Lightwave electronics in condensed matter

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 crystals¹⁰, atomically thin 2D dichalcogenides^11,12,13,14, 2D graphene^15,16, topological insulators^{17,18,19,20,21} and Dirac semimetals²² 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) telecommunication²⁵ 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 resolution¹⁴, 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.

Fig. 1: Scope of lightwave electronics.

a, Different fields of light–matter interaction studies, classified on the basis of the range of field strengths and interaction speeds. Electronics (from computers to 5G technology) operate with weak fields and low (GHz) frequencies; ultrafast optics explores light–matter processes on the femtosecond scale, typically close to resonances with comparatively low field strengths; lightwave electronics (grey box) transiently excites matter on femtosecond (in solids, bottom) to attosecond (in atoms, top) timescales with field strengths up to 100 MV cm^–1. The light emission of atoms can be understood through a three-step model³⁴ (1, ionization; 2, acceleration; 3, recollision), whereas in solids multiple electrons (blue spheres) move across the ionic lattice (red–grey clusters), which has atomic spacings of roughly 5 Å and potential tilts up to 1 eV Å^–1. Laser fusion uses nanosecond pulses and extreme field intensities. Relativistic optics applies femtosecond pulses with peak field strengths exceeding TV cm^–1 to convert light into elementary particles. b, The full scope of lightwave excitations in solids described by time dynamics of Bloch electrons (blue spheres) and holes (red spheres) in terms of momentum and energy bands (conduction band c and valence bands v₁ and v₂). Light pulses (either of the red waves) can be tuned so that they are resonant with an interband transition to generate electrons and holes as a single band pair. Alternatively, non-resonant lightwaves (both red waves) can simultaneously excite multiple band pairs. The lightwaves also create an oscillating force (shaded grey area in the wave vector–time plane) that moves the electrons and holes within the bands (solid and dashed lines). Lightwave excitations can be markedly altered by multi-electron interactions (Feynman diagram with a $V$ vertex) and Bloch oscillations, in which excitations Bragg reflect (bright spots) to the symmetric point at the opposite side of the band if translated far enough.

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.

Fig. 2: Harmonic generation in solids.

a, Lightwave-induced interband transitions illustrated by a simulated two-level system (2LS, right) with a ground state (v) and an excited state (c) with an energy separation $\hbar {\omega }_{{\rm{gap}}}$ (${\omega }_{{\rm{gap}}}$ is the angular frequency of the transition). Right, a flat-top excitation (field $E(t)={E}_{{\rm{p}}{\rm{e}}{\rm{a}}{\rm{k}}}\exp [-10\,({{\omega }_{{\rm{g}}{\rm{a}}{\rm{p}}}t/16\pi )}^{16}\,]\sin ({\omega }_{{\rm{w}}{\rm{a}}{\rm{v}}{\rm{e}}}t)$, peak field strength ${E}_{{\rm{p}}{\rm{e}}{\rm{a}}{\rm{k}}}$ and angular frequency ${\omega }_{{\rm{wave}}}$) is used for both the resonant (${{\omega }_{{\rm{wave}}}=\omega }_{{\rm{gap}}}$, blue arrow) and non-resonant (${{\omega }_{{\rm{wave}}}=\omega }_{{\rm{gap}}}/6$, red arrow) excitation. The corresponding spectrum is indicated next to each arrow. Top, excitation dependence of the final occupation of the c-state after resonant (blue area) and non-resonant (red line) excitation. The scaled excitation strength is defined as $\hbar {\Omega }_{{\rm{peak}}}/{E}_{{\rm{gap}}}$ with the peak Rabi frequency ${\Omega }_{{\rm{peak}}}$ and the energy gap ${E}_{{\rm{gap}}}$. The black line indicates the maximum excited occupation of the c-state. The vertical dotted lines indicate the excitation strengths that are expected to yield inversion, based on the rotating-wave approximation. Bottom, the corresponding 2LS emission spectrum $I(\omega )$ is shown with the blue and red shading for the resonant and non-resonant excitations, respectively. The horizontal dotted lines indicate the expected first, third and fifth harmonic. b, Wave vector–energy diagram for electrons (blue circles) and holes (red circles) under high-harmonic generation (HHG, inset). A strong field induces 2LS-type interband coherences across the gap (grey and black arrows), intraband processes (blue and red arrows) and Bloch oscillations (BO, dotted lines), which together induce HHG. c, Position–time diagram for electron–hole (e–h) pairs under harmonic side band (HSB) generation. Three-step process includes (1) a resonant interband excitation (grey arrow) of coherent e–h pairs, followed by a non-resonant lightwave that (2) displaces and (3) recollides them to produce the HSBs (inset). d, Solid-state HHG in polycrystalline ZnSe driven by mid-infrared pulses. e, HHG in ZnO, the vertical dotted line indicates the band edge of the crystal at which a residual fluorescence signal can be seen. f, Carrier-envelope phase-controlled HHG in GaSe. The red line indicates experimental results and the blue line shows the results of semiconductor Bloch equation computations; the computations not only explained the experiments (shaded area) but also confirmed substantial contributions from dynamic Bloch oscillations (inset). Inset, the calculated dynamics of the distribution of the electrons in the conduction band (n_e); the white line shows the centre of the electron distribution. g, Carrier-envelope phase-controlled HHG in SiO₂. Extreme ultraviolet spectra at different field strengths with the dashed parts of the coloured curves indicating the noise floor. The vertical dashed lines indicate the odd harmonics. h, Up to 18 orders of 1s-exciton HSBs detected in the intensity spectrum (blue circles) of a GaAs quantum well excited by a near-infrared (NIR) laser with frequency ${f}_{{\rm{NIR}}}$ using an intense continuous-wave THz field with a frequency of ${f}_{{\rm{THz}}}=0.58$ THz and ${E}_{{\rm{peak}}}=11.5\,{{\rm{kV\; cm}}}^{-1}$. i, An ultrafast quasiparticle collider; a 10-fs-long resonant NIR pulse (${f}_{{\rm{NIR}}}=392\,{\rm{THz}}$) generated coherent 1s excitons in bulk WSe₂ and a 100-fs THz pulse ($f=23\,{\rm{THz}}$ and ${E}_{{\rm{peak}}}=17\,{\rm{MV\; c}}{{\rm{m}}}^{-1}$) accelerated them to generate HSBs. Measured (top) and computed (bottom) spectrally integrated HSB intensity (red line) as a function of the NIR–THz delay (${t}_{{\rm{ex}}}$) compared with the normalized THz field (E_THz), vertically offset for clarity (blue line). The delay between the global peaks of the I_HSB signals and E_THz (δ_global) is marked and the magnification in the inset demonstrates the subcycle delay between the HSB and THz peaks (δ_SC). Part d adapted with permission from ref. ², American Physical Society. Part e adapted from ref. ³, Springer Nature Limited. Part f adapted from ref. ⁵, Springer Nature Limited. Part g adapted from ref. ⁴⁶, Springer Nature Limited. Part h adapted from ref. ⁴, Springer Nature Limited. Part i adapted from ref. ⁵⁴, Springer Nature Limited.

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 another⁴⁴, 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 oscillations⁴⁵. 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) HHs⁴⁶ (Fig. 2g). The effect of dynamic Bloch oscillations (Fig. 2f) on the shape of HH spectra was eventually demonstrated⁵ 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 analytically⁴⁷ 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), Cd₃As₂ (ref. ²²) and fused silica⁴⁸ 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 control⁴⁹, 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.

Box 1 Maxwell–semiconductor Bloch equations

Elementary lightwave excitations generate microscopic polarization ${P}_{{\bf{k}}}^{\lambda ,\nu }\equiv \langle {a}_{\lambda ,{\bf{k}}}^{\dagger }\,{a}_{\nu ,{\bf{k}}}\rangle$ between electronic states with the same momentum, $\hbar {\bf{k}}$, where ${\bf{k}}$ is the wave vector, but different band indices $\lambda$ and $\nu$. The expectation value, $\langle \cdots \rangle$, contains fermionic creation (${a}_{\lambda ,{\bf{k}}}^{\dagger }$) and annihilation (${a}_{\nu ,{\bf{k}}}$) operators that move electrons between bands, as illustrated in the figure by the arrow between an electronic state (e, filled circle) and a vacancy, called a hole (h, open circle). Once generated, ${P}_{{\bf{k}}}^{\lambda ,\nu }$ induces the microscopic occupation ${P}_{{\bf{k}}}^{\lambda ,\lambda }$ of the state $(\lambda ,{\bf{k}})$. The lowest hierarchy level of quantum-dynamic cluster expansion (Box 2) yields the Maxwell–semiconductor Bloch equations^61,194:

$$\left(\frac{{\partial }^{2}}{\partial {z}^{2}}-\frac{{n}^{2}(z)}{{c}^{2}}\frac{{\partial }^{2}}{\partial {t}^{2}}\right){\bf{E}}({\rm{z}},t)=-{\mu }_{0}\delta (z)\frac{{\partial }^{2}}{\partial {t}^{2}}{\bf{P}},$$

(1)

$${\rm{i}}\hbar \frac{\partial }{\partial t}{P}_{{\bf{k}}}^{\lambda ,\nu }=[{\widetilde{E}}_{{\bf{k}}}^{\nu }-{\widetilde{E}}_{{\bf{k}}}^{\lambda }+|e|{\bf{E}}(0,t)\cdot ({\boldsymbol{\mathscr{A}}}_{{\bf{k}}}^{\nu }-{\boldsymbol{\mathscr{A}}}_{{\bf{k}}}^{\lambda }+{\rm{i}}{\nabla }_{{\bf{k}}})]{P}_{{\bf{k}}}^{\lambda ,\nu }$$

$$\,+\sum _{\alpha }\hbar ({\Omega }_{{\bf{k}}}^{\alpha ,\lambda }{P}_{{\bf{k}}}^{\alpha ,\nu }-{\Omega }_{{\bf{k}}}^{\nu ,\alpha }{P}_{{\bf{k}}}^{\lambda ,\alpha })-{\rm{i}}{\Gamma }_{{\bf{k}}}^{\lambda ,\nu },$$

(2)

with the refractive index n, electric field E, macroscopic polarization P, electron charge $e$, speed of light $c$ and vacuum permittivity ${\mu }_{0}$. The ${\Gamma }_{{\bf{k}}}^{\lambda ,\nu }$ term accounts for two-body correlations (Box 2) and introduces dephasing, screening of the Coulomb-matrix element ${V}_{{\bf{k}},{{\bf{k}}}^{{\prime} }}^{\lambda ,\nu }$, new quasiparticle resonances and quantum-optical effects. $\hbar {\Omega }_{{\bf{k}}}^{\lambda ,\nu }$ is the renormalized Rabi frequency. The ${\nabla }_{{\bf{k}}}$ gradient translates ${P}_{{\bf{k}}}^{\lambda ,\nu }$ within the bands, whereas the Berry connection ${\boldsymbol{\mathscr{A}}}_{{\bf{k}}}^{\lambda }$ takes into account the geometric effects.

For simplicity, we focus on a case in which a planar, classical lightwave propagates perpendicular to an atomically thin 2D semiconductor positioned at $z=0$, described by the Dirac delta function $\delta (z)$, surrounded by passive material with refractive index $n(z)$. The electric field ${\bf{E}}(0,t)$ then excites ${P}_{{\bf{k}}}^{\lambda ,\nu }$ through the renormalized Rabi frequency $\hbar {\Omega }_{{\bf{k}}}^{\lambda ,\nu }\equiv {{\bf{d}}}_{{\bf{k}}}^{\lambda ,\nu }\cdot {\bf{E}}(0,t)+{\sum }_{{{\bf{k}}}^{{\prime} }}\,{V}_{{\bf{k}},{{\bf{k}}}^{{\prime} }}^{\lambda ,\nu }{P}_{{{\bf{k}}}^{{\prime} }}^{\nu ,\lambda }$ containing the dipole-matrix element (${{\bf{d}}}_{{\bf{k}}}^{\lambda ,\nu }$, with ${{\bf{d}}}_{{\bf{k}}}^{\lambda ,\lambda }=0$) and ${V}_{{\bf{k}},{{\bf{k}}}^{{\prime} }}^{\lambda ,\nu }$ that also renormalizes the band energies ${\mathop{E}\limits^{ \sim }}_{{\bf{k}}}^{\lambda }\equiv {E}_{{\bf{k}}}^{\lambda }-{\sum }_{{{\bf{k}}}^{{\prime} }}{V}_{{\bf{k}},{{\bf{k}}}^{{\prime} }}^{\lambda ,\lambda }{P}_{{{\bf{k}}}^{{\prime} }}^{\lambda ,\lambda }$; without ${V}_{{\bf{k}},{{\bf{k}}}^{{\prime} }}^{\lambda ,\nu }$ and ${\Gamma }_{{\bf{k}}}^{\lambda ,\nu }$, equation (2) reduces to independent optical Bloch equations for each k. The macroscopic polarization (per area $S$)

$${\bf{P}}=\frac{1}{S}\sum _{\lambda ,\nu ,{\bf{k}}}{{\bf{d}}}_{{\bf{k}}}^{\lambda ,\nu }{P}_{{\bf{k}}}^{\nu ,\lambda }+\,\frac{|e|}{S}\sum _{\lambda ,{\bf{k}}}\langle {\boldsymbol{\mathscr{A}}}_{{\bf{k}}}^{\lambda }{\,a}_{\lambda ,{\bf{k}}}^{\dagger }\,{a}_{\lambda ,{\bf{k}}}-{\rm{i}}{a}_{\lambda ,{\bf{k}}}^{\dagger }\,({{{\rm{\nabla }}}_{{\bf{k}}}a}_{\lambda ,{\bf{k}}})\rangle $$

(3)

couples the semiconductor Bloch equation excitations self-consistently with equation (1). If currents ${\bf{J}}\equiv -\frac{\partial {\bf{P}}}{\partial t}$ are used instead, ${\boldsymbol{\mathscr{A}}}_{{\bf{k}}}^{\lambda }$ introduces Berry curvature^83,195.

The linearized⁷³ ${\Gamma }_{{\bf{k}}}^{\lambda ,\nu }={\sum }_{{{\bf{k}}}^{{\prime} }}{\gamma }_{{\bf{k}},{{\bf{k}}}^{{\prime} }}^{\lambda ,\nu }{P}_{{{\bf{k}}}^{{\prime} }}^{\lambda ,\nu }$ contains a scattering matrix ${\gamma }_{{\bf{k}},{{\bf{k}}}^{{\prime} }}^{\lambda ,\nu }$, whereas a constant-dephasing approximation with dephasing constant γ replaces it with ${\Gamma }_{{\bf{k}}}^{\lambda ,\nu }\to {\gamma P}_{{\bf{k}}}^{\lambda ,\nu }$, which violates the strict diffusive condition, $\sum _{{\bf{k}}}{\Gamma }_{{\bf{k}}}^{\lambda ,\nu }=0$, for Coulombic scattering⁷⁴ (because $\sum _{{\bf{k}}}{\Gamma }_{{\bf{k}}}^{\lambda ,\nu }\to \gamma \sum _{{\bf{k}}}{P}_{{\bf{k}}}^{\lambda ,\nu }\ne 0$). The correct ${\Gamma }_{{\bf{k}}}^{\lambda ,\nu }$ produces strongly exciton-state-dependent dephasing; therefore, a simple $\gamma$ could not explain recent lightwave experiments^14,76.

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 10¹²W 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 achieved⁵ 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 wells¹. Subsequently, HSB generation was also promoted to the non-perturbative regime⁴ (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 explored⁵⁴ in bulk WSe₂ (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 atomic⁵⁵ and solid-state systems⁵⁶; 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 light⁵⁷ and produce filamentation⁵⁸ or solitons⁵⁹. Conversely, light propagation has been predicted⁶⁰ 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 clusters⁶¹, quasiparticles, such as excitons (atom-like pairs of electrons and holes), exciton molecules⁶² or dropletons²⁶ (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 dichalgogenides⁶⁵ 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 inspired⁷¹ 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 creation⁶¹, dephasing^73,74,75, quantum memory⁷⁶ and interaction screening⁷⁷ 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 model⁷⁹ 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 extensions⁸¹ 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.

Box 2 The hierarchical dynamics of crystal–electron correlations

Many-body problems can be formulated exactly^196,197 by using any complete set of single-body wave functions to span the field operators. For lightwave electronics in solids, Bloch states are a prudent choice to quantify the relevant band energies and matrix elements for interactions. Once those are known, lightwave electronics investigations reduce into a generic many-body problem^61,194,198 described by electron–photon–phonon dynamics^{72,74,77,199,200,201,202,203}:

$${\rm{i}}\hbar \frac{{\rm{\partial }}}{{\rm{\partial }}t}\Delta \langle C\rangle ={F}_{{\rm{c}}{\rm{l}}{\rm{o}}{\rm{s}}{\rm{e}}{\rm{d}}}\,[\langle 1\rangle ,\Delta \langle 2\rangle ,\ldots ,\Delta \langle C\rangle ]+{\rm{H}}{\rm{i}}[\Delta \langle C+1\rangle ],$$

(1)

in which $\Delta \langle C\rangle$ describes correlations between $C$ different particles (or pairs of field operators)^{61,204,205,206}. Physically, $\Delta \langle C\rangle$ determines how particles (entities) cluster together to form quasiparticles or correlated transitions. The functional ${F}_{{\rm{c}}{\rm{l}}{\rm{o}}{\rm{s}}{\rm{e}}{\rm{d}}}$ contains products and sums among $\langle 1\rangle ,\Delta \langle 2\rangle ,\ldots ,\Delta \langle C\rangle$ correlations, yielding the Maxwell–semiconductor Bloch equations^74,194 (Box 1), among others. The functional ${\rm{Hi}}[\Delta \langle C+1\rangle ]$ induces coupling to all higher-order clusters (known as the hierarchy problem¹⁹⁶) and also creates emergent phenomena such as quasiparticles^26,27 and entanglement^207,208 or decoherence^{73,74,75,209,210,211} for the lightwave-controlled transitions. The predictive accuracy of lightwave electronic theory depends on how well the extreme nonlinearities of ${F}_{{\rm{closed}}}$ can be described alongside the hierarchy problem.

In the case of two-particle interactions (such as Coulomb, light–matter or electron–phonon interactions), $\Delta \langle C+1\rangle$ can appear only if $\Delta \langle C\rangle$ correlations already exist owing to the generic source structure (see the figure). In the figure, each red dot corresponds to a particle within a correlation (light-red ovals) and scattering among low-rank correlations (products of ovals within grey rectangles) generates $\Delta \langle C\rangle$, whereas the lonely correlations (black rectangles) induce the hierarchy problem. This structure introduces a strictly sequential build-up of correlations^61,64,74,202 and quantum-dynamic cluster expansion leverages this fundamental property. Classical lightwave excitations always start from single-body states ($C=1$) and extreme-excitation dynamics occur on ultrafast timescales — orders of magnitude faster than high-order cluster formation. Therefore, quantum-dynamic cluster expansion reduces the hierarchy problem to a manageable size by solving the non-perturbative dynamics exactly with a minimal number of existing, low-rank clusters.

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 interaction⁸⁴. 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 HH⁸⁵ and HSB⁵⁶ emission; interpreted HH generation at different excitation regimes⁸⁶; 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 symmetry⁸⁹ and discussed the generation of HHs from topological states⁸⁵.

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 HSB⁹¹ 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 emission¹¹ 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 emission⁹⁹ 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 DFT¹⁰⁰ could help such investigations. Time-dependent DFT has already been used to predict how electronic band-structure properties are connected¹⁰¹ to the laser polarization to enable optimal HH generation in bulk crystals; how a free-standing hBN monolayer could generate HHs¹⁰² 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 magnitude¹⁰³. 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.

Fig. 3: Lightwave characterization of quantum materials.

a, Lightwave access to energy–wave vector (E, k) band structure (black lines). Optimally, a light pulse (resonant or non-resonant) generates (dotted line) electrons (blue circles) and holes (red circles) at a high-symmetry point of the bands. A lightwave moves the electrons and holes to a new k location, which induces (E, k)-dependent high-harmonic (HH) or harmonic sideband (HSB) emission (recombining arrows) as a basis for reconstructing the bands. b, HH spectra measured in ZnO as a function of delay between the HH-generating light and co-propagating second-harmonic field. The blue line and dots mark the experimentally measured and model-based optimum delay of even-order harmonics that are used for band-structure reconstruction. c, Gradient of different electronic bands (${{\rm{\nabla }}}_{{\bf{k}}}{\varepsilon }_{{\rm{g}}}$) (coloured lines, top) and oscillating spectrum of MgO (bottom) at 45° orientation. The spectrum to the right of the dashed line is enhanced by a factor of 50. Spectral intensities are enhanced at van Hove singularities (marked by shaded vertical lines). d, Left, measured (red circles with experimental error bars) and quantum-dynamic cluster expansion (QDCE) computed (shaded) fifth-order HSB intensity (I_HSB) in monolayer WSe₂. Middle, crystal-momentum comb tomography connects I_HSB to the electron–hole energy (black line). The blue line marks k (in units scaled to the distance between high-symmetry points K and K′) of the computed outermost line of the ${\bf{k}}$-comb (colour map) as a function of the sideband energy. The black arrow connects  the rapid increase in I_HSB with the crossing of the outermost line with the electron–hole energy. Right, ${\bf{k}}$-comb tomography of the QDCE-computed (colour map) and measured (black crosses) sixth-order HSB intensity. e, The measured (circles with experimental error bars) dependence of the third, fifth and seventh HHs on the strength of the fundamental electric field (E_f) in a graphene monolayer. The solid lines are calculations of a thermodynamic model and the dashed lines are power 3, 5 and 7 fits to the measured data, which are used to calculate the corresponding effective nonlinear susceptibilities of graphene ${\chi }_{{\rm{ef}}{\rm{f}}\,}^{(n)}$. f, HH spectra of an MoS₂ monolayer as a function of the orientation of the linearly polarized driving field relative to the crystal structure (measured perpendicular to the driving field). g, HH generation in graphene for three pump fields with different ellipticities $\varepsilon$. The spectra are vertically offset for clarity. The seventh and ninth HHs are enhanced for the elliptically polarized pump ($\varepsilon =0.32$) compared with linear polarization ($\varepsilon =0$). The spectra to the right of the dashed line are scaled up by a factor of 10. h, HH intensity spectra I_HH generated from the topologically protected surface state of Bi₂Te₃ as a function of the carrier-envelope phase (CEP) of the driving field. The black dashed lines indicate the slope at which I_HH peaks shift with CEP. Part b adapted with permission from ref. ⁸⁸, American Physical Society. Part c reprinted from ref. ¹¹², Springer Nature Limited. Part d reprinted with permission from ref. ¹³, AAAS. Part e reprinted from ref. ¹⁵, Springer Nature Limited. Part f adapted from ref. ¹¹, Springer Nature Limited. Part g reprinted with permission from ref. ¹¹⁶, AAAS. Part h adapted from ref. ²⁰, Springer Nature Limited.

High-harmonics-based approach

HH generation was used⁸⁸ 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 spectrum¹⁰⁸, including below-gap HHs to avoid absorption-related limitations¹⁰⁹ and intraband harmonics above the gap to reconstruct conduction bands⁴⁶. The Berry-curvature¹¹⁰ information and the interband dipole matrix elements¹¹¹ 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) produce¹¹² 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 measure¹¹³ 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 revealed¹³ 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 used¹³ to assign the band structure of 2D WSe₂ (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 nth-order proportionality to the driving field. In a graphene monolayer, exceptionally large χ⁽ⁿ⁾ were measured (Fig. 3e) in HHs generated by a THz field¹⁵. Purely non-perturbative HHs were measured¹¹ in a monolayer MoS₂ 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 polarization¹¹⁶ 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 Bi₂Te₃, HH generation from the topologically protected surface state was rigorously separated from bulk contributions²⁰. 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 solids¹¹⁷ 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 confirmed¹¹⁸ 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.

Fig. 4: Timing and duration of high-harmonic emission.

a, Timing the excitation (red wave) — translation (red and blue arrows) — emission (blue wave) cycle in high harmonics (HHs), induced by lightwaves. The yellow arrows indicate both the creation (left) and annihilation (right) of electrons (blue circles) and holes (red circles). The temporal features of the HH emission can be deduced from its delay and shape with respect to the crest (grey area) of the half cycle of the lightwave. b, Envelopes of HH intensity measured in GaSe (black line) relative to the driving THz waveform (E_THz, grey shaded area) with a centre frequency of 33 THz and a peak field strength of 44 MV cm^–1. Computed HH-emission envelopes from full quantum-dynamic cluster expansion (QDCE) calculations (red shaded area) and a reduced QDCE calculation that excludes coherences between valence bands (red dashed line). The vertical dashed lines indicate the alignment of E_THz and HH intensity maxima. c, Measured attosecond streaking spectrogram of a HH pulse from fused silica used to reconstruct a femtosecond pulse. d, Reconstruction of an isolated extreme ultraviolet (EUV) HH pulse (blue line). The red line shows the HH-generating femtosecond pulse and the yellow dashed line traces the envelope of the HH pulse. Part b adapted from ref. ¹¹⁷, Springer Nature Limited. Parts c and d reprinted from ref. ¹¹⁸, Springer Nature Limited.

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 controlled¹¹⁹ 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 field¹²⁰. 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 limitations¹²¹ (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 4f-core electrons of a single-crystal tungsten sample. Because the core electrons are essentially immobile, the 4f 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 4f-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 generalized⁴⁸ to monitor the conduction-band streaking signal as a function of excitation delay in SiO₂ to resolve the electronic and lattice responses at attosecond scales.

Fig. 5: Advances in attosecond clocking.

a, Extreme ultraviolet (XUV, purple waves) pulses can be used to time interband excitations (blue–red circles) from resonant light (blue wave) by recording the temporal delay between the photoexcitation of the conduction and core electrons. b, Interpolated 4f and conduction-band photoelectron spectrograms from tungsten as a function of delay between XUV pulse triggering photoionization and a strong near-infrared waveform acting as a streaking field. The red and blue colours mark high and low intensities, respectively. The dashed white lines indicate the small shift (Δτ) in the fringes. c, Map of the photoelectron yields from nickel (111) as a function of the photoelectron energy and pump–probe delay time, excited by s-polarized high-harmonic generation. Sideband (SB) 16 (bottom inset) shows a clear delay between electrons excited from the ${\Lambda }_{3}^{\alpha }$ (red) and ${\Lambda }_{3}^{\beta }$ (blue) bands, whereas sideband 18 (top inset) shows no time delay. The insets show 1D lineouts taken from the regions marked by the white dashed boxes. d, Delay difference between electrons photoexcited from W 4f and Se 4s states (Δt_W4f–Se4s) in a WSe₂ crystal. The left side of the plot shows Δt_W4f–Se4s for two different measurements (red circles and squares) as a function of time after cleaving. The horizontal bars indicate the time period during which data were recorded, and vertical bars indicate the uncertainty of the determined Δt_W4f–Se4s. The shaded bar marks the confidence interval of averaged Δt_W4f–Se4s. The right side of the plot shows averaged Δt_W4f–Se4s (with confidence interval) for different background-subtraction methods. e, Attosecond transient absorption spectroscopy of silicon. Derivative (∂A/∂E) of the transient absorption A with respect to the energy E at the L-edge (100.2 eV) as a function of the delay. The grey lines indicate the timescales of electronic (t_e) and ionic (t_p) dynamics. f, Attosecond transient absorption spectroscopy of GaAs. The square of the pump-pulse vector potential A($\tau$) (left) is compared with the differential absorption ΔAbs(τ) at a photon energy of 45 eV (right) as a function of the pump–probe delay $\tau$. The dots are measured values and the lines are a fit to guide the eye. The shaded vertical lines mark the peak positions closest to zero delay to demonstrate the offset. The error bars in the right part indicate the standard deviation of the measured signal. g, Left, schematic of the generation of electron–hole (e–h) coherences (blue–red spheres) at excitation time ${t}_{{\rm{ex}}}$, followed by a force exerted by a lightwave (shaded area) that separates the electrons (blue spheres) from the holes (red spheres). The relative e–h phase-space coordinate (displacement x, wave vector ${\bf{k}}$) evolves along the solid blue line under the influence of a Coulombic force (purple field lines), which reduces $(x,{\bf{k}})$ and induces correlations. The $(x,{\bf{k}})$ path terminates $(x=0,{\bf{k}})$ when the e–h pairs recollide, generating harmonic sidebands (HSBs) (light burst debris). Dephasing and scattering limit the lifetime of coherences (decreasing number of spheres). Without the Coulombic interactions (dashed line), the same ${t}_{{\rm{ex}}}$ yields a longer $(x,{\bf{k}})$ path that does not reach recollision within the coherence lifetime. Therefore, ${t}_{{\rm{ex}}}$ must be delayed to later times (by $\delta t$) to efficiently generate HSBs. Middle, the measured (circles) and computed (solid line) $\delta t$ as a function of the e–h pair density. Measured data are based on averages of $\delta t$ from five consecutive half cycles and the error bars mark the standard deviation. Right, measured (circles) and computed (lines) spectrally integrated sideband intensity (I_HSB) of bulk (yellow) and monolayer (blue) WSe₂ as a function of ${t}_{{\rm{ex}}}$ showing the $\delta t$ shift between the crests of the bulk and monolayer intensities. Part b reprinted from ref. ¹²¹, Springer Nature Limited. Part c adapted with permission from ref. ¹²⁴, AAAS. Part d adapted with permission from ref. ¹²⁶, AAAS. Part e adapted with permission from ref. ¹²⁹, AAAS. Part f adapted from ref. ¹²⁷, Springer Nature Limited. Part g adapted from ref. ¹⁴, Springer Nature Limited.

Similar pump–probe schemes have since been used to reveal free-electron-like, ballistic transport in tungsten covered with magnesium films¹²³, quantify photoelectron lifetimes for free-electron-like and high-lying conduction states in nickel¹²⁴ (Fig. 5c), resolve the elastic and inelastic scattering in silica nanoparticles¹²⁵ and detect delay shifts¹²⁶ owing to intra-atomic interactions in bulk WSe₂ (Fig. 5d). The injection of carriers from the valence band into the conduction band¹²⁷ in GaAs (Fig. 5f) and screening dynamics¹²⁸ 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 renormalization¹²⁹ in silicon (Fig. 5e), a femtosecond-scale dynamic Franz–Keldysh effect¹³⁰ in polycrystalline diamond, and 300-as charge-transfer rates¹⁶ 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 demonstrated¹⁴ 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 clocking¹⁴ 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 WSe₂. Because exciton binding and correlations are five times stronger in 2D WSe₂ 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 obtained¹⁷ 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 electrons¹⁷ in the topologically protected surface state of Bi₂Te₃. Topological protection supresses back scattering, which can increase¹⁷ 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.

Fig. 6: Lightwave videography in real and momentum space.

a, Schematic of subcycle angle-resolved photoemission spectroscopy (ARPES). An s-polarized THz electric field E_THz (red wave) accelerates the electrons. An ultraviolet (UV) pulse (purple wave) is used to photoemit the electrons. A hemispherical electron detector then images the kinetic energy and emission angle (θ) of the photoelectrons (blue spheres). b, Photoemission map of Dirac fermions in the band structure of the topological surface state of Bi₂Te₃ in the ground state (left) and after the first positive maximum of the THz lightwave (right), which accelerates electrons towards negative k_y. The white dashed line marks the Fermi level. The insets indicate the delay time at which the maps were obtained. ε, energy of the photoelectron; ε_F, Fermi energy; k_y, momentum of the photoelectrons parallel to the Bi₂Te₃ surface. c, Schematic of a THz-driven scanning tunnelling microscope. Inset: the nonlinear current–voltage characteristic (I–V) of the junction between nanotip and sample. d, Temporally resolved excitation and relaxation of an InAs nanodot (black line) and wetting layer (blue line) using the THz-scanning tunnelling microscopy (STM) technique measuring the rectified component of the tunnel current. Inset: an STM scan of a nanodot 500 fs after the excitation, showing electrons trapped into the nanodot. e, Energy diagram illustrating the concept of femtosecond orbital imaging of single molecules. The electric field of a THz pulse allows electrons to tunnel from the substrate to the scanning tunnelling microscope tip via the highest occupied molecular orbital (HOMO) of molecules; LUMO is the lowest unoccupied molecular orbital. Inset: schematic of the experimental setup showing a single molecule in the tunnel junction of a scanning tunnelling microscope, which is irradiated by a THz pulse (red waveform). f, Top, THz-STM image of the THz-induced currents (I_THz) of electrons tunnelling from the HOMO state of a single pentacene molecule to the scanning tunnelling microscope tip. Bottom, THz-pump–THz-probe measurement of the time dynamics of a single pentacene molecule. Coherent oscillation of the pump-induced variation, $\Delta {I}_{{\rm{THz}}}$, about the average current, ${I}_{{\rm{THz}}}$, as a function of delay time (red dots, measurement; solid line, sinusoidal fit). g, The top three slices show THz-STM snapshots of ultrafast motion of electrons photoinjected into C₆₀ multilayer structures. Bottom slice, static STM image of the studied sample. h, Temporal resolution of THz-STM techniques. Fast Fourier transform (FFT) of the THz waveforms retrieved by electro-optic sampling (EOS) and THz-STM. Despite the low-pass filtering antenna effect of the tip, an ultrabroadband incident THz pulse (grey spectrum) can couple frequency components exceeding 15 THz (blue spectrum) into the near field of a metallic scanning tunnelling microscope junction. i, A near-field waveform measured in a THz-STM setup demonstrating a temporal resolution better than 30 fs. Parts a and b adapted from ref. ¹⁷, Springer Nature Limited. Parts c and d adapted from ref. ¹³⁸, Springer Nature Limited. Parts e and f adapted from ref. ²³, Springer Nature Limited. Part g adapted with permission from ref. ¹⁴¹, Hidemi Shigekawa (CC-BY-NC-ND 4.0). Part h adapted from ref. ¹⁹³, CC BY 4.0. Part i adapted with permission from ref. ¹⁴², Hidemi Shigekawa (CC-BY-NC-ND 4.0).

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 STM¹³⁸ (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 nanodot¹³⁸ (Fig. 6d) and also to resolve snapshots of electronic states in carbon nanoribbons¹⁴⁰.

To experimentally achieve videography of single electrons, state-selective tunnelling is required. Low-temperature STM measurements of a single molecule were demonstrated²³ 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 state²⁴. 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 film¹⁴¹ (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 apart⁶ (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 graphene¹⁴⁴ 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.

Fig. 7: Petahertz electronics.

a, Schematic of a metal–dielectric nanojunction used to measure the conductivity and current in SiO₂ induced by an optical driving field (red wave). b, Charge transfer as a function of the delay between the injection and driving fields $\Delta t$ measured with the setup shown in part a, showing a 2.5-fs oscillation cycle. The circles and red line show measured (standard deviation in error bars) and smoothed data, respectively. c, Disentangling of real and virtual carriers in optical current generation. Two few-cycle laser pulses excite a gold–graphene–gold junction either in the bulk section injecting intraband currents in graphene or at a graphene–gold interface creating polarization-induced currents. Measured (circles with error bars indicating standard deviation) and modelled (line) total current as a function of the relative carrier-envelope phase $\Delta {\varphi }_{{\rm{CE}}}$ between the two pulses allows the individual current components injected from each pulse to be inferred. d, Emission pattern of the third harmonic recorded at the sample plane of a gallium-implanted Fresnel-zone plate pattern. Inset: scanning electron microscopic image of this region. e, Inset: a schematic of the experimental setup with a graphene strip on an SiC substrate illuminated by two-cycle laser pulses with a controlled carrier-envelope phase (CEP). The CEP-dependent current induced by linearly (red circles) and circularly (blue circles) polarized light indicates a transition from the weak-field to strong-field regime. The error bars show the standard deviation of the time-integrated signal. f, Inset: the experimental concept in which two single-cycle light pulses with time delay $\Delta t$ illuminate a nanocircuit that acts as an optical antenna. The background image shows a scanning electron micrograph of the gold antenna with a zoom in on the gap region. The current across the gap induced by the single-cycle light pulses is measured as a function of the electric field of the light for CEPs of 0 (red), π/2 (yellow) and π (blue). g, Lightwave valleytronics in a monolayer of WSe₂. Computed distribution of the polarization of coherent excitons (|P_k|², colour bar) in momentum space (k_x–k_y, k is the wave vector and black lines mark the first Brillouin zone) before (top) and after (bottom) lightwave-driven displacement. Coherent excitons are prepared by a circularly polarized near-infrared pulse in the ${\rm{K}}$ valley (red dashed line). A THz pulse with a peak field strength of 23 MV cm^–1 drives the excited coherences to the ${\rm{K}}{\prime}$ valley in less than 7 fs. h, A vision for future lightwave electronics based on continuing improvements in extreme excitations, control, timing and readout of multi-electron correlations, including entanglement. Entangled lightwaves could excite correlated electron–hole (e–h) clusters that are attoclocked (red pulses in the inset) and brought to interact with each other (red waveforms) multiple times during their coherence time (blue curve in the inset) to achieve quantum-information processing. The resulting lightwave quantum processing outcomes (in the multi-electron Fock space) could be detected in quantum-light emission. FWHM, full width at half maximum. Parts a and b adapted from ref. ⁶, Springer Nature Limited. Part c adapted from ref. ¹⁴⁴, Springer Nature Limited. Part d reprinted with permission from ref. ¹⁴⁸, AAAS. Part e adapted from ref. ¹⁵⁰, Springer Nature Limited. Part f adapted from ref. ⁹, Springer Nature Limited. Part g adapted from ref. ¹², Springer Nature Limited.

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 zero¹⁴⁶ 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 achieved¹⁴⁷ 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 HHs¹⁴⁸ (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 monolayer¹⁵⁰ (Fig. 7e) and a nanogap⁹ (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 solids¹². 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 WSe₂ 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 applications¹⁵¹. 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 strong¹⁵ 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 alloys¹⁵⁵ 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 bands¹⁶³. 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 localization¹⁵⁸, 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 graphene¹³² and magnonic space–time crystals¹⁶⁶. 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 revealed¹⁶⁹ 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 materials¹⁷⁸, ultrafast coherent spin¹⁷⁹ and orbital momentum transfer¹⁸⁰, opening the door to spintronic applications in quantum information¹⁸¹. 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 proposed⁹⁹ to expand the scope of lightwave electronics to include quantum spectroscopy. Quantum spectroscopy⁶⁴ 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 shaping¹⁸², 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.