Multi-Sideband RABBIT in Argon

We report a joint experimental and theoretical study of a three-sideband (3-SB) modification of the"reconstruction of attosecond beating by interference of two-photon transitions"(RABBIT) setup. The 3-SB RABBIT scheme makes it possible to investigate phases resulting from interference between transitions of different orders in the continuum. Furthermore, the strength of this method is its ability to focus on the atomic phases only, independent of a chirp in the harmonics, by comparing the RABBIT phases extracted from specific SB groups formed by two adjacent harmonics. We verify earlier predictions that the phases and the corresponding time delays in the three SBs extracted from angle-integrated measurements become similar with increasing photon electron energy. A variation in the angle dependence of the RABBIT phases in the three SBs results from the distinct Wigner and continuum-continuum coupling phases associated with the individual angular momentum channels. A qualitative explanation of this dependence is attempted by invoking a propensity rule. Comparison between the experimental data and predictions from an R-matrix (close-coupling) with time dependence calculation shows qualitative agreement in the observed trends.


I. INTRODUCTION
The reconstruction of attosecond beating by interference of two-photon transitions (RABBIT) is a widely employed technique to measure attosecond time delays in photoionization processes [1][2][3]. The extraction of time information from the RABBIT measurements usually involves retrieving atomic phases encoded in the delay-dependent modulation of the sideband (SB) yield. These SBs are traditionally formed in the photoelectron spectrum by the interaction of two photons (one pump, one probe) with the target. Spectral harmonics from an attosecond pulse train (the pump photons) form discrete photoelectron signal peaks. The presence of a time-delayed infrared field (the probe photon) then creates a signal in between these main peaks that oscillates with the time delay. The so retrieved atomic phase (∆φ at ) from the RABBIT measurement can be separated into a single-photon ionization contribution (∆η, Wigner phase [4]) and a continuum-continuum (cc) coupling phase (∆φ cc ) by applying an "asymptotic approximation" [5][6][7].
Variations of the RABBIT scheme, such as 0-SB, 1-SB, and 2-SB, have been utilized to study dipole transition phases and attosecond pulse shaping [8][9][10]. As the name suggests, in a 3-SB RABBIT scheme, three SBs are formed between two consecutive main photoelectron peaks [11,12]. The delaydependent oscillation in the photoelectron signal of these three SBs requires more than one transition in the continuum, i.e., the absorption or emission of several probe photons. For a hydrogenic system, we recently [12] extended the asymptotic approximation to a decomposition scheme, which expands the phase of the N th -order dipole matrix element M (N ) , describing the absorption of an ionizing extreme ultraviolet (XUV) photon followed by N − 1 infrared (IR) photon * bharti@mpi-hd.mpg.de; Anne.Harth@hs-aalen.de exchange in the continuum, into a sum of the Wigner phase and N −1 cc phases.
For atomic hydrogen, where numerical calculations with high accuracy can be carried out by solving the timedependent Schrödinger equation (TDSE) directly, we verified that the decomposition approximation explains the RABBIT phases in all three SBs qualitatively [12]. As expected, its accuracy improves with increasing energy of the emitted photoelectron. On the other hand, assuming ∆φ cc to be independent of the orbital angular momenta of the continuum states leads to deviations from the analytical prediction, particularly in the lower and the higher SB of the triplet at low kinetic energies.
Even though starting with a 3p electron still limits the information that can be extracted due to the combined effect of at least two Wigner and the cc phases, we decided to perform the present proof-of-principle study on argon due to its experimental advantages, including a significantly lower ionization potential than helium, which may be a viable alternative to atomic hydrogen due to its quasi-one-electron character, as long as one of the electrons remains in the 1s orbital, i.e., away from doubly-excited resonance states. In argon, the intermediate orbital angular momentum after the XUV step is λ = 0 or 2, while λ = 1 in helium. For the latter target, as for atomic hydrogen, the dependence on the Wigner phase would drop out, and the 3-SB setup would provide direct access to the phase associated with higher-order cc transitions [11,12]. Nevertheless, a significant strength of our current setup already lies in the fact that the results within each group are independent of any chirp in the XUV pulse, because the XUV harmonic pair is common to all three SBs. This paper is organized as follows. We begin with a brief review of the basic idea behind the 3-SB setup in Sec. II. This is followed by a description of the experimental apparatus in Sec. III and the accompanying theoretical R-matrix (closecoupling) with time dependence (RMT) approach in Sec. IV. In section V, we first show angle-integrated data (Sec. V A) before focusing on the angle-dependence of the RABBIT phases in the three SBs of each individual group in Sec. V B. We finish with a summary and an outlook in Sec. VI.

II. THE 3-SB SCHEME
In this section, we briefly review the 3-SB scheme introduced in [11] and the analytical treatment presented in [12] as applied to the 3-SB RABBIT experiment.  (Hq−1 and Hq+1) of the frequency-doubled fundamental probe frequency in the XUV pulse, while S q,l , Sq,c, and S q,h are the lower, central, and higher SBs, respectively. These SBs are formed by emission or absorption of probe photons by the quasi-free photoelectrons. |i denotes the initial state and Ip is the ionization potential. Figure 1 illustrates only the two most dominant transition paths for each SB contributing to the oscillation in their respective yields. The lowest-order transition dominates the yield, but its modulation requires interference between at least two distinct paths leading to the same energy. This involves two different XUV harmonics that are aided by absorption or emission of near-infrared (NIR) photons. For the lower (l) and higher (h) SBs, S l and S h , the most important interfering paths are of 2 nd (one harmonic and one NIR) and 4 th (one harmonic and three NIR) order, which results in a weak modulation of the yield. The lowest-order terms contributing to the build-up of the central (c) SB, S c , are both of 3 rd order (one harmonic and two NIR). Consequently, interference between them exhibits the delay-dependent oscillation most clearly.
Mathematically, the angle-integrated yield in the three SBs, considering only two prominent transition paths, can be written as ,m (k l,q )+Ẽ q−1Ẽω M (2,a) ,m (k l,q ) ,m (k c,q ) ,m (k h,q ) Here q labels the SB group, while k l,q , k c,q , and k h,q denote the final linear momenta of the ejected electron in the lower, central, and higher sidebands in each group. The subscript denotes one of generally several allowed orbital angular momenta of the ejected electron in the final state and m labels the magnetic quantum number, which can be 0 or ±1 for the electron starting in the 3p subshell. Note that m is a conserved quantity for all orders n of the transition matrix element M (n) ,m due to our use of linearly polarized light. Furthermore,Ẽ Ω = E Ω e i φΩ andẼ ω = E ω e i ωτ (for absorption) are the complex electric-field amplitudes of the co-linearly polarized XUV-pump (Ω) and NIR-probe (ω) pulses, respectively. ∆φ at ,m = arg[M (a) ,m M * (e) ,m ] is the phase difference between the two matrix elements and a(e) denotes the pathway involving absorption (emission) of the probe photons. Finally, ∆φ q Ω is the spectral phase difference (XUV chirp) of two neighbouring harmonics.
As seen from Eqs. (1), the yield of each SB is separated into an average part I 0 and another term I 1 that oscillates at 4 ω with the delay. As discussed in [12], every dipole transition also adds a phase of π/2. Since the two dominant interfering terms in S l and S h are of different orders (2 nd and 4 th ), this leads to an additional π phase in S l and S h relative to S c , where both interfering terms are of the same (3 rd ) order.
The RABBIT phase (φ R ) includes the spectral phase difference of the two harmonics and the channel-resolved atomic phases weighted according to their transition amplitudes. It is a complex inverse trigonometric function involving many parameters and hence is best determined by fitting the signal to the known analytic form given above. Since the three SBs involve the same pair of harmonics, the contribution of the XUV group delay (i.e., the chirp) to the oscillation phase is the same in all three SBs. This is a key advantage of the 3-SB method, since it removes the influence of the XUV chirp when we compare the phases of the three SBs only within a particular group.  Figure 2 shows the schematic design of our 3-SB RABBIT experimental setup. A commercial fiber-based laser delivers pulses with a duration of approximately 50 fs (FWHM) at a 49 kHz repetition rate with a pulse energy of 1.2 mJ and a center wavelength of 1030 nm. This pulse is split into two parts using a holey mirror (BS) that reflects ≈ 85% of the incoming beam in the pump arm, while the rest passes through the hole into the probe arm. The beam size of the reflected donut beam in the pump arm is reduced by a pair of lenses and passed through a 0.5 mm thick BBO crystal to double its frequency.

III. EXPERIMENTAL SETUP
The conversion efficiency for the Second-Harmonic Generation (SHG) by the BBO crystal is 25 − 30 %. A dichroic beam-splitter (DBS) filters out the fundamental beam, and a lens with a focal length of 12 cm focuses the second harmonic beam inside a vacuum chamber to a focal spot of 30 − 40 µ m on a jet of neon gas, which results in an XUV frequency comb through high-harmonic generation (HHG). The gas nozzle has a diameter of 100 µ m and is operated at a backing pressure of 1.2 bar with a chamber pressure of 5 × 10 −3 mbar. The generated XUV beam is spatially separated from the annular second harmonic with the help of an additional holey dumping mirror (DM). The residual second harmonic passed through the dumping mirror is weak and does not generate any visible sidebands. The beam in the probe arm goes through a retroreflector mounted on a piezoelectric-translation stage that offers a step-resolution of 5 nm with closed-loop position control. Another holey mirror (RM) recombines the NIR (probe) and XUV (pump) beams, which are then focused inside a reaction microscope (ReMi) on a cold gas jet of argon. The ReMi enables coincident detection and the reconstruction of the three-dimensional momenta of the ions and electrons created during the photoionization process [13].
The interferometer was actively stabilized [14] to achieve a stability of ≈ 40 attoseconds over a data acquisition time of seven hours. The stability of the interferometer was critical for the successful realization of the 3-SB scheme since the oscillation period was just 850 attoseconds.

IV. THEORETICAL APPROACH
In the theoretical part of this study, we employ the general R-matrix with time dependence (RMT) method [15] to generate theoretical predictions for comparison with our experimental data. In order to calculate the necessary timeindependent basis functions and dipole matrix elements, we set up the 2-state nonrelativistic model introduced by Burke and Taylor [16] to treat the steady-state standard photoionization process. In this model, multi-configuration expansions for the initial (3s 2 3p 6 ) 1 S bound state and the two coupled final ionic states (3s 2 3p 5 ) 2 P and (3s3p 6 ) 2 S were employed. We checked that the photoionization cross sections at the photon energies corresponding to the various HHG lines was reproduced properly (in agreement with Burke and Taylor [16] as well as experiment [17,18]) by our RMT model.
The probe-pulse duration was chosen as about twice the length of the XUV pulse. We emphasize that the present calculation was meant as a supplement to the current experiment, with the hope of providing additional qualitative insights rather than quantitative agreement, which would require much more detailed information about the actual pulses than what was available. We purposely employed significantly lower NIR peak intensities (10 11 W/cm 2 ) than in the experiment (≈ 6 × 10 11 W/cm 2 ). This reduced the number of partial waves needed to obtain converged results, diminished potential distortions, and thus made it easier to interpret the spectra.

V. RESULTS AND DISCUSSION
Below we present our results. We start with the angleintegrated setup in Sec. V A before going into further detail with angle-resolved measurements and calculations in Sec. V B.
A. Angle-integrated RABBIT phases Figure 3 exhibits the results of our 3-SB RABBIT experiment after integrating the signal over all photoelectron emission angles. To highlight the oscillations, the RABBIT trace in panel (a) is plotted after subtracting the average delayintegrated signal. The delay-integrated photoelectron spectra (normalized to 1 at the highest peak) is plotted in panel (b). Due to the high NIR intensity, some of the main bands are depleted substantially and appear weaker than the SBs in their vicinity. The angle-integrated photoelectron spectrum is integrated over a spectral window of 0.7 eV around the peak of the SBs.
The RABBIT phase (φ R ) is extracted by fitting a cosine function (cf. Eqs. (1)) to this delay-dependent oscillating signals of the sidebands, as seen in Fig. 4. Due to the large dataset available and the excellent stability of the interferometer, the phase retrieval generally resulted in error bars smaller than the symbol size in Fig. 3(c). This gives us confidence in the results obtained from our extraction procedure. The numerical values obtained for the various SB groups, as well the contrast ratio γ ≡ max(SB(τ )) − min(SB(τ )) max(SB(τ )) + min(SB(τ )) are listed in Table I. As expected, the highest contrast is found for the center sideband, due to the same (3 rd ) order of transitions involved. We note that there are several autoionizing resonances with principal configuration 3s3p 6 n in the SB 12 range of photoelectron kinetic energies, which converge towards the (3s3p 6 ) 2 S threshold of the first excited state of Ar + around 13.5 eV [20]. Early measurements of the (3s3p 6 np) 1 P o resonances were reported by Madden et al. [21]. They were also seen by Burke and Taylor [16] in their photoionization work, and further resonances with other configurations, which can be reached by charged-particle or multiphoton impact, were discussed by Bartschat and Burke [22]. More recently, the effect of these resonances on the RABBIT phase in 1-SB setups was reported by Kotur et al. [23] and Cirelli et al. [24].
Since we used the coupled-state description of Burke and Taylor [16], we saw resonance effects in test calculations, but only with appropriate frequencies and sufficiently long pulses, for which the resonance widths could be well resolved. Note that these features are very sensitive to small fluctuations in the frequency and bandwidth of the APT during the XUV generation process. Therefore, these structures were not seen in the three experimental data points presented in the SB 12 region. We hope to generate additional data with tunable high-order harmonic frequencies in the future. This will make it possible to investigate the resonance phenomena in more detail.
As predicted by our generalized decomposition approximation (cf. Eqs. (1)), the lower and the higher SBs oscillate by π out of phase with the central SB. The retrieved RABBIT phases φ R are plotted in Fig. 3(c) after removing the extra π from S l and S h to simplify the comparison. The timedelay axis on the right side of this panel was created via the conversion τ R = φ R /(4ω).
Five SB groups are clearly identifiable in Fig. 3(c). While there are some irregularities in SB 8 and SB 16 , especially with the phase extracted from S l , groups SB 10 , SB 12 , and SB 14 show the expected trend: The RABBIT phases of the three SBs in each group are similar, although a small difference remains visible in SB 10 . That difference, however, essentially vanishes in SB 12 and SB 14 .
The irregularity seen in the SB 8 group is due to a significant contribution of another 4 th -order transition in the absorption path of the lowest SB S l , which involves a transition from M 7 down to the Rydberg states and back up to S l . The Rydberg states enhance the strength of this transition and add a resonance phase that leads to a significant deviation in the RABBIT phase of S l compared to the other members of the SB 8 group. Furthermore, due to the low cut-off of the XUV spectrum based on HHG and the decreasing photoionization cross section of argon with increasing photon energy, the strength of the M 17 peak is very weak compared to the rest of the lower main peaks. As a result, higher-order transitions involving lower main bands also play a significant role in the oscillation of S l in the SB 16 group, which again affects the extracted phase.

B. Angle-differential RABBIT phases
We now further increase the level of detail by investigating angle-dependent RABBIT phases, which is possible due to the angle-resolving capability of the reaction microscope. For the reasons given above regarding the additional complexities associated with the SB 8 and SB 16 groups, we concentrate the remaining discussion on SB 10 , SB 12 , and SB 14 . Figure 5(a-c) shows the RABBIT phases extracted within these groups as a function of the photoelectron emission angle, which is defined relative to the (linear) laser polarization vector. The photoelectron signal is integrated over an angular window of 10 • for each data point. The angleresolved RABBIT phases are shifted to fix the starting phase of the central sideband in each group to zero. According to both our experiment and the calculation (Fig. 5, panels d-f), the phase of S h exhibits a stronger angular dependence compared to that of S c and S l . With increasing photoelectron energy, the differences diminish in both experiment and theory, with theory predicting almost no angle-dependence in the range of SB 14 plotted.
To explain the angle-dependence in the RABBIT phase, we need to consider the interference among all the angularmomentum channels of the sidebands accessed through the absorption and emission paths. We write the signal in compressed form as Here α a and α e are the transition amplitudes involving the various fields and matrix elements, while ( ) denotes the angular-momentum channels accessed through the absorption (emission) path. The dissimilarity in the RABBIT phases (φ R (θ)) of the three SBs can be explained by considering a propensity rule for the transition amplitudes and the dependence of both the Wigner and φ cc phases on the orbital angular momenta. It is well known that the Wigner phase depends on the angular momentum channel. The cc phase has also been shown to depend slightly on whether there is an increase or decrease in the angular momentum, while it appears to remain independent of the target species [25,26]. Therefore, the atomic phases (∆φ at ,m ) arising from the interference between various -channels of emission and absorption paths are also expected to differ. Similar to bound-continuum transitions [27], absorption (emission) within the continuum favors an increase (decrease) in the angular momentum of the outgoing photoelectron, especially for low kinetic energies [26,[28][29][30][31]. The higher SB (S h ) of the group involves the absorption of three probe photons (H q−1 + 3 ω) that, according to the propensity rule, predominantly populate higher angular-momentum states. Along the other path (H q+1 − 1 ω) leading to S h , the emission of one probe photon mainly creates lower angular-momentum states. For S l , emission of three probe photons (H q+1 − 3 ω) primarily leads to the population of lower angular-momentum states. Even though the absorption path (H q−1 + 1 ω) to S l also favors an increase in the photoelectron's angular momentum, the possible values reached by the absorption of a single probe photon remain relatively small.
The interplay of the propensity rule for transition amplitudes to each -channel and the angle-dependent amplitudes of the coupled spherical harmonics determine the angular variation of φ R in the three SBs. In crosschannel interference, = , the angle-dependent spherical harmonics undergo a sign change across their angular nodes, thus resulting in a phase jump by π. If the relative magnitude of these cross-channel interferences is significant compared to that of the same-channel interference terms, = , this can lead to a rapid variation in the angle-dependence of φ R in the vicinity of the nodes [29]. Depending on the value of ∆φ at ,m relative to the average ∆φ at of the interference terms, the additional π-jump at the nodes in Y ,m and/or Y ,m can drive the angle-dependent curve downward or upward.
With increasing value, the position of the first node in the associated Legendre polynomial of the spherical harmonic moves to smaller angles. Due to the propensity rule, the weight of the cross-channel interference term containing large -values is most significant in the higher sideband. This results in a relatively early onset of the descent in the angledependent RABBIT phase in the higher sideband. In the lower sideband, the amplitude of the cross-channel interference term containing large -values is not very strong; hence, the πjump across the node does not produce a substantial change in the overall retrieved phase. With increasing kinetic energy, for both the absorption or emission of the probe photons, the transition amplitudes for increasing and decreasing angular momentum tend to become similar [29]. Hence the contribution of cross-channel interference containing large values decreases with increasing kinetic energy. Thus the πjumps at the nodes of the corresponding spherical harmonics do not change the retrieved phase significantly.
Since the retrieved angle-integrated RABBIT phase is the weighted average of all the channel-resolved RABBIT phases and the weights of these channels in the S l , S c , and S h SBs are different, the angle-integrated RABBIT phase in the three SBs also turns out different. Also, owing to the propensity rule, the unequal transition probabilities of reaching the various angular momentum states of the SBs in absorption and emission of the probe photons may cause incomplete interference in the individual channels, thereby reducing the overall oscillation contrast in the angleintegrated photoelectron signal.
Finally, we notice that the scale of the variations in the angle-dependence of the RABBIT phase depicted in Fig. 5 is smaller in the calculation than in our experiment. Also, the positions of S l and S h relative to S c appear to be switched. In addition to always possible shortcomings in the theoretical model (as sophisticated as it might be) and unknown potential systematic errors in the experiment, the difference in the probe intensities and the pulse details, in general, are likely responsible for at least some of the discrepancies seen here. We hope to be able to investigate this in more detail in the future by performing additional calculations with different intensities and more time delays.

VI. SUMMARY AND OUTLOOK
In summary, we carried out a proof-of-principle 3-SB RABBIT experiment in argon. In contrast to more popular single-SB studies, our technique enables us to focus on the photon-induced transition phases without distortion from a possibly unknown or experimentally drifting XUV chirp. While we confirmed earlier predictions that the angleintegrated RABBIT phases extracted within a SB group become increasingly similar, we enhanced the analyzing power of the setup significantly by resolving the emission angle with a reaction microscope. By doing so, we could identify which of the three sideband phases within a group is most sensitive to a change in the detection angle.
Our experimental efforts were accompanied by numerical calculations performed with the nonperturbative all-electron R-matrix with time dependence method. There is some qualitative agreement between experiment and theory regarding the general trends observed, but significant differences remain in the details. Given the remaining limitations and challenges faced in the present study, especially concerning the details of the pulse and the argon target, the remaining deviations between experiment and theory in the quantitative values of the phases are not too surprising. We hope to address these issues in future improvements of the setup.
As the next step, we plan to repeat this experiment with helium, where the contribution of the Wigner phase for an s → p transition remains the same in all three sidebands. Any differences in the phases within the group then clearly indicate the influence of φ cc . This switch of targets would require extending the harmonic cut-off, which is by no means trivial in our scheme, as the cut-off in the HHG process decreases with the driving frequency. Using helium instead of argon also has the advantage of theory likely being more reliable due to the simplicity of the target. On the other hand, heavier quasi-two-electron targets with an (ns 2 ) 1 S outer-shell configuration (unfortunately, these are metals that would need to be vaporized rather than inert gases) would provide a larger short-range modification of the relevant interaction potential and, therefore, may be more suitable to investigate whether φ cc is indeed nearly universal.
Undoubtedly, many open questions will need to be answered before the effect of the additional continuumcontinuum transitions in single-and multiple-SB RABBIT setups are fully understood. It would be interesting to analyze whether the SB phases always converge to each other with increasing energy, whether or not they cross in a predictable way with increasing emission angle, and how the behavior depends on the target investigated. While we cannot answer these questions at the present time, we hope that other groups will see the work reported in this paper as a worthwhile inspiration to carry out further studies in this field.