Toward Direct Exploration of the Few-Femtosecond Dynamics of Electronic Coherence and Correlation in Quantum Materials Using Time- and Angle-Resolved Photoemission Spectroscopy

Abstract

Over the last two decades, time- and angle-resolved photoemission spectroscopy (trARPES) has become a mature and established experimental technique for the study of ultrafast electronic and structural dynamics in materials. To date, most trARPES investigations have focused on the investigation of processes occurring on time scales of ≳30 fs, in particular, relaxation and thermalization, and have therefore been blind to the initial sub-10 fs dynamics related to electronic coherence and correlation effects. In this article, we illustrate how current trARPES setups reach their limits when it comes to addressing such extraordinarily short time scales and present an experimental configuration that provides the time, energy, and momentum resolutions required to monitor few-femtosecond dynamics on the relevant energy and momentum scales. We discuss the potential capabilities of such an experiment to study the electronic response of materials in the strong-field interaction regime at PHz frequencies and finally review a theoretical concept that may in the future even overcome the competing resolution limitations of trARPES experiments, as imposed by the time–bandwidth product of the probing laser pulse. Our roadmap for ultrafast trARPES indicates a path to break new experimental ground in quantum nonequilibrium electronic dynamics, from which new possibilities for ultrafast control of optical and electronic signals in quantum materials can be explored.

1. Introduction

The time-domain study of photoinduced nonequilibrium phenomena in materials is a highly active and rapidly developing research field that contributes to our understanding of the properties of condensed matter in two important ways [1,2,3]: it provides valuable information about the correlations and dissipations that determine the equilibrium states of matter, but it also provides a novel means to create and explore short-lived transient as well as long-lived metastable nonequilibrium and quasi-equilibrium states that are inaccessible via conventional thermal pathways. By tracking the buildup and decay of the nonequilibrium state, such studies provide direct, real-time insights into the quasiparticle interactions between the involved charge, lattice, and spin degrees of freedom that take place on different characteristic time scales, as partially illustrated in Figure 1 (taken from [1]).

As the most direct probe of momentum-dependent changes in excited carrier populations and electronic band dispersions, trARPES offers an extraordinarily comprehensive view of the temporal evolution of the nonequilibrium charge degrees of freedom. For example, trARPES can monitor carrier–carrier and carrier–phonon interactions in real time, information that is important for optimizing the electronic properties of materials for electronic devices. It can directly study nonequilibrium spin dynamics in magnetic materials on ultrafast time scales, which is central to the development of future ultrafast spintronic devices. In addition, trARPES studies of nonequilibrium dynamics in correlated materials such as high-temperature superconductors or charge-density waves (CDWs) can provide fundamental insights into the driving interactions of these fascinating phenomena, which can contribute to the development of corresponding materials for use in ambient conditions. As a final example, the technique allows the identification of metastable nonequilibrium states of matter and the characterization of their electronic structure, a topic of particular interest for the realization of novel and ultrafast optoelectronic devices. In recent years, several comprehensive review articles have been published on the technique of trARPES and its numerous applications, to which reference is made here instead of a detailed introduction [4,5,6,7]. Under the assumption of an instantaneous removal of the photoelectron from the surface (suddenapproximation), which is typically fulfilled for sufficiently high kinetic energies of the photoelectron, the photoemission signal traces the one-particle spectral function $A(\mathbf{k},\omega )$ [8,9], weighted by the photoemission dipole matrix element and the Fermi–Dirac function, and their changes upon photoexcitation. Here, $\mathbf{k}$ denotes the electron momentum and $\omega$ the electron energy in the initial N-particle state. $A(\mathbf{k},\omega )$ particularly accounts for energy renormalization and lifetime effects due to correlations and interactions in the many-body initial state of the system.

Figure 2 shows a schematic of a typical trARPES setup using extreme ultraviolet (XUV) pulses to probe the sample response in photoemission upon excitation with femtosecond near-infrared (NIR) pulses. Various types of setups have been realized in the recent past and a list of currently installed systems with their main specifications can be found in Ref. [4].

Many different topics have been addressed in numerous trARPES studies over the last decade. These include, for example, the investigation of carrier thermalization and the associated pathways in energy–momentum space [10,11,12], electron–phonon interactions [13,14], photoinduced phase transitions in charge-ordered correlated materials [15,16,17], energy-gap and quasiparticle dynamics in high-temperature superconductors [18,19], a transient Lifshitz transition of Fermi surface topology [20], field-driven carrier dynamics in topological insulators [21], ultrafast and valley-selective excitonic and free-carrier processes in bulk and single-layer semiconducting transition-metal dichalcogenides (TMDCs) [22,23,24,25,26], the formation of Floquet–Bloch states at the surface of a topological insulator [27], and spin and band dynamics associated with ultrafast demagnetization processes [28,29,30]. The time resolutions of these experiments, as usually specified by the full width at half maximum (FWHM) of the pump–probe cross-correlation signal, typically remain in the range of ≳30 fs (for a comprehensive overview of the specifications of trARPES instruments installed worldwide, see again Ref. [4]). This limitation affects the time-domain access to the initial processes during and after photoexcitation, which are governed by electronic correlations and electronic decoherence, typically associated with ultrashort characteristic time scales in the range of a few femtoseconds, especially for strongly correlated materials with their relevant eV energy scales.

Some hints of the wealth of different phenomena that can occur in this time regime come from previous theoretical work. In a photoexcited Mott insulator, carrier relaxation due to electron–electron interaction can drive the system into a nonthermal state with a complicated electronic momentum distribution [31]. The authors suggested that such states might be observable in experiments on sufficiently short time scales where the electronic system is still decoupled from the surrounding thermal bath of phonons. Golež et al. investigated the effect of dynamical screening in photoexcited Mott insulators [32]. They reported a filling and reduction of the Mott gap due to the rapid change in the screening environment on characteristic inverse bandwidth time scales. In addition, a significant shortening of the doublon–hole recombination time was observed, caused by the interaction with the transient formation of low-energy screening modes. Another theoretical study of the pulsed, near-resonance excitation of a Mott-insulator state emphasized the interplay of coherent and incoherent processes in the formation of the final nonequilibrium state [33]. The authors observed that the initial dynamics was due to virtual Floquet states in the presence of the pump, followed by interaction-induced population of the upper Hubbard band from these states. Finally, Shen et al., simulating time-resolved photoemission data within a simplified model for a CDW insulator, observed a transient gap closure that lasted only slightly longer than the duration of the pump pulse [34]. The authors concluded that, on these ultrafast time scales, the electronic CDW order and the gap in the electronic energy spectrum decouple.

Experimentally resolving such processes and disentangling them from other interactions, and establishing a hierarchy of characteristic time scales of purely electronic interactions, requires experimental schemes that simultaneously provide spectral and temporal resolutions that match the relevant characteristic energy and time scales. Depending on the problem, the time resolution of the experiments should allow monitoring of purely electronic processes before coupling to other degrees of freedom occurs or before the electronic coherence in the system is lost.

2. Current Temporal Resolution Limits of trARPES: Few-Femtosecond Dynamics in Two Prominent Layered Quantum Materials

The current limitations of trARPES in entering the interesting few-femtosecond time domain are illustrated below based on results from previous studies of the ultrafast photoexcitation and electronic order dynamics in the layered CDW compounds 1T-TaS₂ and 1T-TiSe₂ [35,36,37,38]. For both compounds, the electronic structures [39,40,41,42,43,44,45,46] and ultrafast spectral responses [15,17,35,36,37,47] indicate the presence of significant electron correlation effects in addition to strong electron–phonon coupling.

2.1. Photoexcitation of a (Possible) Mott-Insulator Phase in 1T-TaS₂

1T-TaS₂ exhibits a complex phase diagram that includes a high-temperature metallic phase, an incommensurate and a nearly commensurate CDW phase at intermediate temperatures, and, finally, below T = 180 K, a low-temperature commensurate CDW (CCDW) phase coexisting with a possible, much-discussed Mott-insulator phase [48,49,50,51]. Figure 3a shows a high-resolution ARPES intensity map of 1T-TaS₂ obtained with synchrotron radiation ($h\nu =96 \mathrm{eV}$) at $T=33 \mathrm{K}$, i.e., in the combined CCDW Mott-insulator phase. The CDW reconstruction of the electronic structure is most pronounced in the clear sub-band splittings along $\overline{\Gamma }-\overline{M}$, the strong band backfolding along $\overline{M}-\overline{K}$, and the large spectral gap opening at the edge of the original Brillouin zone (BZ) (midway between $\overline{M}$ and $\overline{K}$). Here, we are interested in the possible Mott contribution to the phase. It is spectrally evidenced by a Mott–Hubbard-type gap near the $\overline{\Gamma }$ point at the Fermi level $E_{\mathrm{F}}$, which separates the lower Hubbard band (LHB), representing singly occupied sites, and the upper Hubbard band (UHB), representing doubly occupied sites (doublons) (the UHB above $E_{\mathrm{F}}$ is not visible in Figure 3a because it is unoccupied in thermal equilibrium).

Before pump-pulse excitation, the gap is also clearly visible in trARPES data near $\overline{\Gamma }$ [left panel of Figure 3b], despite the limited energy resolution (400 meV) of the trARPES setup used in this study [35]. The right panel of Figure 3b shows trARPES data recorded 65 fs after photoexcitation of the sample with 30 fs NIR pump pulses at a fluence in the range of several mJcm⁻². The data clearly show a closing of the possible Mott–Hubbard gap upon excitation, implying the destruction of the Mott-insulator phase. The temporal evolution of the gap suppression can be followed from the photoemission intensity at $E_{\mathrm{F}}$ as a function of the pump–probe delay [Figure 3c]. The data show a quasi-instantaneous gap collapse within the duration of the pump pulse, with a time constant of less than 20 fs. The result agrees well with the results of a related trARPES study of 1T-TaS₂ which also reported the collapse of the Mott gap on time scales faster than the 30 $\mathrm{f}$s resolution of the experiment [17].

The ultrafast response time is direct evidence for an electronic origin of the gap, strongly supporting the interpretation as a Mott–Hubbard gap. The measured upper limit of the time constant is consistent with an electronic hopping time ${\tau }_{\mathrm{hop}}=\hslash /W$ ≈ 1 fs to 8 fs, which sets the time scale for the response of the correlated electron system to sudden changes [52,53] (where W = 80 meV to 400 meV corresponds to the width of the split-off Ta 5d band that straddles $E_{\mathrm{F}}$ [42]).

The exceptionally fast electronic response of the Mott–Hubbard gap spectral region of 1T-TaS₂ was also confirmed in a more recent trARPES study, which was performed at much lower pump fluences (≈100 μJcm⁻²) to avoid a suppression of the CCDW Mott insulating state [36]. Selected results of the study are shown in Figure 3d,e. Upon photoexcitation, the photoemission spectra show an instantaneous depopulation of the LHB and its repopulation on a few–100 fs time scale [see the green-colored photoemission intensity transient in Figure 3e]. On similar time scales, one can additionally observe the buildup and decay of an excitation continuum above $E_{\mathrm{F}}$. The most interesting feature, however, is a distinct transient spectral peak above $E_{\mathrm{F}}$, visible only at and near the temporal overlap of the pump and probe pulses. The authors interpret this feature as a spectroscopic signature for doublon formation, i.e., a transient population of the UHB across the Mott–Hubbard gap. Interestingly, the decay of the UHB population occurs on much faster time scales than the spectrally overlapping excitation continuum [cf. corresponding photoemission intensity transients in Figure 3e], suggesting that the microscopic processes responsible for the decay of the excitation continuum and the doublon population are different. The authors conclude that the decay of the excitation continuum is due to electron–electron scattering, whereas the observed loss of the UHB must be explained by local electron–electron correlations. Analysis of the UHB photoemission intensity transient gave a value of 20 $\mathrm{f}$s as an upper limit for the doublon recombination time, which in turn is compatible with the expected electronic hopping time. The experimental data also indicated an even faster thermalization of the excited carriers within the transiently populated UHB.

2.2. Ultrafast Suppression of a (Possible) Excitonic Insulator Phase in 1T-TiSe₂

The peculiarity of 1T-TiSe₂ that has attracted some interest in the past is a commensurate $(2\times 2\times 2)$ CDW occurring below a temperature of 200 K [54,55]. It has been proposed that the CDW state arises from an excitonic insulator instability [45,56,57], driven by purely electronic interactions and accompanied by a periodic lattice distortion that is only an accidental byproduct [58].

Recently, time-domain experiments have been used to address this problem, as the observed characteristic time scales in the system response should provide information about the dominant driving interactions [35,37,38,59,60]. TrARPES snapshots from one of our studies of this compound are shown in Figure 4a [37]. The top panel shows a spectrum recorded before the arrival of the 30 $\mathrm{f}$s pump pulse, reflecting the electronic structure of 1T-TiSe₂ in the CDW ground state. The spectral signature characteristic of the CDW phase is the downward dispersing (hole-like) band at $\overline{M}$, which is folded from $\overline{\Gamma }$ due to the formation of the $(2\times 2\times 2)$ CDW superlattice [43,44,45] (in the figure, $\overline{\Gamma }$ and $\overline{M}$ indicate high-symmetry points of the BZ of the undistorted room-temperature phase). When the pump and probe pulses overlap in time (0 $\mathrm{f}$s), an electron-like band appears, indicating the photoinduced population of excited states at and above $E_{\mathrm{F}}$. The additional signal at $\overline{\Gamma }$, indicated by the dashed black line, is a replica of the hole-like band (solid black line) separated by the pump-photon energy of $1.5$ eV due to laser-assisted photoemission (LAPE) [61]. At the same time, the spectral intensity of the hole-like band at $\overline{M}$ is significantly reduced, before disappearing completely 30 $\mathrm{f}$s after excitation [bottom panel in Figure 4a]. The latter observation suggests that the long-range CDW order in the electronic subsystem breaks down exceptionally quickly on sub-vibrational time scales below 30 $\mathrm{f}$s, supporting an electronic origin of the CDW. In principle, an energy shift of the hole-like band towards $E_{\mathrm{F}}$, indicating the closing of the CDW gap as the CDW order is lost [45], should also be visible in the data. However, the limited energy resolution of the trARPES experiment of ≈400 meV prevents the observation of this effect.

The pump fluence dependence of the dynamics shows that CDW quenching accelerates with increasing excitation density. Figure 4b summarizes results on the photoinduced quenching dynamics from time-resolved X-ray diffraction [38] and trARPES studies [35,37,59]. The characteristic fluence dependence indicates a complex interplay of structural and electronic degrees of freedom responsible for the stabilization of the low-temperature CDW-ordered phase in 1T-TiSe₂ [38]. Here, we focus on the high-fluence regime, where comparison with simulations suggests that hot-carrier (Coulomb) screening is important for perturbing order in the electronic subsystem. The characteristic buildup time for carrier screening in response to ultrashort laser excitation is the plasma oscillation period [62,63] which scales with $1/\sqrt{n}$, where n is the free-carrier density generated in the photoabsorption process. At the highest value of n, corresponding to the maximum excitation fluence of ≈5 mJcm⁻² applied in the experiment, the quench dynamics starts to enter the sub-20 fs time domain, beyond the time resolution of the experiment [37].

3. Pushing the Temporal Resolution Limit of trARPES: Entering the Sub-10 fs Time Regime

The above examples illustrate that ultrafast processes, which are mainly governed by electronic interactions or correlations, rely on time-domain experimental techniques with time resolutions of ≈10 fs and below. In this context, trARPES could uniquely provide an energy- and momentum-resolved view of the involved band-structure and carrier-population dynamics. However, this is only possible in a well-balanced experimental configuration, in which, in addition to the dynamical information, the relevant spectral information can be meaningfully resolved. The practical boundary conditions are defined by the characteristic time scales of carrier decoherence and the fundamental inter-relationship between the energy and time domains, as shown schematically in Figure 5a: access to the relevant energy scales associated with coherence times in the range of a few femtoseconds requires energy resolutions of a few 100 meV. From an experimental point of view, the ultimate limit is set by the time–bandwidth product of the probe pulse, which inversely links the time and energy resolution of time-domain spectroscopy.

Recently, our team succeeded in realizing a trARPES experiment that meets all the requirements and approaches the fundamental limit. The key measure to improve the time resolution of an existing system [64] was the implementation of a hollow-core fiber setup in the NIR pump line, which generates broadband light pulses that are then compressed to a few cycles ($6.5$ $\mathrm{f}$s) [65]. The XUV probe line, based on high-harmonic generation, was not affected by this upgrade, so the energy resolution of the instrument remained the same. Details of the experimental setup are described in Ref. [66]. Figure 5b summarizes the experimental verification of the main specifications of the setup. The FWHM of the pump–probe photoemission cross-correlation signal measured at the sample position gives a value of 13 $\mathrm{f}$s, well suited for resolving dynamics in the sub-10 fs regime. At the same time, an ultimate energy resolution of ≈200 meV could be confirmed (mainly limited by the time–bandwidth product of the 11 fs probe pulse).

In a first study of nonequilibrium carrier dynamics in graphite, we demonstrated the ability of the experiment to resolve and discriminate processes occurring on time scales below 10 fs [67]. The two plots in Figure 6a show energy distribution curves (EDCs) around $E_{\mathrm{F}}$ and at the H point of the BZ of graphite, recorded at different time delays after excitation with $6.5$ $\mathrm{f}$s pump pulses and compared to the measured equilibrium distribution at T = 300 K (black line). At the temporal overlap of pump and XUV probe pulse (0 $\mathrm{f}$s), a clear spectral distortion is observed in the EDC, which evolves as a function of time [cf. right panel of Figure 6a]. This indicates the nascent nonthermal character of the carrier distribution on ultrashort time scales [12,68,69,70,71]. The quantitative analysis of the data allows disentangling the different processes involved in the buildup and decay of the nonthermal distribution on the relevant time scales from below 10 $\mathrm{f}$s to several 10 $\mathrm{f}$s [see Figure 6b]. In particular, it was possible to separate different stages during carrier thermalization dominated by either carrier–carrier or carrier–phonon interactions.

Figure 5

Few-femtosecond trARPES: requirements and realization. (a) Schematic representation of the different time regimes of ultrafast processes and the specific challenges with respect to temporal and/or spectral resolution. In the few-femtosecond time regime, spectral resolutions in the 100 meV range are required in order to still be able to resolve spectral peak shifts (bottom left) or gap closings (bottom right) on the characteristic energy scales. (b) Left: Comparison of NIR-pump/XUV-probe cross-correlation signals recorded with 11 fs probe pulses and 6.5 fs (blue) and 34 fs (red) pump pulses, respectively. The open dots show experimental photoemission transients from the transiently populated conduction band of 1T-TiSe₂. The solid lines are the results of fitting an error function to the experimental data. The dashed lines are the derivatives of the fit results, which are interpreted as approximations to the pump–probe cross-correlation curves. Right: EDC of 1T-TiSe₂ (open circles) recorded with 11 fs probe pulses ($h\nu =22.1 \mathrm{eV}$) at $k_y=0.3 {\AA }^{-1}$ along the $\overline{\Gamma }-\overline{M}$ direction 40 fs after excitation with a 6.5 fs pump pulse. A Shirley background (dashed light blue line) was subtracted from the data before linewidth analysis. The red line is a Gaussian fit to the spectral peak in the experimental data, yielding a peak width of 238(10) meV (FWHM). Inset: EDC of polycrystalline Au (full circles) recorded with 11 fs probe pulses ($h\nu =22.1 \mathrm{eV}$) at T = 100 K. The green line is a fit of a Fermi–Dirac distribution function to the data and gives a width of 185(10) meV (12–88% width). For comparison, the time–bandwidth product gives a spectral width of the XUV pulse of 170 meV. (b) reprinted from [66], with the permission of AIP Publishing.

[Figure omitted. See PDF]

Figure 6

Few-femtosecond electron dynamics in graphite. (a) EDCs around $E_{\mathrm{F}}$ for pump–probe delays $\Delta t=0 \mathrm{fs}$ and $\Delta t=13 \mathrm{fs}$ derived from trARPES data near the Dirac cone in graphite compared to an EDC recorded before NIR excitation ($\Delta t=-80 \mathrm{fs}$). The dashed line is a fit of a Fermi–Dirac distribution function to the equilibrium EDC. (b) Different stages of thermalization of a photogenerated nonthermal hot electron distribution in graphite within the first 50 fs after time zero, as derived from a quantitative analysis of a complete series of trARPES snapshots. $\Delta E_{\pi *}$ is a measure of the difference between the nonthermal and thermal electron distributions, $T_{\mathrm{FD}}$ is the apparent electron temperature derived from fitting a Fermi–Dirac distribution function to the EDC of the non-thermalized electron gas. (a,b) reprinted from [67]. Copyright (2018) by The American Physical Society.

[Figure omitted. See PDF]

A very similar trARPES experimental setup, which also achieves a time resolution in the 10 $\mathrm{f}$s range (≈13 fs FWHM of the pump–probe cross-correlation), is described in Ref. [12]. In a study of primary thermalization events in graphene, the authors were able to distinguish between impact ionization and Auger recombination, processes that control the initial carrier relaxation in graphene in a time window up to ≈15 fs after photoexcitation. The measured energy resolution of the setup is worse compared to that reported in [66] ($800 \mathrm{meV}$ at optimum time resolution). To compensate for this limitation, the authors analyzed their data using a special momentum-to-energy conversion method. This takes advantage of the fact that, while energy resolution is intrinsically affected by the time–bandwidth product, the momentum resolution is unaffected. This method may also be applicable to other systems with similarly simple and well-known band dispersions, such as the linear dispersion of the Dirac cone in graphene.

The few-femtosecond trARPES results presented here represent a step toward the study of nonequilibrium electronic structure dynamics in the time domain of electronic coherences and correlations in materials. Together with other complementary time-resolved pump–probe techniques, this may in the future specifically contribute to further elucidate the exotic physical properties of quantum materials that arise from the quantum mechanical properties of their constituent electrons. This particularly includes attosecond to few-femtosecond time-resolved all-optical methods such as transient absorption spectroscopy in the visible [68,71], XUV [72,73], and soft X-ray spectral regime [74]. The exceptional (ultimately few-meV) energy resolution that can be achieved in principle by these techniques, and which is made possible by the independence of the time and frequency domains [75], can complement the unique momentum information of trARPES in such studies. In this context, the advances in the synthesis and investigation of novel low-dimensional and topological quantum materials provide an almost inexhaustible playground.

Finally, one may wonder about the role of photoemission-based attosecond streaking [76,77,78] or RABBIT (reconstruction of attosecond beating by interference of two-photon transitions) [79,80] in the study of electronic interactions in materials. So far, however, both experimental approaches could only be used to study the dynamics of the final states of photoemission but not to probe ultrafast changes in the electronic structure of the initial states. A major limitation, at least within a streaking experiment, is the limited energy resolution, which is in the range of several eV due to the attosecond time profile of the XUV pulses.

4. Future Perspectives of Few-Femtosecond trARPES: Exploring Quantum Interference and Overcoming the Time–Bandwidth Product

4.1. Few-Cycle Strong-Field ARPES at PHz Frequencies

Intense and few-cycle femtosecond pump pulses as part of a trARPES experiment provide unique opportunities for studying changes in the energy–momentum distribution of electronic states and their population due to interactions in the (nonperturbative) strong-field regime. The novel and exciting insights that can be gained from such studies were recently demonstrated in a trARPES experiment on field-driven Dirac currents at the surface of a topological material using THz pump pulses [21]. With a central excitation wavelength in the NIR and visible spectral range, it may become possible to similarly probe processes relevant to electric field-waveform control scenarios of optical and electronic properties in the PHz frequency range, i.e., on characteristic time scales of electronic decoherence in solids [81,82].

As an illustrative example, we propose a strong-field ARPES experiment on Rabi oscillations and Landau–Zehner–Stückelberg interferences in the excited state population in graphene, resulting from the interaction with few-cycle NIR pulses with a peak electric field strength $E_0$ in the Vnm⁻¹ range. The study is motivated by recently performed photocurrent control experiments in graphene using the carrier–envelope phase (CEP) of few-cycle NIR laser pulses as a control parameter [83,84]. Numerical simulations showed that the detected net photocurrent observed for $E_0\gtrsim 2$ Vnm⁻¹ arises from suboptical-cycle Landau–Zener–Stückelberg interferences, consisting of coherent repeated Landau–Zener transitions on the femtosecond time scale [see Figure 7a]. The characteristic signatures of the interferences can be seen in the energy–momentum distribution of the photoexcited carriers within the Dirac cone of graphene after passage of the excitation pulse [Figure 7b]. The black lines in Figure 7b indicate equienergy lines in resonance with multiples of the central photon energy. The quasi-periodic population patterns at the resonance energies are remnants of Rabi oscillations that occur in the transition regime between the (photon-driven) weak-field response and the (field-driven) strong-field response. The off-resonant populations, which show a clear asymmetry with respect to $k_x=0$, are due to CEP-modulated Landau–Zener–Stückelberg interferences. The net photocurrent observed in [83] is a direct consequence of this asymmetry. Note that dissipative processes due to carrier–carrier scattering and carrier–phonon scattering were not considered in the simulations.

Few-femtosecond trARPES would be the ideal tool to map such energy–momentum distributions, which represent the microscopic picture behind the detected mesoscopic photocurrents: the characteristic momentum distribution patterns will only be visible on time scales shorter than the intrinsic dephasing and population relaxation times, which in graphene are of the order of a few femtoseconds (electron–electron scattering) up to several 10 $\mathrm{f}$s (electron–phonon scattering) [12,67,69,85]. A limiting factor for the realization of this experiment can be space-charge effects due to pump pulse-induced multiphoton photoemission processes [86]. In the relevant pump fluence regime, space-charge effects can significantly distort the energy and momentum distribution of the electrons photoemitted by the probe pulse, ultimately preventing a meaningful evaluation of the recorded photoemission intensity maps.

Figure 7c displays the peak electric field strength $E_0$ of few-cycle 800 nm laser pulses as a function of pulse duration and the square root of the pump fluence F. The gray dashed lines mark constant values of $E_0$ in units of Vnm⁻¹. The red line represents the formal boundary between the weak-field and the strong-field regime, corresponding to the Keldish parameter $\gamma =1$ (expressed in the graph by the equality of the Rabi frequency ${\Omega }_{\mathrm{Rabi}}$ and the angular frequency of the driving laser field $\omega$). The three data points in the graph mark the experimental parameters used in two trARPES studies on graphite [67] and graphene [12]. The value of $E_0$ that the electrons experience inside the material was calculated assuming a refractive index of graphite of $n_{\mathrm{Gr}}\approx 2.6$ [87,88] and, in the case of graphene, the refractive index of the underlying SiC substrate of $n_{\mathrm{SiC}}\approx 2.6$ [83]. Both works focused on the time evolution of the energy distribution of the excited carriers but did not consider changes in the momentum distribution. The data still show that few-cycle NIR trARPES in the strong-field regime of interest are within reach. Such an experiment can provide valuable information not only on the energy–momentum dependence of quantum interference pathways but also on how the interferences are affected by decoherence due to dissipative interactions on electronic time scales. In the future, a successful implementation of such an experiment may stimulate further studies such as strong-field photoemission interferometry dealing with topological properties of two-dimensional materials [89], the reciprocal space exploration of coherently controlled electron trajectories [84], or energy- and momentum-resolved CEP-phase and two-pulse few-cycle control experiments in semiconducting monolayer TMDCs and derived heterostructures.

4.2. Coherent Multidimensional ARPES

We have argued above that for spectroscopic access to the regime of electronic time scales in materials it is necessary to realize a trARPES experiment that is balanced in terms of energy and time resolution. Nevertheless, one can wonder if it is possible to realize a trARPES experiment that simultaneously provides high energy resolution and high time resolution beyond the limits imposed by the time–bandwidth product.

A promising concept to achieve this goal is multidimensional spectroscopy, which has already been successfully applied in two-photon photoemission experiments on solids [90,91] and in the gas phase [92]. For example, these studies provided valuable insights into local coherences of plasmonic active nanostructures [90], surface and bulk state coherences at single-crystal surfaces [93,94], excitonic correlations in metals [95], and the dressing of electronic band structures of metals [91]. All these experiments were performed with photon energies in the NIR to near ultraviolet spectral range. XUV pulses would make it possible to apply this concept also to trARPES and thus considerably extend the accessible energy–momentum range. This would open up new possibilities to study electronic processes with simultaneously high time and energy resolution throughout the BZ (and beyond) or via transient spectral changes in (shallow) core-electron levels.

A recent theoretical work described and analyzed a possible experimental configuration that might be able to achieve this goal in the future [96]. The authors proposed to apply the concept of coherent multidimensional optical spectroscopy [97,98] to trARPES. Instead of following the response of the electronic structure to the optical stimulus with a single probe pulse, such types of experiments use two successive probe pulses locked in their relative phase. The additional information compared to conventional pump–probe experiments comes from the interference contribution to the detected probe signal. The effect on the transient photoemission signal is illustrated by the simulations from [96] shown in Figure 8. The authors consider a single occupied energy level that is suddenly shifted in energy. Figure 8a shows the calculated Wigner function N as a function of energy and time, which could be referred to as the time-dependent occupied density of states. The resulting time-resolved photoemission signal for a single Gaussian probe pulse and a phase-locked pair of probe pulses are shown in Figure 8b and c, respectively. Spectral details within the switching time window, which are completely missing in the single-probe experiment due to the limited spectral resolution, are well reproduced in the double-pulse experiment, demonstrating the superior information content of the multidimensional spectroscopy approach. The relative phase stabilization of a probe–pulse pair at a photon energy of about 20 eV (a typical energy range used in common trARPES experiments [11,64,99,100,101]) requires a pulse-to-pulse timing stability in the range of 10 $\mathrm{a}$s, a value that can be achieved with actively stabilized delay stages [102].

5. Conclusions

In materials, the characteristic ≈1 eV energy scales of the electronic degrees of freedom lead to extremely fast dynamics of a few femtoseconds. In order to be able to observe the associated physical processes in detail at the fundamental level, real-time techniques are required that provide not only high time resolution, but also sufficient energy and momentum resolution. Few-femtosecond trARPES schemes, as described in this perspective, have these capabilities and can uniquely provide direct energy- and momentum-resolved information about the dynamics of electronic structures and quasiparticle populations on the characteristic time scales of electronic hopping and ultrafast light–matter interactions. Few-femtosecond trARPES can thus fill the time gap between few–100 as photoemission spectroscopy with ultrahigh time resolution and conventional few–10 fs trARPES with high energy resolution. This time gap is of great fundamental interest because it is here that electronic coherences and correlations arise and decay, and where nonthermal pathways, nonlinear responses, and dynamical interactions dominate. Entering this time regime with few-femtosecond trARPES will enable us to directly image the quantum electronic dynamics in materials in real time and to steer them toward novel nonequilibrium functionalities. Our vision is that we will be able to generate and probe coherent superpositions and correlated many-particle states, and manipulate these states on time scales comparable to or shorter than typical dephasing and interaction times.

Author Contributions

Writing—original draft preparation, K.R. and M.B.; writing—review and editing, K.R. and M.B. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

Not applicable.

Acknowledgments

We would like to thank Eduard Moos for creating Figure 7c of this publication.

Conflicts of Interest

The authors declare no conflicts of interest.

Abbreviations

The following abbreviations are used in this manuscript:

ARPES	angle-resolved photoemission spectroscopy
BZ	Brillouin zone
CCDW	commensurate charge-density wave
CDW	charge-density wave
CEP	carrier–envelope phase
EDC	energy distribution curve
FWHM	full width at half maximum
LAPE	laser-assisted photoemission
LHB	lower Hubbard band
NIR	near-infrared
RABBIT	reconstruction of attosecond beating by interference of two-photon transitions
TMDC	transition-metal dichalcogenide
trARPES	time- and angle-resolved photoemission spectroscopy
UHB	upper Hubbard band
XUV	extreme ultraviolet

Footnotes

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Figures

Figure 1. Schematic temporal evolution of a correlated electron material in a pump–probe experiment with different physical processes taking place on characteristic time scales. Time is shown in units of the intrinsic electronic time scale [Forumla omitted. See PDF.], where W is the (eV scale) electronic bandwidth. Reprinted from [1]. Copyright (2014) by The American Physical Society.

Figure 2. Schematic of an experimental setup for time- and angle-resolved photoemission. The NIR output of a low-repetition Ti:sapphire amplifier system is used for sample excitation (pump) as well as for the generation of XUV pulses in a high-harmonic process for time-delayed probing of the sample response via photoemission. The energy and momentum distributions of the photoemitted electrons are characterized using a hemispherical electron energy analyzer.

Figure 3. Ultrafast dynamics of the Mott-insulator phase in 1T-TaS2. (a) Measured band dispersion along the [Forumla omitted. See PDF.] high-symmetry path for the commensurate CDW phase of 1T-TaS2. Sample temperature and photon energy of the angle-resolved photoemission spectroscopy (ARPES) measurement are indicated. Photoemission intensity is represented in a false-color scale. The arrows mark the Mott–Hubbard gap [Forumla omitted. See PDF.] and the CDW gap [Forumla omitted. See PDF.]. (b) trARPES snapshots taken before optical pumping and at a pump–probe delay of 65 fs at the [Forumla omitted. See PDF.] point of 1T-TaS2 (absorbed pump energy density: 300 Jcm−3; equilibrium sample temperature: 110 K). The arrows indicate the Mott–Hubbard gap. (c) Time dependence of the ARPES intensity at the [Forumla omitted. See PDF.] point. The energy interval over which the intensity has been integrated is [Forumla omitted. See PDF.]. (d) Transient photoemission spectra of 1T-TaS2 obtained for an equilibrium sample temperature of 30 K and a pump fluence of 60 μJcm−2. The curves are offset for better visibility. (e) Temporal evolution of the upper Hubbard band (UHB) spectral signature in direct comparison with the underlying continuum and electronic gap, as well as the lower Hubbard band (LHB) intensity loss. Solid lines are fits to the experimental data using appropriate fitting functions. The temporal derivative −[Forumla omitted. See PDF.] of the least mean square fit of the LHB dynamics is shown as a dashed line for comparison with the UHB signal. Curves are rescaled for better visibility. (a) reprinted, (b,c) adapted from [35]. (d,e) reprinted from [36]. Copyright (2018) by the American Physical Society.

Figure 4. Ultrafast dynamics of the excitonic insulator phase in 1T-TiSe2. (a) trARPES snapshots of 1T-TiSe2 for increasing pump–probe time delays (pump-pulse fluence: 5 mJcm−2). The pump-induced spectral changes at [Forumla omitted. See PDF.] indicate the suppression of the charge-ordered state. (b) Comparison of characteristic signal drop times indicating the suppression of a combined charge- and lattice-ordered state, compiled from different works [35,37,38,59]. In the high-fluence regime (>200 μJcm−2), hot-carrier (Coulomb) screening is sufficiently fast to dominate the overall dynamics. (a) reprinted from [37]; (b) reprinted from [38].

Figure 7. Few-femtosecond trARPES: entering the few-cycle strong-field regime. (a) Illustration of combined interband (Landau–Zener) and intraband transitions in the Dirac cone of graphene, which can act as a beam splitter for an electron wavefunction exposed to a light field in the strong-field regime. In a single oscillation of the electric field, the electron can experience the beam-splitting effect twice, resulting in two possible quantum paths to reach the conduction band. (b) Calculated residual conduction band population in graphene near the K point for different values of a peak electric field strength [Forumla omitted. See PDF.] Vnm−1 of the excitation laser pulse. The data show results for an 800 nm, 5.5 fs laser pulse with a carrier–envelope phase of [Forumla omitted. See PDF.] and linear polarization along the x-direction. The plots show the population after the laser pulse has excited the sample and neglect dissipative processes due to interactions with electrons and phonons. Solid lines indicate the resonances, the energy difference between the valence and conduction bands corresponding to energies [Forumla omitted. See PDF.], [Forumla omitted. See PDF.], and [Forumla omitted. See PDF.] (from inside to outside), where [Forumla omitted. See PDF.] is the laser photon energy. Resonant populations indicate photon-driven processes, off-resonant populations indicate field-driven quantum path interference processes. (c) [Forumla omitted. See PDF.] of few-cycle 800 nm laser pulses as a function of pulse duration and the square root of the incident pump fluence F. Constant values of [Forumla omitted. See PDF.] in units of Vnm−1 are marked by the gray dashed lines. The solid red line marks the boundary between the weak-field and the strong-field regime in graphene ([Forumla omitted. See PDF.] Vnm−1, [Forumla omitted. See PDF.]) reported in Ref. [83]. The blue circled points indicate the experimental laser pulse parameters used in trARPES experiments on graphite [67] and graphene [12]. (a,b) reprinted from [83], Springer Nature.

Figure 8. Two-dimensional spectroscopy applied to trARPES: illustration of the use of different probe pulse profiles in a time-resolved photoemission spectroscopy experiment. The system of interest is a single electronic level that is initially occupied and then suddenly energetically shifted at time [Forumla omitted. See PDF.]. (a) Wigner transform [Forumla omitted. See PDF.] of the Green’s function of the system. (b,c) Calculated momentum-integrated trARPES spectra obtained with different probe–pulse profiles [represented in the inset by the probe–pulse autocorrelation function [Forumla omitted. See PDF.]], as a function of photoelectron energy and time. (b) Classical Gaussian probe pulse. The observed switching is resolution limited. (c) Nonseparable positive definite probe filter [Forumla omitted. See PDF.], which reproduces some of the features of the [Forumla omitted. See PDF.] in the photoemission spectrum due to its off-diagonal structure. (a–c) reprinted from [96]. Copyright (2017) by The American Physical Society.

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