Tag: Atomic and molecular interactions with photons

Strong-field quantum control in the extreme ultraviolet domain using pulse shaping

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Strong-field phenomena play an important part in our understanding of the quantum world. Light–matter interactions beyond the perturbative limit can substantially distort the energy landscape of a quantum system, which forms the basis of many strong-field effects⁸ and provides opportunities for efficient quantum control schemes¹¹. Moreover, resonant strong coupling induces rapid Rabi cycling of the level populations¹², enabling complete population transfer to a target state². The development of intense extreme ultraviolet (XUV) and X-ray light sources has recently led to the investigation of related phenomena beyond valence electron dynamics, in highly excited, multi-electron and inner-shell electron states^{9,10,13,14,15,16,17}. Yet in most of these studies, the dressing of the quantum systems was induced by intense infrared fields overlapping with the XUV and X-ray pulses. In contrast, the alteration of energy levels directly by short-wavelength radiation is more difficult. So far, only a few studies have reported XUV-induced AC-Stark shifts of moderate magnitude (≲100 meV), difficult to resolve experimentally^9,18,19,20.

Another important step in exploring and mastering the quantum world is the active control of quantum dynamics with tailored light fields^21,22,23. At long wavelengths, sophisticated pulse-shaping techniques facilitate the precise quantum control and even the adaptive-feedback control of many light-induced processes, in both weak- and strong-field regimes^{24,25,26,27,28}. Several theoretical studies have pointed out the potential of pulse shaping in XUV and X-ray experiments^29,30,31. As an experimental step in this direction, phase-locked monochromatic and polychromatic pulse sequences have been generated^32,33,34,35. Using this tool, coherent control demonstrations in the perturbative limit^32,35,36 and the generation of intense attosecond pulses were achieved³⁷. Moreover, ultrafast polarization shaping at XUV wavelengths³⁸ and chirp control for the temporal compression of XUV pulses³⁹ were recently demonstrated. However, spectral phase shaping, which forms the core of pulse-shaping techniques, has not been demonstrated for the control of quantum phenomena at short wavelengths. Here we establish spectral phase shaping of intense XUV laser pulses and demonstrate high-fidelity quantum control of the Rabi and photoionization dynamics in helium.

In the experiment, He atoms are dressed and ionized by intense coherent XUV pulses (I > 10¹⁴ W cm⁻²) delivered by the seeded FEL FERMI (Fig. 1a). The high radiation intensity causes a strong dressing of both the bound states in He and the photoelectron continuum, whereas the dynamics of the quantum system are still in the multiphoton regime (Keldysh parameter γ = 11). By contrast, the dynamics of a system dressed with near-infrared (NIR) radiation of comparable intensity would be dominated by tunnel and above barrier ionization (γ = 0.35) (ref. ⁸). Hence, the use of short-wavelength radiation provides access to a unique regime, in which the interplay between strongly dressed bound states and a strongly dressed continuum can be studied.

Fig. 1: XUV strong-field coherent control scheme.

a, Intense XUV pulses dress the He 1s², 1s2p states and the electron continuum. E_± labels indicate the bound dressed states correlated to the 1s2p bare state. Mixing of p- and d-waves in the dressed continuum results in different coupling strengths to the dressed bound states (indicated by the thickness of the arrows). b,c, In the time domain, the AT splitting follows the intensity profile of the XUV field (middle). The dressed-state populations are monitored in the photoelectron eKE distributions (top). XUV pulse shaping enables the control of the non-perturbative quantum dynamics (bottom). For a flat phase ϕ (no chirp), both the excited dressed states are equally populated. For a positive phase curvature (up chirp), the population is predominantly transferred to the lower dressed state and the upper state is depleted, whereas for negative curvature (down chirp), the situation is reversed. d, Principle of XUV pulse shaping at the FEL FERMI. Intense seed laser pulses overlap spatially and temporally with the relativistic electron bunch in the modulator section of the FEL, leading to a modulation in the electron phase space. The induced energy modulations are converted into electron-density oscillations on passing a dispersive magnet section. The micro-bunched electrons then propagate through a section of radiator undulators, producing a coherent XUV pulse. In this process, the phase function of the seed pulse is coherently transferred to the XUV pulse, resulting in precise XUV phase shaping. The FEL pulses are focused on the interaction volume, exciting and ionizing He atoms. The photoelectrons are detected with a magnetic bottle electron spectrometer (MBES).

To dress the He atoms, we induce rapid Rabi cycling of the 1s² → 1s2p atomic resonance with a near-resonant field E(t). The generalized Rabi frequency of this process is $\varOmega ={\hbar }^{-1}\sqrt{{(\mu E)}^{2}+{\delta }^{2}}$, where μ denotes the transition dipole moment of the atomic resonance, δ the energy detuning and $\hbar $ the reduced Planck constant. In the dressed-state formalism, the eigenenergies of the bound states depend on the field intensity and show the characteristic Autler–Townes (AT) energy splitting ΔE = ħΩ (ref. ⁴⁰). The observation of this phenomenon requires the mapping of the transiently dressed level structure of He while perturbed by the external field⁴¹. This is achieved by immediate photoionization over the course of the femtosecond pulses, thus projecting the time-integrated energy level shifts onto the electron kinetic energy (eKE) distribution (Fig. 1b).

Analogous to the bound-state description, the dressed continuum states are obtained by diagonalization of the corresponding Hamiltonian. The hybrid electron–photon eigenstates consist of a mixing of partial waves with different angular momenta, which alters the coupling strength to the dressed bound states of the He atoms (Fig. 1a).

Figure 2 demonstrates experimentally the dressing of the He atoms. The build-up of the AT doublet is visible in the raw photoelectron spectra as the XUV intensity increases (Fig. 2a). The evolution of the AT doublet splitting is in good agreement with the expected square-root dependence on the XUV intensity $\Delta E=\mu \sqrt{2{I}_{{\rm{eff}}}/({{\epsilon }}_{0}c)}$. Here, I_eff denotes an effective peak intensity, accounting for the spatially averaged intensity distribution in the interaction volume, ϵ₀ denotes the vacuum permittivity and c denotes the speed of light. The data can be thus used for gauging the XUV intensity in the interaction volume, a parameter otherwise difficult to determine. At the maximum XUV intensity, the photoelectron spectrum shows an energy splitting exceeding 1 eV, indicative of substantial AC-Stark shifts in the atomic level structure. The large AT splitting further implies that a Rabi flopping within 2 fs is achieved, offering a perspective for rapid population transfer outpacing possible competing intra- and inter-atomic decay mechanisms, which are ubiquitous in XUV and X-ray applications.

Fig. 2: Build-up of the AT splitting in He atoms.

a, Detected photoelectron eKE distribution (raw data) as a function of the XUV intensity (FEL photon energy: 21.26 eV, GDD = 135 fs²). Dashed lines show the calculated AT splitting for an effective XUV peak intensity I_eff accounting for the spatial averaging in the interaction volume. b,c, Photoelectron spectra as a function of photon energy recorded for high XUV intensity (I_eff = 2.92(18) × 10¹⁴ W cm⁻²) (b) and for lower intensity (I_eff ≈ 10¹³ W cm⁻²) (c). In b, an avoided crossing between the lower and higher AT band is visible directly in the raw photoelectron spectra. The photoelectron distribution peaking at eKE = 17.9 eV in a and b is ascribed to He atoms excited by lower XUV intensity (see text).

Figure 2b,c shows the photoelectron yield as a function of excitation photon energy. For high XUV intensity (Fig. 2b), the photoelectron spectra show an avoided level crossing of the dressed He states as they are mapped to the electron continuum (see also Fig. 4). Accordingly, at lower XUV intensity (Fig. 2c), the avoided crossing is not visible anymore. In the latter, the eKE distribution centres at 17.9 eV. In Fig. 2b, a similar contribution appears at the same kinetic energy that overlays the photoelectrons emitted from the strongly dressed atoms. Likewise, a notable portion of photoelectrons at eKE ≈ 17.9 eV in Fig. 2a does not show a discernible AT splitting. We conclude that a fraction of He atoms in the ionization volume are excited by much lower FEL intensity, which is consistent with the aberrated intensity profile of the FEL measured in the ionization volume (Extended Data Fig. 1). This overlapping lower intensity contribution does not influence the interpretation of the results in this work. For better visibility of the main features, we thus subtract this contribution from the data shown in Figs. 3 and 4.

Fig. 3: Strong-field quantum control of dressed He populations.

a, Photoelectron spectra obtained for phase-shaped XUV pulses (see labels for GDD values; photon energy = 21.25 eV; I_eff = 2.8(2) × 1,014 W cm⁻²). The control of the dressed-state populations is directly reflected in the relative change of amplitude in the photoelectron bands. The small peak at 18.13 eV results from imperfect removal of the lower intensity contribution from the aberrated focus. b, Calculations of the time-dependent Schrödinger equation for a single active electron (TDSE-SAE) and a single laser intensity corresponding to the experimental I_eff = 2.8 × 10¹⁴ W cm⁻² (dark colours). Spectral fringes reflect here the temporal progression of the Rabi frequency during the light–matter interaction. The broadened photoelectron spectra (light colours) account for experimental broadening effects caused by the focal intensity averaging and the instrument response function. a.u., arbitrary units.

Fig. 4: Energy-domain representation of the quantum control scheme.

a, Photoelectron spectra as a function of energy detuning for different GDD values as labelled (I_eff = 2.92(18) × 10¹⁴ W cm⁻²). b, TDSE-SAE calculations. Broadening by the instrument response function is omitted in the model. c, Amplitude ratio between the upper and lower photoelectron bands evaluated at the 1s² → 1s2p resonance; hence, δ = 0. Experimental data (red), TDSE-SAE model treating the bound and continuum dynamics non-perturbatively (blue) and TDSE-SAE model applied to the bound-state dynamics, but treating the continuum perturbatively (yellow). d, Dependence of the He ionization rate on the spectral phase of the driving field. Data (red) and TDSE-SAE model (blue). a.u., arbitrary units.

The demonstrated dressing of He atoms provides the prerequisite for implementing the strong-field quantum control scheme (Fig. 1b,c). The main mechanism underlying the control scheme is described in the framework of the selective population of dressed states (SPODS), which is well established in the NIR spectral domain²⁸. Here, we extend SPODS to the XUV domain and include a new physical aspect—that is, the transition of the bound atomic system into a strongly dressed continuum. In SPODS, a flat phase leads to an equal population of both dressed states in the excited state manifold of helium; a positive phase curvature results in a predominant population of the lower dressed state and a negative phase curvature results in a predominant population of the upper dressed state (Fig. 1c). The scheme has been experimentally demonstrated with long-wavelength radiation⁴², in which pulse-shaping techniques are readily available. However, the opportunities for pulse-shaping technologies are largely unexplored for XUV and X-ray radiation.

We solve this problem by exploiting the potential of seeded FELs to allow for the accurate control of XUV pulse properties^39,43. These demonstrations have been so far limited to applications of temporal compression and amplification of the FEL pulses. By contrast, the deterministic control of quantum dynamics in a material system involves many more degrees of freedom, which makes the situation considerably more complex. The seeded FEL FERMI operation is based on the high-gain harmonic generation (HGHG) principle⁴⁴, in which the phase of an intense seed laser pulse is imprinted into a relativistic electron pulse to precondition the coherent XUV emission at harmonics of the seed laser (Fig. 1d). For FEL operation in the linear amplification regime, the phase ϕ_nH(t) of the FEL pulses emitted at the n’th harmonic of the seed laser follows the relationship³⁹

$${\phi }_{n{\rm{H}}}(t)\approx n[{\phi }_{{\rm{s}}}(t)+{\phi }_{{\rm{e}}}(t)]+{\phi }_{{\rm{a}}}.$$

(1)

Here, ϕ_s denotes the phase of the seed laser pulses, which can be tuned with standard pulse-shaping technology at long wavelengths (Methods); ϕ_e accounts for the possible phase shifts caused by the energy dispersion of the electron beam through the dispersive magnet and is negligible for the parameters used in the experiment; and ϕ_a accounts for the FEL phase distortion due to the amplification and saturation in the radiator and has been kept negligibly small by properly tuning the FEL (Methods). Although complex phase shapes may be implemented with this scheme, for the current objective of controlling the strong-field induced dynamics in He atoms, shaping the quadratic phase term (group delay dispersion (GDD)) is sufficient⁴². Therefore, we focus on the GDD control in the following discussion.

Figure 3 demonstrates the quantum control of the dressed He populations. The eKE distribution shows a pronounced dependence on the GDD of the XUV pulses (Fig. 3a). At minimum chirp (GDD = 135 fs²), we observe an almost even amplitude in the AT doublet, whereas for GDD < 0, the higher energy photoelectron band dominates; for GDD > 0, the situation is reversed. These changes directly reflect the control of the relative populations in the upper and lower dressed states of the He atoms. We obtain an excellent control contrast and the results are highly robust (Extended Data Fig. 2), which is remarkable given the complex experimental setup.

The experiment is in good agreement with the theoretical model (Fig. 3b) numerically solving the time-dependent Schrödinger equation for a single active electron (TDSE-SAE; Methods). To account for experimental broadening effects, we calculated the photoelectron spectra for a single intensity (corresponding to the experimental I_eff) and including the focal intensity average present in the experiment (Methods). All salient features of the experiment are well reproduced. The control of the dressed-state populations is in very good qualitative agreement. The different widths and shapes of the photoelectron peaks are qualitatively well-matched between the experiment and the calculations. The difference in the AT energy splitting between the experiment (ΔE_exp ≈ 1.02 eV) and theory (ΔE_theo = 0.74 eV) is in good agreement with the fact that the model underestimates the transition dipole moment of the 1s² → 1s2p transition by a factor of 1.4 (Methods).

The high reproducibility, the excellent control contrast and the good agreement with theory confirm the feasibility of precise pulse shaping in the XUV domain and of quantum control applications, even of transient strong-field phenomena. This is an important achievement in view of quantum optimal control applications at short wavelengths.

The implemented control scheme is not restricted to adiabatic processes²⁸. In our experiment, the dynamics are adiabatic only for the largest frequency chirp (GDD = −1,127 fs²) (Extended Data Fig. 3). However, this also shows that the condition for rapid adiabatic passage² can be generally reached with our approach, offering a perspective on efficient population transfer in the XUV and potentially in the soft X-ray regime.

The active control of quantum dynamics with tailored light fields is an asset of pulse shaping. As another asset, systematic studies with shaped laser pulses can be used to uncover underlying physical mechanisms that are otherwise hidden. Here, we demonstrate this concept for pulse shaping in the XUV domain. The high XUV intensities used in our study lead to a peculiar scenario in which both bound and continuum states are dressed and a complex interplay between their dynamics arises. Hence, for a comprehensive understanding of the strong-field physics taking place, the bound-state dynamics and the non-perturbative photoionization have to be considered. This is in contrast to the strong-field control at long wavelengths, for which the continuum could be described perturbatively⁴².

Figure 4a,b shows the avoided crossing of the photoelectron bands for different spectral phase curvatures applied to the XUV pulses. The experimental data show a clear dependence of the AT doublet amplitudes on the detuning and the GDD of the driving field, in good agreement with the theory. In the strong dressing regime, the bound–continuum coupling marks a third factor that influences the photoelectron spectrum. As predicted by theory, the strong-field-induced mixing of continuum states (Fig. 1a) leads to different photoionization probabilities for the upper and lower dressed states of the bound system⁴⁵. This is in agreement with the prevalent asymmetry of the AT doublet amplitudes observed in our data and calculations (Fig. 4a,b). An analogous effect is observed for the strong-field bound–continuum coupling in solid state systems⁴⁶.

To disentangle this strong-field effect from the influence of the detuning and spectral phase of the driving field, we evaluate the amplitude ratio between the upper and lower photoelectron bands at detuning δ = 0 eV (Fig. 4c). Interpolation to GDD = 0 fs² isolates the asymmetry solely caused by the strong-field bound–continuum coupling. We find reasonable agreement with our model when including the dressing of the ionization continuum (blue curve), in stark contrast to the same model but treating the continuum perturbatively (yellow curve). Hence, the dressing of the He atoms provides a probe of the strong-field dynamics in the continuum. This property is otherwise difficult to access and becomes available through our systematic study of the spectral phase dependence on the photoelectron spectrum.

Another possible mechanism for a general asymmetry in the AT doublet amplitudes could be the interference between ionization pathways through resonant and near-resonant bound states as recently suggested for the dressing of He atoms with XUV^20,47 and for alkali atoms with bichromatic NIR fields⁴⁸. In our experiment, we study the energetically well-isolated transition 1s² → 1s2p, in which the contributions from neighbouring optically active states should be negligible. This provides us with a clean two-level system and greatly simplifies the data interpretation. For confirmation, we performed a calculation with a modified model in which any two-photon ionization through near-resonant states (except for the 1s2p state) was suppressed and, thus, possible photoionization interference effects were eliminated. Still, we observe a pronounced asymmetry in the AT doublet amplitudes (Extended Data Fig. 4). Moreover, owing to the large Keldysh parameter (γ = 11) and the low ponderomotive potential (U_p < 100 meV) in our study, other strong-field effects are expected to play a negligible part in the observed dynamics. We thus assign the experimental observation to the coupling of the dressed atom dynamics with a dressed ionization continuum induced by intense XUV driving fields.

A comprehensive understanding of the strong-field-induced dynamics in the system lays the basis for another quantum control effect, that is, the suppression of the ionization rate of the system, as proposed theoretically⁴⁵. The excitation probability for one-photon transitions is generally independent of the chirp direction of the driving field. However, if driving a quantum system in the strong-field limit, its quasi-resonant two-photon ionization rate may become sensitive to the chirp direction. We demonstrate the effect experimentally in Fig. 4d. A substantial reduction of the He ionization rate by 64% is achieved, solely by tuning the chirp of the FEL pulses while keeping the pulse area constant. The good agreement with the TDSE-SAE calculations confirms the mechanism. This control scheme exploits the interplay between the bound-state dynamics and the above-discussed selective coupling of the upper and lower dressed states to the ionization continuum. We note a stabilization mechanism of the dressed states in He was recently proposed, effectively causing also a suppression of the ionization rate⁴⁷. This mechanism requires, however, extreme pulse parameters, difficult to achieve experimentally. By contrast, our approach based on shaped pulses is more feasible and applies to a broader parameter range.

With this work, we have established a new tool for the manipulation and control of matter using XUV light sources. The demonstrated concept offers a wide pulse shaping window regarding pulse duration, photon energy and more complex phase shapes. In particular, the recent progress in echo-enabled harmonic generation^49,50 promises to extend the pulse-shaping concept to the soft X-ray domain (up to the 600 eV range) in which localized core electron states can be addressed. As such, we expect our work will stimulate other experimental and theoretical activities exploring the exciting possibilities offered by XUV and soft X-ray pulse shaping: first theory proposals in this direction have already been made^29,30,31. The demonstrated scheme already sets the basis for highly efficient adiabatic population transfer^1,2 and an extension to cubic or sinusoidal phase shaping would open up many more interesting control schemes^26,27. This may find applications, for example, in valence-core-stimulated Raman scattering or efficient and fast qubit manipulation with XUV and soft X-ray light. Furthermore, selective control schemes may reduce the influence of competing ionization processes ubiquitous in XUV and X-ray spectroscopy and imaging experiments, for which our work provides experimental demonstration. The generation of coherent attosecond pulse trains, with independent control of amplitude and phases, has been demonstrated at seeded FELs³⁷, bringing pulse shaping applications on the attosecond time scale within reach. This paves the way for the quantum control of molecular and solid state systems with chemical selectivity and on attosecond time scales.

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December 11, 2024
Cooling positronium to ultralow velocities with a chirped laser pulse train

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Ps generation

When a positron pulse is injected into certain media, some positrons form Ps and re-emit as Ps into vacuum. These Ps were used in this laser cooling experiment. Positron pulses, with a width of 16 ns and delivered at a repetition rate of 50 Hz (ref. ⁴⁵), contained approximately 10⁶ positrons per pulse. The positrons were transported with an energy of 5 keV from the positron production unit to the experimental station guided by a typical magnetic field of approximately 10 mT generated by coils. By conducting current in the opposite direction only in the last coil, immediately before the experimental station, we minimized the magnetic field in the experimental region. Iron plates for magnetic shielding and the magnetic lens located downstream contributed to this minimization, further reducing the magnetic field in the experimental region to approximately 0.15 mT. This suppresses the Zeeman effect, resulting in a negligible annihilation rate of Ps in the 1³S₁ state (ortho-Ps) due to spin mixing. The transported positrons were focused onto the Ps formation medium using a magnetic lens. Approximately 1% of the incident positrons passed through the magnetic lens, and the remainder collided with the lens and annihilated. The instability in the number of positrons remained at approximately 1% throughout the experiment.

We used a silica aerogel, a three-dimensional network of SiO₂ (silica) nanograins, as the medium for the formation of Ps at room temperature. The silica aerogel had pores with a diameter of 45 nm and a porosity of approximately 95%. The incident angle of the positron bunch on the aerogel was 0° and approximately half the positrons injected into the silica aerogel formed Ps⁴⁶. Some of the long-lived ortho-Ps atoms diffused and came out of the aerogel towards the experimental region. Ortho-Ps atoms decay into γ-rays with a vacuum lifetime of approximately 142 ns and these γ-rays were detected using a LaBr₃(Ce) scintillator and a plastic scintillator. The time-resolved γ-ray flux was measured by observing the current output of the coupled photomultiplier tubes. No influence was detected on the Ps from possible electrostatic charging of the silica aerogel.

Cooling laser

A pulsed laser at 729 nm, which constitutes the backbone of the cooling laser, is called a chirped pulse-train generator⁴³. This is an injection-locked pulsed laser equipped with an intracavity electro-optic phase modulator. It generates a train of approximately 0.1 ns pulses, each with progressively shifting central frequencies. These pulses are then amplified and frequency tripled, producing the 243 nm cooling laser light. The change in the central frequency over time (chirp) can be adjusted by changing the cavity length of the pulsed laser, the driving frequency and the modulation depth of the electro-optic modulator. The duration of the pulse train (the number of micropulses) can also be controlled up to approximately 600 ns, corresponding to a frequency sweep range of approximately 300 GHz using the chirp rate of the present study. Details on the operating principle of the chirped pulse-train generator can be found in ref. ⁴³, and specifics regarding the design and performance evaluation of the cooling laser in this experiment are described in ref. ⁴⁴.

Although the pulse duration has not yet been precisely measured, based on our measurements with an insufficient temporal resolution and the operating principle of the laser, a 0.1 ns pulse duration is estimated. Although an estimate (rather than a precise measurement), it does not affect the discussions in this paper as it is on a timescale considerably shorter than the natural lifetime of the 2P state.

Laser configuration

Ps atoms emitted into vacuum were irradiated by three different pulsed lasers for chirp cooling in the one-dimensional direction as well as measuring the velocity distribution. The laser beams were incident in a direction orthogonal to the positron beam axis and reflected by a bare aluminium mirror in a counter-propagating configuration. The wavelengths of the light pulses used were 243 and 532 nm, with approximately 93% reflectance off the mirror at these wavelengths. The laser irradiation area was approximately 18 mm in the direction of the positron beam axis and approximately 8 mm in the vertical direction perpendicular to the positron beam axis.

The Ps-cooling chirped pulse-train laser was switched on for approximately 100 ns after the positron pulse impacted the silica aerogel. The fluence of a single pulse in an irradiating pulse train was typically 5 μJ cm^–2. During the irradiation period (approximately 100 ns), the central frequency of the light pulse was varied from 1,233,540 to 1,233,590 GHz, with a spectral width for a single pulse of 8.9 GHz (FWHM). The cooling laser was linearly polarized with a polarization direction orthogonal to the positron beam direction.

Doppler spectroscopy

The velocity distribution of the ortho-Ps in the 1S state was evaluated by Doppler spectroscopy using the 1S–2P transition. The Doppler profile of the 1S state was obtained by measuring the signal associated with the number of positrons produced by ionizing the Ps in the 2P state as a function of the probe pulse frequency that resonantly induces the 1S–2P transition. The Doppler shift is indicative of the velocity of Ps along the propagation direction of the laser beam, allowing the evaluation of the ortho-Ps velocity distribution based on the measured Doppler profile. This measurement was conducted approximately 25 ns after the laser cooling ceased, after the complete de-excitation of the 2³P_J-state Ps through spontaneous emission. Doppler spectroscopy was carried out at a 10 Hz repetition rate, and the laser cooling occurred at a 5 Hz rate. By comparing the Doppler profiles before and after laser cooling, changes in the velocity distribution due to cooling can be assessed.

The second harmonic of an optical parametric oscillator (OPO) excited by the third harmonic of a Q-switched neodymium-doped yttrium aluminium garnet (Nd:YAG) laser was used as the probe laser for Doppler spectroscopy. The pulse duration was approximately 3 ns. The optical frequency of the probe laser was swept at approximately 1.2336 PHz and was measured using a wavelength meter with an accuracy of ±3 pm (corresponding to a frequency accuracy of approximately 15 GHz). Because of the longitudinal multimode nature, the spectral width of the second harmonic of the OPO was approximately 1.1 × 10² GHz. This spectral width was too wide to capture the changes in velocity profile resulting from chirp cooling. Therefore, the second harmonic of the OPO was transmitted through a solid etalon to narrow the spectrum and improve the velocity resolution. The measured transmission spectral width of our custom-made solid etalon available at 243 nm varied from 8 to 16 GHz at FWHM, depending on the angle and position of incidence.

However, the Doppler broadening of Ps without cooling has an FWHM of approximately $27\sqrt{T}$ GHz at a temperature of T K. This corresponds to a frequency width of 470 GHz at room temperature, which is considerably wider than the narrow resolution. To measure the Doppler profile under uncooled conditions and evaluate the temperature, it was unnecessary to spectrally narrow the second harmonic of the OPO. A greater fraction of Ps with a distributed velocity was resonant, resulting in a larger signal. Therefore, we did not use the solid etalon when measuring the Doppler profile under uncooled conditions.

The typical incident fluences of the laser pulse that induced the 1S–2P transition were 0.27 and 2 μJ cm^–2 with and without spectral narrowing, respectively. This resulted in comparable light spectral densities for these two cases. The polarization of the laser pulse that induced the 1S–2P transition was linear and parallel to the positron beam.

For the ionization laser to photoionize Ps in the 2³P_J state, we used the second harmonic (532 nm) of a Q-switched Nd:YAG laser with a pulse duration of 5 ns. This ionizing laser pulse was delivered with the timing of the intensity peak adjusted to approximately 1.4 ns later than that of the ultraviolet nanosecond pulse, which induced the 1S–2P transition. The irradiation fluence of the 532 nm pulse was typically 15 mJ cm^–2. We set the ionization laser to be linearly polarized parallel to the positron beam, similar to the laser that induced the 1S–2P transition. The repetition rate of the ionizing laser was 10 Hz, same as that of the laser inducing the 1S–2P transition.

Ionized positrons, produced with velocity selectivity from the Ps gas by the two-colour pulsed lasers, were drawn into an MCP. The MCP was placed immediately below the interaction region where Ps and the laser beams interacted. We applied a voltage of −2,000 V to the input surface of the MCP to collect the ionized positrons. The MCP was sensitive to the scattered photons of the deep-ultraviolet laser pulses at a wavelength of 243 nm, which resulted in a large background signal. Therefore, a pulsed negative voltage was applied to the MCP input surface after the completion of cooling laser irradiation, with a rise time of approximately 20 ns. Consequently, the MCP gain remained low at the time when the cooling and probe lasers were incident. This reduces the background signal originating from these photons, enabling the highly sensitive detection of ionized positrons. The voltage at the output plane of the MCP was set to 0 V. The amplified electrons were collected at a metal electrode, to which a constant voltage of 1,000 V was applied. The current output from this electrode was converted to a voltage with a 50 Ω resistor, and its time evolution was recorded. Positron signals were observed in the range of approximately 30–80 ns after the pulse voltage was applied, corresponding to the drift time that is dependent on the Ps location at photoionization. Although the background signal originating from the deep-ultraviolet photons was substantially reduced, a residual signal remained. To subtract this contribution, the ionizing laser was switched on and off every 30 s and we evaluated the signal of ionized positrons based on the difference in the integrated signals of the MCP with and without the ionizing laser.

When the number of Ps in the whole velocity distribution was approximately 3 × 10³ immediately after production, the average number of detected positrons in the frequency-resolved measurements was typically 0.5. Consequently, the uncertainty in the excitation signal in the Doppler spectroscopy measurements was characterized by the randomness in the number of ionized positrons, which is governed by Poisson statistics. To achieve an adequate signal-to-noise ratio, it was necessary to set an appropriate measurement time. For the Doppler spectroscopy used to assess the temperature of Ps gas with a frequency resolution of 110 GHz, the integration time for each probe frequency was approximately 20 min. During this integration time, the number of measurement cycles was approximately 1.2 × 10⁴. In the laser cooling experiment, for which the resolution was set to be an order of magnitude greater and thus the signal was weaker, the integration time was approximately 4 h for each probe frequency. During this time, the number of measurement cycles was approximately 7.2 × 10⁴.

Analysis of the measured Doppler profile of uncooled Ps

We estimated the temperature of the Ps emitted from the silica aerogel using the measured Doppler profile. For this purpose, we defined a model function, which was fitted to the data. The model function describes the number of positrons S(ω_R) generated by the photoionization process from the 2P state as a function of the central angular frequency ω_R of the probe pulse that induces the 1S–2P transition. S(ω_R) is written as

$$S({\omega }_{\text{R}})=\int D(v{\rm{;}}\,T)\frac{C}{1+\frac{{I}_{\text{S}}}{I(v{\rm{;}}\,{\omega }_{\text{R}})}}\text{d}v,$$

where D(v; T) is the probability density of Ps with velocity v and temperature T (the Maxwell–Boltzmann distribution function is used); I_S is the saturation intensity at the 1S–2P transition angular frequency ω_eg; I(v; ω_R) is the light intensity at the angular frequency resonant to a Ps atom with velocity v; and C is a constant and free parameter in the fitting. The second term in the integral represents the photoionization probability of Ps at velocity v. The functional form for S(ω_R) was determined using the following relation⁵⁵:

$${P}_{{\rm{e}}}=\frac{1}{2\left(1+\frac{{I}_{\text{S}}}{{I}_{\text{R}}}\right)},$$

which describes the occupation probability of the excited state in a two-level system when irradiated with light of the transition frequency at intensity I_R (see the denominator in the fraction). We used a two-level approximation because we set the spectral width of the probe pulse to be sufficiently wide compared with the splitting in the 1S–2P transition frequency. S(ω_R), determined using P_e, describes the nonlinear responses to the probe laser pulse, such as the Lamb dip and saturation broadening effects in the present Doppler-broadened case.

In our experiment, we directed each laser beam at the Ps in a counter-propagating configuration. Therefore,

$$I(v\,;\,{\omega }_{{\rm{R}}})={I}_{\text{L}}\left({\omega }_{\text{eg}}+\frac{v}{c}{\omega }_{\text{R}}{\rm{;}}\,{\omega }_{\text{R}}\right)+{I}_{\text{L}}\left({\omega }_{\text{eg}}-\frac{v}{c}{\omega }_{\text{R}}{\rm{;}}\,{\omega }_{\text{R}}\right),$$

where I_L(ω; ω_R) is the intensity spectrum, described as a function of ω, of the probe pulse with its central angular frequency ω_R, and c is the speed of light. We adopted the measured spectral width of I_L(ω; ω_R), with the intensity being a free parameter in the fitting. Here we did not include the spatial distributions of light intensity and Ps density. The light intensity of the probe pulse that reproduced the measurement was consistent with the actual light intensity calculated using the fluence, pulse duration and spectral width. This result demonstrates the validity of the proposed model.

The Doppler profile in the direction normal to the surface of the silica aerogel (Extended Data Fig. 1) was measured by the single-path irradiation of the probe pulse and ionization pulse. These optical pulses (diameter, approximately 10 mm) propagated towards the aerogel 125 ns after the peak timing of the positron bunch. The angles of incidence on the aerogel were 0° for the probe beam and 22° for the positron bunch. The peak of the Doppler profile was observed at a relative frequency of approximately −360 GHz, with an FWHM of approximately 390 GHz. In contrast to the velocity components parallel to the aerogel surface, which are randomly distributed, the distribution of the velocity components perpendicular to the surface cannot be effectively described by a simple distribution function that represents gases or beams. The velocity of Ps moving away from the surface depends not only on its velocity in the generating material but also on its work function. Consequently, the velocity distribution can generally differ from the component parallel to the surface. For these reasons, we did not perform an evaluation by fitting the experimental data. Irrespective of the parallel or perpendicular direction to the surface, the emission velocity of Ps changes dynamically with the reduction in momentum due to scattering with the molecules comprising the aerogel, with the velocity distribution also being influenced by the decay due to the Ps lifetime.

Analysis of fractional change in Doppler profile using a phenomenological model

We analysed the fractional change in the velocity distribution induced by the cooling laser by fitting the following phenomenological model to the data: the fractional change for S_on(f) and S_off(f) is defined as

$$\frac{{{S}_{{\rm{on}}}(f)-S}_{\text{off}}(\,f)}{{S}_{\text{off}}(\,f)}.$$

We first used the following raw functions that did not include the frequency resolution in the experiment:

$${S}_{\text{on}}^{\text{raw}}(\,f)=\left\{\begin{array}{cc}\exp \left(-\frac{{m}_{\text{Ps}}\,{c}^{2}{f}^{2}}{2{k}_{\text{B}}{T}_{0}{f}_{0}^{2}}\right), & f < -{f}_{\text{cooled}},\,{f}_{\text{cooled}} < f\\ A\exp \left(-\frac{4\log 2\,{f}^{2}}{\Delta {f}^{2}}\right)+{S}_{\text{cooled}},\, & -{f}_{\text{cooled}}\le f\le \,{f}_{\text{cooled}}\end{array}\right.$$

$${S}_{\text{off}}^{\text{raw}}(\,f)=\exp \left(-\frac{{m}_{\text{Ps}}\,{c}^{2}{f}^{2}}{2{k}_{\text{B}}{T}_{0}{f}_{0}^{2}}\right),$$

where the argument f is the relative frequency, f₀ is the 1³S₁–2³P₂ transition frequency of Ps, m_Ps is the mass of Ps, k_B is the Boltzmann constant and T₀ is the temperature of Ps released from the silica aerogel. On the basis of the experimental results, we assumed that the Doppler profile of the uncooled Ps was a Maxwell–Boltzmann distribution at temperature T₀ = 600 K. The following free parameters used in the fit describe the change in Doppler profile associated with cooling: f_cooled is the Doppler shift corresponding to the optical frequency at the beginning of the cooling laser; Δf is the Doppler width of the decelerated component; S_cooled is the signal level after cooling in the spectral region swept by the chirped cooling laser; A characterizes the magnitude of the decelerated component signal. These raw functions are plotted in Extended Data Fig. 2.

We generated model functions S_on(f) and S_off(f), which correspond to the experimental results obtained, by convolving ${S}_{\text{on}}^{\text{raw}}(f)$ and ${S}_{\text{off}}^{\text{raw}}(f)$ with the frequency resolution due to the linewidth of the probe pulse. The change in Doppler profile associated with cooling was quantitatively evaluated by fitting the modelled fractional change to the measured fractional changes. In Extended Data Fig. 2, ${S}_{\text{on}}^{\text{raw}}(f)$ is plotted using the parameters obtained from the fit.

The fitting parameters varied with the spectral width of the probe pulse, which determined the frequency resolution of the measured Doppler profile. When the spectral width, which varied in the experiment, was set to 8 GHz (narrowest), the widest Doppler spread of the cooling component was evaluated. In the main text, we have shown the corresponding best-fit value (23 GHz) and upper statistical limit (30 GHz) as conservative estimates (the upper statistical limit of the width of the cooled component was evaluated at the 95% confidence level). The population reductions in the cooled spectral region were estimated to be 61% and 49%, respectively. For a spectral width of 16 GHz for the probe pulse, the best-fit value and the upper limit of the width of the cooled component were 18 and 27 GHz, and the corresponding population reductions were 78% and 61%, respectively.

We believe that the estimated population reductions were smaller than those expected from population reduction by laser cooling alone, due to the influence of delayed Ps release from the silica aerogel. Such delayed release from porous materials has been reported previously⁵⁶. Our empirical observations suggest the presence of Ps emitted from the silica aerogel several tens of nanoseconds after the injection of the positron bunch, when we observed the components of zero lateral velocity. However, we cannot quantitatively discuss the delayed fraction due to the absence of available systematic data.

Evaluation of the frequency resolution in the laser cooling experiment

The frequency resolution of the fractional change in the Doppler profile as a result of laser cooling was determined using the spectral width of the probe pulse and intensity-dependent saturation broadening. We evaluated the spectral width of the probe pulse using the optical resolution of the Fabry–Pérot solid etalon used for spectral narrowing. The FWHM optical frequency resolution as a function of the angle of incidence is shown in Extended Data Fig. 3. The resolution was evaluated by measuring the transmission spectrum of single-longitudinal-mode laser pulses at 243 nm. The spectral width of the pulses is expected to be less than 10 MHz, which is considerably narrower than the designed frequency resolution of the solid etalon, thereby enabling the evaluation of the actual resolution. We measured the transmittance as a function of the angle of incidence of the etalon. All the incident angle sweeps designated in the legend were performed in the direction of increasing angle. These three sets of measurements were performed in the experimental period but not consecutively.

The results indicate that although the transmission spectral width tends to increase with the angle of incidence, it varies widely for each measurement. The degree of variation exceeds the measurement uncertainty, suggesting that the conditions of the etalon changed with each sweep. The possible characteristics of the solid etalon that can cause such variations include non-uniform thickness and inhomogeneous strain on the etalon. Variations can then occur because the position of the laser irradiation on the solid etalon cannot be completely fixed. To detect a change in the Doppler profile resulting from laser cooling, the angles of incidence of the probe pulse on the etalon were set in the range tested above, resulting in the same degree of variation in the linewidth of the probe. Therefore, we estimated the spectral width of the probe pulse to be 8–16 GHz, based on the measured range of values shown in Extended Data Fig. 3.

Next, we examined the influence of saturation broadening, which also affects the frequency resolution. Using the effective intensity calculated from the fluence, pulse duration and spectral width of the spectrally narrowed probe pulse, the degradation of the frequency resolution owing to saturation broadening was at most 1 GHz. Thus, saturation broadening can be neglected.

We considered the 8–16 GHz range of the frequency resolution as a systematic uncertainty in the evaluation of the fractional change. Hence, a conservative effective temperature was evaluated.

Allowed 1³
S–2³
P transitions and their intensities

Here we describe the allowed transitions and their intensities among the 1³S–2³P transitions used for laser cooling and Doppler spectroscopy. The transition matrix element is

$$-\langle n=2,L=1,S=1,{J}_{\text{e}}\,,{M}_{\text{e}}\,|\,{\bf{d}}\,|\,n=1,L=0,S=1,{J}_{\text{g}},{M}_{\text{g}}\rangle \cdot {\bf{E}},$$

where d is the electric dipole moment; E is the electric field of light; n, L and S are the principal quantum number, orbital angular momentum and total spin angular momentum, respectively; J and M are the total angular momentum and its projection along the quantization axis, respectively. Subscripts e and g indicate the excited and ground states, respectively.

The electric dipole moments, when we define the quantization axis of the atomic orbitals as the z axis, are shown in Extended Data Fig. 4a,b. The direction of projection of the electric dipole moment is shown at the top of each diagram. The allowed transitions induced by the electric field of light with the corresponding polarization vectors are represented by arrows. The numbers associated with the arrows indicate the square of the absolute value of each component of the electric dipole moment normalized to the following constant:

$${|{d}_{0}|}^{2}={\left(\frac{128\sqrt{2}}{243}2e{a}_{0}\right)}^{2},$$

where e is the elementary charge and a₀ is the Bohr radius. For some transitions, the numbers are omitted because the absolute values of the electric dipole moments coincide with those of the other transitions that differ only in the sign of M. The transition rates are proportional to the values shown in Extended Data Fig. 4a for linearly polarized light parallel to the z axis, and in Extended Data Fig. 4b for orthogonally polarized light. In our experiment, the polarizations of the cooling laser pulse and probe laser pulse in Doppler spectroscopy were linear and orthogonal to each other. Extended Data Fig. 4a,b can be used to evaluate the transition intensity of each pulse.

Extended Data Fig. 4c shows the spontaneous emission rates from the excited states to each ground state normalized by the total decay rate Γ_sp. ≅ 3.13 × 10⁸ s^–1 from each excited state. By symmetry, the spontaneous emission rates from the states with negative M_e values, which are omitted from the table in Extended Data Fig. 4c, are equal to the corresponding rates between the states with the signs for M_e and M_g reversed.

Extended Data Fig. 4a–c shows that in the cooling process, during which the transitions are repeated many times, it is important to use a cooling laser with a spectral width comparable with the splitting in the transition. Otherwise, if we repeat the cooling cycle by transitioning to the 2³P₀ and 2³P₁ states, for example, the 1³S₁ state becomes polarized and eventually makes transitions to these excited states dark. Moreover, the 1³S₁–2³P₂ transition dominated the 1S–2P transitions. Therefore, we present our experimental results as functions of frequency relative to the 1³S₁–2³P₂ frequency difference. Note that the resonance frequency observed at the one-photon transition is approximately 3 GHz higher than this frequency difference owing to the conservation laws of energy and momentum.

Numerical simulation

We evaluated the time evolution of the momentum distribution of Ps under the influence of a cooling laser based on the Lindblad master equation:

$$\frac{{\rm{d}}\rho }{{\rm{d}}t}=\frac{1}{{\rm{i}}\hbar }[H\,,\rho ]+L(\rho ),$$

where t, ħ, H and L(ρ) are the time, Dirac’s constant, Hamiltonian and Liouvillian, respectively. We considered the density matrix ρ in the space spanned by the simultaneous eigenstates of the momentum of Ps and atomic configurations in the L–S coupling scheme. The interaction between Ps and the photon field was incorporated as an electric dipole interaction. This framework can describe the transitions between atomic orbitals through absorption, stimulated emission and spontaneous emission processes, as well as momentum changes because of photon recoil. We incorporated the relaxation of Ps due to annihilation processes into the master equation as a longitudinal relaxation process.

Using the simulated velocity distribution shown in Fig. 3a, we can simulate the fractional change (Fig. 2b). The simulated Doppler profiles with and without the cooling laser irradiation were convolved by the spectral resolution to obtain ${S}_{\text{on}}^{\text{sim}}(\,f)$ and ${S}_{\text{off}}^{\text{sim}}(\,f)$, respectively. The spectral width of the probe pulse determines the spectral resolution. The argument f is the relative frequency, which is the first-order Doppler shift calculated from the velocity of Ps. To express a part of the probed Ps atoms, which interacted with the cooling laser, we introduce an uncooled Ps fraction r. The fractional change can then be calculated as $(1-r)\frac{{S}_{\text{on}}^{\text{sim}}(\,f)-{S}_{\text{off}}^{\text{sim}}(\,f)}{{S}_{\text{off}}^{\text{sim}}(\,f)}$. The parameter r was determined by fitting this function to the measured data. Figure 3b compares the measured and simulated fractional changes. The filled circles are identical to those shown in Fig. 2b. The thickness of the curve was determined on the basis of the frequency resolution in the range of 8–16 GHz. The measured data were well reproduced, with the best estimated r ranging from 0.18 to 0.40 and the spectral resolution shown above. The statistical uncertainty of the estimated r is typically 0.06 at the 1σ confidence level. The resultant fraction r is reasonable under the experimental condition, and its consistency with the measured data supports the successful demonstration of the laser cooling of Ps.

Extended Data Fig. 5 presents the corresponding Doppler profiles mentioned above. Similar to the experiment, the simulation evaluated the Doppler profiles 125 ns after Ps formation, following 100 ns of cooling laser irradiation. Components resonating with the frequency-swept cooling laser were decelerated and concentrated in the frequency domain corresponding to zero velocity. Compared with the case without cooling, the slow components showed a threefold increase. No change was observed in the detuned components, which did not resonate with the cooling laser.

To illustrate the parameter design of the cooling laser in this study, we present typical examples of cooling time dependence and chirp rate dependence based on the simulations constructed here. Extended Data Fig. 6a displays the momentum distribution after cooling, evaluated as a function of cooling time while keeping the chirp rate constant. The optical frequency detuning of the cooling laser at the end of cooling was set to −9 GHz. As the cooling duration is extended, the sweep frequency range of the cooling laser increases, thereby enhancing the contrast in number between the cooled and uncooled components. However, due to the lifetime effects of Ps, the number of cooled atoms is found to decrease compared with shorter cooling times. The maximum number of cooled atoms is achieved at a cooling time of approximately 100 ns, which is the duration used in this study.

Extended Data Fig. 6b shows the momentum distribution after cooling when the chirp rate is varied, with the cooling time fixed at 100 ns. When the chirp rate exceeds the rate characterized by the recoil frequency associated with photon absorption and the natural emission rate, the proportion of atoms that cannot maintain the chirp cooling cycle increases, resulting in decreased efficiency. Note that this calculation assumes that the entire volume of the Ps gas is constantly exposed to the cooling laser.

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September 11, 2024
Attosecond delays in X-ray molecular ionization

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The temporal resolution provided by attosecond technologies developed in the past 23 years gives access to some of the fastest electronic dynamics of matter on their natural timescale. Seminal pump–probe experiments using attosecond light pulses have revealed valence electron dynamics in atoms¹⁶, autoionization dynamics in molecules¹⁷, photoionization delays in solids¹⁸, as well as electron-driven charge migration in simple ionized biomolecules^9,19. In all of these cases, the intrinsically high photon energy of the attosecond light sources inevitably leads to ionization of the target, which has restricted the measurements to ultrafast dynamics of cationic states.

When aiming to investigate the ultrafast light-induced electron dynamics of chiral molecules in their neutral states, the pump pulse must thus have a photon energy below the ionization threshold and a broadband energy spectrum that can trigger coherent electron motion among several electronic states, and a time duration that ensures prompt excitation before any nuclear motion can take place, together with sufficient temporal resolution. The low ionization potential of most molecular systems thus restricts the choice of the pump pulse wavelength to the UV and vacuum-UV ranges, that is, to a spectral region that cannot trigger intricate high-order, strong-field multiphoton-driven processes^20,21 that rarely occur with natural light sources. When these requirements are met, measurements with high time resolution are possible using pump–probe spectroscopic techniques that are highly sensitive to chirality, such as TR-PECD^12,15 recently used to probe nuclear dynamics, internal conversion and photoionization delays in chiral molecules^{11,12,13,14,22}.

We use ultrashort UV pump pulses^23,24 in combination with circularly polarized near-infrared (NIR) probe pulses to study coherent electronic dynamics in chiral neutral molecules with unprecedented temporal resolution. We apply the chiroptical method of TR-PECD to investigate electron-driven chiral interactions in neutral methyl lactate (C₄H₈O₃). Figure 1a,b shows an overview of the experimental approach. First, a linearly polarized UV pulse promptly launches a coherent electronic wave packet just below the ionization threshold in (S)-methyl lactate by means of a two-photon transition. Then, a time-delayed circularly polarized NIR probe triggers ionization from the transient wave packet, providing an exceptional instrument response function of 2.90 ± 0.06 fs (Extended Data Fig. 1). For each pump–probe delay t, the 2D-projected photoelectron angular distributions (PADs) S^(h)(ε, θ, t) are collected with a velocity map imaging spectrometer (VMIS), for both left (h = +1) and right (h = −1) circular polarizations of the probe pulse. ε and θ stand for the kinetic energy and direction of ejection of the photoelectron in the (x, z) VMIS detection plane, respectively. The chiroptical response is characterized by a photoelectron circular dichroism (PECD) image defined as the normalized difference ${\rm{PECD}}\left(\varepsilon ,\theta ,t\right)=2\frac{{S}^{\left(+1\right)}\left(\varepsilon ,\theta ,t\right)-{S}^{\left(-1\right)}\left(\varepsilon ,\theta ,t\right)}{{S}^{\left(+1\right)}\left(\varepsilon ,\theta ,t\right)+{S}^{\left(-1\right)}\left(\varepsilon ,\theta ,t\right)}$, subsequently fitted using a pBasex inversion algorithm¹¹. Snapshots of the measured PECD(ε, θ, t) are presented in Fig. 1c. The signal reaches values of up to 10%, typical for PECD and about two orders of magnitude higher than the analogously defined g factor generally obtained in circular dichroism^25,26. Low-energy electrons (ε ≤ 100 meV; see white dashed circles) are preferentially emitted in the θ = 180° backward hemisphere at t = 5 fs and preferentially ejected forward at t = 11 fs. Their main direction of ejection reverses again at t = 17 fs. Higher-energy electrons (ε > 100 meV) are more likely emitted forward than backward for t ≥ 11 fs, but the magnitude of their asymmetry depends on t.

**Fig. 1: Light-induced chiral dynamics of methyl lactate.**

The PECD(ε, θ, t) images provide quantitative fingerprints of an ultrafast dynamics taking place on the few-femtosecond timescale. To further characterize the temporal evolution of the observed dynamics, we decompose the PAD images in series of Legendre polynomials, ${S}^{\left(h\right)}\left(\varepsilon ,\theta ,t\right)={\sum }_{n=0}^{6}{b}_{n}^{\left(h\right)}\left(\varepsilon ,t\right){P}_{n}\left(\cos \theta \right)$, and calculate the multiphoton PECD (MP-PECD)²⁷, defined as the normalized difference of electrons emitted in the forward and backward hemispheres for h = +1, as ${\rm{MP}} \mbox{-} {\rm{PECD}}\left(\varepsilon ,t\right)=2{\beta }_{1}^{\left(+1\right)}\left(\varepsilon ,t\right)-\frac{1}{2}{\beta }_{3}^{\left(+1\right)}\left(\varepsilon ,t\right)+\frac{1}{4}{\beta }_{5}^{(+1)}\left(\varepsilon ,t\right)$, in which ${\beta }_{n}^{\left(+1\right)}\left(\varepsilon ,t\right)=\frac{{b}_{n}^{\left(+1\right)}\left(\varepsilon ,t\right)}{{b}_{0}^{\left(+1\right)}\left(\varepsilon ,t\right)}$ (see Methods and Extended Data Figs. 2 and 3). ${\beta }_{1}^{\left(+1\right)}\left(\varepsilon ,t\right)$ refers to the isotropic part of the asymmetry in each hemisphere, whereas ${\beta }_{3}^{\left(+1\right)}\left(\varepsilon ,t\right)$ encodes anisotropic features owing to pump excitation^11,28, leading to the angular shaping of the PECD illustrated in Fig. 1c. ${\beta }_{5}^{(+1)}\left(\varepsilon ,t\right)$ has been found to be negligible in our measurements. Figure 2a shows MP-PECD(ε, t) and its ${b}_{1}^{(+1)}(\varepsilon ,t)$ component is shown in Fig. 2b. The results are shown for (S)-methyl lactate and a mirroring symmetric measurement in (R)-methyl lactate clearly confirms the chiral character of the Rydberg-induced dynamics, with minor discrepancies owing to slightly lower enantiopurity and statistics (Extended Data Fig. 4). We observe that the unnormalized MP-PECD(ε, t) behaviour in Fig. 2a closely matches ${b}_{1}^{(+1)}(\varepsilon ,t)$ in Fig. 2b, indicating that the anisotropic effects included in ${\beta }_{3}^{(+1)}\left(\varepsilon ,t\right)$ play a minor role. The MP-PECD can be partitioned into three kinetic energy ranges, as identified in Fig. 2a,b. Between 25 and 100 meV, the photoelectron emission asymmetry reverses in about 7 fs (see Fig. 2c). A clear modulation of the asymmetry remains over several tens of femtoseconds, which is also observed at higher ε between 100 and 300 meV (Fig. 2d) and 300 and 720 meV (Fig. 2e). These modulations are also visible in the time-resolved photoelectron yield b₀(ε, t), albeit their contrast is considerably weaker (Extended Data Fig. 5). This highlights the capabilities of TR-PECD, which relies on differential measurements, over conventional photoelectron spectroscopy. In the following, we aim at assigning the origin of the fast temporal modulation of the asymmetry, which could involve electronic and/or nuclear degrees of freedom.

We modelled the experiment including both the two-photon UV excitation and the NIR photoionization steps as sequential perturbative processes, within the frozen-nuclei approximation. A detailed description of the theoretical model is provided in Methods. The electronic spectrum of methyl lactate and the two-photon excitation amplitudes are obtained through time-dependent density functional theory²⁹. Ionization from the excited states is described using the continuum multiple scattering Xα approach^30,31.

We present the results of our calculations in Fig. 3. The pump pulse populates excited states mainly originating from excitation of the highest occupied molecular orbital (HOMO) of the methyl lactate ground state (see Supplementary Information section 1 and Supplementary Fig. 2). Figure 3a shows the two-photon excitation cross-section associated with almost pure HOMO excitation to Rydberg states. Subsequent photoionization by the probe pulse leads to the emission of photoelectrons with kinetic energies ε = 250 and 500 meV. These ε values are representative of the second and third energy ranges discriminated in the experimental data, respectively—the case of low-energy photoelectron dynamics (ε = 50 meV) is discussed in Supplementary Information section 2.3 and illustrated in Supplementary Fig. 5. Including the HOMO excited states of Fig. 3a in the dynamical calculations yields the time-resolved MP-PECD shown in Fig. 3b,d. The calculations are started at t = 10 fs to ensure no temporal overlap between the pump and probe pulses. The computed asymmetry presents clear modulations as a function of the pump–probe delay. The power spectra of the MP-PECD signals, obtained by Fourier analysis, are compared with their experimental counterparts in Fig. 3c,e. An excellent agreement is found at ε = 250 meV, at which the oscillatory pattern of the MP-PECD is traced back to the pump-induced coherent superposition of 3d and 4p Rydberg states respectively located at E_3d = 8.834 eV and E_4p = 9.120 eV in Fig. 3a. This coherent superposition leads to quantum beatings with approximately 15 fs period, associated with an energy difference between the states of about 300 meV, which survive long after the pump pulse vanishes. We note that the most stable geometries of methyl lactate do not have any vibrational mode in the vicinity of 2,200 cm⁻¹ (about 15 fs)³². Similarly, the coherent superposition of 4p and 4d,f Rydberg states results in the oscillatory feature of the MP-PECD signal at ε = 500 meV. A small mismatch of about 60 meV is observed between the experimental and theoretical power spectra in Fig. 3e. This mismatch is on the order of the error made in quantum chemistry computations of excited-state energies. Overall, Fig. 3b,d unmistakably demonstrates that the electronic coherence of the intermediate Rydberg states, as identified in Fig. 3a, modulates the molecular chiroptical response.

**Fig. 3: Modelling of the experiment.**

In our fixed-nuclei description, the electronic coherences leading to oscillatory MP-PECD do not vanish and even lead to an overestimation of the MP-PECD amplitude at all delays t. By contrast, the oscillations observed in the experimental MP-PECD (Fig. 2c–e) are damped over time, which coincides with the approximately 40 fs lifetime encoded in the time-dependent photoelectron yield (Extended Data Fig. 5). Describing the coupled electron and nuclear dynamics in an energy range in which tens of electronic states lie is beyond state-of-the-art theoretical approaches. Therefore, we alternatively performed ab initio molecular dynamics calculations on the ground state of cationic methyl lactate to which all the HOMO Rydberg states involved in the pump–probe dynamics correlate following ionization (see Supplementary Information section 3 and associated Supplementary Figs. 6–8). These calculations suggest that the most probable source of decoherence in this investigation is non-adiabatic transitions.

The oscillations of the chiroptical response for ε = 250 meV mainly result from the coherent superposition of two states, the 3d and 4p Rydberg states. We now investigate in more detail the role of these states in the chiroptical response. For a single molecular orientation $\widehat{{\bf{R}}}$, the excited electron wave packet reads, at time t after the pump pulse vanishes, $\Phi \left(\widehat{{\bf{R}}},{\bf{r}},t\right)=\sum _{j={\rm{3d,4p}}}{A}_{j}(\widehat{{\bf{R}}}){\Psi }_{j}\left({\bf{r}}\right)\exp (-{\rm{i}}{E}_{j}t/\hbar )$, in which ${A}_{j}(\widehat{{\bf{R}}})$ are the real two-photon transition amplitudes associated with the excited states ${\Psi }_{j}\left({\bf{r}}\right)$. The associated electron density can be partitioned as

$$\rho \left(\widehat{{\bf{R}}},{\bf{r}},t\right)={\rho }_{{\rm{incoh}}}\left(\widehat{{\bf{R}}},{\bf{r}}\right)+{\rho }_{{\rm{cross}}}\left(\widehat{{\bf{R}}},{\bf{r}}\right)\cos \left[\left({E}_{{\rm{4p}}}-{E}_{{\rm{3d}}}\right)t/\hbar \right]$$

(1)

in which ${\rho }_{{\rm{incoh}}}\left(\widehat{{\bf{R}}},{\bf{r}}\right)={A}_{{\rm{3d}}}^{2}\left(\widehat{{\bf{R}}}\right){\Psi }_{{\rm{3d}}}^{2}\left({\bf{r}}\right)+{A}_{{\rm{4p}}}^{2}\left(\widehat{{\bf{R}}}\right){\Psi }_{{\rm{4p}}}^{2}\left({\bf{r}}\right)$ and ${\rho }_{{\rm{cross}}}(\widehat{{\bf{R}}},{\bf{r}})\,=\,$${2A}_{{\rm{3d}}}(\widehat{{\bf{R}}}){A}_{{\rm{4p}}}(\widehat{{\bf{R}}}){\Psi }_{{\rm{3d}}}({\bf{r}}){\Psi }_{{\rm{4p}}}({\bf{r}})$. Figure 4a shows, for one selected orientation $\widehat{{\bf{R}}}$, the coherent part $\rho \left(\widehat{{\bf{R}}},{\bf{r}},t\right)-{\rho }_{{\rm{incoh}}}\left(\widehat{{\bf{R}}},{\bf{r}}\right)$ of the electron density, oscillating back and forth along the molecular structure with a period T = 2πħ/(E_4p − E_3d) of 14.4 fs. Ionization of the 3d and the 4p state superposition leads, after averaging over the orientations $\widehat{{\bf{R}}}$, to the total photoelectron yield, which can be decomposed similarly to equation (1):

$${b}_{0}^{(\pm 1)}\left(\varepsilon ,t\right)={b}_{{0}_{{\rm{incoh}}}}^{(\pm 1)}\left(\varepsilon \right)+{b}_{{0}_{{\rm{cross}}}}^{(\pm 1)}\left(\varepsilon \right)\cos \left[\left({E}_{{\rm{4p}}}-{E}_{{\rm{3d}}}\right)t/\hbar \right].$$

(2)

**Fig. 4: Electron-driven dynamics for a quantum beating of (3d, 4p) Rydberg states monitored at ε = 250 meV.**

The computed yield is presented in Fig. 4b for ε = 250 meV, showing how the coherent state superposition leading to ${b}_{{0}_{{\rm{cross}}}}^{(\pm 1)}\left(\varepsilon \right)\cos \left[\left({E}_{{\rm{4p}}}-{E}_{{\rm{3d}}}\right)t/\hbar \right]$ modulates the incoherent sum ${b}_{{0}_{{\rm{incoh}}}}^{(\pm 1)}\left(\varepsilon \right)$ of individual ionization cross-sections. The unnormalized MP-PECD can in turn be written as:

$$\begin{array}{l}{\rm{MP}} \mbox{-} {\rm{PECD}}\left(\varepsilon ,t\right)={{\rm{MP}} \mbox{-} {\rm{PECD}}}_{{\rm{incoh}}}\left(\varepsilon \right)\\ \,\,\,\,\,\,\,\,+\,{{\rm{MP}} \mbox{-} {\rm{PECD}}}_{{\rm{cross}}}\left(\varepsilon \right)\cos \left[\frac{\left({E}_{{\rm{4p}}}-{E}_{{\rm{3d}}}\right)t}{\hbar }-\Delta \phi \right]\end{array}$$

(3)

in which the extra phase Δϕ arises from the interference of the state-selective continuum partial wave amplitudes building the asymmetry of the photoelectron yield (see Supplementary Information). As usual, this interference is washed out at the level of the total photoelectron yield^30,33. The temporal evolution of the unnormalized two-state MP-PECD is shown in Fig. 4c, from which we extract the time delay Δt = 1.8 fs associated with Δϕ = 0.79 rad. The MP-PECD reverses sign within one period of the oscillation because the asymmetries of single 3d-mediated and 4p-mediated pathways, contributing to the incoherent MP-PECD around which the coherent part oscillates, verify |MP-PECD_cross(ε)| > |MP-PECD_incoh(ε)|. A similar behaviour is observed at lower kinetic energy in the measurement reported in Fig. 2c. Notably, the MP-PECD depends not only on the transient bound resonances—as evidenced in Figs. 2c–e and 3a,b,d—but also on the dichroism encoded by ionization with circularly polarized light. In this respect, we note that a photoexcitation electron circular dichroism (PXECD) configuration³⁴, in which molecules are photoexcited by a circularly polarized pump pulse and subsequently ionized with a linearly polarized probe, would reduce the degrees of freedom to only the transient bound resonances.

The electron dynamics uncovered in this work underlies yet another consequence on the molecular response of photoexcited chiral systems: the coherent superposition of excited states induced by the pump allows to selectively filter—within a few femtoseconds—specific molecular orientations through enantiosensitive photoionization¹⁰. An electron moving within the coherent state superposition creates an electronic current³⁵ ${\bf{J}}(\widehat{{\bf{R}}},{\bf{r}},t)=\frac{\hbar }{m}{\mathfrak{I}}[{\Phi }^{* }(\widehat{{\bf{R}}},{\bf{r}},t){\boldsymbol{\nabla }}\Phi (\widehat{{\bf{R}}},{\bf{r}},t)]$, which reduces to

$$\begin{array}{l}{\bf{J}}(\widehat{{\bf{R}}},{\bf{r}},t)=\frac{\hbar }{m}{A}_{{\rm{3d}}}(\widehat{{\bf{R}}}){A}_{{\rm{4p}}}(\widehat{{\bf{R}}})[{\Psi }_{{\rm{4p}}}({\bf{r}}){\boldsymbol{\nabla }}{\Psi }_{{\rm{3d}}}({\bf{r}})-{\Psi }_{{\rm{3d}}}({\bf{r}}){\boldsymbol{\nabla }}{\Psi }_{{\rm{4p}}}({\bf{r}})]\\ \,\,\,\,\,\sin \left[\frac{({E}_{{\rm{4p}}}-{E}_{{\rm{3d}}})t}{\hbar }\right]\end{array}$$

(4)

when expanding Φ on the (real) 3d and 4p bound eigenstates. The chirality of the molecule induces a curl in the generated electron current, whose rotation direction reverses periodically. Rotating electron currents are known to influence the ionization probability by circularly polarized light: the propensity rules³⁶ establish that one-photon ionization is enhanced when the electrons rotate in the same direction as the electric field. Therefore, the molecules oriented such that their electronic current rotates in the same plane and direction as the ionizing laser pulse are preferentially ionized; see Fig. 4d. Consequently, the produced molecular cations are selectively oriented along the probe polarization rotation axis, corresponding to the light-propagation axis $\widehat{{\bf{z}}}$.

To quantify the degree of orientation of the photoionized molecules, we select a unitary vector ${\widehat{{\bf{e}}}}_{{\rm{mol}}}$ fixed to the internal C–C bond of the methyl lactate cation, as illustrated in the inset of Fig. 4e, and calculate its averaged value over the probe-filtered molecular orientations in the laboratory frame as¹⁰

$${\langle {\widehat{{\bf{e}}}}_{{\rm{lab}}}\rangle }_{\widehat{{\bf{R}}}}^{(\pm 1)}(\varepsilon ,t)=\frac{\int {\rm{d}}\widehat{{\bf{R}}}{W}^{(\pm 1)}(\widehat{{\bf{R}}},\varepsilon ,t){\widehat{{\bf{e}}}}_{{\rm{lab}}}(\widehat{{\bf{R}}})}{{b}_{{0}_{{\rm{incoh}}}}^{(\pm 1)}(\varepsilon )}$$

(5)

in which ${\widehat{{\bf{e}}}}_{{\rm{lab}}}(\widehat{{\bf{R}}})$ is the ${\widehat{{\bf{e}}}}_{{\rm{mol}}}$ vector passively rotated in the laboratory frame and ${W}^{(\pm 1)}(\widehat{{\bf{R}}},\varepsilon ,t)$ is the ionization rate associated to helicity h = ±1 and photoelectrons of energy ε. Notably, the averaged orientation of the cations depends on ε because the ε dependence of the underlying photoionization yield is not the same for all orientations $\widehat{{\bf{R}}}$. The x and y components of ${\left\langle {\widehat{{\bf{e}}}}_{{\rm{lab}}}\right\rangle }_{\widehat{{\bf{R}}}}^{\left(\pm 1\right)}(\varepsilon ,t)$ are found to be zero and only the z component survives the averaging¹⁰ (see Supplementary Fig. 9), leading to

$${\left\langle {\cos \theta }_{{\rm{ion}}}\right\rangle }_{\widehat{{\bf{R}}}}^{\left(\pm 1\right)}\left(\varepsilon ,t\right)={\left\langle {\cos \theta }_{{\rm{ion}}}\right\rangle }_{{\rm{cross}}}^{\left(\pm 1\right)}(\varepsilon )\sin \left[\left({E}_{{\rm{4p}}}-{E}_{{\rm{3d}}}\right)t/\hbar \right]$$

(6)

in which θ_ion is the angle between the internal C–C bond and the probe propagation $\widehat{{\bf{z}}}$ axis (see inset of Fig. 4e). ${\left\langle {\cos \theta }_{{\rm{ion}}}\right\rangle }_{{\rm{cross}}}^{\left(\pm 1\right)}(\varepsilon )$ involves chiral-sensitive products of 3d and 4p excitation and ionization amplitudes. The temporal evolution of ${\left\langle {\cos \theta }_{{\rm{ion}}}\right\rangle }_{\widehat{{\bf{R}}}}^{(+1)}$ is illustrated in Fig. 4e for ε = 250 meV. When ${\langle {\cos \theta }_{{\rm{ion}}}\rangle }_{\widehat{{\bf{R}}}}^{(+1)}(\varepsilon ,t) < 0$, the CO₂CH₃ moiety of the methyl lactate cations preferentially points forward with respect to $\widehat{{\bf{z}}}$, whereas instead it points backward when ${\left\langle {\cos \theta }_{{\rm{ion}}}\right\rangle }_{\widehat{{\bf{R}}}}^{\left(+1\right)}(\varepsilon ,t) > 0$. Such asymmetry could be detected by resolving the direction of fragmentation of the molecular cations (see Supplementary Information section 4.2 and associated Supplementary Figs. 10 and 11). Indeed, the relative numbers of molecules pointing forward and backward at time t, ${N}_{+}^{\left(\pm 1\right)}(\varepsilon ,t)$ and ${N}_{-}^{\left(\pm 1\right)}(\varepsilon ,t)$, respectively, can be linked to ${\left\langle {\cos \theta }_{{\rm{ion}}}\right\rangle }_{\widehat{{\bf{R}}}}^{\left(\pm 1\right)}(\varepsilon ,t)$ (see Methods). This ultrafast filtering of molecular orientation affects the subsequent reactive dynamics of methyl lactate cations, with prompt photoionization dictating the subsequent dissociation along the selected molecular orientation. A forward/backward fragment asymmetry (FBFA) thus naturally arises, which we define as

$${{\rm{FBFA}}}^{\left(\pm 1\right)}\left(\varepsilon ,t\right)=2\frac{{N}_{+}^{\left(\pm 1\right)}\left(\varepsilon ,t\right)-{N}_{-}^{\left(\pm 1\right)}\left(\varepsilon ,t\right)}{{N}_{+}^{\left(\pm 1\right)}\left(\varepsilon ,t\right)+{N}_{-}^{\left(\pm 1\right)}\left(\varepsilon ,t\right)}.$$

(7)

The FBFA is shown in Fig. 4f for h = +1 and ε = 250 meV, reaching absolute values of about 30%, whereas its temporal evolution is dictated by the behaviour of the underlying electron current ${\bf{J}}\left(\widehat{{\bf{R}}},{\bf{r}},t\right)$. Similarly to the MP-PECD, the FBFA switches sign for h = −1 or when the other enantiomeric form of methyl lactate molecules is considered. Because the FBFA is created by the electron current, it vanishes in the case of incoherent population of excited states (see Supplementary Information section 4). This shows that the chiral electronic coherence directly observed in our experiment through TR-PECD is crucial to achieve control over enantioselective dynamics of the nuclei.

We have taken an important step forward by resolving the coherent chiral electronic dynamics of a chiral molecule in the first instants following prompt excitation by an achiral few-femtosecond UV pulse. The results showcase that TR-PECD can provide insights on the role of the primary electron dynamics in the light-induced chiral response of complex molecules. Beyond its impact on the chiroptical properties of the system, the chiral currents generated in our experiment can be exploited for photochemical control, as exemplified by our calculations on enantiosensitive charge-directed reactivity leading to oriented fragmentation. From a broader perspective, our results contribute to the fundamental understanding of electronic chirality at the molecular level and its impact on primary enantiosensitive interactions.

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