Ultrafast decoherence in solid-state high-harmonic generation induced by nuclear-electronic entanglement

Abstract

High-harmonic generation (HHG) has provided groundbreaking insights into nonequilibrium dynamics involving strong light-matter interactions. However, intrinsic quantum effects of condensed matter, such as electronic coherence and its interplay with nuclear degrees of freedom─one of the most fascinating aspects of quantum mechanics─have largely been overlooked. Here, we explore the nuclear quantum effects (NQEs) on electronic coherence in solid-state HHG. Unlike classical nuclei, the strong delocalization of the nuclei caused by zero-point vibrations induces electronic decoherence on an ultrafast (attosecond to femtosecond) timescale through nuclear-electronic entanglement. In this manner, NQEs not only contribute to the ultrafast dephasing of HHG but also suppress the interband electron trajectory and thus switch the dominant mechanism from interband to intraband HHG in solids. This yields measurable signatures in HHG spectroscopy, enabling direct probe of coherence time and nuclear wave packets information in materials.

Nuclear quantum effects reshape solid high-harmonic generation mechanism via attosecond decoherence, enabling new quantum probes.

INTRODUCTION

The rise of attosecond science enables us to explore and understand quantum phenomena involving strong light-matter interactions. This encompasses one of the most fundamental problems in quantum mechanics: electronic coherence and its interplay with nuclear degrees of freedom (1–7). Although lighter electrons move faster on an attosecond timescale, nuclear states affect electron dynamics and quantum coherence through the topology of potential energy surfaces.

Recent photoionization studies (8–13) have revealed a close relationship between electronic coherence/decoherence in ionized molecular systems and electron-nucleus interactions. These interactions, encompassing both classical nuclear dynamics and quantum entanglement, subsequently govern charge migration and reaction pathways (14–17). Understanding the role of nuclear-electronic entanglement (NEE) in electronic decoherence, which remains an open question, is crucial for accessing, manipulating, and potentially performing Bell tests of quantum states (18–21). This is also pivotal in the ongoing quantum technology revolution in the fields such as quantum computing, quantum sensors, photochemistry, and optoelectronics (2, 22–24).

As a prominent area of ultrafast and strong-field science, high-harmonic generation (HHG) occurs on an attosecond timescale before electronic coherence is lost (25, 26). This makes HHG a potential avenue for investigating the interplay between electronic coherence and nuclear dynamics. In molecules and gases, the nuclear autocorrelation has been shown to modulate the electron recombination channels (27–29), thereby shaping the yield of different harmonics and extreme ultraviolet emissions (30).

Nevertheless, because of the multitude collective excitations and the complex interactions among them, the NEE has been largely ignored in solid materials despite that the community starts to realize the correlation between solid-state HHG and classical nuclear dynamics (31–35). This research gap has hindered a comprehensive understanding of solid-state HHG mechanisms, particularly regarding ultrafast dephasing and its impact on the competition between intraband and interband harmonic generation (36–43). In the first case, theoretical HHG spectra agree with experiments only when assuming an extremely fast dephasing time (few femtoseconds) (36–43), whereas experimental measurements of carrier lifetime and scattering rates (44–49) suggest that typical timescales are an order of magnitude longer than those assumed theoretically.

Although various mechanisms have been proposed to resolve this discrepancy—including macroscopic propagation effects (50), spatiotemporal filtering (51), real-space suppressing of long trajectories (52) and large electron-hole separations (53, 54), electron momentum-dependent electron-phonon scattering (EPS) (33, 55) and dephasing (56), and the influence of classical nuclear dynamics (57, 58)—the origin of the ultrafast dephasing in HHG remains an open question.

In this work, we explore the quantum phenomena in attosecond processes via solid-state HHG, with a particular focus on NQEs and their role in driving quantum electronic systems toward statistical ensembles. This process occurs through ultrafast electronic decoherence (~1 fs) within the framework of NEE (59–67). The decoherence induced by NQEs represents a potential physical origin for the ultrafast dephasing observed in solid-state HHG, leading to distinct changes in the relative contributions of intraband and interband harmonics that are different from those in gas-phase systems. Investigations into these dynamics enable not only potential insights in understanding recent HHG experiments (68, 69) but also the direct probing of quantum coherence time and nuclear wave packets (NWPs) in solid materials.

RESULTS AND DISCUSSION

The NQE-induced decoherence of HHG has been first investigated in the high-pressure superconductor Im-3m H3S (Fig. 1A). Im-3m H3S serves as a prototype example where the strong NQEs of hydrogen (H) atoms enhance lattice stabilization and superconductivity. The NQEs, notably the zero-point motions, strongly affect the nuclear delocalization <ΔR²> ^1/2 = <ΔR_x² + ΔR_y² + ΔR_z²> ^1/2. Here ΔR_x,y,z is the nuclei displacement along different directions, which is described by the Gaussian distribution centered around equilibrium position (fig. S9). Figure 1 (A and B) gives the probability distributions of <ΔR²> ^1/2 for sulfur (S) and H nuclei, where the peaks show the most probable nuclear delocalization <ΔR²> ^1/2_max and σ_ΔR denotes the delocalization width. For S nuclei, <ΔR²> ^1/2_max increases slightly from 0.08 Å without NQEs to 0.10 Å with NQEs included. In contrast, the light H nuclei—whose vibration frequency can surpass its thermal energy by an order of magnitude—gives a large superposition of NWPs, resulting in a notable enhancement in nuclear delocalization. Therefore, the <ΔR²> ^1/2_max of H nuclei shifts from 0.12 to 0.22 Å, and σ_ΔR shifts from 0.08 to 0.19 Å when NQEs are considered.

Fig. 1. The NQEs in Im-3m H₃S.

Probability distributions of nuclear delocalization <ΔR²> ^1/2 without NQEs (A) and with NQEs (B) at 100 K. The distribution peaks (labeled by black dotted lines) is the most probable nuclear delocalization <ΔR²> ^1/2_max, and σ_ΔR is the delocalization width. The inset schematics show the lattice structure of Im-3m H₃S without and with NQEs. (C) Spectrum of velocity autocorrelation function of atom displacement oscillations C_vv. The gray dashed line labels the anharmonic phonon peak involving H nuclei. The carrier density dynamics induced by laser at (001) plane without (D) and with (E) NQEs. The central time of the laser envelope t₀ = 12 fs. The without NQEs (w/o NQE) case is based on thermal ensembles, which only include the thermal fluctuations of classical nuclear motions.

On the other hand, the large quantum fluctuations induce the electron rearrangement around the H nuclei, which alters the screening and thus modulates the potential energy surface of the H─S bond (figs. S7 and S8). This ultimately amplifies structural disorder and phonon anharmonicity. As shown in Fig. 1C, the spectrum of velocity autocorrelation function C_vv of atom oscillations, which denotes the phonon density of states, shows the emergence of anharmonic phonon peak involving H nuclei (70–73) at ~0.24 eV, aligning well with recent studies (71–73). As a result, the charge density oscillations exhibit faster decay as shown in Fig. 1 (D and E).

In Fig. 2A, the HHG spectra along the [100] direction of H₃S are depicted both with and without NQEs. For comparison, we plot the HHG spectra without NQEs but with a phenomenological decoherence. This is achieved by artificially setting the decoherence time T₂ to ~1 fs in simulations, where T₂ quantifies the decay time of the off-diagonal elements of the electron density matrix. For subsequent discussions and figures on HHG with/without NQEs, T₂ is set to infinity by default unless stated explicitly, indicating that no intrinsic electronic decoherence is considered. In the absence of NQEs or finite T₂, the harmonics exhibit a noisy spectral configuration, with small splits appearing in the HHG peaks. In contrast, the inclusion of NQEs or a phenomenological T₂ gives visible harmonic peaks.

Fig. 2. HHG dynamics in Im-3m H₃S.

(A) HHG spectra of Im-3m H₃S with and without NQEs. The inset denotes a schematic of HHG process. The normalized intensity of 5th (B) and 11th (C) harmonic as a function of electron doping with and without NQEs. The interband and intraband contributions of 5th and 11th harmonic, without (D) and with NQEs (E). The phase diagram on dominant yielding mechanism of 5th (F) and 11th (G) harmonic with and without NQEs. The dashed line is an extension line. The without NQEs (w/o NQE) case is based on thermal ensembles, which only include the thermal fluctuations of classical nuclear motions. The HHG in (A) to (C) is obtained from the time-dependent density functional theory (TDDFT) calculations, while the analysis in (D) and (F) is obtained by semiclassical Bloch equation (SBE).

As shown in Fig. 2 (B and C), similar to recent experiments (74–76), for low-order HHG, such as the fifth harmonic, the intensity increases with the electron doping level n in all the cases. However, for the high-order interband HHG (e.g., the 11th harmonic), whose intensity decreases with n, the inclusion of NQEs completely reverses the doping dependence.

As illustrated in Fig. 2 (D and E), for intraband fifth harmonics whose intensity is proportional to the carrier concentration n, NQEs do not alter the generation mechanism, thereby maintaining the doping dependence. In contrast, for interband 11th harmonics, whose intensity decreases with n due to the hindered recombination caused by either Pauli-blocking effects (74) or dephasing from incoherent doped-carriers (75, 76), NQEs effectively switch the mechanism from interband to intraband dynamics by largely suppressing the interband contributions. Consequently, the doping dependence is reversed.

By defining r_HH = I_intraband/I_interband, where I_intraband (I_interband) is the intensity of intraband (interband) component of HHG, we obtain the phase diagram of HHG mechanism in Fig. 2 (F and G). Here, r_HH > 1 (r_HH < 1) denotes the intraband (interband) dominant regime of harmonic generation, respectively. The NQE-induced intra- and interband HHG transitions appears to be quite robust across various temperatures. While at high temperatures thermal effects result in a smaller regime for transitions, their impact remains quite limited (fig. S5). These results indicate that, unlike the three-step model of gas-phase HHG (77), NQEs play a decisive role in determining the intraband and interband mechanisms of solid-state HHG.

In solids, the multiple recombination of excited electron-hole pairs generates long and short trajectories of interband HHG emission. As depicted in time-frequency spectrogram in Fig. 3 (A and B), the persistent electronic coherences result in strong quantum interference between the interband trajectories, introducing chirps in post-pulse HHG emission and thus peak splitting in spectrum (36–40). By contrast, the inclusion of NQEs reduces the high-energy (above ninth harmonics, >18 eV) and postpulse HHG emission (Fig. 3B), making the HHG spectrum “clean” (Fig. 2A).

Fig. 3. The NEE and its effects on HHG.

The time-frequency spectrograms of HHG in H₃S without (A) and with (B) NQEs obtained from TDDFT. The gray solid curves label the oscillation of laser field. The black dashed lines, which serve as a guide to the eye, mark the short and long trajectories of HHG emission. (C) Electronic population obtained from TDDFT and coherence (purity) obtained from QMD as a function of time with laser excitation. (D) Time-frequency analysis of 5th and 11th HHG with and without NQEs obtained from TDDFT. The gray lines denote the results for various beads, whereas the green line indicates the results without NQEs but with the presence of the phenomenological decoherence time T₂ ~ 1 fs. The evolution of profile (E) and phase (F) of the NWP obtained from QMD. The black lines mark the distribution of initial NWP. The without NQEs (w/o NQE) case is based on thermal ensembles, which only include the thermal fluctuations of classical nuclear motions.

The result suggests a significant suppression of electronic coherence and, consequently, the interband trajectories (36–40). To illustrate this, we compare the time-frequency trajectories of harmonics associated with NQEs to those without NQEs but with an artificially set decoherence time (T₂ ~ 1 fs) in Fig. 3D. The trajectories show good agreement with each other (also see the similar spectra in Fig. 2A and fig. S6). This scenario is further confirmed by examining the population evolution of the photoexcited electron in Fig. 3C. Different from the phonon-enhanced photocarrier doping mechanism (78) in classical nuclei case, the greater population accumulation in the presence of NQEs originates from suppressed electron-hole recombination, a consequence of the NQE-enhanced electronic decoherence (79).

Recent studies show that propagation effects (50) and EPS (33) also contribute to HHG dephasing. However, the propagation effect requires a large propagation length of ∼10 μm (50), while many HHG spectra are done in atomically thin samples. The dephasing time from EPS is usually several to tens of femtoseconds (33), much longer than our case. At low temperatures, the EPS-induced dephasing is even weaker, while NQEs are more significant. Therefore, NQEs are expected to be a dominant source of electronic decoherence in HHG.

The NQE-induced electronic decoherence can be understood by the NEE effect, where the electron coherence is taken away by the quantum evolution of nucleus subsystem (6, 59–61). The total wave function of entangled electron-nucleus system is written as: $\mid \mathrm{\Psi }(\mathbf{r},\mathbf{R},t)⟩={\sum }_i\mid {\mathrm{\varphi }}_i(\mathbf{r},\mathbf{R})⟩\mid {\mathrm{\chi }}_i(\mathbf{R},t)⟩$, where $\mid {\mathrm{\varphi }}_i(\mathbf{r},\mathbf{R})⟩$ is the ith eigenstate of the electronic Hamiltonian ${\hat{H}}^{\mathrm{e}}\left(\mathbf{R}\right)$, while $\mid {\mathrm{\chi }}_i(\mathbf{R},t)⟩$ is the NWP associated with the electronic state. The electronic coherence is contained in the off-diagonal elements of the electron density matrix:

$${\hat{\mathrm{\rho }}}_{{\mathit{\text{ii}}}^{\prime }}^{\mathrm{e}}\left(t\right)=\mid {\mathrm{\varphi }}_i(\mathbf{r},\mathbf{R})⟩⟨{\mathrm{\varphi }}_{i^{\prime }}(\mathbf{r},\mathbf{R})\mid ⟨{\mathrm{\chi }}_{i^{\prime }}(\mathbf{R},t)\mid {\mathrm{\chi }}_i(\mathbf{R},t)⟩=\mid {\mathrm{\varphi }}_i(\mathbf{r},\mathbf{R})⟩⟨{\mathrm{\varphi }}_{i^{\prime }}(\mathbf{r},\mathbf{R})\mid ⟨\mathrm{\chi }\left(0\right)\mid \text{exp}\{\mathrm{i}[{\hat{H}}_i^{\mathrm{e}}\left(\mathbf{R}\right)-{\hat{H}}_{i^{\prime }}^{\mathrm{e}}\left(\mathbf{R}\right)]t/\hslash \}\mid \mathrm{\chi }\left(0\right)⟩$$

where $\text{exp}\left\{\mathrm{i}[{\hat{H}}_i^{\mathrm{e}}\left(\mathbf{R}\right)-{\hat{H}}_{i^{\prime }}^{\mathrm{e}}\left(\mathbf{R}\right)]t/\hslash \right\}\mid \mathrm{\chi }\left(0\right)⟩$ describes the NWP propagation on the ith electronic surface (59–63). The electronic coherence is lost whenever ${\hat{\mathrm{\rho }}}_{{\mathit{ii}}^{\prime }}^{\mathrm{e}}=0$, which can be achieved either from density matrix of electron part, i.e., $\mid {\mathrm{\varphi }}_i(\mathbf{r},\mathbf{R})⟩⟨{\mathrm{\varphi }}_{i^{\prime }}(\mathbf{r},\mathbf{R})\mid$, or from the NWP when $⟨\mathrm{\chi }\left(0\right)\mid \text{exp}\left\{\mathrm{i}[{\hat{H}}_i^{\mathrm{e}}\left(\mathbf{R}\right)-{\hat{H}}_{i^{\prime }}^{\mathrm{e}}\left(\mathbf{R}\right)]t/\hslash \right\}\mid \mathrm{\chi }\left(0\right)⟩$ vanishes, as discussed here.

We quantify this NQE-induced decoherence by calculating the purity of electron density matrix ${\mathrm{\rho }}^{\mathrm{e}}$ in the presence of NWP obtained by quantum mechanical dynamics (QMD). The purity is defined as $P=\text{Tr}\left({\left[{\mathrm{\rho }}^{\mathrm{e}}\right]}^2\right)$. $P=1$ represents the pure quantum state, while $P=0$ means complete decoherence. As shown in Fig. 3C, after laser excitation, the dynamics of NWP leads to an attosecond electronic decoherence within 1 fs. The NWP collapses during the evolution, resulting in phase mismatch of different components and the nucleus delocalization (Fig. 3, E and F). The electronic decoherence naturally originates from (i) the dephasing and (ii) the decay in overlap of different components of delocalized NWP (62–67). This NQE-induced decoherence is also manifested the electron dynamics in real space (Fig. 1, D and E), and with NQEs, the charge density oscillations induced by laser exhibit faster decay.

This scenario is also reflected in the time-frequency spectrum of HHG, as shown in Fig. 3D. For the fifth harmonics, the phase cancellation of NWP has no effect, as the intraband HHG relies on carrier population (i.e., the diagonal elements of ${\mathrm{\rho }}^{\mathrm{e}}$). The HHG trajectories from different beads exhibit in-phase behaviors. In contrast, for the 11th interband harmonics, which depend on the transition dipole moment (i.e., the off-diagonal elements of ${\mathrm{\rho }}^{\mathrm{e}}$), the HHG trajectories of beads have different phases and cancel each other out, especially in the tail of the laser pulse (after 15 fs). This leads to electronic decoherence and suppression of interband harmonics.

By suppressing the interference between HHG trajectories, NQEs give a one-to-one mapping between harmonic frequency and generation time (at least in each subcycle), with longer times producing higher frequencies (Fig. 4A) (26–28). Because the intensity of interband harmonics changes identically with electronic coherence (purity), as shown in Fig. 4B, by fitting the time-dependent harmonic intensities |I_HH|², the time of NQE-induced decoherence, T₂^NQE, can be directly probed. The retrieved T₂^NQE is ~1.0 fs, consistent with the decoherence time fitting from purity evolution, which is around 0.9 fs.

Fig. 4. Retrieve decoherence time from HHG spectroscopy.

(A) Schematics showing encoding of electronic coherence within time-frequency spectrograms of HHG. The decoherence over time corresponds to higher harmonics yield. (B) Normalized intensity of different harmonics |I_HH|² and electronic coherence (purity) as a function of time in H₃S. The black line is the fitting using exp.[─t/T₂^NQE]. The retrieved T₂^NQE and σ_ΔR^─1 as a function of temperature (C) and H isotope masses (D) obtained from TDDFT. The gray crosses label the T₂^NQE obtained from QMD. The without NQEs (w/o NQE) case is based on thermal ensembles, which only include the thermal fluctuations of classical nuclear motions.

The decoherence dynamics were further investigated from the HHG under different temperatures/hydrogen isotopes in Fig. 4 (C and D). The retrieved T₂^NQE show good agreement with those obtained from purity evolution via QMD. The effect of temperature/isotope in QMD is introduced by changing the size of the initial NWP. Compared to the classical case, NQEs significantly accelerate electron decoherence by an order of magnitude. As the temperature decreases, nuclear delocalization due to thermal effects diminishes, zero-point vibrations that are temperature-independent become dominant, and the decoherence time tends to converge to a constant. A similar dependence is observed for different isotopes.

Moreover, the decoherence time T₂^NQE induced by NQEs shows identical temperature and isotope dependence with the inverse of the nucleus delocalization width σ_ΔR^─1, i.e., T₂^NQE ∝ σ_ΔR^─1 (Fig. 1B). Since the delocalization σ_ΔR indicates the effective width of NWP, this relationship justifies the frozen Gaussian framework (62, 63, 80), which in turn enables the direct measurement of NWP width using HHG spectroscopy.

Given the universality of zero-point vibrations, we extended our study to various materials. The NQEs are found to significantly modify HHG spectra and decoherence dynamics (figs. S10 and S11), demonstrating their broad applicability. A general linear relationship between σ_ΔR, T₂, and √(M_vω) is revealed among different materials, where M_v is the averaged atomic mass and ω is the nuclear vibration frequency. The relationship can be understood by the characteristic length of the quantum harmonic oscillator wave function: $\frac{1}{2}{M}_{\mathrm{v}}{\mathrm{\omega }}^2{\mathrm{\Delta }R}^2\propto (\frac{1}{2}+n)\hslash \mathrm{\omega }$, where $n$ satisfies the Bose-Einstein distribution at a specific temperature and $\mathrm{\Delta }R$ is the nuclear amplitude that is approximately proportional to delocalization width σ_ΔR . Therefore, ${\mathrm{\Delta }R}^{-1}\propto {\mathrm{\sigma }}_{\mathrm{\Delta }R}^{-1}\propto \sqrt{M_{\mathrm{v}}\mathrm{\omega }}$, which allows us to estimate decoherence time through the static properties of the system, as shown in fig. S11.

Last, we discuss the experimental observations of NQEs in HHG. Although more HHG experiments are needed in the future for various materials including H₃S, it is a good alternative to reexamine the readily available HHG experiments of light-element solids with large NQE, e.g., graphene (81, 82). In figs. S10 and S11 and S18, we investigate the NQE and its effects on HHG in graphene. Thanks to high zero-point energy of carbon atoms (~0.1 eV), NQEs contribute to the delocalization of carbon nuclei even at room temperature. NQEs affect the spectroscopic distribution and elliptical polarization angle of various harmonics, producing the HHG spectra that are in good agreement with recent experiments (68, 69). These results suggest that NQEs may constitute an important contribution to the anomalous ellipticity dependence observed in experimental graphene HHG (68, 69).

MATERIALS AND METHODS

The NQE-induced decoherence of HHG has been first investigated in the high-pressure superconductor Im-3m H₃S (Fig. 1A) (70–72, 83, 84). Im-3m H₃S serves as a prototype example where the strong NQEs of H atoms enhance lattice stabilization and superconductivity (70–72). Ring-polymer molecular dynamics (RPMD) (85, 86) and ab initio molecular dynamics (AIMD) are used at different temperatures (50 to 300 K) to account for NQEs and classical thermal effects, respectively. These effects are taken into consideration by sampling the distorted lattice generated from a variety of molecular dynamics trajectories (57).

To study HHG, we combine AIMD and RPMD with first-principles time-dependent density functional theory simulations and semiclassical Bloch equation calculations (87, 88). For Im-3m H₃S system, a linearly polarized laser pulse with Gaussian envelope electric field $\mathbf{F}\left(t\right)={\mathbf{F}}_0\text{cos}\left(\mathrm{\omega }t\right)\text{exp}\left[-{(\mathrm{t}-{\mathrm{t}}_0)}^2/2{\mathrm{\sigma }}^2\right]$ is adopted with ω ~2.0 eV, σ ~6.0 fs, t₀ = 12 fs, and F₀ ~ 0.9 V/Å. For graphene, an elliptical polarized laser pulse with Gaussian envelope electric field $\mathbf{F}\left(t\right)={\mathbf{F}}_0{(1+{\mathrm{\epsilon }}^2)}^{-1/2}\text{exp}\left[-{(\mathrm{t}-t_0)}^2/2{\mathrm{\sigma }}^2\right](\text{cosω}t,\text{εsinω}t)$ is adopted in graphene plane with ω ~0.26 eV, σ ~28.0 fs, t₀ = 100 fs, ε = 0.3, and F₀ ~ 0.2 V/Å. By adding the vector potential $\mathbf{A}\left(t\right)=-c\int \mathbf{F}\left(t^{\prime }\right){\mathit{dt}}^{\prime }$ in the Hamiltonian and evolving the electronic wave functions in real time, the HHG spectra are obtained through the Fourier transform of time-dependent current $\mathbf{j}\left(t\right)=-\frac{e}{m}\int d\mathbf{r}\left\{{\mathrm{\psi }}^{\ast }(\mathbf{r},\mathrm{t})\left[-\mathrm{i}\hslash \mathbf{\nabla }+e\mathbf{A}\left(\mathrm{t}\right)\right]\mathrm{\psi }(\mathbf{r},\mathrm{t})+c\mathit{.}c\mathit{.}\right\}$. To track the NQE-induced electronic decoherence and NEE in H₃S, the time-dependent Schrödinger equation of nucleus-electron–coupled system is solved on the basis of the Born-Huang expansion, and we call it a QMD model (89). See Supplementary Materials for more details.

Supplementary Materials

This PDF file includes:

Notes S1 to S4

Figs. S1 to S18

References