Ultra-stable and versatile high-energy resolution setup for attosecond photoelectron spectroscopy

ABSTRACT

Attosecond photoelectron spectroscopy has opened up for studying light–matter interaction on ultrafast time scales. It is often performed with interferometric experimental setups that require outstanding stability. We demonstrate and characterize in detail an actively stabilized, versatile, high spectral resolution attosecond beamline based on a Mach-Zehnder interferometer. The active stabilization keeps the interferometer ultra-stable for several hours with an RMS stability of 13 as and a total pump-probe delay scanning range of $\sim 400$ fs. A tunable femtosecond laser source to drive high-order harmonic generation allows for precisely addressing atomic and molecular resonances. Furthermore, the interferometer includes a spectral shaper in 4f-geometry in the probe arm as well as a tunable bandpass filter in the pump arm, which offer additional high flexibility in terms of tunability as well as narrowband or polychromatic probe pulses. We demonstrate the capabilities of the beamline via experiments using several variants of the RABBIT (reconstruction of attosecond beating by two photon transitions) technique. In this setup, the temporal-spectral resolution of photoelectron spectroscopy can reach a new level of accuracy and precision.

KEYWORDS:

• Attosecond
• photoionization
• High harmonic generation
• interferometer
• RABBIT
• Density matrix

1. Introduction

The motion of electron wave packets in small quantum systems occurs on the attosecond timescale, and in the past two decades there has been a significant effort made to track these electron dynamics in real-time [1,2]. The discovery of high-order harmonic generation (HHG) in gases [3,4] at the end of the 80s opened the door to the production of isolated attosecond pulses and attosecond pulse trains. Around the millennium shift, isolated attosecond pulses were characterized for the first time by attosecond streaking and the existence of attosecond pulse trains was experimentally demonstrated using the ’Reconstruction of Attosecond Beatings by Interference of Two-Photon Transitions’ (RABBIT) technique  [5–9]. The RABBIT technique is a photoelectron interferometry scheme based on two-photon transitions induced by an attosecond harmonic comb in the extreme-ultraviolet (XUV) spectral range and a phase-locked IR laser field [5,10,11]. The intensity of the interference signal oscillates as, $S(\tau )=\alpha +\beta \text{cos}[2\omega (\tau -{\tau }_{xuv}-{\tau }_A)]$, when thedelay τ between the attosecond pulse train and the IR pulse is scanned. Here, τ_xuv depends on the phase difference between consecutive harmonics in the comb and τ_A contains information on the photoionization transition matrix elements [2,12–15].

The development of attosecond technology, i.e. producing XUV attosecond pulses and controlling the delay between these pump pulses and infrared (IR) probe pulses with attosecond precision, has opened up the world of attosecond science and in particular made possible monitoring of electron dynamics [2,12,13,16–21]. Ionization time delays in atomic [22,23] or molecular [24,25] systems as well as from surfaces [26], the temporal buildup of electron wavepackets from an autoionizing state [27,28] and the electronic rearrangement following ionization have been measured on the attosecond timescale [29–31]. Correlations between electrons during shake-up ionization [32], the interplay between electronic and nuclear degrees of freedom in molecules [25,33], as well as the dynamics of shape resonances in various systems have also been explored [24,31,34,35].

All these attosecond experimental measurements require ultra-stable beamlines for synchronizing the pump and probe pulses to within a few attoseconds, requiring that the optical path lengths need to be stabilized on the nanometer scale. Different beamline configurations, collinear [36,37] or noncollinear [38–41], have been successfully used to achieve attosecond stability for several hours. Collinear configurations, using split-plates or mirrors, are intrinsically stable since the pump and probe pulses are propagating collinearly. However, the scanning range for the pump or probe delay is usually limited to a few tens of femtoseconds. Noncollinear setups, requiring beam splitting and recombination, provide more versatility. However, since the two optical paths propagate separately anywhere from tens of centimeters to tens of meters, the system is more susceptible to fluctuations and drifts. To counteract this, Mach-Zehnder interferometers with copropagating phase-locked continuous wave (CW) lasers have been designed and implemented, allowing for an active stabilization of the optical paths down to tens of attoseconds [38–44]. Other schemes have also demonstrated good stability, e.g. with a femtosecond IR laser stabilization system [45–47] or by stabilizing the delay with the sideband signal from a RABBIT measurement [48].

Many experiments in attosecond science require not only temporal but also spectral resolution [15]. This is the case for studying the dynamics of discrete or quasi-discrete (Fano) resonances in atomic systems [27,28,49,50], for resolving vibrational structure in molecular photoionization [34,51,52] or when multiple processes, e.g. due to shake-up, create spectral congestion [27–31,49,50]. This is especially important for vibration-resolved molecular ionization of molecules. A high spectral resolution can be obtained in photoelectron spectroscopy with highly resolving spectrometers as well as with narrow bandwidth probe fields [50]. Other factors may also affect the temporal stability and/or spectral resolution, such as the frequency chirp of the high-order harmonics or the temporal (spectral) properties of the probe field. In addition, any spatial inhomogeneity, regarding for example, the delay, will reduce the accuracy [14].

Development of attosecond technology promotes novel applications in ultrafast optics [53–55], atomic and molecular physics [2,12,18,56,57], solid-state physics [58–60], and paves the way for new frontiers, such as ultrafast quantum information  [50,61–67] and biological applications [68]. In this work, we present an improved experimental setup for interferometric photoelectron measurements, such as RABBIT [5,6], with long-term stability, and high temporal and spectral resolution. Femtosecond laser pulses are split in a Mach-Zehnder interferometer, which is partly in vacuum and partly in air, thus allowing for flexibility. The delay between the pump and probe pulses is stabilized to around 13 as root-mean-square (RMS) error over hours. A high spectral resolution in the energy-resolved (”rainbow”) RABBIT technique [28] is obtained by using a magnetic bottle electron spectrometer with a 2 m long flight tube and a narrowband wavelength filter in the probe arm [50]. We demonstrate the long-term stability of our interferometer by performing a scan with a 70 fs delay variation over7 hours, and we show the improved spectral resolution due to the narrowband probe (10 nm bandwidth around 790 nm) in a RABBIT measurement across the spin-orbit resolved 3s⁻¹4p resonance in argon. Our experimental setup includes a 4f-shaper, which can be coupled in and out of the probe beam path, which allows shaping the bandwidth of the probe laser into mono-, bi- or polychromatic narrowband components, easily tunable within the laser bandwidth. The 4f-shaper allows the implementation of new, advanced, methods for the characterization of photoelectron wave packets, using, for example, quantum state tomography [64,65].

2. Experimental setup

2.1. Overview

A schematic drawing of the experimental setup is shown in Figure 1. It consists of the pump (red lines), probe (yellow lines) and XUV (purple lines) beam paths, the stabilization system, an HHG chamber, a recombination chamber, and a magnetic bottle electron spectrometer (MBES). Briefly, the beam from a Titanium-Sapphire chirped pulse amplification laser system (1–3 kHz repetition rate, up to 4 mJ pulse energy, tunable 40–80 nm bandwidth and tunable central wavelength around 800 nm) is split by a broadband beam-splitter into the pump (40%) and probe (60%) arms. The beam-splitter is the first optical element in the Mach-Zehnder interferometer and active stabilization system, details of which are presented in the next section.

Figure 1. Schematic representation of the combined attosecond beamline and MBES apparatus. The setup consists of eight main parts: a Mach-Zehnder interferometer, a femtosecond laser stabilization system, a high-order harmonic generation chamber, a recombination chamber, a 2 m long MBES, an XUV spectrometer, a 4f-shaper, and a spectral filter. The probe beam of the interferometer is shown in yellow, the pump beam in red, and the XUV in purple. FS, femtosecond scale delay stage; AS, attosecond scale piezo stage; GC, gas cell; FW, filter wheel; RM, recombination mirror; PM, pick-up mirror; LP, linear polarizer; WP, wedge pair; HOWP, high-order waveplate; C1, C2, cameras; S1, S2, IR spectrometers. Some elements of the active stabilization scheme are presented in more detail in Figure 2.

Figure 2. (a) An illustration of the essential components for measuring the temporal stability of the interferometer, with the probe beam (yellow lines), pump beam (red lines) and XUV (purple lines). FW, filter wheel; RM, recombination mirror; PM, pick-up mirror; $\lambda /2$, half-waveplate; UVFS, UV fused silica; WP, wedge pair; LP, linear polarizer; NF, narrow bandpass filter. (b) Schematic of the aluminum foil mounted on fused silica. (c) Spatial fringes from interfering the femtosecond lasers. (d) Fourier transform of the fringe pattern.

In the pump arm, the beam can pass through an optional spectral filter (discussed below) or directly be apertured by a holey mirror, removing the central part of the beam. The annular beam is focused by a dielectric concave mirror (focal length f = 50 cm) into a gas cell (GC) where high-order harmonics are generated. The cell has a length of 6 mm and is mounted on a stage which allows for fine-tuning of its position along five dimensions (x, y, z, θ_y, θ_z), for alignment relative to the laser and also for the control of the phase matching conditions. The gas is supplied by a pulsed piezo-valve (Amsterdam Piezo Valve) with variable opening time to control the pressure in the cell. Typical conditions imply a backing pressure of around 4–6 bar, resulting in an ambient pressure of some tens of mbar in the gas cell. The collinearly-propagating IR and generated XUV beams pass on a holey fused silica plate, mounted on a rotatable filter wheel (FW). The 1 mm thick fused silica plate with a 2 mm diameter hole, covered by an aluminum foil of 200 nm thickness, blocks the inner part and transmits the outer part of the pump IR. The aluminum foil transmits the XUV beam which then passes through the holey recombination mirror (RM). The toroidal mirror performs the 1-to-1 imaging of the XUV beam from the generation into the gas jet of the MBES. The outer part of the pump IR is reflected by the RM and is used for the temporal stabilization of the system. The RM is drilled in two directions to transmit and reflect parts of both the pump- and probe arms of the interferometer, which is integral to the active delay stabilization. The stabilization will be discussed in detail in section 2.2.

The probe IR laser either passes through the 4f-shaper (discussed in more detail below) or is directly focused into the recombination chamber by a silver coated concave mirror (f = 50 cm) mounted on a piezo delay stage (AS) for the purpose of controlling the delay between the two pulses with attosecond scale precision. The outer part of the probe beam is reflected by the RM and imaged into the interaction region of the MBES by a toroidal mirror, where it is spatio-temporally overlapped with the XUV in the experiment and detection gas is provided with a diffusive jet. The inner part of the probe beam passes through the hole in the RM, to be used for stabilization, achieving attosecond temporal resolution. There is a long–delay stage (FS) in the beam path of the pump for coarse adjustments of the delay between the two arms on the femtosecond scale, while the piezo delay stage in the probe arm is used to control the delay from tens of attoseconds to hundreds of femtoseconds.

The MBES has a 2 m long flight tube with a 4π sr collection angle. The energy resolution is better than 80 meV for low energy photoelectrons, and retarding lenses can slow down fast electrons to enhance the resolution of high energy electrons. The measured response function of the MBES has a width of approximately 40 meV for electrons with kinetic energy below 1 eV, which increases to around 80 meV for those with kinetic energy below 5 eV. After the interaction chamber, the XUV and IR beams reach an XUV spectrometer chamber where the harmonic spectrum and divergence, or the IR spatial profile, can be monitored. The home-made flat-field XUV spectrometer is based on a Hitachi concave grating, which images the focus in the interaction chamber onto the plane of the detector, consisting of multichannel plates and a phosphor screen. The grating is placed on a linear translation stage so that it can be moved in or out of the beam path. Moving the grating out and rotating the aluminum filter out of the pump arm, the spatial overlap of the two arms in the interaction can be imaged onto a camera (C2). When coupling the probe into the 4f-shaper, a polychromatic narrow bandwidth spectrum can be obtained by placing slits in the Fourier plane of the shaper. The increased beam path length due to the 4f-shaper is compensated by extending the pump arm by the same distance (4f-extension), hence preserving the temporal overlap and, to a large extent, stability.

2.2. Active stabilization

To stabilize the pump-probe setup on the attosecond timescale, we derive an error signal from the interference of parts of the IR from the pump- and probe arms of the interferometer (AS in Figure 1). The layout of the stabilization system is shown in Figure 2(a). The outer part of the pump laser (red lines), passing through the fused silica filter plate [shown in Figure 2(b)], is reflected by the RM, whereas the inner part of the probe laser (yellow line) propagates through it. The two beams are transversely shifted after the RM, and when focused by a lens and overlapped in time, give rise to spatial fringes detected by camera C1, as shown in Figure 2(c). Fast Fourier transform (FFT) is used to identify the spatial frequency component corresponding to the fringes [shown in Figure 2(b)] and to find their phase. An error signal, generated from the difference between the desired and the measured phases, is used to stabilize the interferometer with a feedback loop controlling the piezoelectric translation stage in the probe arm (see AS in Figure 1). The fringes are recorded at 13 Hz. A long illumination time of 60 ms per frame, averaging 180 consecutive shots at 3 kHz repetition rate of the laser, is chosen to compensate for the slow delay drifts during RABBIT measurements [14].

RABBIT measurements require that the XUV and IR pulses overlap in time in the interaction region in the MBES and at the same time the two IR pulses used for temporal stabilisation must overlap at the camera C1. In this sense, our stabilization scheme is conceptually different from those based on CW lasers, which neither give any information on the actual XUV and IR temporal overlap in the interaction region nor are sensitive to the gas target density used for HHG [69]. Once the temporal overlap is achieved both in the MBES interaction region and at camera C1, it can easily be maintained, on a daily basis, by checking the spatial fringes on camera C1. A rough estimate of the temporal overlap in the interaction region of the MBES can be obtained by using a pickup mirror (PM) to reflect the generating and probe IR pulses towards a spectrometer (see also S1 in Figure 1), when the aluminum filter is removed, and by tuning the coarse stage in the pump arm (FS in Figure 1) until the beam paths have the same length. Then, the difference between the XUV and probe IR pulses results only from the transmission of the XUV through the aluminum filter. The fine adjustment needed to find exact temporal overlap is done with the help of a RABBIT measurement (see below).

To make sure that the temporal overlap in the interaction region and at the stabilization camera C1 is obtained at the same time, path length differences in the stabilization part (Figure 2a), originating (i) from the fused silica plate that holds the metallic filter and (ii) from geometrically different beam paths at the RM for the pump- and probe pulses, are compensated by an additional piece of glass in one arm and a pair of glass wedges in the other arm. The contrast of the fringes recorded with camera C1 is optimized with a $\lambda /2$-plate in one arm and a polarizer before the camera. Finally, a narrow spectral bandpass filter is used to extend the temporal overlap region. The stabilization system works up to 400 fs scanning range when using a $(10\pm 2)$ nm filter centered at 790 nm. This filter can be easily changed to different central wavelengths to adjust for the tunability of our laser spectrum, and different bandwidths can be used to increase or decrease the scanning range.

2.3. Long-term stability

Often, attosecond experiments require stable delay conditions over several hours, especially for high-quality photoelectron spectroscopy measurements. We characterized both the short- and long-term stability of the interferometer in different conditions. In-loop measurements, i.e. the phase error extracted from the stabilization camera C1, are instructive for obtaining an estimate of the overall stability, but do not necessarily reflect the actual stability achieved in the interaction region of the MBES, which is not easily accessible for diagnostics. For out-of-loop measurements, we therefore use the PM to reflect both pump- and probe IR beams while misaligning the RM slightly to produce spatial fringes in the detector plane. For these measurements, the metallic filter was removed, including the fused silica plate carrying it. The delay shift in the stabilization beam path was adjusted with the wedge pair. Two detectors were used, i.e. a camera at 13 Hz frame rate, again with an illumination time of 60 ms, for long-term measurements, and a fast CCD line detector (Basler L101k) for shot-to-shot measurements at the full 3 kHz repetition rate of the laser. Figure 3(a) shows the relative long-term time delay when the stabilization system is locked and unlocked, respectively. When unstabilized, the maximum drift of the interferometer measured during 1 h is in the range of 400 as with an RMS around 112 as. With the stabilization on, the obtained variation of the measured delay is around 13 as RMS. The true shot-to-shot instability, obtained with the CCD line detector, is around 50 as with 10 min recording time [Figure 4(a,b)]. Full repetition rate shot-to-shot measurements reveal the limiting factors for the stability via Fourier decomposition in time [Figure 4(c)]. The spectrum contains several sources of vibration in our system; most prominent are mechanical resonances between 200 and 400 Hz and a vibration peak resulting from a turbomolecular pump at 980 Hz. A small peak at 100 Hz originates from the Roots pumps of the central vacuum system, located in another room, on a separate, mechanically vibration-isolated, platform. Without these precautions, the vibrations of the pumps would dominate the instability spectrum. Finally, Figure 3(c) shows in-loop measurements from a delay scan taken over a total of 70 fs in 7 h scanning time. The stabilization system remained for 90 s at each delay step. The inset illustrates the change of delay by moving the piezoelectric stage (AS in Figure 1). The measurement demonstrates the stability during long-term scans.

Figure 3. (a) Variation of the long-term relative delay over time when the interferometer is unlocked and locked. (b) Standard deviation of the delay, recorded over 1 hour, with and without active stabilization. The RMSs are around 13 as and 112 as, respectively. (c) Variation of the delay during a long scan; the inset shows the effect of the delay control.

Figure 4. (a), (b) Variation and standard deviation of the relative delay by shot-by-shot measurement at 3 kHz when the interferometer is locked. (c) Spectral composition of the short-term stability of the interferometer with active stabilization.

2.4. Spectral shaping of the probe pulse

Besides its stability, our setup offers a large flexibility for both pump and probe spectral shaping. Narrowband spectral filters can be inserted in the probe path, improving considerably the spectral resolution of RABBIT measurements, as described in the next section (see also [50]). The IR spectra from our femtosecond laser before and after a 10 nm narrow bandpass filter are compared in Figure 5(a). Both the bandwidth and central wavelength of the probe can easily be changed by inserting different band pass filters. To further improve the flexibility, the 4f-shaper was implemented into the probe arm of the interferometer, as shown in Figure 1. The shaper consists of two silver coated concave mirrors (f = 50 cm) and two gratings with 1200 lines per mm, purchased from Spectrogon. The spectral components are spatially dispersed in the Fourier plane of the first mirror, where various spectral shaping can be performed by combinations of slits of different widths. An identical focusing mirror and grating collimate the beam and remove the angular chirp.

Figure 5. (a) Spectrum from the femtosecond laser before (black) and after (blue) passing through a 10 nm narrowband filter centered at 790 nm. (b) Spectrum after passing through the 4f-shaper with two slits in the Fourier plane. Measured d-scan trace, based on second harmonic generation, before (c) and after (d) passing through the 4f-shaper.

This technique allows creating, for example, bichromatic probe pulses, as shown in Figure 5(b). Additionally, a spatial light modulator placed in the Fourier plane (currently not installed) can give full tunability and shaping of the probe pulses in terms of amplitude and phase.

The temporal dispersion of the 4f-setup is minimized with the help of dispersion scan (d-scan) measurements [70,71]. The principle of the d-scan technique to characterize ultrashort pulses is the sensitivity of second harmonic generation (SHG) to the spectral phase of the pulse. Manipulating the spectral phase in a controlled way, by translating wedges into the beam, while recording the SHG spectrum, a d-scan trace is obtained (see Figure 5(c)). An iterative numerical procedure retrieves the generating pulse in the spectral and temporal domains. Comparing the measured d-scan traces before and after the 4f-shaper, as shown in Figure 5(c,d), indicates that the 4f-shaper does not increase second-order dispersion. However, some slight third-order dispersion, indicated by the tilt above 400 nm in Figure 5(d), is added, which most likely is related to the large beam size and compact configuration of the shaper.

2.5. Spectral shaping of the pump pulse

The femtosecond laser chain is equipped with acousto-optic programmable gain and dispersive filters (DAZZLER and MAZZLER), which allow tuning the central wavelength and bandwidth of the laser, before being split into pump and probe. This tunability is essential in spectroscopic applications, for example, to excite a specific resonance by absorption of a high-order harmonic with a spectrum centered on the resonance central frequency. It is, however, sometimes incompatible with the requirements for the probe pulse. For example, the generation of a bichromatic probe pulse and a broad input spectrum to the 4f-shaper is required. To nonetheless enable independent pump and probe spectral shaping, a compact, inline spectral filter capable of tuning the bandwidth and central wavelength was designed and placed in the pump pathway before HHG (Spectral filter in Figure 1).

The filter that is inspired by a single-stage Lyot filter is comprised of a high-order wave plate, oriented at 45° to the incoming linear polarization, and a polarizer behind. The inherent chromaticity of such arrangement results in a sinusoidal transmission as a function of wavelength, with the peaks, when the thickness corresponds to a high-order, full λ-plate and zeros, when the thickness corresponds to a $\lambda /2$-plate. If the order (i.e. the thickness) of the filter is chosen carefully, mostly one lobe within a selected bandwidth is transmitted (see Figure 6), where the bandwidth (FWHM) is approximately $\mathrm{\Delta }\lambda \approx \frac{{\lambda }_0}{2N}$. Here, λ₀ is the selected transmission peak and N is the integer order of the wave plate. In practice, the thickness of the wave plate must be tunable to choose the transmitted peak and bandwidth. This is achieved by a pair of anti-reflection coated quartz-wedges, cut in the c − z-plane, with a thickness of 0.5 mm at the thin edge, 1.4 mm at the thick edge and a length of 30 mm. Due to the beam size in the probe arm, the minimum thickness of the plate formed by the wedges is 1.45 mm, which corresponds to too high order (N ≈16 for ${\lambda }_0=800$ nm). We therefore combine the wedges with a second 1 mm thick high-order quartz wave plate, oriented with its axes crossed in respect to the first one. The combination of the two plates allows for any order between approximately 5 and 15. For N = 10 (i.e. $d\approx 900\mu$m), we obtain a transmission bandwidth of about 40 nm (see Figure 6) and by slightly tuning the wedge position, the transmission window can be moved.

Figure 6. Transmission spectrum of the pump pulses for different wedge positions. The thickness of the birefringent filter changes with −30 µm per 1 mm translation.

3. Experimental results

In the following, we will demonstrate the capabilities and flexibility of our experimental setup with a series of experiments based on ultrafast photoelectron interference. In particular, we show ordinary RABBIT for direct photoionization, rainbow-RABBIT to study resonant ionization from Rydberg and autoionizing states as well as a new technique, KRAKEN (Kvanttillstånds tomogRafi av AttoseKund ElektroNvågpaket) enabling quantum state tomography of photoelectron wavepackets.

3.1. Temporal stability: long-term RABBIT experiment

In the RABBIT experiments presented in this article, high-order harmonics were generated with a transform limited pulse of 40 fs, centered at 800 nm wavelength, focused into a gas cell filled with argon. Due to the centrosymmetry of the generating gas, only odd order harmonics are generated. After passing through the aluminum filter, the XUV radiation includes harmonics from the 11th to the 25th order. We denote these HHq in the following, where q is the harmonic order. The delay between the attosecond pulse train and the infrared pulse in the interaction region of the MBES is varied by 70 fs with a step size of 200 as.

The recorded RABBIT spectrogram is shown in Figure 7. Besides photoelectron peaks corresponding to absorption of harmonics, sideband peaks are visible. These peaks, labeled by the net number (n) of involved laser photons (SBn), are due to two-photon transitions as indicated in the right panel, absorption of Hq (H(q +2)) and absorption (emission) of one IR photon from the probe laser. Due to the interference of these two pathways, the sideband signal oscillates at frequency 2ω, i.e. twice the angular frequency of the driving laser, with a period of ${\tau }_0=1.33$ fs for 800 nm [5,6]. The results presented in Figure 7 exhibit an excellent contrast, over a long delay range of 70 fs, and for a total acquisition time of 7 hours. This demonstrates the high stability and reliability of our interferometric setup.

Figure 7. Photoelectron signal (in color) in He as function of harmonic photon energy and delay in a RABBIT measurement. The delay step is 200 as. The total acquisition time of the scan is $\sim 7$ hours.

3.2. Tunability: RABBIT experiment of resonant ionization via Rydberg state

For studying resonant ionization with the RABBIT technique [72], one harmonic (Hq) is tuned to excite a bound state close to the ionization threshold of the studied gas, which is ionized by absorption of an additional IR photon. The created wavepacket is probed by a nonresonant-ionization process involving absorption of H$(q+2)$ and emission of an IR photon. This technique allows probing attosecond dynamics of resonant ionization from Rydberg states [72–75]. In Ref.  [76], we study two-photon resonant ionization of He via the 1s4p¹P₁ state which is reached by absorption of H15, after tuning the central laser wavelength to 794 nm. The oscillation of SB16 is shown in Figure 8. The analysis is performed spectrally resolved, corresponding to the rainbow-RABBIT technique, to extract the amplitude and phase of the resonant electron wavepacket [28,49,50]. The results show a smooth π-rad phase variation across the resonance in SB16, and a relative π-rad phase shift between SB16 and H17, as marked by the white lines in Figure 8(a). To extract the amplitude and phase of oscillations from a RABBIT trace, either Fourier analysis or cosine fitting can be used  [14]. In the present case, cosine fitting was applied. Furthermore, a comparison between the results obtained with the probe polarization parallel to, and perpendicular to, the polarization of the XUV light is shown in Figure 8(b). As explained in [76], by changing the polarization of the probe field, different ionization channels can be selected. Both s and d channels coexist in the parallel configuration, while only the d channel is allowed when the probe polarization is perpendicular to that of the pump since only Δm $=\pm 1$ transitions are possible. The measured relative phase shifts are well reproduced by calculations using the two-photon random-phase approximation with exchange (RPAE) method [77].

Figure 8. (a) Photoelectron signal (in color) of SB16 and H17 from He in a below-threshold RABBIT measurement, where H15 excites the 1s4p state. (b) The measured phase of SB16 when the probe IR polarization is parallel (red dots) and perpendicular (black dots) to XUV. The red and black lines are calculated for IR parallel to XUV and d-wave only [76].

3.3. Spectral resolution: RABBIT experiment via autoionizing state with a narrowband probe

Figure 9 shows a RABBIT spectrogram obtained in Ar. The central wavelength and bandwidth of the femtosecond laser is tuned (before the interferometer) so that the energy of H17 is close to that of the autoionizing state 3s⁻¹4p (${\mathrm{E}}_{\text{res}}=26.61$ eV). A 10 nm bandpass interference filter centered at 790 nm [see Figure 5(a)] is placed in the probe arm. By using a probe with a narrower bandwidth, the spectral phase convolution in rainbow-RABBIT measurements can be decreased [50,78,79], leading to an enhancement in spectral-phase resolution in resonant RABBIT experiments. A deconvolution algorithm is used to account for the spectrometer resolution [49]. The photoelectron peaks present two components, due to the spin–orbit interaction, which leads to two final ion states (3p${}^5{}^2$P${}_{1/2}$ and 3p${}^5{}^2$P${}_{3/2}$) separated by 0.18 eV. Both spin-orbit splitting and autoionization resonance, with the window-like spectral profiles, are visible in H17 as well as in SB16 and SB18. Thus, the amplitude and phase across autoionization states of atoms after photoexcitation can be extracted by this type of attosecond photoelectron interferometry [28,49,50,80].

Figure 9. (a) Photoelectron signal (in color) in Ar as function of harmonic photon energy and delay in a RABBIT measurement. Spectral intensity (blue) and phase (black) of the 2ω oscillation component for (b) SB16 and (c) SB18. The delay step is 120 as. Recording this scan took $\sim 3$ hours. Absorption of H17 is resonant with the 3s⁻¹4p autoionizing state in Ar.

The oscillations of the sidebands are analyzed by Fourier transform applying the rainbow-RABBIT technique [28]. The spectral amplitude and phase of the Fourier components oscillating at frequency 2ω are shown in Figure 9(b,c) for SB16 and SB18. Both plots present two separate and similar components which are due to the spin-orbit splitting of the Ar⁺ ground state and separated in energy, as expected, by 0.18 eV. Our energy resolution is better than in previous data, where the two spin-orbit components partially overlapped in energy [80]. Furthermore, in the case of SB16, the phase varies by close to 2π rad across the Fano resonance, which is larger than previously measured values [27,80], thus indicating improved spectral resolution and measurement accuracy. An explanation of the 2π rad phase jump will be presented in a future publication.

3.4. Flexibility: KRAKEN experiment with a bichromatic probe

Finally, our setup allows us to manipulate the pump and probe pulses independently in a simple manner. This is important for the implementation of new excitation schemes, which go beyond the ordinary RABBIT technique. One of these is the proposed KRAKEN (English: quantum state tomography of attosecond electron wave packet) protocol [65], enabling quantum state tomography of photoelectron wavepackets, i.e. the measurement of the density matrix. A pair of synchronized, spectrally narrow probe pulses with central frequencies ω₁ and ω₂ are employed to induce interference of different continuum states of the one-photon electron wave packet, resulting in a delay-dependent interference pattern. The amplitude of the oscillations allows to draw conclusions on the degree of coherence of the different continuum states with each other. By repeating the measurement with different bichromatic probes, i.e. varying the energy difference between ω₁ and ω₂, the full one-photon electron density matrix can be reconstructed experimentally.

The need for a tunable bichromatic probe motivated the development of the 4f-shaper in the probe path. Figure 10 presents results of a 115 fs-long scan in argon, with 200 as delay steps. The spectrum of the bichromatic probe, shown in Figure 10(b), comprises two components with ≈7 nm widths, centered at 770 nm (ω₁) and 820 nm (ω₂). In the KRAKEN configuration, the active stabilization system operates by setting the narrow bandpass filter before the stabilization camera (C1 in Figure 1) to be centered at one of the spectral components of the bichromatic probe. The observed sidebands oscillate rapidly at frequencies $2{\omega }_1$, $2{\omega }_2$ as well as ${\omega }_1+{\omega }_2$ and slowly at the beating frequency ${\omega }_1-{\omega }_2$. The amplitudes and phases of individual oscillations can be extracted via Fourier transform. The different frequency components of the SB24 oscillations are shown as black line in Figure 10(b). Meanwhile, the blue line displays the frequency component of the electron signal from H23, wherein only a very weak beating signal can be observed.

Figure 10. (a) Photoelectron signal (in color) in Ar as function of harmonic photon energy and delay in a KRAKEN measurement, i.e. with a bichromatic probe (≈ 7 nm bandwidths centered at 770 nm (ω₁) and 820 nm (ω₂)). The delay step is 200 as. The total acquisition time of the scan is $\sim 13$ hours. (b) Frequency components of oscillations in SB24 (black line) and H23 (blue line).

The KRAKEN protocol requires a tunable bichromatic IR probe at a variable delay relative to the XUV pulses. The density matrix of electron wave packets can be reconstructed by performing several long delay scans (typically a few hundreds of fs) with different frequency differences (${\omega }_1-{\omega }_2$) for the bichromatic probe [65]. This motivated the design and commissioning of the ultrastable and flexible setup presented in this work. As indicated in the simulated results presented in Figure 11 [65], both the purity of the quantum state and the degree of electron-ion entanglement (concurrence) can be measured by performing KRAKEN measurements. The purity, $\text{tr}({\rho }_{XUV}^2)$, is calculated by summing over the density matrix, as given in Figure 11(b),(c), and (d), while the concurrence, $C=\sqrt{2[1-tr({\rho }_{XUV}^2)]}$, describes the entanglement between ions and electrons after ionization (see ref [65] for details). Figure 11 also shows that these quantities can be controlled in a simple way, by varying the bandwidth of the XUV radiation, which can easily be done in our setup using the spectral filter implemented in the pump path. The experimental realization of the KRAKEN method will be discussed in more detail in a future article.

Figure 11. Entanglement between ion and photoelectron in argon. (a) The purity and concurrence of electron from a direct calculation (blue and red lines) and KRAKEN reconstruction (blue and red crosses). (b)-(d) are the reconstructed density matrices of photoelectron from KRAKEN protocol with XUV bandwidths at 0.14 eV, 0.21 eV, and 0.35 eV [65].

4. Summary

In this article, a versatile, ultra-stable, high-spectral resolution setup for attosecond experiments based on a Mach-Zehnder interferometer was presented. An active stabilization system, which uses parts of the pump and probe femtosecond laser beams, minimizes the effects of fluctuations and drifts. It allows for the control of the delay between the pump- and probe arms with an RMS error of 13 as over several hours. Novel experimental schemes are enabled by independent spectral tuning of the femtosecond pulses in the pump- and probe arms, in particular allowing for bi- and polychromatic probes.

The capabilities of the setup aredemonstrated in three different types of experiments. The first one is a long RABBIT (interferometric) measurement showing high stability for several hours over a large delay range of 70 fs. The second demonstrates the high spectral resolution that can be obtained by using a 10 nm bandpass spectral filter in the probe path. A RABBIT measurement, performed in argon, with H17 close to the 3s⁻¹4p autoionizing state, shows excellent spectral resolution allowing for the observation of spin-orbit splittings and Fano resonances. The analysis of the data using the Rainbow RABBIT method allows us to precisely map out the phase through resonances.

Finally, the flexibility of our setup is exemplified by performing a scan using a bichromatic probe field, obtained with a 4f-shaper. This will be useful for the implementation of new characterization techniques, for the complete characterization of mixed photoelectron quantum states on the attosecond timescale. In the future, full shaping capability of the probe pulses in terms of amplitude and phase can be obtained by the integration of a spatial light modulator.

Acknowledgments

AL and MA are partly supported by the Wallenberg Center for Quantum Technology (WACQT) funded by the Knut and Alice Wallenberg foundation.

Funding

The work was supported by the H2020 European Research Council [advanced grant QPAP, 884900]; Knut och Alice Wallenberg Foundation [Wallenberg Center for Quantum Technology (WACQT)]; Knut och Alice Wallenberg Foundation [Attosecond chronoscopy of electron wave-packets probing entanglement and time-ordering of quantum processes]; Swedish Research Council [2020-0520, 2020-06384, 2018-03731, 2016-04907, 2013-8185, 2021-04691].

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The work was supported by the H2020 European Research Council [QPAP, 884900]; Knut och Alice Wallenbergs Stiftelse [Wallenberg Center for Quantum Technology (WACQT)]; Knut och Alice Wallenbergs Stiftelse [Attosecond chronoscopy of electron wave-packets probing entanglement and time-ordering of quantum processes]; Vetenskapsrådet [2020-0520]; Vetenskapsrådet [2020-06384]; Vetenskapsrådet [2018-03731]; Vetenskapsrådet [2016-04907]; Vetenskapsrådet [2013-8185]; Vetenskapsrådet [2021-04691].