Introduction

Coherent phenomena are an intriguing aspect of the quantum world. Their dynamics reveal rich information on a quantum system including ultrafast decay mechanisms and couplings to its environment1. To probe coherent dynamics in real-time, specialized ultrafast methods are necessary which rely on interferometric measurements, mapping the oscillatory phase evolution of excited coherences1. This is most challenging for the electronic degrees of freedom where oscillation periods scale inversely with excitation energy and thus require extremely high timing stability in the atto- to zeptosecond range. Yet, to capture the full dynamical picture of a system, it is essential to track the evolution of all degrees of freedom including electronic coherences2.

The extension of coherent time-resolved spectroscopy to extreme ultraviolet (XUV) photon energies is highly desirable as it offers unprecedented site-specificity by accessing localized inner-shell states and opens the door to attosecond time resolution3,4,5. However, these experiments are extremely challenging due to the required ultra-high phase stability and the lack of phase-matching/cycling schemes necessary to isolate subtle coherence signals6. For these reasons, a real-time study of the evolution of an XUV electronic coherence has not been reported and only few examples of quantum beat and vibrational wave packet (WP) studies have been demonstrated so far7,8,9.

Coherent control in the XUV spectral range is a second field of high interest, which likewise relies on the preparation and probing of coherent states of matter. In a bichromatic approach, coherent control was achieved by controlling the relative delay between two XUV pulses10. Another, more general approach, relies on phase manipulation of the XUV pulses. Pulse shaping technology, only available in the infrared (IR) to ultraviolet (UV) spectral range, has enabled advanced control schemes, with applications in nonlinear optics and the steering of chemical reactions11. In the XUV, phase manipulation has been indirectly shown by varying the chirp12 or polarization13 of the driving field. However, direct and independent manipulation of the relative phase and delay of XUV pulses in a pulse sequence has not been accomplished so far.

In this work, we implement a phase modulation technique for XUV pulse sequences, which facilitates both flexible coherent control schemes, and advanced coherent nonlinear spectroscopy techniques. To this end, phase-locked XUV pulse pairs with full control over delay and relative phase were prepared at the FERMI free-electron laser14 (FEL) using twin seeding15 combined with phase-manipulated seed pulses (Fig. 1). The intense phase-locked UV seed pulse pairs are created with a highly stable interferometer based on a monolithic design16, and are used to seed the FEL process by means of the high-gain harmonic generation (HGHG) scheme. This leads to the emission of fully coherent XUV pulse pairs at a specific harmonic of the seed wavelength17.

Results

Independent phase and timing control of XUV pulse pairs

Previous experiments indicated the possibility of XUV phase manipulation through the seed pulse properties12,15. In the current work, we significantly advance this approach by implementing high precision, decoupled phase and timing control of XUV pulse pairs while avoiding the challenge of phase shaping at XUV wavelengths. For this purpose, two phase-locked acousto-optic modulators (AOMs) control the relative phase ϕ21 = ϕ2 − ϕ1 of the seed pulses (Fig. 1). Upon HGHG, the imprinted phase transfers to a well-defined phase shift of 21 for the XUV pulses at the nth harmonic, enabling flexible phase manipulation (Fig. 2a), while the applied twin-seeding concept allows for even higher pump-probe timing control than advanced XUV split-and-delay units18 and is not restricted to bichromatic pump-probe schemes10.

The XUV phase control is demonstrated by manipulating the phase of XUV Ramsey-type interference fringes for photon energies up to 47.5 eV (Fig. 2b, c). The high quality of timing and phase control is directly reflected in the interferograms. The extremely stable seed laser interferogram indicates that the phase stability is only limited by fluctuations picked up during the HGHG process. Here, timing jitter between seed pulses and the electron bunch (≈42 fs RMS) leads to phase jitter of the produced XUV pulses due to the residual longitudinal energy chirp of the electron bunch12,15 (see Supplementary Fig. 1). However, when using echo-enabled harmonic generation, this noise source can be efficiently reduced, as in this process the spectrotemporal properties of the XUV pulses are significantly less sensitive to electron-bunch jitter than in HGHG19. Note that our concept allows for the independent control of both timing and phase properties of the XUV pulses. Combined with multiple-pulse seeding, many pulse shaping and nonlinear spectroscopy applications can be realized now at XUV wavelengths.

Tracking XUV electronic coherences in helium

One type of applications of this concept are advanced coherent spectroscopy schemes based on phase-cycling concepts20. Coherent multipulse spectroscopy schemes (for example two-dimensional spectroscopy, transient grating spectroscopy) require selective, background-free detection of the weak nonlinear signals. Phase-cycling is an effective concept for highly selective detection which has, in the IR to UV spectral range, drastically improved the detection sensitivity and paved the way for a plethora of nonlinear spectroscopy applications21,22. In the XUV, these experiments are extremely challenging, as they involve sequences of multiple phase-locked XUV pulses with independently controllable timing and phase parameters. Here, we demonstrate such an experiment by tracking the temporal evolution of attosecond electronic coherences with phase-modulated XUV pulse sequences.

The model system examined is the 1s² → 1s4p transition in helium (Fig. 3a). The first XUV pulse creates a coherent superposition of ground and excited state (electronic WP), denoted |ψ(t)〉. The second XUV pulse, delayed by τ, projects this WP onto a stationary population state, which is probed by ionization of the 1s4p state with a near infrared (NIR) pulse, yielding the signal

$$\begin{array}{*{20}{c}} {S \propto \left\langle {\psi \left( \tau \right){\mathrm{|}}1{\mathrm{s}}4{\mathrm{p}}} \right\rangle = A\left( \tau \right)e^{i\phi \left( \tau \right)}} \end{array}$$
(1)

where A(τ) denotes the amplitude and ϕ(τ) the phase evolution of the WP. According to the 1s² → 1s4p transition energy E= 23.74 eV, the signal oscillates with a period of h/E = 174 as, requiring an extremely high pump-probe timing stability of δτ < 20 as (corresponding to an optical path stability of δOP < 6 nm) for adequate sampling.

Here, we apply a specialized phase-cycling scheme to solve this problem. By combining phase modulation of both XUV pulses with phase-synchronous lock-in detection, we are able to downshift the signal oscillation period by a factor of >50 (rotating frame detection20) and remove most of the phase jitter from the signal while improving the overall sensitivity through efficient lock-in amplification (details on the phase-modulation scheme and the rotating frame detection can be found in the Methods section). Figure 3b shows the time-domain interferogram recorded for the helium excitation, exhibiting clean periodic oscillations of the induced attosecond electronic WP in excellent agreement with theory (Fig. 3c). The data quality is remarkable considering the applied low FEL energy (≤30 nJ) and the probed low atom density in the sample, which is several orders of magnitudes lower than in typical transient absorption experiments23,24. This shows the high signal-to-noise performance and sensitivity of the method, even with the challenging XUV wavelength conditions.

The signal quality furthermore allows for direct Fourier analysis to gain spectral information (Fig. 3d), yielding a frequency spectrum with a signal to noise ratio of 10. Careful preparation of the electron bunch combined with the greatly reduced acquisition times due to rotating frame sampling allowed us to monitor the WP oscillations up to 700 fs, leading to a high spectral resolution in the Fourier domain (Gaussian RMS width σ = 6 meV). This is even slightly better than achieved in XUV transient absorption experiments23,24. In addition, the Fourier transform approach combines spectral resolution with highly selective detection types, here demonstrated for mass-resolved photoion and energy-gated photoelectron detection. Furthermore, the phase-cycling lock-in detection scheme enables highly efficient filtering of background signals in addition to mass/energy gating. As such, we were able to uncover WP signals that are up to a factor of 200 smaller than the background count rates inside the selected ion-mass/electron-energy windows.

At short delays where both seed pulses overlap temporally (<150 fs), twin-pulse seeding is expected to break down due to the nonlinear response of the electron density modulation initiating the FEL process to optical interference effects, causing shot-to-shot amplitude modulation of the FEL radiation. This explains the amplitude drop of our data at <150 fs. Interestingly, we can still observe a clear oscillation and a stable signal phase (not shown), indicating that information could also be gained from this region.

Phase-resolved real-time dephasing of a Fano resonance

The helium study served as a model for an unperturbed quantum system exhibiting long-lived electronic coherences and negligible dephasing. In argon, we investigate a different situation for the 3s23p6 → 3s13p66p1 transition. The 6p valence orbital couples to the Ar+ continuum via configuration interaction (Fig. 4a), opening-up an autoionizing pathway that introduces significant dephasing. Thereby, the phase shift between direct and autoionization pathways leads to characteristic Fano profiles25. Recently, a two-electron WP was fully reconstructed with the imprinted phase information in the Fano line shape24. Moreover, the scattering behavior of an electron WP in the vicinity of a Fano resonance has been investigated26,27. On the other hand, the dephasing of the WP in its quasi-bound state as it tunnels into the continuum has not been experimentally accessible so far. Here, we provide a full characterization of this case by tracking directly the time evolution of the Ar 3s-6p electronic coherence and resolving the amplitude A(τ) and phase function ϕ(τ) of the excited WP. Note, that ϕ(τ) is here directly obtained from experimental data without the need of an iterative algorithm or a theoretical model (details in Supplementary Note 1).

Figure 4b shows the recorded time domain transients from which the complex-valued WP signal $$S(\tau ) = A\left( \tau \right)e^{i\phi \left( \tau \right)}$$ is constructed. Here, the signal decay directly reflects the tunneling of the WP into the continuum. Its Fourier transform S(ω) is connected to the susceptibility of the sample28 $${\upchi}\left( \omega \right) \propto iS(\omega )$$. Note the imaginary unit in the previous equation, a consequence of the phase modulation and detection scheme (see also ref. 28). Hence, we simultaneously obtain the absorption $$\left( { \propto Re\left[ {{\mathrm{S}}\left( \omega \right)} \right]} \right)$$ and dispersion $$\left( { \propto - Im\left[ {{\mathrm{S}}\left( \omega \right)} \right]} \right)$$ curves of the resonance (Fig. 4c). The absorptive part shows the typical Fano line shape (theoretical derivation in Supplementary Note 2 and Supplementary Fig. 2) in good agreement with synchrotron measurements29. Note that absorptive and dispersive parts of the susceptibility are independently retrieved in the measurement without requiring the Kramers-Kronig relation. Such full characterization of the susceptibility is particularly challenging in the XUV spectral range and can be beneficial in spectral regions where many resonances are congested, for instance at absorption edges30.

From χ(ω) we obtain the spectral phase ϕ(ω). It resembles the characteristic of a damped driven harmonic oscillator (Fig. 4c), implying the measurement’s sensitivity to the bound character of the electron WP while it coherently oscillates in the Coulomb potential of the atomic core, in contrast to refs. 26,27. This is in accord with the fact that we track the electronic dephasing relative to the system’s steady ground state. This dynamic has been so far not accessible due to the required high phase stability to probe the attosecond coherent oscillations between excited and ground state. In refs. 26,27 the electron WP generated at a Fano resonance was characterized relative to a reference continuum WP which makes these experiments sensitive to the continuum state of the generated WP. This explains the different shape of the spectral phase retrieved in these experiments. We note that in the presented measurement, the uncertainty of the absolute phase is still ≈1 rad, but with appropriate calibration of the apparatus, uncertainties < 0.1 rad are feasible in future experiments (for more details on the phase calibration see “Methods” section).

Discussion

To the best of our knowledge, resolving electronic coherences spanning over 28.51 eV (i.e., 145 as oscillation period) in a time-resolved experiment has not been achieved so far, which in the studied example enabled direct time-domain observation of the dephasing of an inner subshell—valence shell coherence. While electronic dephasing is studied in many examples in the visible, in the XUV range it may provide selective information about the coupling of a specific site to the environment or real-time information about intra- and inter-particle decay mechanisms. Furthermore, the phase-sensitive detection provides information about the spectral phase of electron WPs without the need to prepare a second, known reference WP in the system. This will be of advantage when studying the WP evolution in complex molecular and solid state systems.

In conclusion, we introduced a highly sensitive phase-cycling-based coherent spectroscopy technique at XUV wavelengths and tracked the attosecond time evolution of electronic WPs in a phase-resolved fashion. Scaling photon energies using echo-enabled harmonic generation19 will allow localized core states to be addressed. In addition, the unprecedented sensitivity of our method provides ideal conditions for applications with tabletop high harmonic generation (HHG) sources. The generation of phase-locked XUV pulse pairs with HHG using phase-locked seeding has already been demonstrated31. Thus, an extension of our method to HHG sources is experimentally at hand. Our phase-cycling method combined with coherent XUV light sources, introduces a versatile XUV spectroscopy toolbox for the wide range application of specialized nonlinear spectroscopy schemes, so far only accessible at visible wavelengths. Specifically designed detection protocols to address fundamental problems of ultrafast non-adiabatic dynamics32, many-body phenomena6, or the control of chemical reactions along complex energy landscapes may now be realized in the XUV regime.

Methods

FEL setup and sample preparation

The experiments were performed at the low-density matter (LDM) beamline endstation33 at the FEL FERMI. We tuned the FEL photon energy to  = 23.74 eV (fifth harmonic) for helium, and 28.51 eV (sixth harmonic) for argon. The FEL pulse duration estimated from the spectral bandwidth was 57 ± 4 fs and 45 ± 3 fs for the fifth and sixth harmonic respectively, which is in good agreement with the predictions from literature34. The helium beam was generated by supersonic expansion through a pulsed nozzle, yielding a density on the order of 1013 cm−3 in the interaction region. The argon beam was effused from a gas cell, yielding a much lower density of ≈109 cm−3. In order to avoid saturation of the detection electronics and space charge effects, the FEL energies were adjusted using metal filters, yielding Epulse ≤ 30 nJ and Epulse ≈ 10 µJ per pulse for the helium and the argon study, respectively. The FEL repetition rate was 50 Hz. In order to provide similar parameters of the electron bunch for both seed pulses at all delays, the longitudinal beam properties should be as uniform as possible. This was achieved by reducing the compression of the electron bunch, yielding a sufficiently constant region along the bunch, allowing for a delay range of ≈1 ps. The XUV pulses were focused into the atomic beam to a diameter of 70 µm FWHM.

Phase-modulated coherent time domain XUV spectroscopy

The seed laser pulses (λ ≈ 261 nm, Δt = 100 fs FWHM) are split in a Mach-Zehnder interferometer (MZI), creating phase-locked pulse pairs with controllable delay τ (Fig. 1). Two AOMs are driven with continuous-wave radio frequencies in phase-locked mode. Each driving frequency can be controlled with 0.12 Hz frequency and 380 µrad phase resolution. By manipulating frequency and phase of the driving fields high precision control of the relative phase ϕ21 between the seed pulses is achieved. We note that in comparison to established pulse-shaping technologies, this phase control approach can handle the required high seed pulse energies (>10 µJ) and provides a high purity, artifact-free phase manipulation. This is crucial for the application in strongly nonlinear processes such as HHG, where any phase shaping distortion is amplified by the nonlinear process.

To generate the phase-controlled XUV pulse sequences, the phase-manipulated twin-seed pulses interact with a relativistic electron bunch in the modulator of the FEL. Thereby amplitude and phase of the twin-seed pulse are imprinted on the longitudinal phase space of the electron bunch. This transfers to an electron density modulation in a dispersive chicane, finally leading to the emission of an XUV pulse pair at a specific harmonic of the seed wavelength in subsequent undulators.

We use the phase modulation scheme developed by Marcus and coworkers28, to perform phase-modulated coherent time domain XUV spectroscopy. Here, ϕ21 is incremented on a shot-to-shot basis, leading to a quasi-continuous modulation of ϕ21 at a frequency of Ω ≈ 3 Hz. Via the HGHG process, the relative phase of the XUV pulses transfers to 21, leading to a phase modulation at a frequency of nΩ. The XUV pulse pair creates an electronic WP |ψ〉 between the ground state |g〉 and excited state |e〉. We monitor the temporal evolution of the WP by ionization out of the excited state. In the case of helium, this was realized with an additional synchronized NIR laser, while argon ions were obtained by autoionization. The evolution of the WP gives rise to delay-dependent oscillations in the ion yield $$S\left( \tau \right) = A\left( \tau \right){\mathrm{cos}}( {\omega _{{\mathrm{eg}}}\tau + \phi _0} )$$. Here ωeg is the transition frequency from the ground to the excited state (ωeg = 3.61 × 1016 rad s−1 for the excited helium resonance), and ϕ0 is a phase shift determined by the parameters of the WP. The phase modulation imparts an additional real-time modulation on the ion yield: $$S( {\tau ,t} ) = A( \tau ){\mathrm{cos}}( {\omega _{{\mathrm{eg}}}\tau + \phi _0 - n{\mathrm{\Omega }}t} )$$. Simultaneously, we collinearly trace the MZI with a narrowband 266 nm continuous-wave laser, which undergoes the same phase modulation and records all jitter inside the MZI, yielding the reference signal for lock-in detection: $$R( {\tau ,t} ) = {\mathrm{cos}}(\omega _{{\mathrm{ref}}}\tau - {\mathrm{\Omega }}t)$$, where ωref = 7.08 × 1015 rad s−1 is the optical frequency of the reference laser. Referencing the lock-in amplifier to a harmonic of R (same harmonic as in the HGHG process) and demodulating S therewith yields the in-phase (real component) X(τ) and in-quadrature components (imaginary component) Y(τ) of S = X + iY, from which the signal amplitude $$A = \sqrt {X^2 + Y^2}$$ and phase ϕ = arctan (Y/X) are readily constructed. From this we reconstruct the complex-valued WP signal $$S\left( \tau \right) = A(\tau ){\mathrm{exp}}( {i\bar \omega _{{\mathrm{eg}}}\tau + i\phi _0} )$$, with $$\bar \omega _{{\mathrm{eg}}} = \omega _{{\mathrm{eg}}} - n\omega _{{\mathrm{ref}}}$$.

Note, that we detect here the attosecond oscillations ωeg of the WP signal in a reference frame rotating at frequency eg, a technique commonly referred to as rotating frame detection20. This leads to a downshift of the signal frequency: $$\omega _{{\mathrm{eg}}} \to \bar \omega _{{\mathrm{eg}}}$$, which, for the case of the helium measurement $$( {\bar \omega _{{\mathrm{eg}}} = 6.93 \times 10^{14}{\mathrm{rad}}\,{\mathrm{s}}^{ - 1}} )$$ corresponds to a downshift factor of 52. This explains the $$2\pi /\bar \omega _{{\mathrm{eg}}} \approx 9\,{\mathrm{fs}}$$ period of the signal in Fig. 3b, c instead of a 2π/ωeg = 174 as period and allows for much larger delay steps. Similar numbers apply for the argon measurment. Furthermore, all phase jitter arising from the MZI cancels efficiently during the demodulation process, which considerably improves the signal quality.

Spectral information is deduced by Fourier transforming the downshifted quantum oscillations FT(S(τ)). Here the real and imaginary parts correspond to the absorptive and dispersive parts of the line shape, respectively28. The phase calibration of the apparatus was done with the helium measurements following the routine in ref. 28. In between the helium and argon measurements, the delay zero position of the seed interferometer shifted due to a technical failure. The shift was inferred in the post processing from the seed laser Ramsey-type interference fringes which lead to a larger uncertainty than for the helium measurements, where time zero was deduced from seed laser autocorrelation measurements. This explains the relatively large uncertainty of the absolute phase mentioned in the main text. More details of the optical seed laser phase modulation setup and the harmonic lock-in demodulation scheme can be found in refs. 16,35, respectively.