Attosecond-resolved quantum fluctuations of light and matter

TL;DR

The paper advances Attosecond science by merging quantum optics with XUV high-harmonic generation, showing that quantum fluctuations from bright squeezed vacuum can be imprinted onto attosecond pulses driven by a strong coherent field. By exploiting an in-situ attosecond interferometry scheme, the authors reconstruct the quantum state of XUV high harmonics via Husimi distributions and time-resolved quadrature analysis, revealing squeezed, non-Gaussian, and cat-like features in half-integer and even harmonics. They also resolve sub-cycle tunneling fluctuations induced by squeezed light, extracting complex phase perturbations $(\alpha_1,\alpha_2,\beta_1,\beta_2)$ on a shot-by-shot basis and correlating them across half-cycles. Collectively, the work establishes a foundation for attosecond quantum electrodynamics, enabling sub-cycle control and tomography of both electron and photon quantum states with potential for quantum-enhanced metrology in the XUV regime.

Abstract

Until recently, attosecond optical spectroscopy and quantum optics evolved along non-overlapping directions. In attosecond science, attosecond pulses have been regarded as classical waves, applied to probe electron dynamics on their natural time scale. Here, we transfer fundamental concepts of quantum optics into attosecond physics, enabling control of both the properties of the XUV attosecond pulses and the quantum fluctuations of matter on attosecond time scales. By combining bright squeezed vacuum (BSV) with a strong laser field to drive high-harmonic generation, we transfer the quantum properties of the BSV onto the resulting XUV attosecond pulses. Applying advanced attosecond interferometry, we reconstruct the quantum state of the XUV high harmonics and their associated attosecond pulses with attosecond precision. Finally, we resolve the squeezing of the electron's wavepacket during one of the most fundamental strong-field phenomena - field induced tunneling. The ability to measure and control quantum correlations in both electrons and XUV attosecond pulses establishes a foundation for attosecond quantum electrodynamics, manipulating the quantum state of electrons and photons with sub-cycle precision.

Figures (10)

• Figure 1: Generation of squeezed attosecond XUV pulses. a, HHG is driven by the combination of a strong coherent field of frequency $\omega$ and a weak bright squeezed vacuum (BSV) field of frequency $\omega/2$, producing a train of quantum attosecond pulses. The $\omega-\omega/2$ geometry acts as a temporal analogue of a four-slit interferometer, where four attosecond bursts are emitted per $\omega/2$ cycle. Each burst has a fluctuating amplitude and phase described by a complex phase shift $\sigma_1(\tau)$ and $\sigma_2(\tau)$, representing the perturbation of the BSV to the quantum action of the electron. Their interference gives rise to a comb of integer and half-integer squeezed harmonics. b, Schematic illustrations of phase space diagrams for harmonics of different orders. Odd harmonics are approximately in a coherent state, half-integer harmonics exhibit squeezed fluctuations, and even harmonics exhibit phase-space displacement and squeezed fluctuations. c, Experimentally resolved HHG spectra generated by a coherent field only (blue) and the two-color field, composed of coherent and BSV sources (red). The perturbative field leads to the appearance of half-integer (blue) and even (pink) harmonics.
• Figure 2: Photon statistics of the XUV harmonics for a fixed and scanned two-color delay.a, Second order correlation function ($g^{(2)}$ ) as a function of harmonic order. Odd harmonics exhibit Gaussian statistics, with $g^{(2)} \approx 1$. Half-integer and even harmonics exhibit a long-tailed distribution, with $g^{(2)}$ values clustering around 2.3 and 4.8, respectively. These values correspond to $g^{(2)}$ and the photon number kurtosis $\langle\hat{n}^4\rangle/\langle\hat{n}^2\rangle^2$ for the input BSV, as indicated by dashed gray lines in a (Supplementary Information). b Intensity distributions of three types of harmonics: $2N+1$ (17), 2N (16), $2N + \frac{1}{2}$ (16.5). c, One and three BSV-photon (orange dashed arrows) processes interfere to generate half-integer harmonics. d, Mean value of different optical frequencies as a function of the two-color delay. e, $g^{(2)}$ and mean oscillations of selected harmonics (11,11.5, 14.5 ,15.5) as a function of the two-color delay (Supplementary Information). The oscillations of $g^{(2)}$ are in phase with the mean value oscillations. Notably, very weak, yet clearly resolved, $g^{(2)}$ oscillations are observed in harmonic 11.
• Figure 3: Reconstruction of sub-cycle quantum fluctuations of tunneling in atoms driven by squeezed lighta, The coherent field (blue) induces pairs of trajectories, labeled 1 and 2, every half cycle of the fundamental field. The correlation between $\beta_1$ and $\beta_2$ reflects a correlation between two instantaneous tunneling events: when the first exhibits excess noise (a broader $\beta_1$ spread), the subsequent event, half a cycle later, becomes quieter (a narrower $\beta_2$ spread). b, The correlation between $\alpha_1$ and $\alpha_2$ reflects a correlation between two successive trajectories when the first exhibits excess noise (a broader $\alpha_1$ spread), the subsequent event, half a cycle later, becomes quieter (a narrower $\alpha_2$ spread).
• Figure 4: Quantum state tomography of high-harmonic emission. (a–c) Experimentally reconstructed Husimi distributions for harmonics at $\lambda=66$nm (harmonic 12), 64nm (harmonic 12.5), and 61.5nm (harmonic 13). (d–f) Corresponding theoretical calculations (supplementary information). a,d Even harmonic orders (12) appears as a squeezed-like state elongated along one quadrature. b,e Half-integer harmonic orders (12.5) exhibits a non-Gaussian, two-lobed structure with a central hole, reflecting the nonlinear mapping of the Gaussian BSV fluctuations into the harmonic quadratures. c,f Odd harmonic orders (13) show a dominant coherent-like lobe with a trailing depletion toward the origin of phase space, consistent with energy redistribution into neighboring harmonics half-integer and even harmonics. The close correspondence between experiment and theory highlights the robustness of our interferometric quantum-state tomography scheme.
• Figure 5: Time-domain reconstruction of attosecond emission. Reconstructed attosecond waveforms of the two-color HHG source showing the mean electric field $\langle E(t)\rangle$ (black), the instantaneous variance $\Delta E^2(t)$ (dashed purple), and its smoothed envelope (solid purple). The rapid sub-cycle oscillations of $\Delta E^2(t)$ reveal attosecond-scale noise bursts --direct evidence of quantum noise upconversion from the driving squeezed light into the XUV regime. The slower envelope follows the field-variance profile of the input bright squeezed vacuum, indicating that the underlying quantum noise structure is preserved through the high-harmonic generation process.