Introduction

Excitation by strong light fields can be used for various significant processes in solids, such as optical manipulation of electronic structure1,2,3,4,5,6,7,8,9,10,11,12, or generation of high harmonics (HHG)13,14. These phenomena are very promising for the development of ultrafast optoelectronic devices15,16,17,18, and are intensively investigated19,20. Revealing microscopic details of laser-driven electron dynamics is essential for a deeper understanding of strong-field-induced processes in solids15,21,22,23,24. New advances to generate few- and sub-femtosecond x-ray pulses enable a real-time microscopic view into laser-driven electron dynamics24,25,26,27,28,29,30,31,32,33,34 that is essential for revealing their mechanisms.

When light interacts with a material, charges rearrange within a unit cell and microscopic electron currents are formed. The widely-used concept that optically-induced charge separation merely gives rise to a dipole moment fails on the atomic scale. Induced charge distributions have a rich structure and various symmetry features35,36. In this article, we develop a method that employs ultrashort nonresonant x-ray pulses to probe in real time charge and electron current distributions within a unit cell of a crystal during the interaction with an optical field. The method benefits from a short wavelength of hard x rays, which provides sufficient spatial resolution to access the microscopic optical response on the atomic scale.

We build on the idea of x-ray-optical wave mixing (XOWM), in which an x-ray and an optical field simultaneously interact with matter leading to sum and difference generation that encodes microscopic optical response37,38. XOWM has been realized in several experiments at free-electron laser facilities39,40,41. Here, we go beyond this concept and suggest a subcycle-resolved measurement, in which the duration of an x-ray pulse is shorter than a period of an optical cycle. X-ray diffraction using ultrashort x-ray pulses provides much more information on laser-driven electron dynamics than can be provided by a time-unresolved XOWM.

Results and Discussion

Microscopic nonlinear optical response and x-ray-optical wave mixing

We consider an α-quartz crystal driven by a periodic electromagnetic field with a photon energy of 1.2 eV, an intensity of 1012 W cm−2 and polarized along the (1,1,0) direction. We selected α-quartz for illustration of our results, since this material has recently attracted interest in the context of strong-field-light-induced phenomena42,43,44,45,46,47.

According to the second-order susceptibility tensor for the crystal class 32, the second-order polarization of α-quartz is aligned along (0,1,0) for a driving electric field aligned along (1,1,0)48. This nontrivial alignment was the motivation to select this particular electric-field polarization. HHG in α-quartz has been demonstrated for similar driving laser parameters as those considered here45,49.

Figure 1a–l shows the first-, second- and third-order microscopic optical response of α-quartz at different phases during an optical cycle calculated with the Floquet-Bloch formalism combined with the density functional theory (DFT)36. Namely, oscillations of the electron density and the electron-current density with frequencies ω, 2ω and 3ω in the presence of the optical electric field evolving as \(\sin ( \omega t)\) are shown. Blue and yellow isosurfaces indicate positively- and negatively-charged regions relative to the field-free electron density of α-quartz. Except for the first-order response, optically-induced charge rearrangements and microscopic electron currents have a nontrivial structure. A charge distribution due to nonlinear response cannot be characterized in terms of microscopic dipoles, and the microscopic currents form complex vortices. Our method provides a tool to reconstruct these complex structures.

Fig. 1: Microscopic optical response and the corresponding signal.
figure 1

The first-, second, and third-order microscopic optical response of an α-quartz crystal at different phases of the driving electromagnetic field polarized along the (1, 1, 0) direction and the corresponding x-ray-optical wave mixing signal (XOWM), in which side peaks that appear due to an optical response of the related order are highlighted. A primitive unit cell is shown and outlined by gray lines. Silicon and Oxygen atoms are shown by blue and red spheres, respectively. a–d Oscillations of the electron density (light blue and yellow surfaces) and the electron-current density (brown arrows) with frequency ω at phases ωt equal to (a) 0; (b) π/2; (c) π and (d) 3π/2. e–h Oscillations of the electron density and the electron-current density with frequency 2ω at phases 2ωt equal to (e) 0; (f) π/2; (g) π and (h) 3π/2. (i)–(l) Oscillations of the electron density and the electron-current density with frequency 3ω at phases 3ωt equal to (i) 0; (j) π/2, (k) π and (l) 3π/2. Electron-current density above a threshold of 3 a.u., 0.003 a.u. and 0.002 a.u. for the first, second and third orders, respectively, is shown and lengths of the vectors are scaled logarithmically. XOWM signal at G = (1, 2, 0) with highlighted side peaks that correspond to a certain order of optical response are shown in the same rows as the oscillations of this order, namely, (m) the side peaks shifted by  ± ω, (n) the side peaks shifted by  ± 2ω and (o) the side peaks shifted by  ± 3ω relative to ωin are highlighted.

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In the following, we consider that an α-quartz crystal is probed by an ultrashort x-ray pulse during the time it is driven by the optical field. In ref. 50, we developed a general formalism to describe the interaction of an electronic system simultaneously with an optical field and x-ray pulse, and applied it to a description of a time-unresolved experiment as an example. Here, we use this framework to develop a method to reveal insights into laser-driven electron dynamics by means of ultrafast x-ray scattering from a laser-driven material.

The total Hamiltonian of a crystal, an optical field and a probe x-ray pulse is given by

$$\hat{H}={\hat{H}}_{{{{\rm{el}}}}}+{\hat{H}}_{{{{\rm{into}}}}}+{\hat{H}}_{{{{\rm{intx}}}}}+{\hat{H}}_{{{{\rm{o}}}}}+{\hat{H}}_{{{{\rm{x}}}}},$$
(1)

where \({\hat{H}}_{{{{\rm{el}}}}}\) is the Hamiltonian of the crystal, \({\hat{H}}_{{{{\rm{into}}}}}\) and \({\hat{H}}_{{{{\rm{intx}}}}}\) are the Hamiltonians that describe the interaction of the crystal with the optical driving field and with the x-ray pulse, respectively. \({\hat{H}}_{{{{\rm{o}}}}}\) and \({\hat{H}}_{{{{\rm{x}}}}}\) are Hamiltonians of the optical and the x-ray fields, respectively. The interaction Hamiltonian \({\hat{H}}_{{{{\rm{into}}}}}\) is given by \(\frac{1}{c}\int{d}^{3}r{\hat{\psi }}^{{{\dagger}} }({{{\bf{r}}}})({\hat{{{{\bf{A}}}}}}_{{{{\rm{o}}}}}\cdot \hat{{{{\bf{p}}}}})\hat{\psi }({{{\bf{r}}}})\), where \(\hat{\psi }\) (\({\hat{\psi }}^{{{\dagger}} }\)) are electron annihilation (creation) field operators, \({\hat{{{{\bf{A}}}}}}_{{{{\rm{o}}}}}\) is the vector potential of the optical field, \(\hat{{{{\bf{p}}}}}\) is the momentum operator, c is the speed of light. This and the following expressions are in atomic units. We neglect the term proportional to \({\hat{{{{\bf{A}}}}}}_{{{{\rm{o}}}}}^{2}\) in the interaction between the electronic system and the optical field. We describe the interaction with the optical field within the dipole approximation.

We assume that the photon energy of the x-ray pulse is in the hard x-ray region and is nonresonant with any core-excitation energy of the crystal. In this case, resonant x-ray scattering governed by the term proportional to \({\hat{{{{\bf{A}}}}}}_{{{{\rm{x}}}}}\cdot \hat{{{{\bf{p}}}}}\) becomes negligible and the interaction of the electronic system and the x-ray pulse is given by the Hamiltonian \({\hat{H}}_{{{{\rm{intx}}}}}=1/(2{c}^{2})\int{d}^{3}r{\hat{\psi }}^{{{\dagger}} }({{{\bf{r}}}}){\hat{{{{\bf{A}}}}}}_{{{{\rm{x}}}}}^{2}\hat{\psi }({{{\bf{r}}}})\) that describes nonresonant x-ray scattering. Elastic nonresonant x-ray scattering forms the basis for x-ray diffraction techniques, since it connects the scattering signal to the Fourier transform of the electron density51,52,53. The \({\hat{{{{\bf{A}}}}}}_{{{{\rm{x}}}}}^{2}\)-interaction term can also lead to inelastic scattering.

In the presence of the optical field, both elastic and inelastic x-ray scattering signals get modulated and the total x-ray scattering probability is given by Ptot. = Pq.e. + Pinel.. We refer to the scattering signal that results due to the optically-induced modulation of elastic scattering, as quasielastic scattering and its probability as Pq.e.. As we will show, it contains the relevant information about the optically-driven dynamics in a material. It is still necessary to compute the probability of inelastic scattering, Pinel., to ensure that it does not suppress the quasielastic scattering signal. For α-quartz, we find that it is two orders of magnitude smaller than the quasielastic scattering signal in the spectral range that we will consider.

Let us briefly review the case when the duration of the x-ray probe pulse is longer than one cycle of the optical driving field. The resulting x-ray scattering signal can be understood by analogy to conventional x-ray-diffraction signals. An x-ray-diffraction signal from a crystal consists of Bragg peaks resulting from elastic scattering. The intensity of the Bragg peak at reciprocal-lattice vector G is proportional to the corresponding Fourier component of the electron density of the crystal squared, ∣∫d3reiG⋅rρ(r)∣2. When x-ray and optical fields simultaneously interact with a crystal, wave mixing occurs, leading to sum and difference frequency generation37,38. Sum and difference frequency generation manifest themselves as side peaks to Bragg peaks that are energetically separated from them by multiples of the optical driving frequency, μω. The intensity of the side peaks of the Bragg peak at G of order μ is given by the Fourier transform \(| \int{d}^{3}r{e}^{i{{{\bf{G}}}}\cdot {{{\bf{r}}}}}{\widetilde{\rho }}_{\mu }({{{\bf{r}}}}){| }^{2}\), where \({\widetilde{\rho }}_{\mu }({{{\bf{r}}}})\) is the associated Fourier amplitude of the time-dependent electron density oscillating in a periodic optical field50

$$\rho ({{{\bf{r}}}},t)=\sum\limits_{\mu }{\widetilde{\rho }}_{\mu }({{{\bf{r}}}}){e}^{i\mu \omega t}.$$
(2)

Figure 1m–o show the calculated x-ray-optical wave mixing signal for the Bragg peak at G = (1, 2, 0) and its side peaks up to the third order. Here, G is represented in terms of integer multiples of the primitive reciprocal lattice vectors and is parallel to \((\sqrt{3},\sqrt{2},0)\). This Bragg peak is just one possible example to illustrate our results. Different subfigures show the same signal, in which μth-order side peaks that are related to the Fourier transform of the density oscillations shown in the same row in Fig. 1 are highlighted.

Subcycle-resolved x-ray-optical wave mixing

Generally, the probability of quasielastic x-ray scattering of a Gaussian-shaped x-ray pulse with duration τp is given by50

$${P}_{{{{\rm{q}}}}.{{{\rm{e}}}}.}({ \omega }_{{{{\rm{s}}}}},{{{\bf{G}}}})= {P}_{0}{\sum}_{\mu }{{{{\mathcal{E}}}}}_{\mu }^{2}{\left| \int{d}^{3}r{e}^{i{{{\bf{G}}}}\cdot {{{\bf{r}}}}}{\widetilde{\rho }}_{\mu }({{{\bf{r}}}})\right| }^{2}\\ +{(-1)}^{\Delta \mu }{P}_{0}{\sum}_{\mu ,\Delta \mu > 0}2{{{\rm{Re}}}}\left[{{{{\mathcal{E}}}}}_{\mu }{{{{\mathcal{E}}}}}_{\mu +\Delta \mu }{e}^{i\Delta \mu \omega {t}_{{{{\rm{p}}}}}}\int{d}^{3}r{e}^{i{{{\bf{G}}}}\cdot {{{\bf{r}}}}}{\widetilde{\rho }}_{\mu }({{{\bf{r}}}})\int{d}^{3}r{e}^{-i{{{\bf{G}}}}\cdot {{{\bf{r}}}}}{\widetilde{\rho }}_{\mu +\Delta \mu }^{* }({{{\bf{r}}}})\right].$$
(3)

Here, \({P}_{0}={\sum }_{{s}_{{{{\rm{s}}}}}}| ({{{{\boldsymbol{\epsilon }}}}}_{{{{\rm{in}}}}}\cdot {{{{\boldsymbol{\epsilon }}}}}_{x,{{{{\boldsymbol{\kappa }}}}}_{{{{\rm{s}}}}},{s}_{{{{\rm{s}}}}}}^{* }){| }^{2}{ \omega }_{{{{\rm{s}}}}}^{2}/(4{\pi }^{2}{\omega }_{{{{\rm{in}}}}}^{2}{c}^{3})\), where ϵin is the mean polarization vector of the incoming x-ray beam, ωs is the energy of a scattered photon with momentum κs, the sum over ss refers to the sum over polarization vectors of the scattered photons \({{{{\boldsymbol{\epsilon }}}}}_{x,{{{{\boldsymbol{\kappa }}}}}_{{{{\rm{s}}}}}{s}_{{{{\rm{s}}}}}}^{* }\) and ωin is the mean photon energy of the incoming x-ray beam. tp is the time of x-ray-pulse arrival relative to a reference time t = 0, when the phase ωtp and the optical electric field are zero. \({{{{\mathcal{E}}}}}_{\mu }=\sqrt{\frac{{\tau }_{p}^{2}\pi }{2\ln 2}}{e}^{-{({\omega }_{{{{\rm{s}}}}}-{\omega }_{{{{\rm{in}}}}}-\mu \omega )}^{2}{\tau }_{p}^{2}/8\ln 2}\) are Gaussian-shaped functions centered at the positions of the side peaks. Their width is inversely proportional to the x-ray-probe pulse duration. A product of such two Gaussians is also a Gaussian function centered in the middle between their centers.

If the duration of the probe pulse is longer than one cycle of the optical driving field, the Gaussian-shaped functions are spectrally narrow and peaks centered at different positions do not spectrally overlap. Any product \({{{{\mathcal{E}}}}}_{\mu }{{{{\mathcal{E}}}}}_{\mu +\Delta \mu }\) for Δμ ≠ 0 is negligible and the second term in Eq. (3) vanishes. A quasielastic scattering signal then consists of isolated peaks with a time-independent intensity as shown in Fig. 1m–o.

When the duration of the probe pulse decreases, the spectral peaks get broader and start interfering. The second term in Eq. (3) becomes nonzero and the intensity of the signal becomes time-dependent. We adjust the x-ray-pulse duration such that only nearest-neighbor side peaks can spectrally overlap. This is achieved for an x-ray pulse duration of 2.35 fs for the case of an optical field with a photon energy of 1.2 eV, corresponding to an optical period of 3.4 fs. In this case, the interference terms are proportional to Gaussian functions centered in the middle between nearest-neighbor side peaks, i.e. at ωs = ωin + (μ + 1/2)ω.

Figure 2 shows the subcycle-resolved x-ray-optical wave mixing signal by a probe x-ray pulse with duration of 2.35 fs at G = (1, 2, 0) and G = ( −1, −2, 0). The signal evolves in time with the period T = 2π/ω in agreement with Eq. (3). We observe that the signal is no longer centrosymmetric in momentum space in contrast to the time-independent case. The quasielastic scattering signal is also not symmetric with respect to the scattered energy.

Fig. 2: X-ray scattering signal from the laser-dressed α-quartz crystal.
figure 2

X-ray scattering signal at (a) G = (1, 2, 0) and (b) G = ( −1, − 2, 0) at the probe-pulse arrival time tp = 0 as a function of ωs − ωin. (c, d) are the same as (a) and (b), respectively, but at tp = T/4. (e) and (f) are the same as (a) and (b), respectively, but at tp = T/2. (g, h) are the same as (a) and (b), respectively, but at tp = 3T/4. The signal is normalized to the intensity of the main Bragg peak of α-quartz at G = (1, 2, 0). The filled yellow areas highlight the differences between the signals at G = ( −1, − 2, 0) and G = (1, 2, 0) at the same tp. The gray vertical lines are situated at the positions of the side peaks, ωs − ωin − μω, and their heights correspond to the relative intensities of the side peaks in a time-unresolved case.

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Let us analyze the time and momentum dependences of the signal to relieve the origin of centrosymmetry breaking. The momentum dependence of the signal is due to the Fourier components of the amplitudes of the electron density. The Fourier components of the density amplitudes can be represented as a sum of a centrosymmetric and an antisymmetric function \(\int{d}^{3}r{e}^{i{{{\bf{G}}}}\cdot {{{\bf{r}}}}}{\widetilde{\rho }}_{\mu }({{{\bf{r}}}})=\int{d}^{3}r\cos ({{{\bf{G}}}}\cdot {{{\bf{r}}}}){\widetilde{\rho }}_{\mu }({{{\bf{r}}}})+i\int{d}^{3}r\sin ({{{\bf{G}}}}\cdot {{{\bf{r}}}}){\widetilde{\rho }}_{\mu }({{{\bf{r}}}})\). In materials with time reversal symmetry, which includes α-quartz, the even-order density amplitudes \({\widetilde{\rho }}_{{\mu }_{{{{\rm{even}}}}}}({{{\bf{r}}}})\) are real functions and the odd-order density amplitudes \({\widetilde{\rho }}_{{\mu }_{{{{\rm{odd}}}}}}({{{\bf{r}}}})\) are purely imaginary, i.e. they have a phase that is either nπ or (n + 1/2)π, where n is an integer36. The phases of the \({\widetilde{\rho }}_{\mu }({{{\bf{r}}}})\) determine the phases of the symmetric and antisymmetric parts of the Fourier components. We employ these dependencies to express the antisymmetric part of the signal as

$$[{P}_{{{{\rm{q}}}}.{{{\rm{e}}}}.}({\omega }_{{{{\rm{s}}}}},{{{\bf{G}}}})-{P}_{{{{\rm{q}}}}.{{{\rm{e}}}}.}({\omega }_{{{{\rm{s}}}}},-{{{\bf{G}}}})]/2 = \, {P}_{0}\cos (\omega {t}_{{{{\rm{p}}}}})\sum\limits_{\mu }2{{{{\mathcal{E}}}}}_{\mu }{{{{\mathcal{E}}}}}_{\mu +1} \left[i\int{d}^{3}r\sin ({{{\bf{G}}}}\cdot {{{\bf{r}}}}){\widetilde{\rho }}_{\mu }({{{\bf{r}}}}) \right. \\ \left. \int{d}^{3}r\cos ({{{\bf{G}}}}\cdot {{{\bf{r}}}}){\widetilde{\rho }}_{\mu +1}({{{\bf{r}}}})-i\int{d}^{3}r\cos ({{{\bf{G}}}}\cdot {{{\bf{r}}}}){\widetilde{\rho }}_{\mu }({{{\bf{r}}}}) \right. \\ \left. \int{d}^{3}r\sin ({{{\bf{G}}}}\cdot {{{\bf{r}}}}){\widetilde{\rho }}_{\mu +1}({{{\bf{r}}}})\right].$$
(4)

We find that it evolves in time as \(\cos (\omega {t}_{{{{\rm{p}}}}})\). At the same time, the phases of the density amplitudes also determine the oscillation phases of the electron density that evolves as

$$\rho ({{{\bf{r}}}},t)={\widetilde{\rho }}_{0}({{{\bf{r}}}})-{\varrho }_{1}({{{\bf{r}}}})\sin (\omega t)+{\varrho }_{2}({{{\bf{r}}}})\cos (2\omega t)-{\varrho }_{3}({{{\bf{r}}}})\sin (3\omega t)+\cdots \,,$$
(5)

and, via the continuity equation, those of the electron-current density,

$${{{\bf{j}}}}({{{\bf{r}}}},t)=-{{\mathfrak{j}}}_{1}({{{\bf{r}}}})\cos (\omega t)-{{\mathfrak{j}}}_{2}({{{\bf{r}}}})\sin (2\omega t)-{{\mathfrak{j}}}_{3}({{{\bf{r}}}})\cos (3\omega t)-\cdots$$
(6)

Here, \({\varrho }_{{\mu }_{{{{\rm{even}}}}}}({{{\bf{r}}}})=2{{{\rm{Re}}}}[ \, {\widetilde{\rho }}_{{\mu }_{{{{\rm{even}}}}}}({{{\bf{r}}}})],{\varrho }_{{\mu }_{{{{\rm{odd}}}}}}({{{\bf{r}}}})=2{ \, {{\rm{Im}}}}[ \, {\widetilde{\rho }}_{{\mu }_{{{{\rm{odd}}}}}}({{{\bf{r}}}})]\) and \({{\mathfrak{j}}}_{\mu }\) are real-valued amplitudes of the electron density and electron-current density, respectively. Thus, we find that the antisymmetric part of the subcycle-resolved x-ray-optical wave mixing signal evolves in time as the first-order oscillation of the electron-current density and is maximal at tp = 0 and tp = T/2. At these two times during the period, the first-order oscillation of the electron-current density has a maximal magnitude and the first-order oscillation of the electron density is zero (see Fig. 1a–c). The quasielastic scattering signal is centrosymmetric in momentum space at times tp = T/4 and tp = 3T/4, when the first-order electron-current density vanishes.

The signal in Fig. 2 is sensitive only to the first-order oscillation of microscopic optical response, since we selected a probe-pulse duration that is longer than the period of the second-order oscillation with the frequency 2ω. If the probe-pulse duration is chosen short enough to resolve an nth-order oscillation with the frequency nω, the signal would have oscillating terms with frequencies ω, 2ω, ⋯  , nω. The oscillation with the frequency Δμω would be due to interference terms between μ and μ + Δμ side peaks. Even in this case, the connection of the antisymmetric part of the signal to oscillations of the electron-current density still holds. The antisymmetric part of the signal would consist of time-dependent terms that oscillate either as \(\cos (\Delta \mu \omega {t}_{{{{\rm{p}}}}})\) for odd Δμ or as \(\sin (\Delta \mu \omega {t}_{{{{\rm{p}}}}})\) for even Δμ. Thus, the time evolution of the antisymmetric part of the signal correlates with the time evolution of the electron-current density in Eq. (6). A similar effect of momentum-symmetry breaking of time- and momentum-resolved x-ray signals from a nonstationary electron system that correlates in time with currents has been predicted for other types of pump-probe experiments as well54,55,56,57.

Analyzing the centrosymmetric part of the signal, [Pq.e.(ωs, G) + Pq.e.(ωs, − G)]/2, we find that it oscillates as \(\sin (\omega {t}_{{{{\rm{p}}}}})\), which is in phase with the first-order oscillation of the electron density. This connection can also be generalized for high-order oscillations of the signal, if the temporal resolution is sufficient to resolve them. The centrosymmetric part of the signal would consist of a time-independent part and a time-dependent part due to interference terms between μ and μ + Δμ side peaks. They would oscillate as \(\sin (\Delta \mu \omega {t}_{{{{\rm{p}}}}})\) for odd Δμ or as \(\cos (\Delta \mu \omega {t}_{{{{\rm{p}}}}})\) for even Δμ, which correlates with oscillations of the electron density. Thus, we find that the phases of temporal oscillations of the density and those of the electron-current density are encoded in the temporal behavior of the centrosymmetric and antisymmetric parts, respectively.

Retrieval of optically-induced charge distributions and directions of electron currents

Let us analyze the signal in Fig. 2 from a different perspective. We now focus only on the time dependence of the quasielastic scattering signal and find that it can be represented as

$${P}_{{{{\rm{q}}}}.{{{\rm{e}}}}.}({\omega }_{{{{\rm{s}}}}},{{{\bf{G}}}})= \frac{{P}_{0}}{4}{\sum}_{\mu }{{{{\mathcal{E}}}}}_{\mu }^{2}| {{{{\mathcal{P}}}}}_{\mu }({{{\bf{G}}}}){| }^{2}\\ +\frac{{P}_{0}}{2}{\sum}_{\mu }{(-1)}^{\mu +1}{{{{\mathcal{E}}}}}_{\mu }{{{{\mathcal{E}}}}}_{\mu +1}| {{{{\mathcal{P}}}}}_{\mu }({{{\bf{G}}}})| | {{{{\mathcal{P}}}}}_{\mu +1}({{{\bf{G}}}})| \sin (\omega {t}_{{{{\rm{p}}}}}\\ +{\alpha }_{\mu }({{{\bf{G}}}})-{\alpha }_{\mu +1}({{{\bf{G}}}})),$$
(7)

where we expressed the Fourier transform of the real-valued amplitudes of the electron density as \(\int{d}^{3}r{e}^{i{{{\bf{G}}}}\cdot {{{\bf{r}}}}}{\varrho }_{\mu }({{{\bf{r}}}})={{{{\mathcal{P}}}}}_{\mu }({{{\bf{G}}}})=\left\vert {{{{\mathcal{P}}}}}_{\mu }({{{\bf{G}}}})\right\vert {e}^{i{\alpha }_{\mu }({{{\bf{G}}}})}\). The optical electric field oscillates as \(\sin (\omega t)\) and the time evolution of the subcycle-resolved x-ray-optical wave mixing signal comprises oscillations that display a phase shift with respect to the electric-field oscillation. The phase difference in time is determined by the momentum-dependent phases of the Fourier components of the optically-induced charge distributions, αμ(G).

The quasielastic signal provides both the phases αμ(G) and the amplitudes \(| {{{{\mathcal{P}}}}}_{\mu }({{{\bf{G}}}})|\) of the Fourier components of the optically-induced charge distributions ϱμ(r) of all orders μ as we will show with the expression in Eq. (7). This is a big advantage of a time-resolved measurement over a time-unresolved one. Knowledge of the amplitudes and phases of the Fourier components of a density at all G provides the complete information to reconstruct it. Especially, the phases are crucial for the reconstruction; the “phase problem”, i.e., the lack of phase information, is a well-known problem in crystallography58. A time-unresolved measurement that leads to separated side peaks (see Fig. 1m–o) provides only the amplitudes \(| {{{{\mathcal{P}}}}}_{\mu }({{{\bf{G}}}})|\) of the Fourier components. A time-resolved measurement provides also the phases as we show below.

For the phase retrieval, we need to know the phase of the Fourier component of the unperturbed density α0(G), which is not a problem for an experiment performed on a material with a known structure. Figure 3 shows the time-dependent part of the signal at different tp in the spectral range centered around ωin + ω/2. From Eq. (7), we know that it is given by the interference term between the main peak and the first-order side peak, which oscillates as \(-\sin (\omega {t}_{{{{\rm{p}}}}}+{\alpha }_{0}-{\alpha }_{1})\). Here, we subtracted the time-independent part of the signal from the total signal. The time-independent part is given by the centrosymmetric part of the signal at time tp = 0, [Pq.e.(G, tp = 0) + Pq.e.( −G, tp = 0)]/2. The time-dependent signal in Fig. 3 oscillates as \(-\sin (0.012\pi +\omega {t}_{{{{\rm{p}}}}})\) and we determine that α1(G) = α0(G) − 0.012π. The interference term between the first-order and the second-order side peak is centered at ωin + 3ω/2 and oscillates as \(\sin (\omega {t}_{{{{\rm{p}}}}}+{\alpha }_{1}-{\alpha }_{2})\) and this is how we can find α2(G). This procedure can then be repeated for the subsequent orders and various G.

Fig. 3: Time-dependent part of the signal.
figure 3

Time-dependent part of the x-ray scattering signal at G = (1, 2, 0) at different probe-pulse arrival times as a function of ωs − ωin. The signal is normalized to the intensity of the main Bragg peak of α-quartz at G = (1, 2, 0).

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The amplitudes \(| {{{{\mathcal{P}}}}}_{\mu }({{{\bf{G}}}})|\) of the Fourier components can be reconstructed either from a time-independent measurement or from the time-independent part of the signal that is given by Gaussians centered at μωin with the intensity proportional to \(| {{{{\mathcal{P}}}}}_{\mu }({{{\bf{G}}}}){| }^{2}\). This way, we obtain the Fourier components \({{{{\mathcal{P}}}}}_{\mu }({{{\bf{G}}}})\) including the amplitude and the phase. Measuring the subcycle-resolved x-ray-optical wave mixing signal at various G, one can reconstruct charge distributions ϱμ(r) for all orders of microscopic optical response μ. In a real experiment, orders with small amplitudes like the third and the fourth ones in our case would be very challenging to detect and reconstruct, but there are no fundamental restrictions to reconstruct them.

The directions of electron currents can also be retrieved once the charge distributions have been reconstructed. The amplitudes of the electron density and the electron current densities are connected via the continuity equation as \({{{\rm{div}}}}{{\mathfrak{j}}}_{\mu }({{{\bf{r}}}})=-\mu \omega {\varrho }_{\mu }({{{\bf{r}}}})\). Their Fourier transforms are connected via

$$\int{d}^{3}r{e}^{i{{{\bf{G}}}}\cdot {{{\bf{r}}}}}{\varrho }_{\mu }({{{\bf{r}}}})=\frac{i}{\mu \omega }{{{\bf{G}}}}\cdot \int{d}^{3}r{e}^{i{{{\bf{G}}}}\cdot {{{\bf{r}}}}}{{\mathfrak{j}}}_{\mu }({{{\bf{r}}}})$$
(8)

due to the Green’s first identity, if the surface integral of the electron current density vanishes. For example, the first-order electron-current density is unidirectional and parallel to the light polarization axis ϵo. The intensity of the first-order side peak is given by ∣∫d3reiG⋅rϱ1(r)∣2 and must then scale as ∣(ϵo ⋅ G)∣2, which agrees with the experimental observation in ref. 39. With our method, we can reconstruct the projection of the Fourier components of the electron-current density including phases. The sign of the projections reveals the direction of the current and the phases reveal spatial symmetry features of the electron-current density.

The amplitudes at different reflections depend on the intensity of the driving field50. We find that the intensities of the first- and second-order side peaks decrease by one and two orders of magnitude, respectively, when the intensity is decreased from 1012 W cm−2 to 1011 W cm−2. But the proportionality factors deviate considerably from ten and one hundred, respectively, which indicates that the interaction with the driving field is nonperturbative at the intensity of 1012 W cm−2 and the selected photon energy.

When the photon energy of the driving field is selected to be 9 eV and approaches resonant energies, the intensities of the first-order side peaks overall moderately increase in comparison to the ones at the off-resonant driving frequency of 1.2 eV. At the same time, the intensities of the second-order side peaks decrease by one order of magnitude. This indicates that first-order absorption dominates at resonant energies.

We find that the difference between phases α0(G) and α1(G), and between α1(G) and α2(G) is close to zero at the reflection (1, 2, 0) in α-quartz. This is different for other reflections and orders, which show a variety of values. We also find that the phases αμ(G) change when reducing the intensity to 1011 W cm−2 at the same photon energy of the driving field, as well as when increasing the photon energy to 9 eV at the same intensity of the driving field. Hence, some orders and reflections appear to be relatively insensitive to changes in driving-field parameters, whereas other orders and reflections display a much more pronounced sensitivity to the intensity and photon energy of the driving field. This indicates that certain spatial and temporal features of optically induced charge distributions can depend strongly on parameters of the driving field.

Outlook

We considered a system with time-reversal symmetry driven by a linearly polarized field. If time-reversal symmetry is broken, for example, because a material is driven by a circularly polarized field or is magnetic, the analysis must be extended. It is then necessary to explicitly consider spin-resolved amplitudes of the electron density, \({\widetilde{\rho }}_{\mu \uparrow }\) and \({\widetilde{\rho }}_{\mu \downarrow }\). The scattering probability is then \({P}_{{{{\rm{q}}}}.{{{\rm{e}}}}.}({\omega }_{{{{\rm{s}}}}},{{{\bf{G}}}})={P}_{{{{\rm{q}}}}.{{{\rm{e}}}}.}^{\uparrow }({\omega }_{{{{\rm{s}}}}},{{{\bf{G}}}})+{P}_{{{{\rm{q}}}}.{{{\rm{e}}}}.}^{\downarrow }({\omega }_{{{{\rm{s}}}}},{{{\bf{G}}}})\), where \({P}_{{{{\rm{q}}}}.{{{\rm{e}}}}.}^{\uparrow }({\omega }_{{{{\rm{s}}}}},{{{\bf{G}}}})\) and \({P}_{{{{\rm{q}}}}.{{{\rm{e}}}}.}^{\downarrow }({\omega }_{{{{\rm{s}}}}},{{{\bf{G}}}})\) are given by Eq. (3), but with \({\widetilde{\rho }}_{\mu }\) replaced by \({\widetilde{\rho }}_{\mu \uparrow }\) and \({\widetilde{\rho }}_{\mu \downarrow }\), respectively. The connection of the time-dependence of the scattering probability to the time-dependence of the electron density and the electron current density should not hold for broken time-reversal symmetry. However, \({P}_{{{{\rm{q}}}}.{{{\rm{e}}}}.}^{\uparrow }({\omega }_{{{{\rm{s}}}}},{{{\bf{G}}}})\) and \({P}_{{{{\rm{q}}}}.{{{\rm{e}}}}.}^{\downarrow }({\omega }_{{{{\rm{s}}}}},{{{\bf{G}}}})\) would still be connected to the Fourier components of spin-resolved amplitudes of the electron density and their phases. This connection follows directly from Eq. (3) and does not rely on time-reversal symmetry.

The time-independent part of the signal is symmetric with respect to the scattered energy also due to time-reversal symmetry. If time-reversal symmetry is broken, this symmetry should also be broken and it would be interesting to analyze which information can be extracted from its energy dependence.

Our theoretical framework based on the Floquet-Bloch formalism combined with a DFT band structure cannot capture excitonic effects. Excitonic effects can be captured if the Floquet-Bloch Hamiltonian is constructed taking particle-hole interaction into account. For example, one can use two-particle wave functions obtained using the Bethe-Salpeter equation instead of one-particle Kohn-Sham wave functions for the Floquet-Bloch Hamiltonian to account for excitonic effects.

Conclusion

When light interacts with a material, charges rearrange within its unit cell forming complex structures. Microscopic electron currents that can point in different directions are induced. Such microscopic optical response can be detected with x rays due to their short wavelength. Here, we proposed a method to reconstruct oscillations of charge and electron-current density distributions employing ultrafast x-ray diffraction during the time a material is driven by light, namely, by subcycle-resolved x-ray-optical wave mixing.

The subcycle-resolved x-ray-optical wave mixing signal Pq.e.(G, tp) is notably noncentrosymmetric with respect to the scattering vector G. The temporal evolution of the antisymmetric part of the signal, (Pq.e.(G, tp) − Pq.e.( − G, tp))/2 reveals the phases of the temporal oscillations of the electron-current density. The temporal evolution of the centrosymmetric part of the signal, (Pq.e.(G, tp) + Pq.e.( − G, tp))/2 reveals the phases of the temporal oscillations of the electron density. Furthermore, the signal encodes the Gth Fourier components of charge distributions and projections of the Gth Fourier components of electron-current density distributions. The amplitudes of the Fourier components can be reconstructed from the time-independent part of the signal. The phases of the Fourier components can be reconstructed from the phases of the temporal oscillations of the time-dependent part of the signal. If both phases and amplitudes of the Fourier components at various G can be measured, then the optically-induced charge distributions can be completely reconstructed. An atomically-resolved view into light-matter interactions will provide a deeper understanding of optically-driven electron dynamics and prompt further developments of nonlinear optics towards technological applications.

Methods

Kohn-Sham wave functions were calculated using the ABINIT software package59,60,61,62 with Perdew–Burke–Ernzerhof functionals63,64. We took into account 24 valence and 216 conduction bands on a 16 × 16 × 16 k-point grid. The calculated Kohn-Sham wave functions were used to construct the Floquet-Bloch Hamiltonian as described in refs. 36,50. The infinite Floquet-Bloch Hamiltonian was approximated by a matrix with 141 blocks, each containing 240 states. The number of k-points, conduction bands and blocks were increased until convergence was reached. We applied the scissors approximation65 to correct the band gap from the calculated 6.1 eV to the experimental value of 8.9 eV66. Optically-induced charge oscillations and electron current densities were calculated as described in ref. 36 and visualized using VESTA67. α-quartz is chiral and has two structural types; we considered the type with space group 152.