Attosecond Dynamics in Liquids
H. J. Wörner, A. Schild, D. Jelovina, I. Jordan, C. Perry, T. T. Luu, Z. Yin
CChapter 1
Attosecond Dynamics in Liquids
H. J. Wörner, A. Schild, D. Jelovina, I. Jordan, C. Perry, T. T. Luu, Z. Yin
Abstract
Attosecond science is well developed for atoms and promising resultshave been obtained for molecules and solids. Here, we review the first steps in de-veloping attosecond time-resolved measurements in liquids. These advances provideaccess to time-domain studies of electronic dynamics in the natural environmentof chemical reactions and biological processes. We concentrate on two techniquesthat are representative of the two main branches of attosecond science: pump-probemeasurements using attosecond pulses and high-harmonic spectroscopy (HHS). Inthe first part, we discuss attosecond photoelectron spectroscopy with cylindrical mi-crojets and its application to measure time delays between liquid and gaseous water.We present the experimental techniques, the new data-analysis methods and the ex-perimental results. We describe in detail the conceptual and theoretical frameworkrequired to fully describe attosecond chronoscopy in liquids at a quantum-mechanicallevel. This includes photoionization delays, scattering delays, as well as a coherentdescription of electron transport and (laser-assisted) photoemission and scattering.As a consequence, we show that attosecond chronoscopy of liquids is, in general,sensitive to both types of delays, as well as the electron mean-free paths. Throughdetailed modeling, involving state-of-the-art quantum scattering and Monte-Carlotrajectory methods, we show that the photoionization delays dominate in attosec-ond chronoscopy of liquid water at photon energies of 20-30 eV. This conclusionis supported by a near-quantitative agreement between experiment and theory. Inthe second part, we introduce HHS of liquids based on flat microjets. These resultsrepresent the first observation of high-harmonic generation (HHG) in liquids ex-tending well beyond the visible into the extreme-ultraviolet regime. We show thatthe cut-off energy scales almost linearly with the peak electric field of the driverand that the yield of all harmonics scales non-perturbatively. We also discuss theellipticity dependence of the liquid-phase harmonics, which is systematically broad-ened compared to the gas phase. We introduce a strongly-driven few-band modelas a zero-order approximation of HHG in liquids and demonstrate the sensitivity ofHHG to the electronic structure of liquids. Finally, we discuss future possibilities formodelling liquid-phase HHG, building on the methods introduced in the first part ofthis chapter. In the conclusion, we present an outlook on future studies of attoseconddynamics in liquids.
H. J. WörnerLaboratorium für Physikalische Chemie, ETH Zürich, 8093 Zürich, Switzerlande-mail: [email protected]
A. Schild, D. Jelovina, I. Jordan, T. T. Luu, Z. YinLaboratorium für Physikalische Chemie, ETH Zürich, 8093 Zürich, Switzerland 1 a r X i v : . [ c ond - m a t . o t h e r] S e p H. J. Wörner, A. Schild, D. Jelovina, I. Jordan, C. Perry, T. T. Luu, Z. Yin
Attosecond science has the potential to address fundamental open questions in chem-ical and physical sciences. By directly accessing the electronic dynamics of matter ona time scale that naturally freezes any structural dynamics, attosecond spectroscopytargets the most fundamental electronic processes that define the properties of mat-ter. The methods of attosecond science have been well established and most broadlydemonstrated on atoms [1–4]. Very promising results have also been obtained onmolecules [5–9] and solids [10–14]. The field of attosecond molecular dynamics hasrecently been reviewed in Ref. [15].Attosecond dynamics in liquids have so far not been accessible to time-resolvedexperiments. However, they are expected to play a particularly important role inchemical and biological processes that dominantly take place in the liquid phase.The primary processes underlying charge and energy transfer in the liquid phasetake place on few-femtosecond to attosecond time scales [16] and have eluded directexperimental access to date. These processes include e.g. the ionization of liquidwater, electron scattering during its transport through the liquid phase, as well asintermolecular Coulombic decay (ICD) [17] and electron-transfer-mediated decay(ETMD) [18], to name only a few examples.Among these, the ionization of liquid water plays a particularly important rolesince it initiates the dominant pathway of radiation damage in living matter (see Ref.[19] for a recent review). The ionization of liquid water is inherently an attosecondprocess and therefore offers a target of primary relevance for attosecond science.However, the primary ionization event itself is not the only attosecond processinvolved in the ionization of liquid water. The subsequent electron scattering duringits transport through the liquid phase also takes place on a sub-femtosecond timescale. It controls the energy deposition during electron scattering and thereby thechemistry initiated by ionizing radiation in liquid water. Important new insights intothe early femtosecond time scale of water ionization have very recently been obtainedusing both one-photon ionization [20] and strong-field ionization [21].In this book chapter, we discuss two recently demonstrated experimental ap-proaches to attosecond science in the liquid phase: attosecond photoelectron spec-troscopy [22] and high-harmonic spectroscopy [23]. We present the experimentalmethodology and discuss the first experimental results, along with the developedconceptual framework and the related theoretical methods.In the first part, we describe the principles and methods of attosecond photo-electron spectroscopy of liquids [22]. This technique has been applied to measurethe time delay between photoemission from liquid and gaseous water. Photoion-ization delays from isolated molecules in the gas phase have only recently beenmeasured [9]. These measurements have been interpreted through the developmentof a general theoretical and computational methodology [24]. Here, we use photoe-mission from gaseous water molecules as a reference against which photoemissionfrom liquid water is clocked. We find that photoemission from liquid water is delayed by ∼ ∼ Attosecond Dynamics in Liquids 3 quantum-scattering calculations on water clusters of variable size, as well as Monte-Carlo trajectory simulations benchmarked against the time-dependent Schrödingerequation, that the measured delays are dominated by the local solvation structurein liquid water. Electron scattering during transport, in contrast, makes a negligiblecontribution to the measured delays.In the second part, we describe the experimental methods and first results onnon-perturbative high-harmonic generation in liquids [23]. This recent discoveryrepresented a paradigm change in attosecond science because a series of previousexperimental works on water droplets [25–27] concluded that high-harmonic gen-eration was impossible at the natural density of liquid water. Previous observationsof harmonic generation from liquids were indeed limited to low-order harmonics inthe visible range [28] and to the coherent-wake-emission regime [29], where high-harmonic generation takes place in a plasma created by the ultra-intense laser pulse(peak intensity > W/cm ) rather than in the original target. The main observa-tions from our work [23] are a linear scaling of the cut-off energy with the electricfield, a pronounced sensitivity of the observed spectra on the electronic structureof the liquids and a broadened ellipticity dependence of the high-harmonic yieldcompared to gas-phase HHG. High-harmonic spectroscopy of liquids offers a com-plementary and remarkably different approach to the same fundamental problemsdiscussed above. It provides access to the attosecond dynamics of the ionizationstep. It is sensitive to the dynamics of electron transport through the liquid phase,and thereby to electron scattering dynamics. Finally, it might also be sensitive to thespatial shape of the created electron hole through the recombination step. Liquid-phase high-harmonic spectroscopy therefore offers several promising prospects forexploring the electronic structure and dynamics of liquids, solutes and solvents.In the outlook that concludes this book chapter, we discuss possible future avenuesof attosecond science in liquids. These include time-resolved studies of ICD andETMD, which are the prototype elementary processes of charge and energy transferin living matter, but have unknown time scales. The most direct way to probe electron dynamics in liquid water is to perform atime-resolved measurement on the attosecond time scale. For this purpose, we havechosen attosecond photoelectron spectroscopy. It consists in creating freely movingelectrons in the conduction band of liquid water through ionization with an attosecondpulse and probing their dynamics through interaction with a temporally delayed near-infrared (NIR) pulse before detecting them through photoelectron spectroscopy.For the first experiment, described herein, we have used attosecond interferometry,which combines an extreme ultraviolet (XUV) attosecond pulse train (APT) with afemtosecond NIR laser pulse (see Fig. 1.1). This method has previously been appliedto measure photoionization delays in atoms [3] and molecules [9]. The particularadvantage of attosecond interferometry over e.g. attosecond streaking is the high
H. J. Wörner, A. Schild, D. Jelovina, I. Jordan, C. Perry, T. T. Luu, Z. Yin spectral resolution that is achieved [30], in addition to the high temporal resolutionthat is common to both methods. Whereas the spectral resolution in attosecondstreaking is limited by the temporal duration of the attosecond pulse through thetime-bandwidth product, the spectral resolution in attosecond interferometry is not.Instead, the resolution is limited by the spectral width of the individual harmonicorders, which are limited by the length of the APT and its chirp over the (femtosecond)time scale of the APT. The high spectral resolution is a crucial aspect of the presentexperiments because it enables the spectral distinction of photoelectrons originatingfrom the gas and liquid phases. These give rise, respectively, to the narrow and broadspectral features illustrated in Fig. 1.1.Experiments based on attosecond interferometry or attosecond streaking directlymeasure time delays in photoionization, but all such measurements are inherently rel-ative at present. It is therefore essential to identify a suitable reference against whichthese delays can be measured. In the present experiment, we turn the presence of theevaporating gas-phase molecules, usually perceived as a nuisance in liquid-microjetphotoelectron spectroscopy, into an advantage. By measuring photoionization timedelays of liquid water relative to those of isolated water molecules, we eliminatethe ”trivial” single-molecule contributions to the photoionization delays. The word”trivial” is consciously put in quotation marks because the measurement and inter-pretation of molecular photoionization delays is itself a highly active and intriguingarea of research [9, 24, 31–33]. However, the goal of the present work goes muchbeyond this, by aiming at identifying the specific contributions of the liquid phaseto the attosecond photoemission delays. These include two types of effects, i.e. themodification of the electronic structure and therefore the scattering potential of anisolated water molecule caused by solvation, as well as electron scattering dynamicsduring its transport through the liquid phase.In the following sections, we discuss the experimental techniques, which includethe realization of liquid-phase attosecond interferometry and the methods devel-oped to retrieve photoionization delays from overlapping photoelectron spectra. Wealso discuss in detail the novel theoretical methods, which comprise the integra-tion of laser-assisted scattering in the formalism of attosecond interferometry, thedevelopment of a general three-dimensional Monte-Carlo code for simulating suchexperiments and the identification of condensation effects on the photoionizationdelays from water clusters. This extensive development of experimental and theo-retical methods has culminated in the quantitative interpretation of the measuredphotoemission time delays of liquid water.
The experimental setup consists of an attosecond beamline delivering XUV APTsand a femtosecond NIR pulse, a liquid microjet delivering a micrometer-thin streamof liquid into a high-vacuum chamber, and a photoelectron spectrometer. In thiswork, we have used the attosecond beamline described in [34], which uses nested
Attosecond Dynamics in Liquids 5 e - e - IR XUV li qu i d j e t ga s p h o t o e l e c t r o n d e t e c o r nozzleH O (gas)H O (liq) vacuumlevel IRIR H (q-1) H (q+1) kinetic energyintensitymainbandssidebands Fig. 1.1 Attosecond time-resolved photoelectron spectroscopy of liquid water . A spectrallyfiltered attosecond pulse train, composed of a few high-harmonic orders (H ( q − ) ,H ( q + ) , etc.) su-perimposed with a near-infrared femtosecond laser pulse interacts with a microjet of liquid water.Photoelectrons are simultaneously emitted from the liquid and the surrounding gas phase. The re-sulting photoelectron spectra are measured as a function of the time delay between the overlappingpulses. Adapted from Ref. [22]. white-light and monochromatic interferometers to actively stabilize the time delaybetween NIR and XUV pulses to extreme accuracy. High-harmonic generation froma NIR laser pulse centered at 800 nm in an argon gas cell is used to generate theAPT. The created APT is spectrally filtered by thin metallic foils to reduce its spectralbandwidth and thereby the spectral overlap in the photoelectron spectra. Specifically,we have used Sn filters to isolate the harmonic orders 11, 13 and 15, or Ti filtersto isolate the harmonic orders 17, 19 and 21. A particular advantage of these metalfilters is their sharp spectral truncation at the high-energy side of the spectrum, whichmakes the remaining spectral overlap manageable (see Fig. 1.2). The liquid microjetis formed by expanding high-purity liquid water (with the addition of NaCl to aconcentration of 50 mM) through a ∼ µ m inner-diameter capillary with a high-performance liquid-chromatography pump. The photoelectron spectrometer is a 1-mlong magnetic-bottle time-of-flight spectrometer previously described in Ref. [35].The experimental results are shown in Fig. 1.2, where the left (right) columnsshow the data acquired with the Sn (Ti) filter. Correspondingly, the data on theleft-hand side is mainly composed of photoelectron spectra generated by H13 andH15, whereas the data on the right-hand side mainly originates from H19 and H21.These data have been analyzed through a principal-component analysis (PCA) basedon photoelectron spectra of gaseous and liquid water that were measured with amonochromatized high-harmonic light source [36] and the same photoelectron spec-trometer [35]. This PCA provides a unique decomposition of the measured spectrainto its constituents. The panels a) in Fig. 1.2 show photoelectron spectra acquired H. J. Wörner, A. Schild, D. Jelovina, I. Jordan, C. Perry, T. T. Luu, Z. Yin ωω A BC DE FG HJI
Fig. 1.2 Attosecond photoelectron spectra of liquid and gaseous water . Data acquired with aSn-filtered APT (left) or a Ti-filtered APT (right). a) Photoelectron spectra in the absence (blue) andpresence (orange) of the IR field with their principal-component fit (full lines) and decomposition(filled curves). b) Difference spectra (circles), principle-component fit (line) and decomposition(filled curves) into side bands (orange) and depletion (blue). c) Difference spectra as a function ofthe APT-IR time delay. d) Fourier-transform power spectrum of c. e) Amplitude and phase of the2 ω component of the Fourier transform. Adapted from Ref. [22]. with the XUV APT only (blue) or with XUV-APT and NIR pulses (orange). Thepanels b) show the difference spectra, obtained by subtracting the XUV-only spectrafrom the XUV+NIR spectra. In the panels a) and b) the circles represent the exper-imental data and the full lines the PCA fit. The contributing principal componentsare shown as filled areas, whereby the XUV-only contributions are shown in blueand the side-band contributions are shown in orange. The perfect agreement betweenthe PCA fit and the experimental data shows that the measured spectra have beensuccessfully decomposed into the contributing photoelectron spectra from liquid andgaseous water of the relevant high-harmonic orders.The panels c) show the difference spectra (from panels b) as a function of thetime delay between the overlapping XUV-APT and NIR pulses. Positive signals (inorange) are dominated by side-band contributions and negative signals (in blue) aredominated by the depletion of the XUV-APT photoelectron spectra. The Fouriertransform of the data in panels c) along the time-delay axis is shown in panels d)and the phase and amplitude of the Fourier transform, integrated over the dominant2 ω frequency component is shown in panels e). These complex-valued Fourier Attosecond Dynamics in Liquids 7 transforms have been analyzed within the PCA with minimal assumptions, i.e. theentire photoelectron bands corresponding to the highest-occupied molecular orbital(HOMO) of the gas or liquid contributions are assigned a characteristic phase ( φ gas or φ liq ). This minimal assumption is sufficient to reproduce the complete electron-kinetic-energy-dependent complex-valued Fourier transforms, as illustrated in panelse. As a consequence, it is possible to determine the photoemission time delaysbetween the HOMO bands of the liquid and gas phases: ∆ τ = τ liq − τ gas = φ liq − φ gas ω . (1.1)We find τ liq − τ gas = ±
20 as at a photon energy of 21.7 eV (SB14) and τ liq − τ gas = ±
16 as at a photon energy of 31.0 eV (SB20).In addition to robust delays, the PCA also enables the determination of reliablemodulation depths of the side-band intensities. This additional observable of attosec-ond interferometry has received very little attention to date. In an idealized situationwhere the amplitude of the two-photon-ionization pathways leading to the same side-band state have the same amplitude, the modulation contrast will be 100 %. The ex-perimentally observed modulation contrast will in general be smaller than 100 % forvarious possible reasons. In the gas phase, these include i) different amplitudes of thetwo-photon-ionization pathways, ii) contributions of several two-photon-ionizationpathways that do not interfere, e.g. because they correspond to different initial eigen-states of they leave the system in different final states, iii) contribution of ionizationchannels (e.g. emission angle or target orientation) corresponding to different timedelays, iv) fluctuations in XUV-IR path-length difference, etc. In condensed-phaseattosecond interferometry, additional effects include v) contributions from differ-ent local environments around the photoionized entity and vi) decoherence of thephotoelectron wave packet during transport through the condensed phase.Given the many possible contributions to a finite modulation contrast, it is thusimportant to find a meaningful reference. In the present case, the best referenceis our previously published measurement of molecular photoionization delays inwater vapor [9], which was carried out with the same apparatus, under the sameexperimental conditions as the liquid-phase experiments. This experiment revealeda modulation contrast of HOMO SB14 indistinguishable from 100 % within theexperimental signal-to-noise ratio (see Fig. 4.3 in [37]).Using the PCA discussed above, we have determined a relative modulation depthof the liquid- compared to the gas-phase signal M r = M liq / M gas of 0 . ± .
03 ata photon energy of 21.7 eV and 0 . ± .
06 at a photon energy of 31.0 eV. Thedetermination of relative modulation depths has the advantage that it eliminates thecontributions i)-iv), at least as far as single-molecule contributions and the exper-imental imperfections are concerned. Since a deviation from a perfect modulationcontrast was not observable in water vapor [37], the finite modulation depths re-ported above can mainly be attributed to the effects v) and vi). As further discussedin Section 1.2.3, contribution v) is significant and contribution vi) might also beimportant.
H. J. Wörner, A. Schild, D. Jelovina, I. Jordan, C. Perry, T. T. Luu, Z. Yin
In this section, a detailed theoretical treatment of attosecond photoelectron spec-troscopy (APES) in liquids is given. Although a significant amount of work on APESof solids has previously been published (see e.g. [38–41]), a comprehensive treatmentthat includes both photoionization delays and scattering delays has been missing un-til our recent work [42]. Motivated by experimental results [10–12, 43–46], mostprevious theoretical treatments were designed to describe experiments on metallicsamples. The description of APES in liquids fundamentally differs from the situ-ation encountered in metals. This is due to the different penetration depths of theXUV and NIR fields. Whereas the XUV fields penetrate more deeply into metalsthan the NIR, the situation is reversed in liquids, which are practically transparentto the NIR radiation. Since the temporal information originates from the interactionof the XUV-induced photoelectron wave packet with the NIR radiation, APES onmetals is mainly sensitive to the transport time of the photoelectron wave packetfrom the point of ionization to the metal-vacuum interface. In APES of liquids, thesituation is reversed and the NIR field is present throughout the medium. ThereforeAPES in liquid is sensitive to other aspects of attosecond photoionization dynamics.The following sections present a detailed conceptual and theoretical analysis of thissituation.
Time delays in quantum scattering were first analyzed by Wigner and Smith [47,48].They studied conventional scattering events, corresponding to the collision of twoparticles. Photoionization can be understood as a half collision and can be describedwith a very similar quantum-mechanical formalism. Consequently, it is not surprisingthat photoionization delays are closely related to scattering delays, both conceptuallyand in terms of their magnitudes [41, 49].Both scattering and photoionization delays typically lie in the attosecond domain.They have therefore become accessible in the time domain only recently [3, 50].Attosecond science has so far focused on the measurement of time delays in pho-toionization because they are directly accessible to APES. As a consequence, timedelays in (conventional) scattering, i.e. full collisions have so far not been discussed.Here, we show that they can play a significant role in condensed-phase APES. Moreimportantly, we show that laser-assisted electron scattering is a general phenomenonthat needs to be included into a comprehensive treatment of condensed-phase APES.We initiate our discussion with a classical analysis that provides a transparentbasic understanding of the problem. Figure 1.3 illustrates the definition of theseclassical delays. It is easy to understand that an attractive (repulsive) potential resultsin a negative (positive) delay because the particle is locally accelerated (decelerated)compared to the potential-free propagation before returning to the same asymptoticvelocity. As a consequence the time required to cover the same horizontal distance issmaller (larger) compared to potential-free propagation, which results in a negative
Attosecond Dynamics in Liquids 9
Particle is «advanced» t < 0Classical result: t tot = t photoionization + t scattering< 0 > 0 Particle is «delayed» t > 0Reference Fig. 1.3 Time delays in photoionization and scattering in the classical limit . The time delaysare defined as the differences in the arrival times of the classical particles that have propagatedthrough different potentials with that of a ”reference” particle propagating through potential-freespace with the same final (asymptotic) velocity. (positive) delay, as illustrated. In these classical considerations, it is also easy tounderstand that such classical scattering delays are additive. As long as the potentialsdo not overlap, the delay contributed by a particular potential is only a property ofthe local potential and does not depend on the previous trajectory of the particle.The same considerations hold true for a classical treatment of photoionizationdelays. This situation is illustrated at the bottom of Fig. 1.3. The classical definitionof a photoionization delay corresponds to a situation where the particle starts fromthe bottom of an attractive potential with a finite velocity. This velocity must bechosen such that the final (asymptotic) velocity of the particle is the same as for thereference particle. This classical photoionization delay will be negative for the samereason as the classical scattering delay for an attractive potential is negative. Theclassical photoionization delay is equal to half of the classical scattering delay forthe same potential.Continuing on this analogy, it is easy to show that the total classical delay forone photoionization and one scattering event is simply the sum of the two classicaldelays, as long as the potentials do not overlap. This situation is illustrated at thebottom of Fig. 1.3.
The translation of the results from the previous section to observables of attosecondspectroscopy requires at least two modifications. First, electrons are not classicalobjects and should therefore be described quantum-mechanically. When this is notpossible, the classical approximation has to be justified, as we further discuss be- low. Second, the situation analyzed in classical terms in the previous section wouldcorrespond to a time-of-flight measurement with attosecond resolution, which isimpractical. Instead, attosecond spectroscopy measures such time delays throughtwo-color photoionization. Whereas the relation of photoionization delays measuredwith such techniques with the (quantum-mechanical) Wigner delays has been es-tablished [41, 49], the treatment of full-scattering events itself and full-scattering incombination with photoionization has only been addressed in our recent publica-tion [42].In this section, we present a complete quantum-mechanical analysis of attosecondtime delays caused by photoionization and scattering as measured by attosecond in-terferometry. We note that these results are not specific to attosecond interferometry,but also apply to attosecond streaking, which will be the subject of future publi-cations. Starting from the numerical solution of the time-dependent Schrödingerequation, we show that the obtained delays remarkably deviate from the classi-cal expectation. Building on analytical descriptions of laser-assisted photoemission(LAPE) and laser-assisted electron scattering (LAES), we develop a complete ana-lytical description of the quantum-mechanical problem at hand. This has led to thediscovery of a novel phenomenon that we call ”non-local” attosecond interferome-try. In contrast to the traditional understanding of attosecond interferometry, wherethe interaction of the photoelectron wave packet with the XUV and NIR radiationtakes place in a common spatial volume, the non-local pathways involve interac-tions at spatially separated regions. Specifically, the XUV absorption is localizedto the position of the initially bound electronic wave function, whereas the NIRinteraction takes place at the position where scattering occurs. These ”non-local”XUV/NIR interactions lead to the interesting result that the total delays measuredin the presence of photoionization and scattering are, in general, sensitive to bothphotoionization and scattering delays, and to the distances travelled between XUVand NIR interactions, i.e. to the mean-free paths of electron scattering.We solve the one-electron time-dependent Schrödinger equation (TDSE) in onedimension using a model potential consisting of an attractive and a repulsive potentialwell (see Fig. 1.4). Details of this calculation are given in Ref. [42]. The system isinitiated in the lowest bound electronic state of the attractive potential. The TDSEis solved in the presence of an XUV field consisting of two harmonic orders (H q − and H q + ) and a NIR field. The photoelectron spectrum is calculated by projectingthe total wave function at the end of the propagation onto the continuum eigenstatesof the model system. This gives rise to a photoelectron spectrum as shown on thebottom left of Fig. 1.4. The calculated photoelectron spectra reveal an oscillation ofthe intensity of SB q as a function of the delay δ between the XUV and NIR pulses. Inthe calculation we actually vary the carrier-envelope phase of the NIR pulse insteadof varying the time delay δ . This is equivalent for the present purpose but avoidsunwanted effects caused by the envelope functions of the two fields. The temporalshift of the SB q maximum with respect to δ = τ .Solving the TDSE under the described conditions for different separations x between the attractive and repulsive potentials provides the result shown in thebottom-right part of Fig. 1.4. The total delay displays an oscillatory dependence Attosecond Dynamics in Liquids 11 -70-60-50 -40 -30-20 D e l a y / a s TDSE theory t photoion. x / nm ( t ) t scatt x H q+1 H q-1 SB q Photoelectron spectrum d d / fs t H q+1 H q-1 SB q Fig. 1.4 Time delays in photoionization and scattering as measured by attosecond interfer-ometry . This figure shows (schematically) the employed XUV and NIR fields (top left), the usedmodel potential (top right), the calculated photoelectron spectrum as a function of the time delay δ and the calculated delay τ as a function of the separation x between the attractive and repulsivepotentials. on x . This is in remarkable contrast to the naive classical expectation discussed inthe previous section. For our model system, the photoionization delays amounts to-44 as, as indicated by the red dashed line, and the scattering delay to +21 as. Withinour classical considerations this would lead to a total delay of -23 as, independentof x as indicated by the dashed green line. The result of the TDSE (green circles)departs markedly from this classical prediction. The calculated delay τ is found tooscillate around the value of the photoionization delay with a maximal amplitudegiven by the scattering delay. More precisely, the total delay τ can be expressed as asum of the photoionization delay τ PI and a non-local delay τ nl , τ = τ PI + τ nl , (1.2)where τ nl oscillates with the distance x .This result significantly contrasts with the previous understanding of attosecondinterferometry on isolated particles, where only local XUV and NIR interactions areimportant. In such a framework, the observed oscillations cannot be explained. Theycould, however, be quantitatively explained in Ref. [42] by developing an analyticalformalism that includes both local and non-local pathways.Figure 1.5 shows the relevant paths that an electron can take so that it endsup in the sideband q . The paths L1 and L2 are “local” paths that are the basis ofAPES in gas-phase molecules, the paths L3 and L4 are paths where interaction with q+1q-1q q+1q-1qq+1q-1qq+1q-1q q+1q-1qq+1q-1q L1 L2L3 L4NL1 NL2
Fig. 1.5
Schematic depiction of the four “local” (L1, L2, L3, L4) and the two “non-local” pathscontributing mainly to the attosecond interference signal if one scattering event is possible afterionization. the assisting IR field is still at the ionization site but the outgoing electron wavepacket undergoes scattering, and NL1 and NL4 are “non-local” paths where photonexchange happens at the scattering center. In words, those paths are:• L1: Absorption of an XUV photon of frequency ( q − ) ω and absorption of oneIR photon at the ionization site,• L2: absorption of an XUV photon of frequency ( q + ) ω and emission of one IRphoton at the ionization site,• L3: absorption of an XUV photon of frequency ( q − ) ω and absorption of oneIR photon at the ionization site, and scattering at the perturber without photonexchange,• L4: absorption of an XUV photon of frequency ( q + ) ω and emission of oneIR photon at the ionization site, and scattering at the perturber without photonexchange,• NL1: absorption of an XUV photon of frequency ( q − ) ω at the ionization siteand absorption of one IR photon at the perturber, and• NL2: Absorption of an XUV photon of frequency ( q + ) ω at the ionization siteand emission of one IR photon at the perturber.While L1 and L2 are the two paths that explain APES for gas-phase molecules, allsix paths are needed reproduce the delay of Fig. 1.4. Each of these paths contributesto the second-order amplitude (i.e., the amplitude for two-photon transitions) f , i.e., f = f L1 + f L2 + f L3 + f L4 + f NL1 + f NL2 , (1.3) Attosecond Dynamics in Liquids 13 and the sideband intensity is then I ∝ | f | . The amplitudes are f L1 = F q − q e ik q x (1.4) f L2 = F q + q e ik q x (1.5) f L3 = F q − q e ik q x f qq (1.6) f L4 = F q + q e ik q x f qq (1.7) f NL1 = F q − q − e ik q − x f q − q (1.8) f NL2 = F q + q + e ik q + x f q + q . (1.9)Here, F mn indicates the amplitude of transition from state m to state n at the pho-toionization site and f mn indicates the amplitude of transition from state m to state n at the perturber. An additional phase factor is due to propagation to the scatteringsite with momentum k q (for L1, L2, L3, and L4), with momentum k q − (NL1), orwith momentum k q + (NL2).Numerical values for the amplitudes F mn and f mn can be obtained within what isknown as strong-field approximation, soft-photon approximation, or Kroll-Watsontheory, and which is referred to as laser-assisted photoelectric effect (LAPE) andlaser-assisted electron scattering (LAES), respectively [51–53]. For LAPE, it isassumed that the initial state is a bound state, the final state is that of a laser-dressedfree electron in the IR field, the ionizing XUV pulse can be treated as perturbation,and the IR field is approximately a continuous wave. Similar approximations aremade for LAES, with the initial and final states being laser-dressed states of the freeelectron in the IR field and the scattering potential is the perturbation. These theoriesprovide the amplitudes F q + nq = e in ( π + ω ∆ t ) J n (cid:32) k q F IR0 ω (cid:33) f PI q + n (1.10) f q + nq = e in ( π + ω ∆ t ) J n (cid:32) ( k q − k q − n ) F IR0 ω (cid:33) f ES q + n , q (1.11)where F IR0 is the maximum field strength of the IR pulse, f PI q + n is the field-freephotoionization amplitude, and f ES q + n , q is the field-free elastic scattering amplitude. J n are the Bessel function of the first kind.The side band intensity in this multi-path model is calculated as I ∝ | f | = A cos ( ω ( ∆ t + τ PI + τ nl )) , (1.12)where the photoionization delay is given by the phase differences of the LAPEamplitudes, τ PI = arg ( F q + q ) − arg ( F q − q ) ω . (1.13) With some algebra, the non-local delay can be approximated as τ nl ≈ ω A nl A l sin (cid:18) (cid:0) k + x s + φ + (cid:1)(cid:19) × sin (cid:18) ( k − x s + φ − ) (cid:19) , (1.14)where A l and A nl correspond to the (approximately equal) amplitudes of the processeswhere absorption or emission of the IR photon happens during photoionization andduring scattering, respectively, i.e., A l ≈ | f L1 + f L3 | ≈ | f L2 + f L4 | (1.15) A nl ≈ | f NL1 | ≈ | f NL2 | , (1.16)and the phase shifts of the two sine functions are φ + = arg ( f q + q ) − arg ( f q − q ) (1.17) φ − = arg ( f q + q ) − ( + f qq ) + arg ( f q − q ) . (1.18)The period of oscillation with x s is characterized by the two wave numbers k + = k q + − k q − (1.19) k − = k q + − k q + k q − . (1.20)Clearly, the non-local delay (1.14) exhibits a beat pattern, i.e., an interference of twowaves with slightly different oscillation frequencies. In this case, the two frequenciesgiving rise to the beat pattern are k q + − k q and k q − k q − , whose sum is k + (thefast oscillation with x s ) and whose difference is k − (the slow oscillation with x s ). Wealso note due to the phase shifts the non-local delay starts close to its maximum andthe oscillation is cosine-like.The first step in translating these results from a one-dimensional model systemto the condensed phase consists of including the effect of path-length distributions.The distance travelled by an electron between photoionization and the first collisionor between two collisions is, in general, a distributed quantity, the average of whichis defined as the mean-free path. For simplicity, we first consider only one scatteringevent and we assume an exponential distribution of path lengths, which is the standardassumption in condensed-phase electron-transport simulations (see, e.g. Ref. [19]).Figure 1.6 shows the effect of the path-length distribution on the calculated delay.The top panel shows the results obtained when the distance between photoionizationand scattering has a fixed value x , whereas the bottom panel shows the case wherethe distance is a distributed quantity following an exponential distribution, as afunction of the average path-length MFP. We find that the oscillatory behavior fromthe fixed-distance case is turned into a monotonic decay from τ photoion + τ scatt to τ photoion . This result has an intuitive explanation. When the MFP is long, specificallyMFP ≥ L , where L was defined above, the contribution of scattering cancels because The 1 in 1 + f qq comes from combination of the paths L1 and L3 or L2 and L4. Attosecond Dynamics in Liquids 15 Path length (nm)
Path-length distribution
MFP D e l a y / a s fixed distance x / nm t photoion. t scatt. D e l a y / a s distributed path lengths MFP / nm t photoion. t scatt. Fig. 1.6 Effect of path-length distribution . The top panel illustrates the case of a fixed distance x between photoionization and scattering, whereas the bottom panel represents the case where thepath lengths follow an exponential distribution. In the bottom panel, the delay is given as a functionof the average path length (MFP). of the oscillatory nature of the delay dependence on the path length. When MFP (cid:28) L the delays behave additively, i.e. the classical situation is recovered.In a second step, we now include the treatment of multiple collisions. Figure 1.7shows the results obtained for n = − τ nl ≈ τ sca n (cid:213) j = cos ( nk x s ) , (1.21)where τ scatt is the scattering delay. Thus, at x =
0, the non-local delay is ca. τ photoion + n τ scatt . In the case of distributed path lengths (dashed lines)These results are in line with the same intuitive explanation given above. For longMFPs the contribution of scattering vanishes, such that the calculated delay is equalto the photoionization delay. For very short MFPs ( (cid:28) L ), the delays again behaveadditively, i.e. the classical limit is reached. The previous section has established the conceptual foundations of attosecond inter-ferometry in condensed matter on the basis of simple model potentials. The purpose x s in nm0 τ sca τ sca τ sca τ − τ P I fixed distance, one collisionfixed distance, two collisionsfixed distance, three collisionsdistributed distance, one collisiondistributed distance, two collisionsdistributed distance, three collisions Fig. 1.7 Effect of multiple collisions . Calculated delay for the case of a finite number of collisionsseparated by fixed distances x (full lines) or path lengths that follow an exponential distribution ofaverage x (dashed lines). of this section is to present the results of state-of-the-art quantum scattering calcu-lations on water clusters as a computationally tractable model of liquid water.Beginning with an isolated water molecule, we calculate the complex scatteringfactor for the fully elastic scattering process of a free electron with a water moleculein its rovibronic ground state. In our calculations, we use orientational averagingof the scattering target. As a consequence, the scattering problem has cylindricalsymmetry, such that the scattering factor f ( θ, E ) depends only on the polar angle θ (see Fig. 1.8) and the electron kinetic energy E . This scattering factor defines boththe differential scattering cross section (DCS) d σ d Ω and the scattering delay τ scatt asdefined in Fig. 1.8.The DCS has a global maximum at θ =
0, corresponding to forward scattering anda local maximum at θ = π , corresponding to back scattering. The scattering delayin the forward direction is positive, corresponding to the naive classical expectation.The scattering delay displays a rapid variation around θ ≈ Attosecond Dynamics in Liquids 17 e - 𝜏 𝑠𝑐𝑎𝑡𝑡 = 𝜕arg(𝑓 𝜃, 𝐸 ) 𝜕𝐸 q 𝑑𝜎𝑑Ω = 𝑓(𝜃, 𝐸) q (rad) D C S ( - c m ) E k =10.5eV Differential scattering cross section -150-100-500 Scattering delay q (rad) d e l a y ( a s ) Fig. 1.8 Calculations of DCS and scattering delay for water monomer . These calculations weredone for an electron kinetic energy of 10 eV using the commercial R-matrix code Quantemol, inwhich the calculation of the DCS is implemented. The calculation of the scattering delay from thescattering factor provided by Quantemol was developed in house [54].
DCS without backscattering. The width of the quasi-Gaussian peak converges veryrapidly with cluster size.The rapid convergence of the DCS with the size of water clusters suggests thatthe highly accurate quantum-scattering calculations presented in this section can beused to model the description of electron scattering in liquid water. In addition toproviding the crucial scattering factors required to model attosecond interferometry,they additionally enable us to determine both elastic and inelastic mean-free path forelectron scattering in liquid water [56], as discussed in the following section.
Given the DCS for electron scattering at a certain kinetic energy, we are now in theposition to determine the elastic and inelastic mean-free paths (EMFP and IMFP)[56]. For this purpose, we turn to two liquid-jet measurements of the photoionizationof water: A measurement of the effective attenuation length (EAL) [57] and ameasurement of the gas-phase and liquid-phase photoelectron angular distributions(PAD) [58] for oxygen 1s photoionization of water molecules. The underlying ideasof the experiments and of our simulations are depicted in Figure 1.10. For ionizationat a depth | z | below the surface the measured photoelectron signal at the kineticenergy corresponding to direct photoionization behaves as S ( z ) ∝ e −| z |/ r EAL , (1.22) q (rad) D C S ( no r m a li s ed ) MonomerDimerTrimerTetramerPentamerHexamerHeptamer
Fig. 1.9 Effect of condensation on the DCS of electron scattering . Calculated DCS for electronscattering with water clusters of the indicated size using Quantemol. The equilibrium structures ofthe most stable isomers of the water clusters of each size have been taken from [55]. All DCS havebeen normalized at θ = i o n i z a t i o n d e p t h | z | intensity e -|z|/r EAL
D D D e ff ective attenuation length photoelectron angular distribution Fig. 1.10
Left: Schematic depiction of the effective attenuation length (EAL) for ionization in aliquid. The signal of photoelectrons with kinetic energy corresponding to direct ionization decreasesexponentially with the depth of the ionization site due to inelastic scattering. The EAL is the widthparameter of the exponential function. Right: Idea of the measurement of the photoelectron angulardistribution (PAD) of a liquid: Molecules are ionized with a beam and the photoelectron signal ismeasured with a detector above the surface that has a relatively small detection angle (indicated as“D”). Variation of the beam polarization axis yields the PAD of the liquid. where r EAL is the EAL. The loss of signal with | z | is due to inelastic scattering, andthe EAL is also a lower bound for the IMFP. We can simulate the experiment withour Monte Carlo trajectory code by using the field-free quantities and by simply Attosecond Dynamics in Liquids 19 measuring the number of electrons exiting the liquid depending on the assumedionization depth.The PAD can, for achiral molecules, be described to a good approximation withan energy-dependent asymmetry parameter β asPAD ( θ ) ∝ + β P ( cos θ ) , (1.23)where θ is the polar angle and P is the Legendre polynomial of second order. β -parameters were measured in [58] for the gas phase and for the liquid by rotating thepolarization direction of the ionizing beam relative to a detector. In particular, thedetector is not moved relative to the surface of the liquid jet, as indicated in Figure1.10. We can simulate also this experiment with our Monte Carlo trajectory codeby rotating the PAD for ionization, which we take to be that of gas-phase water, asoxygen 1s-electrons are considered and we do not expect the corresponding orbitalsto be influenced by solvation, except for a small shift of the binding energy. Thedetected signal is obtained by counting the electrons which exit the surface under asmall polar angle.The program code for these simulations, CLstunfti, is publicly available [59]. Weindeed find the analytical behaviors given by (1.22) and (1.23) numerically. We alsofind the following dependencies of the EAL and the β -parameter β liq of liquid wateron the EMFP and on the average number of elastic scatterings (cid:104) N ela (cid:105) = IMFP / EMFP:• For fixed EMFP, the EAL increases linearly with (cid:112) (cid:104) N ela (cid:105) . This is because the EALis essentially the root-mean-square translation distance of a random walk [60].• For fixed (cid:104) N ela (cid:105) , the EAL increases linearly with the EMFP. This happens because,for fixed (cid:104) N ela (cid:105) , both the EMFP and the IMFP are scaled by the same amount andhence essentially all trajectories are also just scaled accordingly.• For fixed (cid:104) N ela (cid:105) , β liq does not depend on the EMFP. This independence is alsoa result of the discussed scaling, as only trajectories are scaled but angles areinvariant to changing both EMFP and IMFP by the same amount.• For fixed EMFP, β liq decreases to zero with increasing (cid:104) N ela (cid:105) , i.e., the PADbecomes closer and closer to the isotropic PAD with each additional collision.From these dependencies follows that a set of parameters ( r EAL , β liq ) corresponds toa unique set of mean free paths (EMFP, IMFP). We use this fact to find the EMFPand IMFP needed for our simulations. For sideband 14 we obtain an EMFP of 0.6 nmand (cid:104) N ela (cid:105) ≈
7, and for sideband 20 we obtain a slightly longer EMFP of 0.8 nm and (cid:104) N ela (cid:105) ≈
6. The values for the EMFP and IMFP from 10 eV to 300 eV determinedwith this approach can be found in [56].
In this section, we combine the results described in the preceding subsections toa comprehensive model of APES in liquids. Starting from the discussion of theone-dimensional model system introduced in section 1.2.2.2, we generalize these results to three dimensions and include the output from ab-initio quantum scatteringcalculations on realistic models of liquid water. For this purpose, we need the elasticand inelastic mean-free paths for electron transport in liquid water. These weredetermined based on two independent experiments on liquid-water microjets that wehave inverted within our new formalism described in the previous section. The resultis a fully self-consistent description of liquid-phase attosecond spectroscopy thatincorporates the newly discovered non-local processes, as well as a state-of-the-artdescription of time delays in photoionization and scattering. + A B C Fig. 1.11 Contributions of photoionization and scattering to the measured delays . Schematicrepresentation of the potentials used in the TDSE calculations (top) modelling the photoionizationand scattering as attractive and repulsive potentials, respectively. The resulting delays ( τ tot ) for thecase of a single collision (middle) oscillate between τ PISB + τ scaSB and τ PISB − τ scaSB as a function of thedistance r (lines). In the case of an exponential path-length distribution with average r , a monotonicdecay is obtained (symbols). In the case of n elastic collisions, sampled according to an exponentialpath-length distribution with average r , the delay decays from τ PISB + n τ scaSB to τ PISB (bottom). Adaptedfrom Ref. [22] Attosecond Dynamics in Liquids 21
As a first step towards interpreting the experimentally measured time delays, wereturn to the one-dimensional model system discussed in section 1.2.2.2. The at-tractive potential is now chosen to mimic the ionized water molecule whereas therepulsive potential represents electron scattering with a neutral water molecule, asillustrated in Fig. 1.11a. We additionally choose the photon energies to be thoseused in the experiment, i.e. harmonic orders 13/15 and 19/21 of an 800-nm drivingfield. Panel b shows the calculated delays as a function of the distance r between thepotentials representing photoionization and scattering. The full lines show the resultsof calculations done with fixed distances, whereas the symbol correspond to calcu-lations assuming an exponential distribution of path lengths. These results confirmthe conclusions of section 1.2.2.2. Specifically, the total delay decays monotonicallyfrom τ PI q + τ sca q to τ PI q with increasing mean-free path.With the availability of the EMFP and IMFP for liquid water at the relevantelectron-kinetic energies from section 1.2.2.4, we can now address the role of scat-tering delays in attosecond spectroscopy of liquid water within the one-dimensionalmodel. The ratio IMFP/EMFP defines the number of elastic collisions that take place(on average) before one inelastic collision occurs. This ratio amounts to ∼ ∼ r . Al-though the variation of the calculated delay with r is no longer strictly monotonic, asa consequence of the more rapid variations of the delay with r in the case of multiplecollisions (see Ref. [42] for details), the rapid convergence of both delays to τ PI q isstill obvious. Most importantly, when using the EMFP values determined in section1.2.2.4, we find that both delays have practically converged to τ PI q . In other words,the one-dimensional calculations already suggest that the contributions of electronscattering during transport in liquid water are negligible under our experimentalconditions. Although these results were obtained with simple model potentials, theyalready show that any possible contribution of scattering delays will be stronglyreduced by the nature of the non-local processes.Before attempting an interpretation of the experimental results, we need to gen-eralize this formalism to three dimensions and include a realistic description ofphotoionization and scattering in liquid water. Since the fully quantum-mechanicaltime-dependent description of electron transport in liquid water is computationallyintractable, we opted for a hybrid quantum-classical approach. The analysis of theTDSE calculations in section 1.2.2.2 has shown that a quantum-mechanical descrip-tion of photoionization and scattering within the Kroll-Watson formalism, combinedwith a classical-trajectory representation of electron transport, produces results ofquantitative accuracy for electrons with kinetic energies E k (cid:39)
10 eV. Independentearlier work has shown that electron-transport simulations in liquid water based on aclassical-trajectory Monte-Carlo description also reach quantitative agreement witha fully quantum simulation of electron transport, provided that the kinetic energy islarger than ∼
10 eV [61]. We therefore based our description of liquid-phase attosec-ond spectroscopy on the same approach as section 1.2.2.2, i.e. a quantum-mechanical description of photoionization, scattering and their laser-assisted counterparts and aclassical-trajectory Monte-Carlo description of the electron transport steps.This is done as follows:• We assume a flat interface between the liquid and the vacuum at z = z < z <
0. Those trajectories, taken together,represent the photoelectron wave packet originating from one ionization event,thus each trajectory has a phase.• Half of these trajectories are initialized in state n = q − n = q +
1, where a “state” corresponds to the kinetic energy n (cid:126) ω − E b .• The generalization of formula (1.10) to three dimensions reads F q + nq = e in ( π + ω ∆ t ) J n (cid:32) k q · F IR0 ω (cid:33) f PI q + n , (1.24)which describes the amplitude of changing from state q + n to state q . We knowthe photoelectron angular distribution | f PI q + n ( θ )| depending on the polar angle θ from photoionization calculations described in Section 1.9 and we do not needthe phase arg f PI q + n ( θ ) , because we are only interested in the non-local contribution τ nl to the total measured delay τ = τ PI + τ nl , cf. (1.2) Equation (1.24) can be used to model the interaction of the electron with theassisting IR field during photoionization. This is done by randomly selectingtrajectories to change to states q − q , or q +
2, according to the | F q − q − | (emission), | F q − q − | (no photon exchange), and | F q − q | (absorption) if the trajectory is initiallyin state q −
1, and according to the amplitudes | F q + q + | (absorption), | F q + q + | (nophoton exchange), and | F q + q | (emission) if the trajectory is initially in state q + If a state change happens, the corresponding trajectories obtain a phase as givenby (1.24) (without the unknown phase due to f PI q + n ). We note that as the laser fieldpolarization is always assumed to be in z -direction, F q + nq depends only on theoffset ∆ t between ionizing XUV and assisting IR pulse, the ionization direction,and the initial/final states.• In the end, only trajectories reaching the surface in state q contribute to the side-band of interest and hence are the only trajectories included in the determinationof τ nl . The probability of ending up in state q is negligible if the trajectories endup once in q ±
2, hence those trajectories are discarded. We use two sets of angles: The “lab” frame with polar angle θ and azimuthal angle φ , where θ isthe angle relative to the z -axis define via the surface normal, and the relative frame with polar angle ϑ and azimuthal angle (never explicitly used here), where ϑ is the angle relative to the directionof motion of the electron/trajectory. For the simulations described here the polarization of the laserpulses is assumed to be in z -direction, hence there is no difference between lab frame and relativeframe for ionization. This is different for scattering. The probability to stay in the respective states is obtained from | F q − q − | and | F q + q + | . For therelevant parameters, absorption or emission of two IR photons is irrelevant. Attosecond Dynamics in Liquids 23 • We use mean-free paths r MFP based on the general definition P ( r ) = r MFP e − r / r MFP (1.25)to model random scattering in the medium. Each trajectory is attributed a ran-domly chosen maximal path to travel according to an IMFP r IMFP . This IMFPreflects (electronically) inelastic scattering such that the scattered electrons looseenough energy not to be detected at the kinetic energy of the considered sideband.If a trajectory reaches this maximum path length before reaching the surface, it isdiscarded. • The trajectories are moved in the direction chosen randomly according to | f PI q + n ( θ )| and for a distance r chosen randomly according to (1.25) with EMFP r EMFP . At this distance, elastic scattering is assumed to happen. A phase k n r isadded to the phase of the trajectory, where k n is the momentum that the electronhas in its current state.• The generalization of formula (1.11) for laser-assisted scattering to three dimen-sions is f q + nq = e in ( π + ω ∆ t ) J n (cid:32) ( k q − k q − n ) · F IR0 ω (cid:33) f ES q + n , q (1.26)where f ES q + n , q ( ϑ ) is the field-free scattering amplitude depending on the polar angle ϑ relative to the current direction of motion, k q − n is the momentum vector of theincoming electron, and k q is the momentum vector of the outgoing electron. Foreach elastic scattering, a probability to change states as well as new directions forthe motion of the electrons is chosen according to the amplitudes f q + nq . If a statechange happens, the corresponding phase is added to the phase of the trajectory.• From | f q + nq | new directions for the outgoing electron trajectories are determinedand the process of elastic scattering is repeated. • Trajectories are stopped if they end in states q ±
2, if their path is longer than thepreviously sampled corresponding maximum path, or if they reach the surface at z = q which originate from the same startingpoint, are added coherently. The contribution to the total second-order amplitudeoriginating from a given point is calculated by coherent summation over alltrajectories, which reach the detector with total acquired phases γ j : There is another way to implement the IMFP which is computationally more efficient if manyIMFP values should be tested: No maximum path length is set but the contributions of the trajectoriesreaching the surface in state q to the total signal are weighted as e − r tot / r MFP , where r tot is the totalpath that the respective trajectory travelled until reaching the surface. We note that f q + nq depends on three angles, for example on the polar angle of the incomingelectron trajectory θ in and the polar and azimuthal angles θ ou , φ ou of the outgoing electron trajectoryin the lab frame, as well as on the offset ∆ t and the initial and final state of the electron. If thelaser field polarization axis was not aligned with the z -axis of the lab frame (defined by the surfacenormal), f q + nq would also depend on the azimuthal angle φ in of the incoming electron trajectory.4 H. J. Wörner, A. Schild, D. Jelovina, I. Jordan, C. Perry, T. T. Luu, Z. Yin f w k ( ∆ t ) = n det k (cid:213) j f t j , k = n det k (cid:213) j e γ j ( ∆ t , z ini k ) . (1.27)• Simulations are run for a large number of initial positions z ini j . As it is assumedthat electrons originating from different initial positions z ini j in the liquid do notinterfere, these contributions are summed incoherently, so that the total signalfor a given value of the offset ∆ t is ρ ( ∆ t ) = (cid:213) k n det k n tot | f w k ( ∆ t )| . (1.28)Figure 1.12 illustrates the simulation procedure schematically. In the figure, twoionization sites (at z ini1 and z ini2 ) are depicted, from which three trajectories each arestarted. Let us first concentrate on one trajectory, i.e., the one with amplitude f t3 , .This trajectory starts in state q −
1, scatters once at a distance r without photonexchange, scatters a second time at a distance r and absorbs a photon to change tostate q , and scatters one last time without photon exchange after a distance r beforeit reaches the surface in state q and is thus counted to the signal. The amplitude forthe trajectory is thus f t3 , = F q − q − e ik q − r f q − q − e ik q − r f qq − e ik q r f qq e ik q r s , (1.29)where the field-dressed ionization amplitude F q − q − and all field-dressed scatteringamplitudes depend on the directions along the path of the electron trajectory. Thefinal signal obtained from the six trajectories is obtained as given in the figure.We note that the actual simulation uses millions to billions of trajectories for eachionization site, hundreds to thousands of ionization sites for convergence of τ nl ,and most of the trajectories do not end in state q as the transition probabilities arerelatively small.Using the input parameters described in Sections 1.2.2.3-1.2.2.4, we can simulatethe contribution of the non-local delay to the total delay of APES for liquid water. Wefind that the non-local delay is almost independent of the average number of elasticcollisions (cid:104) N ela (cid:105) ∈ [ , ] and decays with an increasing EMFP ∈ [0.1,1.3 ]nm. Adecrease of τ nl with increasing EMFP is expected from the one-dimensional model,as we see a decay of τ nl with the mean free path, cf. 1.2.2.2. The independence of τ nl on (cid:104) N ela (cid:105) in the tested range is surprising but, as shown below, τ nl is also rathersmall. This independence may originate from a suppression of coherence by randomscattering, or from the dominance of relatively short trajectories originating fromclose to the surface.Figure 1.13 shows the dependence of the non-local delay (for EMFP=0.55 nm and (cid:104) N ela (cid:105) =
11) on the initial position of the trajectories below the surface. We can seethat τ nl becomes larger with starting depth, but is only ca. 5 as for an initial depth of There is, however, no significant change to the result if the summation is done coherently. Attosecond Dynamics in Liquids 25 q+1q-1q q+1q-1qq+1q-1q q+1q-1qq+1q-1q q+1q-1q q+1q-1qq+1q-1qq+1q-1q q+1q-1q coherent sum coherent sumincoherent sum F q-1q-1 f qq-1 f qq r r r f q-1q-1 r Fig. 1.12 Three-dimensional model of attosecond interferometry in the condensed phase
Sketch of possible trajectories and how they are detected, i.e., how the intensity in the sideband q is calculated as a function of the offset ∆ t between XUV and IR pulse (the experimental controlparameter). On the left side, three trajectories are shown which originate from a depth z ini1 , two ofwhich end up in state q ( f t1 , and f t3 , ; for the latter, also path lengths r jk and amplitudes of theindividual processes are given). The coherent sum of the corresponding amplitudes provides thecontribution from z ini1 . On the right side, three trajectories originating from z ini2 , where only onetrajectory ends in state q ( f t3 , ). Contributions for different z ini j are summed incoherently to obtainthe sideband intensity ρ ( ∆ t ) . τ nl is small compared to the experimentallymeasured value of the delay. We find indeed that according to our simulations, thecontribution of τ nl to the total delay is only about 2 as.Thus, we conclude that the non-local delay is negligible in our attosecond ex-periments on liquid water. Nevertheless, there are situations where the non-localdelay may play a role, for example if the EMFP is significantly shorter than in thepresent case or if the considered medium has long-range order, such as e.g. a crystal.For these cases, the non-local delays may become measurable, which would provideadditional information regarding the scattering delays and mean-free paths. z ini in nm0246810 n o n - l o c a l d e l a y n l ( a s ) z ini in nm051015202530 s q u a r e d n o n - l o c a l d e l a y n l ( a s ) Fig. 1.13
Dependence of the non-local delay τ nl for liquid water on the initial position z ini ofthe ionized electron below the surface ( z = τ nl ( z ini ) is approximately a linear function. The deeper the electron starts, the less trajectories reach thesurface, hence the accuracy of τ nl decreases with | z ini | if the number of trajectories is not increased. We now return to the interpretation of the experimental results described in section1.2.1. These measurements have yielded time delays between photoemission fromliquid and gaseous water of 69 ±
20 as at 21.7 eV photon energy and 48 ±
16 as at31.0 eV photon energy. In addition, relative modulation contrasts of 0.17 ± ± Attosecond Dynamics in Liquids 27 structure of water molecules changes with the addition of neighboring molecules.This is a consequence of the relatively strong dipole moment of the water molecule,which causes the formation of strong hydrogen bonds between water molecules. Theconsequence is a significant change in the electronic structure, which becomes visiblein the hybridization of the molecular valence orbitals. In addition to the local changeof the electronic wave function, condensation also causes a partial delocalization ofthe one-electron wavefunctions (orbitals) over more than one molecule. These effectsconcern the initial bound electronic wave function. They will all have an influenceon the photoionization delay. Second, the final continuum wave function is alsoaffected by condensation, probably even more than the initial state. The continuumwavefunction for bulk liquid water is very complex. It consists of a complicatedconduction-band part inside the liquid bulk and a simpler part in the vacuum regionoutside the liquid. However, all that is needed to accurately describe photoionizationis the continuum wave function over the spatial region where the initial-state wavefunction has a significant amplitude. Therefore, our calculations on water clusterscan be expected to converge to the results for liquid water when the water clustersbecome sufficiently large as to describe the relevant spatial extension of the initial-state wave function. Both effects discussed in this paragraph are naturally includedin our quantum-scattering calculations. H O (H O) (H O) photoionization delays t PI
49 as
97 as
18 as
110 as48 as
29 as experiment t liq - t gas t (H O) - t H O theory
30 as
61 as
71 as
32 as
107 as
36 as
111 as
47 as
157 as
41 as stretched stretchedbent bent photon energy(side-band order)
62 as
29 as
108 as
23 as stretched bent
PI PI
Fig. 1.14 Effect of solvation and hydrogen bonding on photoionization delays . Calculatedphotoionization delays for H O, (H O) and (H O) , representing water molecules with zero, onecomplete or one complete and one partial solvation shells, respectively. The bottom row indicatesdelays obtained by stretching (blue arrow) or bending (green arrow) one hydrogen bond in theclusters. Adapted from Ref. [22] Figure 1.14 shows the calculated photoionization delays of water clusters of in-creasing size at the two photon energies relevant for this work. These calculationswere done with ePolyScat [62,63] using the methods described in Ref. [24]. The firsttwo lines of the table show the photoionization delays of isolated water molecules, a water molecule with a complete first solvation shell (pentamer) and with a partialsecond solvation shell in which all dangling hydrogen bonds of the first shell havebeen coordinated (undecamer). Interestingly, the photoionization delays systemat-ically increase with increasing size of the water cluster. Whereas the calculationsat the lower photon energy show evidence of convergence with cluster size, theconvergence is less obvious at the higher photon energy. Since converged quantum-scattering calculations on (H O) reach the limit of current computational methods( ∼
180 CPU days for one calculation), these calculations could not yet be extended tolarger water clusters. However, the relative photoionization delays between the waterclusters and the water monomer are in rewarding agreement with the experimentalresults. Whereas the calculated relative delay of 61 as at 21.7 eV agrees with theexperiment within the error bar, the relative delay of 30 as at 31.0 eV is close to theerror interval of the experiment.In addition to reaching near-quantitative agreement with the experiment, the re-sults in the two bottom lines of Fig. 1.14 shed light on the other experimentalobservable introduced in this work: the modulation contrast in attosecond interfer-ometry. Whereas the calculations shown in the upper two lines assumed a tetrahedralcoordination geometry of each water molecule with O-O distances fixed to 2.8 Å,the local solvation environment is known to be partially distorted in liquid waterand subject to fluctuations on picosecond time scales. To simulate the effect of thestructural distortions of the hydrogen bonds, we have stretched or bent one of thehydrogen bonds by 0.7 Å or 50 ◦ , respectively, following the distortions used inRef. [64]. In the case of the pentamer, we have moved one of the water molecules,and in the case of the undecamer, we have moved a group of 3 water moleculeswhile keeping its internal geometry unchanged. The effects of these local distortionsof the hydrogen-bond structure is remarkable. The delays calculated at 21.7 eV dis-play a pronounced sensitivity to the structure with variations reaching -16 as in thepentamer and +47 as in the undecamer. The corresponding changes at 31.0 eV aremuch smaller and do not exceed ± ± ± Attosecond Dynamics in Liquids 29 particularly promising, as well as the investigation of other attosecond time-scaleprocesses in the liquid phase.
High-harmonic generation (HHG) in gases has been extensively studied over the lastthree decades. Apart from its quintessential significance as a coherent light source,it has also been developed into a unique spectroscopic technique. High-harmonicspectra indeed contain a wealth of information about the structure and dynamics ofthe medium from which they are emitted. In the gas phase, HHG can be understood asa process that involves strong-field ionization, electron propagation in the continuumand photorecombination. As a consequence, high-harmonic spectra indeed containinformation about the electronic structure of the medium which is encoded in theorientation-dependence of the strong-field-ionization and photorecombination dipolematrix elements. They also contain dynamical information as a consequence ofthe unique mapping from the transit time of the electron in the continuum to theemitted photon energy, when the contributions of the ”short” electron trajectoriesare recorded. The applications of high-harmonic spectroscopy (HHS) have led tomany important results, such as the tomography of molecular orbitals [65], theobservation of structural and electronic dynamics on attosecond time scales [6, 66],the identification of two-center interference minima [67, 68], the observation ofCooper minima [69], the determination of ionization and recombination times inHHG [70] and the measurement and laser control of attosecond charge migration[71]. The field of HHS has recently been reviewed in Ref. [72].High-harmonic generation in solids is a much younger and highly dynamic re-search field. Although harmonic generation up to the 7 th order of a MIR driverin ZnSe has already been observed in 2001 [73], the rapid development of HHSof solids has started with the observation of high-order harmonic generation inbulk solids in 2011 [74], the analysis of inter- and intraband contributions toHHG [14, 75, 76] and the observation of extreme-ultraviolet HHG from solids [13].HHS of solids has developed very rapidly, including impressive developments withterahertz drivers [77,78], observations of crystal-structure effects [79], the measure-ment of the Berry curvature of solids [80] and the prediction of the possibility toobserve phase transitions in solids [81]. The field of solid-state HHS has recentlybeen reviewed in Ref. [82].In remarkable contrast to these developments, high-harmonic generation fromliquids has barely been developed so far. Although relatively low-order harmonicgeneration from liquids in the visible domain has been observed in 2009 [28], it tookuntil 2018 before HHG from bulk liquids was reported [23]. This situation is certainlyto some extent the consequence of the technological challenges associated with HHGfrom liquids. Early efforts to observe liquid-phase HHG indeed date back to 2003,when HHG from liquid-water droplets was attempted [25]. Instead of observing theexpected coherent HHG emission, the authors only observed incoherent emission from a plasma formed by multiple ionization of water molecules. Coherent XUVemission was only observed following the action of a pump pulse preceding the HHGpulse that caused a hydrodynamic expansion of the droplets [25]. These resultswere confirmed in later experiments and combined with a model of the dropletexpansion [26, 27], which led to the conclusion that HHG was not observed at thedensity of liquid water but only appeared at densities that were significantly lower.The results reported in Ref. [23] contrast with this interpretation by showing thatcoherent HHG does occur at the density of liquid water. Finally, high-harmonicemission from a plasma created through the interaction of a laser with a liquidmicrojet has been observed in Ref. [29]. These results were obtained in the so-called coherent-wake-emission regime, which is reached for extremely-high laserintensities ( I > W/cm ). In this regime, HHG takes place from laser-inducedplasma dynamics, which are insensitive to the properties of the target used to createthe plasma. In this section, we discuss the methods, results and interpretations ofHHG from bulk liquids, which form the basis of liquid-phase HHS, and introducea new type of targets for strong-field science and the development of new high-harmonic light sources. The first successful observation of extreme-ultraviolet HHG from bulk liquids wasrealized with the experimental setup illustrated in Fig. 1.15 [23]. The key enablingtechnology for this experiment was the flatjet technique [83,84]. The flatjet is createdby colliding two cylindrical microjets under an impact angle of 48 ◦ . In the presentexperiment, cylindrical jets with a diameter of 50 µ m were used, resulting in a flatjetwith a thickness of ∼ µ m. Since the thickness of the flatjet scales quadratically withthe diameter of the colliding jets under reasonable assumptions [83], the thicknesscan easily be reduced, which however comes at the cost of a smaller cross section ofthe flatjet. High-harmonic generation is realized by focusing a short-wave infrared(SWIR) femtosecond pulse centered at 1.5 µ m onto the flatjet in normal incidence,as shown in Fig. 1.15a. The high-harmonic emission is detected by a flat-fieldspectrometer consisting of an entrance slit, a concave variable-line-spacing gratingand a microchannel-plate detector backed with a phosphor screen and a charge-coupled device camera. A photograph of the flatjet under operating HHG conditionsis shown in Fig. 1.15b. The green light originates from scattering of the third-harmonic radiation. Figure 1.15c shows the simultaneously observed high-harmonicspectra emitted from bulk liquid ethanol and the surrounding gas-phase ethanol.The two spectra were independently normalized to their maximal intensity. Thecomparison of the two spectra immediately reveals several characteristic differences,i.e. the liquid-phase harmonics have i) a much lower cut-off, ii) a larger divergence andiii) a rapidly decreasing intensity distribution compared to the gas-phase harmonics.The most important aspect of liquid-phase HHG is the separation of the signalsfrom the liquid and gas phases. The operation of a liquid jet in vacuum necessarily Attosecond Dynamics in Liquids 31 cb Slit MCPGrating
30 fs λ ~ 1.5 µ m1.2 mJ, 1 kHz Flat Jet
Interactionspot a xx X: Second order diffraction
Liquid-phaseGas-phase255075100125
ZYX
Wavelength (nm)
Fig. 1.15 Observation of HHG from liquids . a) schematic representation of the experimentalsetup without the vacuum chambers, b) photograph of the flatjet under operating conditions,c) simultaneously recorded high-harmonic spectra of liquid and gaseous ethanol. Adapted fromRef. [23]. entails the presence of a surrounding gas phase created by evaporation from the liquidjet. The challenge of separating HHG from the gas and liquid phases is elegantlysolved by the natural shape of the flatjet, as illustrated in Fig. 1.16. Since the thicknessof the flatjet decreases from top to bottom across the first sheet, the interfaces arenot parallel to each other but form a small angle with respect to each other. Thisleads to two consecutive refractions of the fundamental driving field upon entranceand exit of the liquid phase. The XUV radiation generated within the liquid jet isnot (significantly) refracted when it exits the liquid because the index of refractionchange is negligible in the XUV. This leads to a natural separation of liquid- andgas-phase HHG on the detector. The observed gas-phase HHG originates entirelyfrom the gas phase located behind the flatjet because the high-harmonic radiationcreated in front of the jet is completely absorbed, given the typical absorption lengthsof ∼
10 nm at 20 eV [85]. This natural separation of HHG has several immediatebenefits, which include the possibility to perform HHS experiments on the gas andliquid phases simultaneously and the separation of generated high harmonics from thedriving fields, which could become relevant for high-average-power attosecond [86]or XUV-frequency-comb [87] experiments.One of the most fundamental characteristics of HHG is the scaling of its cut-offenergy with the intensity of the driving field. In gases, the cut-off scales quadraticallywith the peak electric field of the driver. This can intuitively be understood as the
100 200 300 400 500 600 50100150200250300 Y p i x e l s liquid gas X Y Z n i = 1 n t ~ 1.0 n i = 1 @ 70 nmn i = 1 n t ~ 1.33 αα gas density gas densityn i = 1 @ 1500 nm flat-microjetflat-microjet Fig. 1.16 Separation of HHG from the liquid and gas phases . A schematic representation ofthe optical path of the fundamental beam across the liquid jet is shown on the left. The curvatureof the jet surfaces has been exaggerated for clarity. A spatio-spectrally resolved far-field image ofthe high-harmonic spectra simultaneously emitted from the liquid and gas phases is shown on theright. Adapted from Ref. [23]. signature of the maximal kinetic energy that the continuum electron can acquire fromthe driving field. The experimental results obtained on liquid-phase HHG are shownin Fig. 1.17. They reveal a quasi-linear scaling ( H cut − off ∝ E . ) of the cut-off photonenergy with the peak electric field of the driver. This result points at a fundamentallydifferent mechanism of HHG in liquids compared to gases. The obtained result issimilar to observations made in solids, where a quasi-linear scaling of the cut-offenergy was also observed [74]. This cut-off scaling is common to both types ofmechanisms discussed for solids, i.e. intra-band currents (Bloch oscillations) andinterband polarization (generalized recollision) [13, 14, 76].We next turn to the scaling of the high-harmonic yield with the driving-fieldintensity. Figure 1.17 shows the dependence of the yield of H13 and H21, generatedfrom a 1.5 µ m driver in liquid ethanol on the peak electric-field strength. An electricfield strength of 1 V/Å corresponds to a peak intensity of 2.65 × W/cm . Thedashed lines of the same color indicate the corresponding perturbative scaling laws.All observed harmonic orders thus follow non-perturbative scaling laws with adeviation from the perturbative scaling that strongly increases with harmonic order.These results demonstrate the strongly non-perturbative character of the observedHHG from liquids.Figure 1.18 compares the ellipticity dependence of the HHG yield from liquid-and gas-phase ethanol. The ellipticity dependence in general contains important in-formation about the properties of the medium and the strong-field-driven electrondynamics. We find that the ellipticity dependence of all harmonic orders emittedfrom liquid ethanol is clearly broadened compared to the gas-phase emission. In thegas phase, the ellipticity dependence narrows down with increasing harmonic order,which is a signature of the laser-driven electron dynamics in the continuum: thelonger the trajectory of the electron is, the more its trajectory is influenced by the Attosecond Dynamics in Liquids 33 C u t - o ff h a r m o n i c o r d e r S p e c t r a l i n t e n s i t y ( a r b . u . ) Peak electric field (V/Å)
Peak electric field (V/Å) ba E E H cut-off ~ E Fig. 1.17 Scaling of the liquid-phase HHG cutoff and harmonic yields . Scaling of the cut-offharmonic order (a) and the yield of selected harmonic orders (b) with the peak electric field of thedriver ( ∼
30 fs, centered at 1.5 µ m) in liquid ethanol. Adapted from Ref. [23]. laser field, which results in a larger sensitivity to ellipticity. The same observationapplies to the liquid-phase ellipticity dependence, which also narrows down withincreasing harmonic order. This observation suggests that a trajectory-based under-standing of high-harmonic generation in liquids might be adequate. Figure 1.18ccompares the ellipticity dependence of H13 from water and several alcohols to thatof gas-phase ethanol. The ellipticity dependencies of all liquids are very similar andconsiderably broader than those of the gas phase. The broadening of the ellipticitydependence in the liquid phase can have several origins, which include i) differencesin the continuum-electron propagation, ii) electron scattering in liquids, iii) a differ-ent spatial extension of the electron-hole wavefunctions. Explanation i) is insufficientto rationalize the large observed effects because the differences in the ionization en-ergies between the isolated and condensed molecules are on the order of 1 eV andthe electron mass is not noticeably reduced for electron propagation in the liquids.Explanation ii) is more difficult to quantify because electron elastic mean-free pathsare not available for alcohols and only recently became available in liquid water(see section 1.2.2.4 and Ref. [56]). In Ref. [23], we have used the mean-free paths determined for amorphous ice [88], for a lack or reliable data on liquid water. Onthis basis, we have assumed that electron scattering was negligible because the totalpropagation length of the electron trajectory emitting the cut-off harmonic (20 eVphoton energy, emitted by a 1.5 µ m driver with a peak electric field of 1.5 V/Å)amounts to ∼ ∼ ∼ The theoretical description of HHG in liquids meets with several challenges. First,liquids have a density comparable to solids, such that the typical extensions ofcontinuum-electron trajectories known from gas-phase HHG would correspond tothe electron wave packet encountering several neighboring molecules on its trajec-tory. Consequently, the methods developed to understand HHG in solids might bethe better starting point. However, and this is the second challenge, liquids are intrin-sically disordered, which prevents the rigorous application of a momentum-spacedescription of HHG, which has enabled rapid progress in the understanding of HHGin solids. Nevertheless, the absence of long-range order does not completely excludethe application of momentum-space methods because liquids possess short-rangeorder. As long as the spatial extension of the continuum-electron trajectory is com-
Attosecond Dynamics in Liquids 35
Ellipticity
Ellipticity S p e c t r a l i n t e n s i t y ( a r b . u . ) ab gas gasliquid Harmonic13 c Water - LiquidEthanol - LiquidMethanol - LiquidIsopropanol - LiquidEthanol - Gas ε = 0.5 ε = 0 liquid ε = 0.5 ε = 0 Fig. 1.18 Ellipticity dependence of liquid- vs- gas-phase HHG . a) Schematic representationof the effect of ellipticity on the propagation of the electron wave packet. b) Dependence of theyield of selected harmonics on the ellipticity of the driving field in liquid and gaseous ethanol.c) Comparison of the ellipticity dependence of H13 from different liquids to gaseous ethanol. Allmeasurements were recorded with a ∼
30 fs driver, centered at 1.5 µ m. Adapted from Ref. [23]. parable to the length scale of this short-range order, the latter can be expected to playa role in HHG.Based on these considerations, we have chosen the semiconductor Bloch equations(SBE) as a starting point for modelling liquid-phase HHG. The electronic structuresof liquids are indeed commonly understood as those of large-bandgap semiconduc-tors [90]. The presence of local order, which is enhanced by the existence of stronghydrogen bonds, additionally justifies the use of an effective band structure [89].The advantage of the SBE is that they naturally include both interband and intrabandcontributions to HHG, such that additional assumptions regarding the mechanisms atplay are not needed [76]. The solution of the SBE with a realistic band structure, suchas that derived from a calculation of a slab of e.g. 128 molecules however quicklyreaches the limit of computational feasibility, in addition to the limits imposed bythe accuracy of density-functional-theory calculations.We have therefore chosen, as a zero-order approximation, an approach that circum-vents all of these challenges. We have used the density of occupied and unoccupiedstates, measured by state-of-the-art X-ray spectroscopies, to derive a highly simpli-fied model band structure that reflects the known densities of states, arising from the three occupied outer-most valence bands and the two lowest-lying unoccupiedconduction bands. The main goals of this approach were to identify i) the sensitivityof the calculated HHG spectra to the electronic structure of the liquids, i.e. the bandgap and the width of the bands, ii) the number of contributing bands, and iii) thecapability of such a simple model to explain the characteristic differences in theHHG spectra of liquid water as compared to the alcohols. -15-10-5051015 11 15 1913 17 21 2523Harmonic order ba Energy (eV) S p e c t r a l i n t e n s i t y ( a r b . u . ) - π /d 0 π /d k DOS (arb.u.) E n e r g y r e l a t i v e t o o n s e t o f u n o cc u p i e d s t a t e s ( e V ) Ethanol - ExpWater - ExpEthanol - SimWater - SimEthanolWater c CB2CB1(E)CB1VB1VB3VB2
Fig. 1.19 Sensitivity of HHG to the electronic structure of liquids . a) Densities of states ofliquid water and ethanol obtained form X-ray spectroscopy, b) model ”band structure” chosen toreflect these densities of states, c) measured HHG spectra of liquid water and ethanol using a ∼ µ m and spectra calculated by solving the SBE for the model ”bandstructure” shown in b. Adapted from Ref. [23]. Figure 1.19a shows the densities of states (DOS) of liquid water and ethanolas measured by X-ray spectroscopy [91–94]. In the case of water, these consist ofthree outer-valence bands, which are labelled according to the symmetries of theorbitals of the isolated water molecule (1b , 3a and 1b ). The unoccupied statesconsist of two conduction bands, that are labeled according to the symmetries ofthe unoccupied orbitals of the isolated water molecule (4a and 2b ). The DOS ofethanol is comparable to that of water, with an important difference being the absenceof a local maximum at the position of the 4a band of liquid water. Based on theseDOS and the results of detailed ”band-structure” calculations [89], we have derivedthe highly simplified model band structure shown in Fig. 1.19b. The difference inthe DOS of liquid water and ethanol is accounted for by choosing a larger width forthe lowest conduction band of liquid ethanol compared to liquid water.The solution of the SBE using these model band structures yields high-harmonicspectra that are in very good agreement with the observed spectra (full and dashedlines, respectively, in Fig. 1.19). Most importantly, these calculations fully reproducethe main characteristics of the high-harmonic spectra, i.e. the monotonic decrease inthe spectrum of ethanol and the plateau observed at the lowest three harmonic ordersin liquid water, followed by a cut-off. We have verified that the calculated spectra Attosecond Dynamics in Liquids 37 do not change significantly when the two lower-lying valence bands and the highest-lying conduction band are removed from the calculation. Therefore the calculatedhigh-harmonic spectra are mainly sensitive to the properties of the highest-lyingvalence and the lowest-lying conduction band.Within our model, the characteristic differences between the HHG spectra ofwater and ethanol therefore exclusively originate from the different widths of theconduction bands, for which a cosinusoidal shape was assumed. The larger width inthe case of ethanol causes the monotonic decay of the HHG spectrum, whereas thenarrower width in the case of water causes the appearance of the plateau and cutoff.An additional characteristic of the HHG spectrum of liquid water is the appearanceof a local minimum at H13 (10.7 eV). This local minimum is also reproduced in theSBE calculations. As we showed in Fig. 7 of Ref. [23], the position of this minimumshifts in energy according to the size of the bandgap.The comparison between our measurements and calculations therefore showsthat high-harmonic spectroscopy of liquids is sensitive to the electronic structure ofliquids, particularly to the properties of the conduction band, as well as the bandgap.In the future, the methods for describing liquid-phase HHG can be improved in dif-ferent ways. The most accurate computationally tractable approach would probablyconsist in running real-time, real-space time-dependent density-functional-theory(TDDFT) calculations on slabs of liquids sampled from molecular-dynamics calcu-lations. This approach has the advantage of requiring minimal assumptions beyondthose inherent to the TDDFT approach. It has the disadvantage that the physicalinsight into the mechanisms at play remains limited. Therefore, it might be desirableto develop complementary approaches, which offer additional insights. One suchapproach would consist in Monte-Carlo trajectory calculations of the strong-fielddriven electron dynamics. These calculations could rely on the principles describedin Section 1.2.2 and thereby incorporate electron scattering and laser-assisted elec-tron scattering. The additionally required strong-field-ionization and photorecombi-nation matrix elements could be obtained by extending the methods established forisolated molecules to the condensed phase by using the cluster approach describedin Section 1.2.2. This method would offer additional insights into liquid-phase HHGbecause the role of electron scattering, orbital delocalization and many other effectscould be disentangled. The challenge for this approach will be to achieve a sufficientaccuracy to reach agreement with the experimental data.The results presented in this section set the foundations for the development ofliquid-phase high-harmonic spectroscopy. In the future, the detailed mechanism ofliquid-phase HHG will be studied with attosecond temporal resolution by using both in-situ methods such as two-color HHG [14, 70] and attosecond transient absorption[95] based on water-window high-harmonic sources [96] or ex-situ methods, such asattosecond photoelectron spectroscopy [97]. Of particular interest are the capabilitiesof HHS to resolve attosecond electron dynamics in the liquid phase, such as electron-liquid scattering dynamics and the spatial characteristics of electron-hole dynamicsin liquids. Targets of primary interest will be water itself, but also aqueous solutions,which offer the possibility to study solvated species in their natural environment,and other liquids.
In this chapter, we have described two novel experimental approaches that enable at-tosecond time-resolved experiments to be performed on liquid samples. The crucialinnovations have been the development of attosecond photoelectron spectroscopywith cylindrical liquid microjets and the demonstration of high-harmonic spec-troscopy with flat microjets.In the first case, a general novel approach to analyze and interpret overlappingattosecond photoelectron spectra has been developed. This technique generalizesattosecond photoelectron spectroscopy not only to the liquid phase, but actually toany complex sample. On the conceptual side, a general theoretical framework forcondensed-phase attosecond photoelectron spectroscopy has been developed thatincludes, for the first time, the treatment of photoionization and scattering delays,as well as a coherent treatment of all processes involved. This framework has beenvalidated first by benchmarking against the time-dependent Schrödinger equationand finally by reaching quantitative agreement with attosecond time delays of liquidwater.In the second case, a general approach to high-harmonic spectroscopy in liquidshas been established. This result represents a change of paradigms in a field whereXUV high-harmonic generation from liquids was previously assumed to be impos-sible. Our novel experimental approach has enabled the first observation of XUVHHG from liquids, the unequivocal separation and simultaneous measurement ofliquid- and gas-phase HHG and their detailed comparison. We have found a linearscaling of the high-harmonic cut-off energy with the peak electric field of the driverand a highly non-perturbative scaling of the HHG yield. We have found a systemat-ically and substantially broadened dependence of the HHG yields on the ellipticityof the driving field, compared to the gas phase. Finally, based on the solution of thesemiconductor Bloch equations of a strongly-driven model band system, we havefound a pronounced sensitivity of the HHG spectra on the electronic structure ofliquids, particularly the properties of the conduction band and the band gap.These developments establish two major avenues for developing attosecond time-resolved spectroscopy of liquids and solutions. They open a myriad of future possi-bilities to explore the role of electronic dynamics in solvated atoms, ions, moleculesand nanoparticles in their natural environment. Of particular interest for such workare the prototypical processes of charge and energy transfer. We will briefly discusstwo examples of such processes. Intermolecular Coulombic decay [17] in liquidwater is the prototype for energy transfer in the liquid phase and a source of slowelectrons in liquid water. ICD occurs when the inner-valence (2a ) or the core (1a )orbitals of water (clusters or liquid) are ionized. So far, ICD has only been observedin water dimer [98] and water clusters [99, 100], but not in bulk liquid water. Time-resolved experiments on ICD in water are equally missing. Recent theoretical workpredicts a time scale of 12-52 fs for ICD in small water clusters [100]. However, adifferent type of calculations predicts ICD lifetimes of 3.6-4.6 fs for isoelectronic(HF) clusters [101], suggesting that the ICD lifetimes in water (clusters) could alsobe much shorter. Electron-transfer-mediated decay [18] is a prototype of ultrafast Attosecond Dynamics in Liquids 39 charge transfer reactions in liquids. ETMD occurs when the inner-valence or corehole created by ionization is filled by an electron from a neighboring particle. Theenergy made available in this process then serves to eject a second electron from theparticle that provided the first electron to fill the hole (ETMD(2)) or a third particle(ETMD(3)). ETMD has only recently been observed in the liquid phase [102]. Itstime scale is presently unknown.These two examples illustrate the possible future of liquid-phase attosecond sci-ence. They represent two relatively simple and very fundamental mechanisms ofelectronic dynamics in liquids. They are representative of a broad variety of elec-tronic processes that play a role in most chemical reactions and biological transfor-mations. Looking forward, the methods described in this chapter will provide accessto the study of many important, but poorly understood electronic processes in theliquid phase.
Acknowledgements
We gratefully acknowledge the contributions of many co-workers and collaboratorswho have contributed to this work over several years. This work was financiallysupported by an ERC starting grant (project no. 307270-ATTOSCOPE), an ERCconsolidator grant (project no. ATTOLIQ) and the Swiss National Science Founda-tion (SNSF) via the National Center of Competence in Research Molecular UltrafastScience and Technology and projects no. 200021_159875 and 200021_172946. A.S. is grateful for financial support from an Ambizione grant of the SNSF. D. J. thanksthe FP-RESOMUS fellowship program.
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