## Abstract

Key properties of quantum materials stem from dynamic interaction chains that connect stable electronic quasiparticles through short-lived coherences, which are difficult to control at their natural time and length scales. Lightwave electronics sculpts the quantum flow of electrons and coherences faster than an oscillation cycle of light by using intense optical-carrier waves as fast biasing fields, which can access multi-electron interaction chains. In this Review, we summarize the key functionalities and the latest advances in lightwave electronics for both fundamental and technological explorations. For example, lightwave-driven ballistic electron transport through dynamically changing band structures has already led to the demonstration of phenomena such as high-harmonic emission and dynamic Bloch oscillations. Lightwave electronic control could also seamlessly convert quantum states between light and matter to create quantum chips that simultaneously exploit electronics for efficient interactions and optics for speed or long coherence lifetimes. Additionally, we present an outlook towards applications of lightwave electronics including quasiparticle colliders to explore quantum phenomena; all-optical band-structure reconstruction in ambient conditions; attoclocks to measure the interaction dynamics of diverse quantum phenomena; ultrafast electron videography to watch electronic reactions unfold; efficient light sources to create compact integration; and petahertz electronics to speed up traditional semiconductor electronics.

## Introduction

Traditionally, light–matter interactions generate emissive or absorptive processes, which occur when a photon induces an efficient transition between two matter states that have an energy difference that is resonant with the photon energy. Such processes are often relatively slow, lasting for multiple cycles of light, because the time–energy uncertainty prevents a light pulse with too few oscillations from matching the energetic width of a matter resonance. However, when the intensity of a lightwave becomes strong enough, its momentary peak field can efficiently translate electrons much faster than its half cycle through the force it exerts on electrons, irrespective of the photon energy. This concept has led to lightwave electronics in which the carrier wave (not the photon energy) drives electronic coherences much faster than electronic scattering to coherently drive complex quantum states of condensed-matter systems. The first steps have already been demonstrated in systems ranging from bulk semiconductors^{1,2,3,4,5}, metallic antenna structures^{6,7,8,9}, van-der-Waals-bonded atomic crystals^{10}, atomically thin 2D dichalcogenides^{11,12,13,14}, 2D graphene^{15,16}, topological insulators^{17,18,19,20,21} and Dirac semimetals^{22} to individual molecules on substrates^{23,24}.

Despite the many fundamental and technological challenges, lightwave electronics could optimally interface optics and electronics by simultaneously leveraging ultrafast oscillations of light, enduring photon coherence, and extremely strong electron–photon and electron–electron many-body interactions. The main challenge is the mismatch between the characteristic clock cycles of optical and electronic systems: the oscillation period of electromagnetic waves is approximately 5 fs at telecom wavelengths. This period is five orders of magnitude faster than the clock rates of state-of-the-art electronics in computers or fifth-generation (5G) telecommunication^{25} technology, which have 1-ns cycles. Additionally, such gigahertz technologies involve electrons that scatter thousands of times per clock cycle, destroying any useful quantum information.

To use electronic quantum information in future chips, an electronics–optics interface must perform operations within a few femtoseconds and time them with attosecond precision, before scattering occurs and destroys the quantum information. However, attosecond durations correspond to tens of electronvolts in energy, which matches poorly with the millielectronvolt energy scales used in electronics. This disparity can be addressed by using a lightwave that changes electronic states much faster than the oscillation cycle of light or any disturbing scattering events. In this case, the light and electron energies do not need to match and the resulting coherent nonlinear processes can be timed with extreme subcycle resolution. For example, Coulombic interactions in 2D quantum materials were clocked with 300-as resolution^{14}, corresponding to 1/130th of a cycle, when driven by a lightwave. This resolution is sufficient to illuminate how many-body effects unfold to create emergent quantum correlations, ranging from forming electronic clusters^{26,27} to phase transitions^{28,29,30}. Lightwave electronics can be used to access and control such correlations (Fig. 1) as one of its many prospects.

In this Review, we outline how lightwave electronics could be used to overcome deleterious scattering and leverage beneficial extreme nonlinearities for moving, flipping, timing, and exploring electronic quantum-information and many-body phenomena even within room-temperature condensed-matter systems. We first summarize the foundational aspects of lightwave electronics and discuss how these ideas led to the understanding of high-harmonic (HH) generation in solids. These initial studies indicated intriguing condensed-matter effects beyond simple descriptions; we overview the key many-body aspects needed in a predictive theory and summarize most common methods used so far. We then identify challenges and the latest developments in lightwave electronics, including new light sources, attoclocking, electron videography and quantum technology. We also provide a list of open challenges and a vision of how future lightwave electronics might impact quantum science and technology.

## The concept of lightwave electronics

Lightwave electronics was initially proposed in the context of atomic and molecular systems, in which the motion of electrons can be controlled by using the oscillating carrier wave of strong light pulses to induce HH emission^{31,32,33,34,35,36,37}. The first steps towards harmonics in semiconductors are summarized in Fig. 2. Atomic systems and semiconductors share similar semiclassical features but differ in their quantum aspects, as we elaborate subsequently.

### Semiclassical features

The key aspects of HH excitations can be understood semiclassically through the so-called three-step model (3SM)^{38} (Fig. 1a, shaded box). First, a strong lightwave ionizes an electron from a state bound to the ionic core. Second, the light force accelerates the effectively free electron away from the ion and increases its kinetic energy. Third, as the sign of the oscillating field flips, the electron returns to its ion core and releases its excess kinetic energy as HH emission when it collides with the ion. The 3SM has brought much clarity to attosecond science^{39,40} and has guided concepts such as the tomographic imaging of molecular orbitals^{41,42,43}. In atomic systems, the maximum energy (cut-off) of HH emission scales with the ponderomotive energy, \({U}_{{\rm{pond}}}\propto {E}_{{\rm{peak}}}^{2}/{\omega }_{{\rm{wave}}}^{2}\), which is the cycle-averaged quiver energy of an electron accelerated by a lightwave with a peak field \({E}_{{\rm{peak}}}\) and angular frequency \({\omega }_{{\rm{wave}}}\).

In crystalline solids, ionic cores are arranged in a crystal structure (Fig. 1a, shaded box), which modifies the electronic response to be different from that of isolated atoms. In solids, electrons are no longer bound to a single ion core, but instead occupy delocalized Bloch states that extend over the entire crystal. The electron dispersion as a function of the wave vector \({\bf{k}}\) is described by energy bands \({E}_{{\bf{k}}}^{\lambda }\) separated by an energy gap \({E}_{{\rm{gap}}}\); \(\lambda \) is a discrete band index (Fig. 1b). Treating electronic intraband motion semiclassically indicates that HH emission takes place when electrons and holes re-encounter one another^{44}, establishing an interesting similarity to the atomic 3SM.

Nevertheless, when HH emission was experimentally observed in solids^{2,3,5} (Fig. 2), the HH characteristics differed from those of atoms. Instead of exhibiting only odd-order HHs (as in atomic systems), bulk ZnSe produces both even-order and odd-order HHs (Fig. 2d) owing to broken symmetry in the \({E}_{{\bf{k}}}^{\lambda }\) dispersions. Additionally, electrons that accelerate towards the edge of the Brillouin zone Bragg reflect to the opposite side of the Brillouin zone (Fig. 1b), which induces Bloch oscillations^{45}. These Bloch oscillations and the non-parabolic \({E}_{{\bf{k}}}^{\lambda }\) produce a cut-off energy that depends linearly on \({E}_{{\rm{peak}}}\) (ref. ^{3}) (Fig. 2e) (not quadratically as predicted by \({U}_{\text{pond}}\) and the 3SM). This linear dependence persists all the way to extreme ultraviolet (UV) HHs^{46} (Fig. 2g). The effect of dynamic Bloch oscillations (Fig. 2f) on the shape of HH spectra was eventually demonstrated^{5} by comparing experimental and theoretical results using few-cycle mid-infrared pulses with a controlled carrier-envelope phase (CEP).

### Two-level features

To fully understand HH generation in solids, it is important to consider that lightwave electronics can also induce interband transitions between valence (\({E}_{{\bf{k}}}^{{\rm{v}}}\)) and conduction (\({E}_{{\bf{k}}}^{{\rm{c}}}\)) bands (moving electrons between bands instead of within a band). Such excitations become possible when light–matter interactions change electronic wave functions on the unit-cell-level, whereas the 3SM explains only the semiclassical \({\bf{k}}\) motion. Valence-to-conduction band dynamics can be mapped into a two-level system (2LS), which is one of the foundational models for light–matter interactions. Specifically, excitations in 2LSs couple occupations in the valence and conduction bands, \({f}_{{\rm{v}}}(t)\) and \({f}_{{\rm{c}}}(t)\), respectively, with the polarization \(P(t)\) between them. Excitations of 2LSs are driven by the Rabi frequency, \(\Omega (t)\equiv {\bf{d}}\,\cdot \,{\bf{E}}(t)/\hbar \), which is the dot product of the dipole matrix element \({\bf{d}}\) between the bands and the electric field \({\bf{E}}(t)\) that oscillates as \(\cos {(\omega }_{{\rm{w}}{\rm{a}}{\rm{v}}{\rm{e}}}t)={\rm{R}}{\rm{e}}[\exp (\,-{\rm{i}}{\omega }_{{\rm{w}}{\rm{a}}{\rm{v}}{\rm{e}}}t)].\) For classical light fields, the optical Bloch equations (OBEs) determine the dynamics of \((P,{f}_{{\rm{v}}},{f}_{{\rm{c}}})\), which introduce pure quantum aspects into lightwave electronics because the quantized energy levels of a 2LS have no classical-particle counterpart.

For resonant excitations, the photon energy \(\hbar {\omega }_{{\rm{wave}}}\) is equal to the transition energy, \({E}_{{\rm{g}}{\rm{a}}{\rm{p}}}\equiv {E}_{{\bf{k}}}^{{\rm{c}}}-{E}_{{\bf{k}}}^{{\rm{v}}},\) which is the bandgap at \({\bf{k}}=0\). If \(\Omega (t)\) is sufficiently low, the induced \(P(t)\) oscillates dominantly as \(\exp (\,-{\rm{i}}{\omega }_{{\rm{w}}{\rm{a}}{\rm{v}}{\rm{e}}}t),\) although the lightwave \(\Omega (t)\) contains both co-rotating \((\exp (\,-{\rm{i}}{\omega }_{{\rm{w}}{\rm{a}}{\rm{v}}{\rm{e}}}t))\) and counter-rotating (\(\exp (\,+{\rm{i}}{\omega }_{{\rm{w}}{\rm{a}}{\rm{v}}{\rm{e}}}t)\)) components. The OBEs can be solved analytically^{47} with the rotating-wave approximation (RWA) in which the envelope of the co-rotating part of \(\Omega (t)\) defines the excitations. For the resonant case (\({\hbar \omega }_{{\rm{wave}}}={E}_{{\rm{gap}}}\)) and a continuous \(\Omega (t)={\Omega }_{{\rm{peak}}}\cos ({\omega }_{{\rm{wave}}}t)\) with amplitude \({\Omega }_{{\rm{peak}}}\), the RWA solution predicts Rabi oscillations, in which \({f}_{{\rm{v}}}(t)\) and \({f}_{{\rm{c}}}(t)\) interchange periodically with a Rabi-flop period of \(T=2{\rm{\pi }}/{\Omega }_{{\rm{peak}}}\). As long as \({f}_{{\rm{c}}}(t)\) remains small, the lightwave is absorbed linearly at a single frequency (\({\hbar \omega }_{{\rm{wave}}}={E}_{{\rm{gap}}}\)). But when \({f}_{{\rm{c}}}(t)\) grows, the absorption becomes nonlinear with \({f}_{{\rm{c}}}(t)\) and is accompanied by features such as the Rabi-splitting bands at \(\hbar {\omega }_{{\rm{wave}}}\,\pm \,{\hbar \Omega }_{{\rm{peak}}}\). The strongest nonlinearities appear at the completely inverted 2LS (\({f}_{{\rm{v}}}=0\) and \({f}_{{\rm{c}}}=1\)).

Lightwave electronics can push excitations in 2LSs to an extremely nonlinear regime, in which \(|\hbar \Omega (t)|\) becomes comparable with \({E}_{{\rm{gap}}}\). This nonlinearity causes the RWA to break down and the OBEs can only be solved numerically, and both the co-rotating and counter-rotating parts of \(\Omega (t)\) must be accounted for. In solids, the magnitude of electronic \({\bf{d}}\) is typically around 5 eÅ and is limited by the atom–atom distance (unit cell) of the crystal. Ultrafast lightwave excitations with peak field strengths of up to 100 MV cm^{–1} have been applied in solids such as ZnO (ref. ^{3}), GaSe (ref. ^{5}), Cd_{3}As_{2} (ref. ^{22}) and fused silica^{48} without inducing sample breakdown. For a typical dipole of 5 eÅ and a peak field strength of 100 MV cm^{–1}, \(\hbar {\Omega }_{{\rm{p}}{\rm{e}}{\rm{a}}{\rm{k}}}\) becomes as large as 5 eV, with a Rabi period of \(T\) = 827 as (if the RWA is valid). This \(\hbar {\Omega }_{{\rm{p}}{\rm{e}}{\rm{a}}{\rm{k}}}\) exceeds \({E}_{{\rm{gap}}}\) of most semiconductors, leading to extremely nonlinear excitations with transitions that are orders of magnitude faster than electronic scattering.

Solving the OBEs numerically for pulsed excitations with a flat-top envelope and a spectral width that is approximately 13 times narrower than the transition energy demonstrates that lightwave electronics induces ultrafast inversion even for non-resonant (\(\hbar {\omega }_{{\rm{wave}}}\ll {E}_{{\rm{gap}}}\)) excitations (Fig. 2a, right). The final excitation \({f}_{{\rm{f}}{\rm{i}}{\rm{n}}}\equiv {f}_{{\rm{c}}}(t={\rm{\infty }})\) after the pulse for resonant (\(\hbar {\omega }_{{\rm{wave}}}={E}_{{\rm{gap}}}\)) excitations exhibits regular Rabi flops up to about \({\hbar \Omega }_{{\rm{peak}}}={0.5E}_{{\rm{gap}}}\), followed by increasingly irregular oscillations as the system transitions into the extremely nonlinear regime in which the 2LS mixes co-rotating and counter-rotating contributions (Fig. 2a, top).

At low excitation strengths (\({\hbar \Omega }_{{\rm{peak}}}/{E}_{{\rm{gap}}}\ll 1\)), \({f}_{{\rm{fin}}}\) is close to zero for non-resonant excitations (\({\omega }_{{\rm{wave}}}={\omega }_{{\rm{gap}}}/6\)), which is in agreement with RWA predictions (Fig. 2a, top). As the non-RWA contributions increase, \({f}_{{\rm{fin}}}\) starts to exhibit transitions for \({\hbar \Omega }_{{\rm{peak}}}\) above \(0.5\,{E}_{{\rm{gap}}}\) (unlike the RWA solution). For elevated excitation strengths \(\hbar {\Omega }_{{\rm{p}}{\rm{e}}{\rm{a}}{\rm{k}}}\), \({f}_{{\rm{fin}}}\) shows an irregular series (unlike the regular Rabi flopping) of peaks approaching unity as non-RWA contributions invert the system. However, the maximum \({f}_{{\rm{c}}}\) excitation (\({f}_{\max }\)) onsets much earlier and is much larger than \({f}_{{\rm{fin}}}\). Because \({f}_{{\rm{fin}}}\) also quantifies absorption by matter, \({f}_{\max }\gg {f}_{{\rm{fin}}}\) implies that there is a reversible energy exchange between light and matter, enabling non-resonant lightwaves to transiently excite matter at extreme field strengths without inducing breakdown.

Comparing the intensity spectrum, \(I(\omega )\equiv {|\omega P(\omega )|}^{2}\), of the light emitted by resonant and non-resonant excitations shows that only the resonant case produces Rabi splitting around the \({\omega }_{{\rm{gap}}}\) resonance (Fig. 2a, bottom). To exploit Rabi splitting for resonant state control^{49}, all broadening and dephasing mechanisms must be smaller than \({\hbar \Omega }_{{\rm{peak}}}\). As \({\Omega }_{{\rm{peak}}}\) increases, resonantly driven \({E}_{{\rm{gap}}}-{\hbar \Omega }_{{\rm{peak}}}\) can become negative, which induces new frequency branches through extreme nonlinearities. By contrast, the strongly non-resonant excitation produces HHs at \(n{\omega }_{{\rm{wave}}}\) (\(n=1,3,5,\ldots \)), implying that HHs in condensed-matter systems can originate from both quantum (2LS) and semiclassical (3SM) processes because each \({\bf{k}}\) state can simultaneously be the centre of 2LS excitations and the starting or recollision point of 3SM translations. Light-induced HH and harmonic sideband (HSB) generating processes in solids (Fig. 2b,c) can both be used to exploit extreme excitations to optimally mix the coherent interplay of electronic 3SM motion, 2LS transitions (in HHs), Bloch oscillations and many-body effects to control and access the quantum character of electronic transport and transitions, as means to explore quantum phenomena and quantum-information processing in solids. To describe these effects fully, the OBEs must be generalized to semiconductor Bloch equations (SBEs; Box 1) that systematically include the key many-body effects.

### First steps towards lightwave electronics

Initially, relatively strong-field investigations were used to explore Rabi flopping^{50,51,52,53} between excitonic \(1s\) and \(2p\) states when Coulomb-enhanced Rabi frequency and splitting became larger than dephasing-induced broadening. The path to HHs was reached in ZnSe in which HHs up to the seventh order were observed by applying strong mid-infrared pulses with peak intensities of around 10^{12 }W cm^{−2} (ref. ^{2}) (Fig. 2d). Subsequently, up to 25 HHs were detected in ZnO (ref. ^{3}) extending even above \({E}_{{\rm{gap}}}\) (Fig. 2e). Solid-state HH generation with phase-locked light pulses was achieved^{5} using THz-driving fields. The phase-stable HH transients of both even and odd orders cover more than 13 optical octaves (Fig. 2f) and exhibit a clear plateau and a cut-off region that strongly depends on the CEP of the driving field. The SBEs attribute the appearance of even harmonics to multiband-interference effects and identify clear dynamic Bloch oscillations when the system is driven with a THz pulse (Fig. 2f, inset). These Bloch oscillations introduce the experimentally observed, strong CEP dependence of the HHs.

HHs originate from multiple processes that could compromise the **k** precision of lightwave excitations (Fig. 2b). A more controlled way to study the motion of electrons in solids is to pump the system resonantly and drive the excited state with a second, strong lightwave. This process resembles atomic HH generation more than the generation of solid-state HHs because it starts with a well-defined state (excitonic coherence, that is, an electron–hole atom) and forces the electrons and holes constituting the excited quasiparticles to undergo an ionization–acceleration–recollision process, as in the 3SM, inducing the emission of HSBs (Fig. 2c). This HSB modality was observed in the perturbative regime in GaAs quantum wells^{1}. Subsequently, HSB generation was also promoted to the non-perturbative regime^{4} (Fig. 2h) by using a free-electron laser as a strong THz source to excite sidebands up to the 18th order. The weak intensity decay with increasing sideband order confirmed the non-perturbative nature of HSBs, which are strongest for a linearly polarized THz field and vanish for circularly polarized radiation. The dynamic nature of quasiparticle collisions was explored^{54} in bulk WSe_{2} (Fig. 2i) by recording the strength of HSB emission as a function of the delay time, \({t}_{{\rm{ex}}}\), between the resonant near-infrared (NIR) excitation and the THz wave. The NIR pulse used was shorter than the THz half cycle to resolve whether \({t}_{{\rm{ex}}}\) affected the HSB intensity. As in any particle collider, the debris of a collision — here HSB emission — was detected, producing a modulated HSB intensity as a function of \({t}_{{\rm{ex}}}\) (Fig. 2i, top). Distinct peaks in this intensity were associated with excitation times that produce efficient electron–hole recollisions, which was verified with SBE computations (Fig. 2i, bottom). A delay with respect to THz field crests was also observed (Fig. 2i, inset magnification), as expected from an atomic-like 3SM.

### Single-body and many-body features

At a single electron level, lightwave electronics in atomic, molecular and solid-state systems have similar extreme scale and propagation features. In particular, extreme nonlinear excitations (Fig. 2a) mix resonant and non-resonant transitions to all orders (scales), which invalidates perturbative approaches and introduces time–energy and position–momentum scales that span multiple orders. For example, up to 100 orders of harmonics have been demonstrated in both atomic^{55} and solid-state systems^{56}; therefore, theories must also capture this number of scales. For optically thick samples (compared with the HH wavelength), a theory must include self-consistent lightwave propagation to predict phase-matching conditions for each harmonic. The resulting nonlinear light propagation can also focus light^{57} and produce filamentation^{58} or solitons^{59}. Conversely, light propagation has been predicted^{60} to induce effective dephasing for matter coherences. Although such wave propagation effects in HH and HSB generation have not yet been thoroughly investigated, they will be crucial to optimizing lightwave electronics technology.

At a single-electron level, crystals introduce nontrivial bands, interferences between them, geometric and topological effects and Bloch oscillations as unique solid-state features. Prospects for quantum-information applications emerge when utilizing the crystal–electron correlations (CeCs) of solids because lightwaves typically excite millions of electrons and holes, which are strongly coupled by the Coulomb force (Fig. 1b, white lines). Oppositely charged electrons and holes attract one another and can form bound electron–hole clusters^{61}, quasiparticles, such as excitons (atom-like pairs of electrons and holes), exciton molecules^{62} or dropletons^{26} (tiny electron–hole droplets with quantized energies). If such quasiparticle states can be sufficiently isolated from one another, they could serve as quantum-information units with potential quantum-logic applications^{12,63,64}. As promising quantum materials, transition metal dichalgogenides^{65} and III-nitride^{66,67} (BN, GaN and AlN) monolayers produce strong enough exciton binding to achieve quantum effects at room temperature. Coulomb and phonon interactions can also create emergent phenomena such as phase transitions^{28,68,69,70} or couple the targeted quantum states with unwanted degrees of freedom, which induce dephasing and limit the lifetime of quantum information. The full scope of HH and HSB many-body correlations remains largely unexplored, although harnessing them could lead to optimal optics–electronics interfaces for controlling quantum information.

## Predictive lightwave electronics theory

Many theoretical approaches have been applied to predict lightwave phenomena. The most comprehensive and predictive approaches simultaneously address the band structure, extreme scales, pulse propagation and CeC challenges. This section briefly overviews the key assumptions and achievements of these approaches.

### Quantum-dynamic cluster expansion

Several of the studies described in this Review (Fig. 2) were inspired^{71} and quantitatively explained by quantum-dynamic cluster expansion (QDCE)^{12,13,14,54,72} (Box 2) and/or its first hierarchical level: the Maxwell–SBEs (Box 1). The predictive prowess of QDCE stems from its ability to efficiently and accurately compute the extreme non-equilibrium excitation dynamics from the ground state with minimal but comprehensive input (only experimental lightwaves and fundamental material inputs such as the electronic bands and interaction matrix elements are needed). Non-perturbative quasiparticle excitations and their systematic creation^{61}, dephasing^{73,74,75}, quantum memory^{76} and interaction screening^{77} features are self-consistently included in QDCE (Box 2). Additionally, QDCE computations can be accurately limited to the optically active bands^{77,78}, which considerably reduces the computational effort involved compared with all-electron investigations. The material input can then be settled with any well-defined approach. For example, the Hubbard model^{79} or its extensions have been used to describe correlated materials, whereas semiconductor studies have relied on tight-binding models, density-functional theory (DFT)^{80} or its perturbative GW extensions^{81} to describe a broad range of quantum materials. Thus, QDCE can predictively address the band structure, extreme length, time and energy scales, pulse propagation and CeC challenges. Being a comprehensive approach, QDCE is primarily limited by computational resources; therefore, the addition of new mechanisms or clusters often requires meticulous investment to develop efficient implementations.

### Reduced SBEs

Omitting the Coulomb interaction from the Maxwell–SBEs considerably reduces the computational complexity because the resulting reduced SBEs couple the Bloch states only via light (Fig. 2b). This structure also generalizes the OBEs^{82,83} and is often used in situations in which strong-field effects dominate other effects such as the Coulomb interaction^{84}. The reduced SBEs fully address the extreme scale and propagation challenges but they only capture correlation aspects, such as dephasing and quasiparticle effects, phenomenologically. Several reduced-SBE studies have revealed intriguing extreme scale and crystal–electron insights. For example, this body of work established the influence of geometric aspects on HH^{85} and HSB^{56} emission; interpreted HH generation at different excitation regimes^{86}; suggested new schemes to tomographically map band structures through HH emission^{87,88}; predicted two-colour excitation schemes to control valley-dependent excitation in 2D graphene-like systems with broken inversion symmetry^{89} and discussed the generation of HHs from topological states^{85}.

### Effective single-particle approaches

The CeC aspects of lightwave electronics can be further simplified with effective single-particle approaches to explore HH and HSB features semiclassically, including the generalization of the atomic 3SM for HH^{44,90} and HSB^{91} generation. By integrating an effective single-body Schrödinger equation with the saddle-point method, the interband dynamics can be solved analytically^{44,83,92,93,94} to connect nonlinear emission with recolliding electron–hole trajectories. Selection rules for the intraband emission^{11} can also be deduced from such an analysis. Analytical insights about the intraband emission have also been obtained with the Floquet–Bloch theory^{86,95,96}, which was used to solve electronic bands induced by a time-periodic modulation. This approach is guiding efforts in explaining lightwave electronic effects in topological quantum materials^{97,98} with extended coherence times owing to topological protection. The intraband emission was connected to Fourier components of the band structure^{3,46} by solving the classical dynamics of a single electron in a solid. Promising steps towards exploring quantum-optical HH emission^{99} have been made by quantizing the light field. Although effective single-particle approaches sometimes lack quantitative results, their strength lies in providing intuitive pictures about the underlying physical processes, making them ideal for inspiring further lightwave electronic studies.

### Time-dependent DFT

Developing a systematic description of how dynamic lattice changes affect lightwave electronics remains challenging, and time-dependent DFT^{100} could help such investigations. Time-dependent DFT has already been used to predict how electronic band-structure properties are connected^{101} to the laser polarization to enable optimal HH generation in bulk crystals; how a free-standing hBN monolayer could generate HHs^{102} with cut-off behaviour similar to that of atomic systems; and that the topology of edge states can change the intensity of HHs by up to 14 orders of magnitude^{103}. Although combining the description of light-propagation effects with those of correlations remains challenging, time-dependent DFT could provide invaluable guidance for explaining what lightwave features will emerge when the lattice is changed, either reversibly^{104,105} or irreversibly^{106,107} (leading to breakdown).

## Quantum-material characterization

Lightwave-driven HH and HSB processes can easily transport carriers (for HHs) and quasiparticles (for HSBs) through the Brillouin zone (Fig. 3a). Therefore, both HH and HSB emission must contain signatures related to the band structure, Berry curvature, scattering and other microscopic matrix elements. Because the HSB-generating and HH-generating dynamics tracks the excited electronic basis states behind all quantum phenomena, it is desirable to connect distinct HH or HSB emission with electronics — with the goal of achieving all-optical reconstruction of the band, geometric and interaction details.

### High-harmonics-based approach

HH generation was used^{88} to achieve all-optical band reconstruction by measuring the HH as a function of the delay between a strong NIR driving pulse (pump) and a weak second harmonic pulse (probe) to create an oscillatory delay structure for even-order HHs (Fig. 3b). The optimal delay for maximal HH emission was then compared with computational models to identify the most likely band structure. This idea was subsequently extended to optimize the full power spectrum^{108}, including below-gap HHs to avoid absorption-related limitations^{109} and intraband harmonics above the gap to reconstruct conduction bands^{46}. The Berry-curvature^{110} information and the interband dipole matrix elements^{111} can also be retrieved this way. As the most direct connection between the band structure and HH details, van Hove singularities (points in the band structure at which the group velocity vanishes) produce^{112} spectral caustics (enhanced HHs) at photon energies matching these singularities (Fig. 3c). Nonetheless, HH generation has not matched the ability of angle-resolved photoemission spectroscopy (ARPES) to directly map details of the electronic band structures. This finding is not unexpected because HH generation unrestrictedly initiates interband excitations (Fig. 2a) anywhere in the bands and even excites multiple bands, which scrambles the electronic details. However, ARPES cannot be used in ambient conditions to quantify the properties of quantum materials.

### Quasiparticle-collider approach

HSB generation addresses all of the challenges described earlier because it works in ambient conditions, and its resonant pulse (Fig. 2c) excites a quasiparticle wave packet (such as excitons) at a precise wave vector of a single conduction–valence band pair. As the lightwave moves the quasiparticle wave packet through the band pair, HSB radiation is emitted only if the photon energy, \(\hbar {\omega }_{{\rm{HSB}}}\), matches the energy difference, \({E}_{{\bf{k}}}\), between the studied band pair. In the same way, the polarization direction of HSBs tracks the geometric phase of the Bloch electrons. These connections were used to measure^{113} dynamic Jones matrices to reconstruct geometric phases in the heavy and light holes in GaAs quantum wells.

Although HSBs generate narrow wave packets, their widths are still much broader than **k**-dependent changes in the band structure. Detailed QDCE predictions revealed^{13} that lightwaves can move the wave packet roughly sinusoidally in momentum space to emit two bursts of HSBs per cycle. These bursts create a distinct crystal-momentum comb (\({\bf{k}}\)-comb, Fig. 3d, middle) through interference (in close analogy to frequency combs^{114,115}). Comparing theoretical and experimental results verified (Fig. 3d, left and right) that the intensity of HSB emission, \({I}_{{\rm{HSB}}}\), at energy \(\hbar {\omega }_{{\rm{HSB}}}\) suddenly increases when the outermost \({\bf{k}}\)-comb passes \({E}_{{\bf{k}}}\) (Fig. 3d, middle) along the direction of the lightwave. By recording \({I}_{{\rm{HSB}}}\) as a function of the frequency (\({\omega }_{{\rm{wave}}}\)) and field strength (\({E}_{{\rm{peak}}}\)) of the lightwave, the onset of emission was used^{13} to assign the band structure of 2D WSe_{2} (Fig. 3d, right) in ambient conditions with higher resolution than the wave-packet size. This tomographic approach is well suited to direct, high-precision characterization and the control of quantum materials under operational conditions.

### Characterizing 2D and topological materials

The perturbative response of materials can be quantified in terms of nonlinear optical susceptibilities *χ*^{(n)} defining the *n*th-order proportionality to the driving field. In a graphene monolayer, exceptionally large *χ*^{(n)} were measured (Fig. 3e) in HHs generated by a THz field^{15}. Purely non-perturbative HHs were measured^{11} in a monolayer MoS_{2} crystal. The polarization state of the HHs revealed the details (Fig. 3f) of the relative strength of anomalous intraband currents and interband transitions as a function of the polarization direction of the driving field. For a graphene monolayer, an elliptical driving polarization^{116} maximized the seventh and ninth HH (Fig. 3g) when the pump ellipticity was set to *ε* = 0.32, unlike gases, which often exhibit reduced HHs with increasing ellipticity as elliptical 3SM paths do not terminate with a recollision. Topological materials also exhibit HHs with unusual features^{20,21,85}. For example, by tuning the mid-infrared driving frequency below the bulk bandgap of the topological insulator Bi_{2}Te_{3}, HH generation from the topologically protected surface state was rigorously separated from bulk contributions^{20}. Lightwave-driven ballistic acceleration of surface electrons through the Dirac point allowed the CEP of the driving field to tune HH peaks to any non-integer value (Fig. 3h), whereas the Berry curvature was imprinted on the polarization state of the HH radiation.

## Temporally tunable light sources

HH generation in solids also offers possibilities to create coherent and tunable light sources for ultrafast pulses ranging from UV to soft X-rays. Such sources could be much more compact and use less power than their atomic or molecular counterparts because nonlinear materials (solids) are more densely packed than gases, which reduces the pump-laser intensities by orders of magnitude owing to enhanced light–matter interactions.

The temporal tunability of HH sources depends on the duration of the HH burst with respect to the driving lightwave and the time delay between them (Fig. 4a). In 2015, temporally resolved HH emission from solids^{117} was recorded using multi-THz-driven GaSe, detecting a 7-fs HH burst, which was synchronized with the positive field crest of the driving field. The HH emission (Fig. 4b) was not delayed with respect to the driving field, in contrast to predictions from the 3SM, which implies an inherent delay between the excitation (THz crests) and the emission. Furthermore, the emission bursts only emerged at the positive field maxima. The synchronized HH emission was confirmed^{118} by temporally resolved measurements of solid-state HH emission from fused silica, this time reaching extreme UV frequencies with attosecond pulse lengths (Fig. 4c). The reconstructed HH pulse (Fig. 4d) was not only synchronized with the crest of the femtosecond pulse but it was also 472-as long. Both the lack of delay and unipolarity of the emission were fully explained by a QDCE computation (Fig. 4b); five bands and multiple excitation pathways yield constructive (destructive) interference and amplified (suppressed) emission with positive (negative) field crests, which perfectly synchronized the emission.

Additionally, the efficiency of HH generation depends on the orientation of the crystal with respect to the lightwave, including changes in the polarization direction of the HHs^{3,5,11}. Also, both the timing and polarization of HH emission can be controlled^{119} by the orientation, which implies that solid-state HH sources could be used to shape light pulses. Unlike in atomic HH generation, solids can produce harmonic waveforms that do not depend on either the CEP or the intensity of the driving field^{120}. Thus, HH pulses from solids could have great potential as a robust light source for ultrashort pulses.

## Attosecond clocking

Many-body correlation dynamics is a central temporal aspect (Box 2) that influences emergent quantum phases as well as petahertz (PHz) control of quantum devices. Therefore, optics–electronics interfaces should optimally clock correlations with attosecond precision both to regulate when lightwave-driven processes start and to observe what then happens. Such attoclocking would make it possible to synchronize multiple quantum processes so that multiple quantum signals can share the same excitation spot. Besides creating the backbone of PHz technology, attoclocking could provide a direct view on fundamental many-body correlations in condensed matter.

Because correlation-driven quantum processes in solids operate on 100-meV energy scales, time–energy uncertainty limits the temporal resolution of a single event to no less than a few femtoseconds. Conversely, attosecond resolution in detecting a single event cannot measure the related energy with resolution below roughly 10 eV, which seems to render attosecond clocking unsuitable for solids. However, this uncertainty does not limit the timing of two independent events. Additionally, lightwave electronics amplifies extreme nonlinear effects the most at the crests of the driving field, creating peaks that are much sharper than an oscillation cycle. By timing two events, such as the crest of the lightwave and the peak of any extreme nonlinear effect, subcycle resolution can be achieved.

### Attoclocking in the UV range

A two-pulse scheme was used to overcome these energy–time uncertainly limitations^{121} (Fig. 5a). First, a few-cycle NIR pulse excites electrons to the conduction band to be clocked with an extreme UV pulse (centre photon energy, 81 eV and energy band width, 6 eV) that photoexcites both the conduction and 4*f*-core electrons of a single-crystal tungsten sample. Because the core electrons are essentially immobile, the 4*f* signal serves as a reference clock for timing the moving conduction electrons. The photoelectron-emission-streaking trace of the NIR pulse from conduction (Fig. 5b, top, around 85 eV) and 4*f*-electron bands (Fig. 5b, bottom, around 59 eV) exhibited a \(\Delta \tau =110\,{\rm{as}}\) delay difference between the signal crests such that \(\Delta \tau \) clocked the conduction band dynamics at attosecond scales (these results were later confirmed by ref. ^{122} and others). This approach was generalized^{48} to monitor the conduction-band streaking signal as a function of excitation delay in SiO_{2} to resolve the electronic and lattice responses at attosecond scales.

Similar pump–probe schemes have since been used to reveal free-electron-like, ballistic transport in tungsten covered with magnesium films^{123}, quantify photoelectron lifetimes for free-electron-like and high-lying conduction states in nickel^{124} (Fig. 5c), resolve the elastic and inelastic scattering in silica nanoparticles^{125} and detect delay shifts^{126} owing to intra-atomic interactions in bulk WSe_{2} (Fig. 5d). The injection of carriers from the valence band into the conduction band^{127} in GaAs (Fig. 5f) and screening dynamics^{128} in the transition metals Ti and Zr were also observed with sub-femtosecond resolution. Additionally, several switching schemes have been demonstrated with this approach, including 450-as bandgap renormalization^{129} in silicon (Fig. 5e), a femtosecond-scale dynamic Franz–Keldysh effect^{130} in polycrystalline diamond, and 300-as charge-transfer rates^{16} between 2D graphene and bulk silicon carbide. Despite these breakthroughs in attoclocking, UV techniques have not yet directly detected quantum-phenomena-driving quasiparticle correlations within the conduction and valence bands alone because quasiparticle energies (100-meV scale) are three orders of magnitude smaller than UV photon energies (100-eV scale).

### Attosecond clocking with a quasiparticle collider

Attoclocking of quasiparticle correlations was demonstrated^{14} with a quasiparticle collider (Fig. 2g). An NIR pulse precisely excites only a single quasiparticle coherence at an excitation time \({t}_{{\rm{ex}}}\). An intense multi-THz wave then induces a force (\({F}_{{\rm{THz}}}\), Fig. 5g, left) that moves coherent electron–hole pairs by increasing both the relative electron–hole distance \(x\) and momentum \(\hbar k\). As \({F}_{{\rm{THz}}}\) flips its sign, \(x\) decreases all the way to recollision (\(x=0\)) when the excess energy (\(|\hbar k| > 0\)) is released as HSB emission. Only optimal \({t}_{{\rm{ex}}}\) with respect to the \({F}_{{\rm{THz}}}\) peaks maximizes the HSB emission; this optimal \({t}_{{\rm{ex}}}\) is altered by the effects of Coulombic correlations (Fig. 5g, left).

Quasiparticle attosecond clocking^{14} was used to measure \({t}_{{\rm{ex}}}\) with respect to the field crests of a multi-THz waveform, defining the timing change, \(\delta t\), as a function of Coulomb correlations. The 300-as resolution achieved represents massively subcycle precision (7.5 millicycle of the THz period) and was sufficient to time Coulombic effects in bulk and 2D WSe_{2}. Because exciton binding and correlations are five times stronger in 2D WSe_{2} than in the bulk, the HSB intensities peaked 1.2 ± 0.3 fs later, on average, for bulk than for the 2D material (Fig. 5g, right). This approach was also used to directly observe that exciting more electrons and holes weakened Coulomb interaction, changing \(\delta t\) by up to 500 as (Fig. 5g, middle).

## Electron videography

Lightwave electronics also offers videographic possibilities to directly visualize intricate subcycle electron motion either within the band structure of a crystalline solid or in real space. Slow-motion movies in momentum space were obtained^{17} by combining ARPES with a single-cycle THz pulse and a time-delayed UV pulse (shorter than a THz half cycle), which accelerate and photoemit the electrons, respectively (Fig. 6a). The momentary electronic occupations of the energy bands (Fig. 1b) were then recorded by detecting the excess kinetic energy of the photoemitted electrons as a function of the exit angle and time. The resulting band-structure movies (Fig. 6b) directly recorded how the THz field accelerates Dirac-like electrons^{17} in the topologically protected surface state of Bi_{2}Te_{3}. Topological protection supresses back scattering, which can increase^{17} the coherence lifetime of the accelerated electrons beyond 1 ps, introducing the possibility for lightwave Floquet–Bloch band engeneering^{131,132} with topological insulators^{133,134,135} and Weyl semimetals^{136,137}.

Although lightwave ARPES arguably offers the most direct view of Bloch electrons in lattice-periodic crystalline solids, the carrier wave of light can also be used to control electron dynamics on atomic and molecular length scales, which are orders of magnitude below the optical diffraction limit. By combining scanning tunnelling microscopy (STM) with a lightwave bias, extreme spatial and subcycle temporal resolution can be achieved simultaneously^{23,138,139}. Focusing intense THz field transients into the tunnelling junction of a room-temperature scanning tunnelling microscope creates a transient voltage pulse to replace the usual static electric bias between the tip and the sample used in STM^{138} (Fig. 6c). The nonlinear current–voltage characteristic of the junction between the tip and the sample (Fig. 6c, inset) causes a rectified net current, which is read out with conventional STM electronics. This approach was used to measure the ultrafast dynamics of collective charges in a single InAs nanodot^{138} (Fig. 6d) and also to resolve snapshots of electronic states in carbon nanoribbons^{140}.

To experimentally achieve videography of single electrons, state-selective tunnelling is required. Low-temperature STM measurements of a single molecule were demonstrated^{23} by tuning the junction so that the peak of a THz waveform opens the tunnelling channel through a selected molecular orbital (Fig. 6e) to controllably add a single electron to a specific orbital. This highly selective approach enabled snapshot images of the single-electron orbital to be obtained with a subcycle resolution of 100 fs and sub-angstrom precision (Fig. 6f). Pump–probe experiments further revealed coherent molecular vibrations at THz frequencies in the time domain (Fig. 6f, bottom). In subsequent studies, the THz near field in the junction was used as a femtosecond atomic force to control structural dynamics while leaving the system in its electronic ground state^{24}. By adding a visible pump pulse, an electron movie was produced featuring the picosecond-scale spreading of photoexcited carriers in the potential landscape of a molecular film^{141} (Fig. 6g). For even higher temporal resolution, spintronic THz emitters with tips that support ultrabroadband near-field coupling could be used (Fig. 6h). In principle, the lightwave–STM approach is widely scalable in frequency, and mid-infrared bias fields (Fig. 6i) can achieve a temporal resolution below 30 fs (ref. ^{142}). The ongoing work to achieve the highest possible time resolution combined with atomic spatial resolution opens a pathway for exploring the motion of the elementary building blocks of matter at fundamental length and timescales in slow-motion movies.

## Quantum technology

Lightwave electronics can also rapidly drive electron transport in solids. The development of PHz technology follows a hierarchy of steps whereby a single process, such as flipping a state, becomes fast enough (1-fs scale), timed accurately enough (100-as scale, Fig. 5) and chained to multicycle operations to achieve processing with PHz clock rates. Each step poses different fundamental and technological challenges; however, the majority of PHz research still focuses on achieving the first two steps. The first step — rapidly flipping a state — was demonstrated by measuring lightwave-induced transport in bulk-fused silica between gold electrodes, patterned 500 nm apart^{6} (Fig. 7a). The charge transfer induced by the lightwave had a 2.5-fs oscillation cycle (Fig. 7b), confirming that lightwaves can be used to directly detect charges coherently moving over mesoscopic distances at PHz rates. The range of transport was later extended to multiple microns^{105,143}. Similar experiments using two gold electrodes on graphene^{144} resolved intraband currents excited in bulk from polarization-induced currents at the graphene–gold interfaces (Fig. 7c). Subsequently, we discuss the developments that have been made towards achieving PHz lightwave control, integration, processing and transduction.

### Lightwave control

Completely directional electron transport can be achieved^{7,143,145} by making the lightwaves sufficiently short and asymmetric. The lightwave-induced extreme nonlinear effects synchronize the coherent charge transport to the field maximum. Optimal current control could be achieved using a direct current pulse that consists of exactly one half cycle. However, conventional sources can only produce pulses with negative and positive parts that sum to zero^{146} in the far field. The second-best approach is to use lightwaves with two small negative dips and a large positive peak. For example, in 2022, a peak-to-dip ratio of 4:1 was achieved^{147} in an intense THz field transient.

### Integrated sources and electronics

PHz devices require the integration of lightwave sources as well as electronics to decrease the power consumption and focus HH and HSB emission on the active components. Nanostructuring can address these challenges. Integrated Fresnel zone plates can achieve diffraction-limited self-focusing of HHs^{148} (Fig. 7d). A simple array of bar-shaped metal antennas increased HH emission by a factor of 10 (ref. ^{8}) to reach UV harmonics with a 2.1-µm driver. Patterning the shape of the HH-generating material can further increase the HH emission (through a Fano-type resonance)^{149} by two orders of magnitude compared with the unpatterned material. Electron currents have also been driven in diverse conductive nanostructures such as a graphene monolayer^{150} (Fig. 7e) and a nanogap^{9} (Fig. 7f) with few-cycle pulses. In both examples, the currents were strongly dependent on the CEP of the waveforms, which is a key step towards achieving nanoscale electronics, possibly at attosecond timescales. However, further work is still needed to achieve chains of several PHz processing steps.

### Towards quantum-information processing

Extended HSB investigations demonstrated the flipping of an excitonic coherence between two pseudo-spin states with a THz wave within 5 fs — much faster than scattering — which is a step towards controlled quantum information in solids^{12}. Specifically, QDCE computations predicted that a 5-fs, right-circularly polarized NIR pulse, resonant with the \(1s\)-A exciton state, can create a localized initial state (Fig. 7g, top) at the \({\rm{K}}\) valley of a WSe_{2} monolayer, with a positive-helicity pseudospin. These calculations suggested that a 40-THz wave with a peak field strength of 23 MV cm^{–1} could move the excitonic packet to the opposite \({\rm{K}}{\rm{\mbox{'}}}\) valley with an inverted helicity, leading to the emission of left-circularly polarized light (Fig. 7g, bottom). The predicted valleytronic qubit flip was experimentally verified by observing the left-circularly polarized HSBs.

### Towards quantum transducers

The process of converting quantum information between two modalities — quantum transduction — is one of the pending challenges for quantum-information technology. For example, the quantum efficiency *η* of quantum transduction from microwave (GHz) to optical (PHz) photons is still well below (\(\eta \ll 1\)) what is needed for practical applications^{151}. At the same time, it is likely that as *η* approaches unity, strong-to-ultrastrong coupling challenges will be introduced, similar to those that lightwave electronics already handles. Thus, harmonics in lightwave electronics offer excellent testbeds for exploring which quantum materials and designs could amplify \(\eta \). Most suggested approaches for GHz-to-PHz transduction^{151,152} are based on exploiting nonlinear optical responses of matter, characterized by the \({\chi }^{(n)}\) proportionality of the response of the material to an \(n{\rm{th}}\) power of the field. To this end, graphene was demonstrated (Fig. 3e) to produce exceptionally strong^{15} third, fifth and seventh harmonics yielding \(\eta =2\times {10}^{-3}\) (680 GHz), \(\eta =2.5\times {10}^{-4}\) (370 GHz) and \(\eta =8\times {10}^{-5}\) (300 GHz fundamental source), respectively. These efficiencies encourage further work to engineer new 2D-material heterostructures^{153,154} and digital alloys^{155} to develop quantum integration, efficient HH sources and quantum transduction.

## Summary and outlook

Lightwave electronics is becoming one of the most promising approaches to characterize and control complex quantum phenomena in condensed-matter systems. These developments could open the door to new quantum materials and methodologies to discover emergent quantum-dynamic effects within them.

### Challenges in lightwave electronics

Yet, multiple challenges remain for lightwave electronics to reach its full potential. Subsequently, we elaborate some of the exciting challenges and directions that could enrich the research and impact of lightwave electronics further.

#### Floquet engineering

From a single-electron perspective, monochromatic lightwaves modulate electronic potentials periodically with frequency \({\omega }_{{\rm{wave}}}\), in addition to the potential of the regular ion lattice. Analogous to spatial lattices, temporal periodicity induces energy bands by replicating the regular band structure as Floquet copies^{156,157,158} that are shifted in energy by multiples of \(\hbar {\omega }_{{\rm{wave}}}\). Floquet engineering is realized when electrons are moved coherently and the lightwave potential (Fig. 1a) approaches the strength of the ionic potential to hybridize the Floquet–Bloch sidebands with one another^{131,156,159,160,161,162} and with the ground-state bands^{163}. Although it is not yet clear how many-body aspects and decoherence modify this intriguing scenario, Floquet engineering could create interesting quantum effects such as dynamic localization^{158}, photon-dressed topological states^{131,159,160,161,162} and time crystals^{164,165}. Besides proof-of-concept experiments in atomic systems^{164,165}, quasi-stationary Floquet–Bloch states in topological insulators have been studied^{131,160}, leading to observations of emergent quantum phenomena such as a light-induced anomalous Hall effect in graphene^{132} and magnonic space–time crystals^{166}. However, detecting the dynamics of Floquet states including their transient emergence and the transfer of electrons between sidebands^{167,168} has remained challenging. In 2023, subcycle ARPES measurements in the strong-field regime revealed^{169} that Floquet–Bloch band structures emerge after a single oscillation cycle of the driving field. This observation sparks hope for optical band-structure engineering and that the interplay of interband and intraband excitation dynamics during HH generation might be revealed.

#### Quantum materials

Quantum-material-enhanced^{15,116,169,170,171} and cavity-enhanced (Fig. 7d) HH generation suggests that there could be a path towards achieving perfect quantum efficiency (*η* → 1) to optimize HH sources, quantum transduction and phonon–photon–electronic interfaces. Approaching *η* → 1 in nonlinear interactions implies that very few photons can be converted to different photons across a broad energy range, or a photon can switch another photon. Thus, besides providing options for quantum transduction, lightwave electronics could also support the development of all-optical quantum logic^{144,172}. Integrating quantum transducers and photon-logic units onto semiconductor circuits would offer a natural quantum–classical interface and enable the distribution of tasks between existing quantum and semiconductor technologies. We expect that multiple breakthroughs are needed to achieve \(\eta \to 1\), which will involve resolving how strong coupling affects lightwave excitations.

Besides pure material innovations, ideas have also emerged^{173,174,175,176,177} to transiently synthesize phase transitions only when lightwaves sculpt the flow of electrons and correlations. These new frontiers introduce a multitude of intriguing challenges such as how dynamic phases can be synthesized, maintained and switched in any material. In addition to the electronic aspects, lightwaves have been demonstrated to control magnetic materials^{178}, ultrafast coherent spin^{179} and orbital momentum transfer^{180}, opening the door to spintronic applications in quantum information^{181}. In this context, systematically combining the latest generation of experiments that can disentangle different degrees of freedom (such as subcycle and attosecond spectroscopy, lightwave ARPES and lightwave STM) with theory will be equally challenging as the highly correlated states and the full dynamics creating them must be described simultaneously.

Future lightwave electronics research could also explore the phenomena that emerge close to material breakdown. In this regime, the lattice of the Bloch electrons will change so abruptly that the dynamics of the ionic lattice and core electrons will also affect the key processes. As the mechanical motion of the lattice approaches breakdown, it would also create extreme phonon excitations that could be leveraged to optimize and extend quantum interfaces. For example, extreme phonon excitations could be seamlessly coupled to extreme optical and electronic excitations to broaden the scope of next-generation quantum investigations and devices. We envisage that satisfactory theoretical descriptions must combine several first-principles approaches such as time-dependent DFT and QDCE.

#### Quantum spectroscopy

Additionally, steps to exploit the quantum fluctuations and correlations of quantized light have been proposed^{99} to expand the scope of lightwave electronics to include quantum spectroscopy. Quantum spectroscopy^{64} accesses and specifically controls the nonlinear responses of matter, and lightwave electronics generates such effects extending to ultrastrong coupling. If delicate quantum states can be incorporated into extremely strong sources using approaches such as quantum-light shaping^{182}, it could be possible to control excited states using the quantum states of light alone. Conversely, quantum nonlinearities in harmonic emission could provide a route to produce efficient quantum-light sources that would be a valuable resource for quantum information and lightwave electronics. It would also be interesting to see electron videography evolve into correlation videography. This extension could benefit from the combination of excitation and probing with multiple nanotips and quantum spectroscopy, with the goal of detecting and controlling individual quasiparticle clusters at their natural length scale and timescale.

Although developing a full quantum-optical description of lightwave electronics is challenging, QDCE has already described quantum-optical aspects through semiconductor luminescence equations^{183,184}, resonance fluorescence correlations^{185,186,187,188} and quantum spectroscopy^{26,64,189}. Such quantum-optical aspects of lightwave excitations remain underexplored and we expect that QDCE-type approaches could offer insights into the use of quantum materials as hosts for diverse quantum-information processing applications.

#### Free electrons

The dynamics of photoexcited electrons offers an intriguing extension of lightwave electronics (Fig. 6). Several experiments have photoexcited free electrons outside a solid to manipulate, control and characterize the coherence of a single electron by demonstrating attosecond-scale electron pulses^{190,191,192}. The next challenge could be to leverage multi-electron correlations inside and outside condensed-matter systems and especially transfer electronic correlations in solids or molecules to multiple free electrons. To achieve this, new concepts are needed to reduce the Coulomb repulsion between nearby free electrons; extremely fast and strong lightwaves could possibly accomplish this task.

#### Multi-electron systems

Although many of the open challenges described earlier can be pursued independently, all of them must be holistically addressed to drive multi-electron quantum information with lightwave electronics (Fig. 7h). Such quantum-information operations would extend over the entire Fock space of multi-electron states being excited, switched and detected on timescales that are much faster than their decoherence. The robustness of traditional semiconductor devices stems largely from the very low error rate when millions of electrons are flipped in a transistor or a semiconductor laser. By contrast, a single-state flip of a qubit will inevitably introduce fragility into quantum-information systems. Conversely, multi-electron states could encode quantum information at multiple levels of correlations, possibly with topological protection, to become much more robust than qubits. Multi-electron dynamics also drives emergent effects in condensed matter, and directly detecting and controlling these dynamics at multi-electron levels could lead to new types of quantum technologies.

Such fundamental and technological aspirations must simultaneously overcome the challenges of quantum input, quantum processing and quantum output. Quantum input could be realized by using next-generation lightwave sources to excite the desired multi-electron correlations (Fig. 7h) while also taking advantage of quantum fluctuations, improved quantum transduction and new routes to achieve quantum spectroscopy of extreme nonlinear excitations. As multi-electron states are extremely transient, attosecond precision is needed to orchestrate suitable entanglement-processing sequences. Major investment will be required to discover how a given correlation can successfully feed another correlation to produce functional entanglement-processing chains (Fock-space circuit). Once processed, the quantum information should be extracted to visualize the output multi-electron correlations.

### The potential of lightwave electronics

Lightwave electronics has already taken major steps towards achieving multi-electron quantum-information processing. Demonstrations for quantum input include attosecond sources (Fig. 4), excitation of diverse dynamic processes (Fig. 2) and improvements in quantum transduction and topological protection (Fig. 3), among others. Quantum processing has been used to achieve quasiparticle attoclocking (Fig. 5a–f), to control quasiparticle dynamics (Fig. 5g) and to build elements for PHz quantum electronics (Fig. 7). The outputs already include producing coherent emission (Figs. 2 and 3), constructing band-structure details (Fig. 3a–d), detecting the properties of quantum materials (Fig. 3e–h) and filming electronic processes as they unfold in diverse nanostructures (Fig. 6). Altogether, these concepts can control, switch or drive quantum properties (Figs. 5–7) much faster than scattering occurs, which is the first step towards the processing of quantum information (Fig. 7g) even in room-temperature solids. With these encouraging developments, lightwave electronics is poised to lead to advances in multi-electron phenomena, material science and quantum-information technology.

We expect that these developments could increase the number of operations that can be achieved within the coherence time from few to thousands. This breakthrough alone will enable an electronics–optics interface, in which semiconductor-compatible technology can be a million times faster than existing electronics with ever increasing accuracy in detecting and controlling multi-electron effects in condensed-matter systems. We also anticipate that lightwave electronics will eventually combine, control, clock and characterize concepts both to explore emergent quantum phenomena in condensed-matter systems and to build quantum-information technology that is integrable, scalable and compatible with traditional semiconductor chips.

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## Acknowledgements

M.B. and M.K. received support from ARO through Award W911NF1810299, W.M. Keck Foundation and College of Engineering Blue Sky Research Program. M.M. and R.H. have been supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through Project ID 422 314695032-SFB 1277 (Subproject A05) as well as Research Grants HU1598/7 and HU1598/8. The authors acknowledge F. Langer and M. Knorr for discussions on early versions of this Review and J. Freudenstein for his help in generating some of the 3D graphics shown in the figures.

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Borsch, M., Meierhofer, M., Huber, R. *et al.* Lightwave electronics in condensed matter.
*Nat Rev Mater* **8**, 668–687 (2023). https://doi.org/10.1038/s41578-023-00592-8

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DOI: https://doi.org/10.1038/s41578-023-00592-8