. Lett. , 133002 (2010).
ACKNOWLEDGEMENTS
This work was funded by the German Research Foundation (DFG) through priority program SPP 1840 QUTIF.
Figure 1 | Overview of the experimental quantities that give access to changes of the Wigner time delay. a , Electric field (cid:12)(cid:28)⃗(cid:19)(cid:30)(cid:21) and negative vector potential – %⃗(cid:19)(cid:30)(cid:21) for one cycle of the co-rotating two color (CoRTC) field comprised of a high-intensity pulse (central wavelength of 390 nm) and a low-intensity pulse (central wavelength of 780 nm). The helicities of the two pulses are indicated with arrows. Using that the ion’s momentum vector ⃗ ¤ ' always points along the molecular axis allows for the measurement of the orientation of the molecular axis. ⃗ ¤ ' is measured in coincidence with the electron momentum vector ⃗ (cid:27)"(cid:27) . As illustrated, β is the electron emission angle relative to the molecular axis in the polarization plane. b , Measured electron momentum distribution in the polarization plane of the laser’s electric field: ATI and SB peaks are half-rings which are spaced by the energy of a photon of the weaker laser pulse at 780 nm (1.6 eV). The most probable electron angle, ](cid:19)/(cid:21) , is indicated for the first ATI peak. c, Changes in the most probable electron angle, (cid:141)] , as a function of / are presented for each ATI/SB peak separately. ] is determined for every energy peak independently as the mean of ](cid:19)/(cid:21) over all / (see Fig. S1). The values for the vector potential and the electron momentum are in atomic units (a.u.). The error bars show the standard deviation of the statistical errors. Figure 2 | Changes of the Wigner time delay in the molecular frame. a, Experimentally retrieved changes of the Wigner time delay, Δ5 (cid:24)(cid:25)(cid:18)(cid:26)(cid:27)(cid:17) , as a function of the electron energy and / (relative angle between electron momentum vector ⃗ (cid:27)"(cid:27) and the molecular axis of (cid:1) (cid:2) in the polarization plane). b, Theoretically modeled changes of Δ5 (cid:24)(cid:25)(cid:18)(cid:26)(cid:27)(cid:17) shown in analogy to the experimental data. The molecular axis is shown schematically. Gray arrows indicate light’s helicity. Figure 3 | Origin of the observed changes of the Wigner time delay for strong-field ionization. a , Two-dimensional distribution of the molecular orbital of (cid:1) (cid:2) in position space. Different tunneling directions (orange arrows) and the corresponding perpendicular directions (black lines labeled with @ ⟂ ) are indicated. The tunnel exit position that is used in a is chosen unrealistically short (about a factor of 10) for illustrational purposes (see Methods). b, One-dimensional cuts of the square of the wave function in position space along the four exemplary lines labeled with @ ⟂ from a . The position offset Δ@ ⟂ is zero for tunneling parallel or perpendicularly to the molecular axis ( / = 0° and / = ±90° ) due to the symmetry of (cid:1) (cid:2) . Maximally positive [negative] values of Δ@ ⟂ appear for / = −45° [ / = 45° ]. c, Position offsets that are anti-parallel [parallel] to the final electron momentum lead to positive [negative] Δ5 (cid:24)(cid:25)(cid:18)(cid:26)(cid:27)(cid:17) . In good approximation, ⃗ (cid:27)"(cid:27) is perpendicular to the tunneling direction. Note, that the width of the illustrated wave packets in c is not to scale (see b for a realistic width of the distribution). @ M(cid:17)yM sketches the distance the electron wave packet travels in position space for / = 0° . Supplementary Figure 1 | Illustration of the data analysis that is performed to extract ^_ from the measured electron momentum distributions. a, Measured electron momentum distribution in the polarization plane (only electron momenta corresponding to a kinetic energy «(cid:25)(cid:26) < 11 (cid:128)“ are considered). b, The envelope of the electron momentum distribution is retrieved from the full momentum distribution in a by filtering higher Fourier components along the radial direction. The obtained envelope is shown in b . c, Normalized electron momentum distribution obtained by elementwise division of a by b . This normalization procedure is done for all values of / , that are shown in d , separately. d , The most probable electron angles, α , are plotted as a function of β (for each energy peak independently). The difference between ATI and SB peaks is approximately 180°. e, f, show the same data as d but using a different range on the vertical axis. ] is determined for every energy peak independently as the mean of ](cid:19)/(cid:21) over all / . The values ] are depicted as horizontal lines in e and f . The changes of the most probable electron emission angle as a function of β are defined as Δ](cid:19)/(cid:21) = ](cid:19)/(cid:21) − ] . The error bars show the standard deviation of the statistical errors.
Supplementary Figure 2 | Linking the experimentally accessible quantity ^_ to changes of the Wigner time delay and to changes of the phase gradient of the initial momentum distribution. a , The procedure that leads to Fig. 9(a) from Ref. 12 is used. Here, we use the time-dependent electric field from our experiment (see Fig. 1(a)) and calculate for each desired energy peak and every desired Δ] the change of the Wigner time delay. The result is shown in a and can be read like a look-up-table. This allows for the generation of Fig. 2(a) from the data shown in Fig. 1(c). b, The change of the phase gradient of the initial momentum distribution (cid:141)m (cid:133)yzz is calculated in full analogy but not used and only shown for the sake of completeness (see Methods and the discussion of Fig. 6(e) from Ref. 12 for details regarding (cid:141)m (cid:133)yzz ). upplementary Figure 3 | Quantitative presentation of measured and calculated changes of the Wigner time delay. a , Experimentally obtained changes of the Wigner time delay, (cid:141)5 (cid:24)(cid:25)(cid:18)(cid:26)(cid:27)(cid:17) , as a function of the electron’s emission angle relative to the molecular axis ( β ). The overall shape of the curves is very similar for the ATI peaks and the SB peaks. The amplitude decreases for higher electron energies. For the first ATI peak in the experiment the values for (cid:141)5 (cid:24)(cid:25)(cid:18)(cid:26)(cid:27)(cid:17) and the corresponding error bars have been scaled as indicated in the legend (the modulation amplitude of the first ATI peak is approximately 3 times higher in the experiment compared to theory (see b )). The deviations for the first ATI peak comparing experiment and theory might be due to Coulomb interaction after tunneling that is not included in our theoretical model. All error bars show the standard deviation of statistical errors (see Methods). b, Shows the result from the theoretical model in analogy to a . Apart from the error bars, a and b show the same data as Fig. 2(a) and 2(b).show the same data as Fig. 2(a) and 2(b).