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Generation of bright quantum high-order harmonic driven by combined coherent and bright squeezed vacuum light

This study demonstrates that combining strong coherent and weak bright squeezed vacuum lasers to drive high-order harmonic generation in atomic gas enables the creation of bright quantum high-order harmonics, where macroscopic propagation effects and gas pressure modulate the phase and quantum properties of the resulting attosecond bursts, paving the way for attosecond quantum spectroscopy.

Original authors: Sizuo Luo, Wentao Wang, Yaoshun Sun, Liyuan Wang, Lingrui Hu, Dajun Ding, Xiangyu Tang, Mingxuan Li, Jianmin Yuan

Published 2026-07-17✓ Author reviewed ⓘ

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CC BY 4.0

Original authors: Sizuo Luo, Wentao Wang, Yaoshun Sun, Liyuan Wang, Lingrui Hu, Dajun Ding, Xiangyu Tang, Mingxuan Li, Jianmin Yuan

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, bustling dance floor where electrons are the dancers. Usually, when we want to watch these tiny dancers move, we shine a bright, steady spotlight on them. This spotlight is a laser, and when it hits atoms, it can knock electrons loose or make them spin so fast they emit their own tiny flashes of light. Scientists have figured out how to make these flashes incredibly fast—so fast they last only "attoseconds." To put that in perspective, an attosecond is to a second what a second is to the entire age of the universe. These super-fast flashes, called "high-order harmonics," act like a high-speed camera, letting us freeze-frame the motion of electrons and see chemistry and physics happening in real-time.

But there's a catch. The lasers we use to create these flashes are usually "coherent," meaning they are perfectly orderly, like a marching band where everyone steps in perfect unison. While this is great for taking clear photos, it doesn't tell us much about the messy, jittery, quantum nature of the universe, where things are often uncertain and fluctuating. Recently, scientists started asking: What if we could make these super-fast flashes using light that isn't perfectly orderly? What if we used light that has a bit of "quantum jitter" built into it? This would allow us to take photos that not only show where the electrons are, but also reveal the hidden quantum secrets and correlations of the dance itself. This is the frontier of "attosecond quantum spectroscopy," a field trying to capture the spooky, probabilistic nature of reality at the fastest timescales possible.

In this study, a team of researchers decided to mix the orderly with the jittery to see what happens. They combined a strong, steady laser (the marching band) with a weaker, "squeezed" light source that has wild quantum fluctuations (the jittery dancer). They fired this mixed light into a cloud of krypton gas to generate high-order harmonics. What they found is that the "jittery" part of the light doesn't just add a little noise; it completely changes the rules of the game when the light travels through the gas.

The researchers discovered that different types of light flashes produced by this mixture—some that are perfectly even, some that are half-steps between the usual beats, and some that are the standard odd beats—each need a different amount of gas pressure to shine their brightest. It's as if the orderly laser and the jittery light are trying to march in step, but the jittery light keeps tripping the marchers in different ways depending on how crowded the dance floor (the gas) is. When the gas is too thin or too thick, the different types of flashes get out of sync and fade away.

Through detailed computer simulations that matched their real-world experiments, the team showed that the weak, jittery light messes with the "action phase" of the electrons. Think of the action phase as the internal clock of the electron's journey. The jittery light tweaks this clock just enough that the electrons arrive at the finish line at slightly different times. This tiny shift changes how the light waves from different atoms line up (a process called phase matching). Because the jittery light fluctuates from shot to shot, the perfect pressure to make the light bright also fluctuates.

The result is a new way to create "bright quantum high-order harmonics." The team showed that by carefully tuning the gas pressure, they could control not just the brightness of the light, but also its quantum personality. They found that some of the new light flashes (the even and half-integer ones) kept the "bunching" behavior of the jittery source, meaning the photons tended to arrive in groups rather than one by one. This proves that the quantum weirdness of the driving light was successfully transferred to the new, super-fast X-ray light. This work suggests that we can now build brighter, more controllable sources of quantum light, paving the way for a new kind of spectroscopy that can probe the deepest, most quantum secrets of matter with unprecedented clarity.

Technical Summary: Generation of Bright Quantum High-Order Harmonic Driven by Combined Coherent and Bright Squeezed Vacuum Light

Problem Statement
High-order harmonic generation (HHG) driven by intense coherent lasers has established attosecond pulses as a primary tool for tracing electron dynamics. Recent theoretical and experimental efforts have explored driving HHG with quantum light sources, such as bright squeezed vacuum (BSV), to probe quantum-mechanical correlations and transfer quantum states to emitted radiation. While single-atom responses to quantum driving fields have been investigated, the macroscopic propagation effects of quantum high-order harmonic generation (QHHG) in dense media remain largely unexplored. Specifically, it is unclear how the large photon-number fluctuations inherent to BSV fields influence phase matching, coherence, and the resulting quantum properties of bright XUV sources during propagation through an atomic gas.

Methodology
The study employs a combined experimental and theoretical approach to investigate QHHG in krypton (Kr) gas.

Experimental Setup: A 35-fs, 800-nm coherent laser pulse is split into two arms. One arm serves as the strong coherent driving field. The other arm generates a weak 1600-nm BSV field via spontaneous parametric down-conversion (SPDC) in a $\beta$ -barium borate (BBO) crystal. The two fields are recombined to form a quantum two-color field interacting with Kr gas in a 3-mm cell. The gas pressure is precisely controlled (5–60 Torr) to study phase-matching conditions. Harmonic spectra are recorded using a home-built XUV spectrometer. The BSV field is characterized by a second-order correlation function $g^{(2)}(0) = 2.324$ , confirming its non-classical nature.
Theoretical Modeling: The authors perform macroscopic propagation simulations by solving the one-dimensional time-dependent Schrödinger equation (1D-TDSE) coupled with Maxwell's equations. The driving field is modeled as a superposition of a noiseless 800-nm fundamental component and a 1600-nm BSV field characterized by a Husimi distribution. To account for quantum fluctuations, 1,000 single-shot realizations of the BSV field are randomly sampled. A wave-mixing model is established to analyze the phase-matching mechanism, decomposing the electron action phase into a coherent component and a perturbation term induced by the BSV field.

Key Contributions and Results

Generation of Quantum Harmonic Components:
The experiment successfully generates a spectrum containing not only the dominant odd harmonics ( $H_{2N+1}$ ) driven by the coherent field but also new spectral features induced by the BSV field:
- Half-integer harmonics ( $H_{(2N+1)\pm 1/2}$ ): Resulting from the absorption or emission of a single BSV photon.
- Even-order harmonics ( $H_{2N}$ ): Resulting from the absorption and emission of two BSV photons.
  The presence of these components confirms the transfer of quantum properties from the driving BSV field to the XUV radiation.
Distinct Pressure-Dependent Yields:
A critical finding is that different harmonic components exhibit significantly different optimal gas pressures for maximum yield, indicating distinct phase-matching conditions:
- Odd harmonics ( $H_{2N+1}$ ) peak around 20–24 Torr.
- Half-integer harmonics ( $H_{2N \pm 1/2}$ ) show pressure dependencies that vary by component (e.g., $H_{20.5}$ peaks at 27 Torr, while $H_{21.5}$ peaks at 13 Torr).
- Even harmonics ( $H_{2N}$ ) require much higher pressures, peaking between 30 and 44 Torr.
  This contrasts with the behavior of the odd harmonics driven solely by the coherent field, which remain consistent with standard phase-matching expectations.
Mechanism of Phase Mismatch Perturbation:
Theoretical analysis reveals that the BSV field perturbs the action phase ( $\phi_{act}$ ) of the electron wave packet. While the BSV field is too weak to significantly alter the ionization degree (and thus the refractive index contributions from neutral atoms and plasma), it introduces a perturbation term $\sigma$ to the action phase.
- The phase mismatch $\Delta k$ is modified by the gradient of this perturbation ( $\nabla \sigma$ ).
- Because the BSV field has a fluctuating amplitude and a 0- $\pi$ phase ambiguity, the perturbation $\sigma$ varies shot-to-shot.
- This results in different optimal gas pressures for the four distinct emission pathways (labeled P1–P4) that contribute to the final spectrum. The macroscopic yield is an integration of these fluctuating contributions.
Quantum Fluctuations and Correlation Functions:
The study analyzes the shot-to-shot intensity fluctuations and the second-order correlation function $g^{(2)}(0)$ .
- Odd harmonics ( $H_{2N+1}$ ) exhibit Gaussian-like statistics with $g^{(2)}(0) \approx 1.1$ , similar to coherent light.
- Half-integer harmonics ( $H_{2N \pm 1/2}$ ) show super-bunching characteristics with $g^{(2)}(0) \approx 2$ .
- Even harmonics ( $H_{2N}$ ) exhibit strong super-bunching with $g^{(2)}(0) > 4$ .
- The $g^{(2)}(0)$ values are found to be highly sensitive to gas pressure, particularly in phase-mismatched regimes, where macroscopic propagation effects significantly influence the observed quantum statistics.

Significance
The paper claims that these results lay a foundation for the establishment and application of attosecond quantum spectroscopy. By demonstrating that macroscopic propagation conditions (specifically gas pressure) can be used to control the relative intensity and quantum properties (fluctuations and bunching) of bright quantum XUV sources, the work provides a pathway to generate tailored attosecond pulses with specific quantum states. The ability to manipulate the phase-matching of sub-cycle attosecond bursts via a weak quantum perturbation offers a new degree of freedom for probing quantum-mechanical correlations in matter.

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