🔬 optics

Symmetry breaking by quantum light in solid-state high-harmonic generation

This paper demonstrates that quantum fluctuations in circularly polarized light can break dynamical symmetry in solid-state high-harmonic generation, thereby enabling the production of classically forbidden harmonics in materials like graphene and MoS $_2$ while preserving crystal symmetry.

Original authors: P. Stammer, J. Rivera-Dean, D. Kim, A. Chacón, W. Gao, C. Granados

Published 2026-05-28

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: P. Stammer, J. Rivera-Dean, D. Kim, A. Chacón, W. Gao, C. Granados

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

Imagine you are trying to bake a very specific type of cake. In the world of physics, this "cake" is a beam of light made of extremely high-energy waves, created when a material (like a crystal) is hit by a powerful laser. This process is called High-Harmonic Generation (HHG).

Usually, the rules of the kitchen (the laws of physics and the shape of the crystal) are very strict. They act like a bouncer at a club: "You can only enter if you are an odd-numbered guest." If the crystal is shaped like a hexagon (like graphene), the bouncer is even stricter: "Only guests who are 1, 5, 7, 11, etc., can enter. No multiples of 3 allowed."

For decades, scientists treated the laser light hitting the crystal like a perfect, predictable machine—a "classical" field. They assumed the light was as orderly as a marching band. But this new paper asks: What happens if the light isn't a perfect marching band, but a slightly chaotic jazz band with random fluctuations?

Here is the breakdown of what the researchers found, using simple analogies:

1. The Strict Bouncer (Classical Rules)

In the old way of thinking, the light and the crystal have a "dance" together. The crystal has a specific symmetry (like a snowflake or a honeycomb). The laser light also has a symmetry (like spinning in a circle).

The Rule: When they dance, the crystal only allows certain "steps" (harmonics) to happen. If the crystal says "No multiples of 3," then no multiples of 3 appear in the light coming out. It's a hard rule.

2. The Chaotic Jazz Band (Quantum Light)

The researchers decided to stop using the "perfect marching band" laser. Instead, they used quantum light.

The Analogy: Imagine the laser light is a river. A classical laser is a smooth, straight river. A quantum laser is a river with ripples and eddies (fluctuations). Even if the river flows in the same direction on average, the water is constantly jiggling and swirling in unpredictable ways.
The Experiment: They shone this "jiggly" quantum light onto two different materials:
- Graphene: A flat sheet of carbon atoms shaped like a honeycomb (6-sided symmetry).
- MoS2 (Molybdenum Disulfide): A crystal with a 3-sided symmetry.

3. Breaking the Rules (Symmetry Breaking)

Here is the magic trick the paper discovered:
The "ripples" in the quantum light act like a subtle thief. They sneak in and break the strict dance rules without actually changing the shape of the crystal.

What happened: Because the light was "jiggling" (fluctuating), it created a new pathway for the energy. It opened a "back door" that the strict bouncer didn't see coming.
The Result: The materials started producing harmonics that were previously forbidden.
- In Graphene, which usually blocks multiples of 3, the quantum light allowed them to appear.
- In MoS2, the quantum light allowed harmonics that the classical rules said should be impossible.

The Metaphor: Imagine a lock that only opens with a key turned exactly 90 degrees. The classical light is a hand turning the key perfectly. The quantum light is a hand that is slightly shaking while turning. That shake allows the tumblers in the lock to wiggle just enough to open the door, even though the key wasn't turned perfectly.

4. Why This Matters (The "New Tool")

The paper claims this is a new way to control light.

Tuning the Chaos: The researchers found that by adjusting how much the light "jiggles" (the intensity of the quantum fluctuations), they could control exactly how many of these "forbidden" harmonics appear. It's like turning a dial to decide how many extra guests get into the club.
Making Shorter Pulses: Because the quantum light allows more types of harmonics to appear (filling in the gaps that were previously empty), the resulting light pulses become shorter and cleaner.
- Analogy: If you are trying to make a short, sharp sound (like a drum beat), you need many different musical notes playing together. If the rules only let you play the 1st, 3rd, and 5th notes, your sound is long and wobbly. If the quantum rules let you play the 2nd, 4th, and 6th notes too, you can pack all the notes together to make a much sharper, shorter "click."

5. The "Ghost" in the Machine

The paper also looked at the "personality" of the light coming out.

Classical Light: Acts like a calm, predictable crowd (Poissonian statistics).
Quantum-Generated Light: The forbidden harmonics created by the quantum light act like a rowdy, super-enthusiastic crowd (super-bunching). They are "bunched" together in a way that classical light never is. This proves that the light coming out carries the "fingerprint" of the quantum jiggles that created it.

Summary

The paper shows that by using quantum light (light with inherent random fluctuations) instead of perfect classical light, scientists can break the strict symmetry rules that usually govern how crystals interact with lasers.

This allows them to:

Generate light frequencies (harmonics) that were previously impossible.
Control these new frequencies by adjusting the "amount of jiggling" in the light.
Create shorter, sharper bursts of light (attosecond pulses) by filling in the gaps in the light spectrum.

It's like discovering that if you shake the ingredients just right, you can bake a cake with flavors that the recipe book said were impossible to make.

Technical Summary: Symmetry Breaking by Quantum Light in Solid-State High-Harmonic Generation

Problem Statement
In solid-state high-harmonic generation (HHG), the emission spectra are strictly governed by the interplay between the driving laser field and the crystal's spatial symmetries. These dynamical symmetries impose selection rules that dictate which harmonic orders are allowed or forbidden. For instance, in materials with specific rotational symmetries (e.g., $C_3$ for MoS $_2$ or $C_6$ for graphene) driven by circularly polarized light, certain harmonic orders (such as $q=3j$ ) are classically forbidden. While previous research has explored how quantum fluctuations in light can alter selection rules in atomic gases, the role of quantum light in breaking dynamical symmetries within the complex environment of solid-state systems remains an open question. Specifically, it is unclear how the intrinsic quantum fluctuations of a driving field modify the symmetry-imposed constraints on harmonic emission in crystals.

Methodology
The authors investigate the nonlinear response of two distinct solid-state materials—graphene (exhibiting $C_6$ rotational symmetry and inversion symmetry) and Molybdenum disulfide (MoS $_2$ , exhibiting $C_3$ rotational symmetry and lacking inversion symmetry)—to circularly and bicircularly polarized quantum light.

Theoretical Framework: The study employs a quantum optical approach where the driving field is decomposed into a classical mean field ( $E_{cl}$ ) and quantum fluctuations ( $\delta \hat{E}$ ). The total Hamiltonian includes the crystal Hamiltonian, the classical dipole interaction, and a fluctuation-induced interaction term.
Symmetry Analysis: The authors analyze the dynamical symmetry operations $U_n = R_{2\pi/n} T_{\tau/n}$ (spatial rotation combined with time translation). They demonstrate that while the mean-field Hamiltonian preserves these symmetries, the higher-order field correlations (specifically the variance $\Delta E^2(t)$ ) do not remain invariant under the same transformations, leading to dynamical symmetry breaking.
Numerical Simulation: The nonlinear response is calculated by solving the time-dependent density matrix equations in the Wannier gauge using tight-binding Hamiltonians (Haldane model for graphene and a three-band model for MoS $_2$ ). The quantum driving field is modeled as a squeezed state (amplitude-squeezed and phase-squeezed cases are considered). The total HHG spectrum is obtained by averaging the coherent-state-driven spectra over the Husimi $Q$ distribution.
Pulse Synthesis and Quantum Signatures: The temporal structure of the emitted radiation is reconstructed from the complex field amplitudes. Additionally, the quantum optical nature of the emitted harmonics is characterized using Wigner functions and the second-order correlation function $g^{(2)}(0)$ .

Key Contributions and Results

Dynamical Symmetry Breaking by Quantum Fluctuations:
The central finding is that quantum fluctuations in the driving field break the dynamical symmetry of the system, even when the average field remains symmetric. This occurs because the quantum fluctuations introduce a "helicity-zero" interaction channel (associated with the expectation value $\langle \hat{a}_R \hat{a}_L \rangle$ ) that is absent in classical descriptions. This channel relaxes the conventional selection rules.
Generation of Classically Forbidden Harmonics:
- Graphene ( $C_6$ ): Under classical circular driving, only harmonics $q = 6j \pm 1$ are allowed; $q=3j$ and even harmonics are forbidden. With quantum light, the $q=3j$ harmonics (e.g., $q=3, 9$ ) emerge, while even harmonics remain forbidden due to the preservation of the crystal's inversion symmetry.
- MoS $_2$ ( $C_3$ ): Classically, $q=3j$ harmonics are forbidden, while $q=3j \pm 1$ are allowed (both even and odd due to broken inversion symmetry). Quantum fluctuations enable the generation of the forbidden $q=3j$ harmonics.
- Control: The intensity of these forbidden harmonics scales with the squeezing intensity ( $I_q$ ) of the driving field, providing a tunable control parameter for symmetry engineering.
Temporal Pulse Synthesis:
The inclusion of forbidden harmonics ( $q=3j$ ) fills spectral gaps, resulting in a more continuous and uniform spectral distribution. This leads to the synthesis of shorter pulse trains with simpler temporal structures. Specifically, the central pulse duration is reduced by approximately 25% (from 1.6 fs to 1.2 fs in the simulated MoS $_2$ case) compared to coherent driving, moving closer to the Fourier limit without requiring single-cycle driving fields.
Quantum Optical Signatures:
The emitted harmonics driven by quantum light exhibit distinct quantum statistical properties. The Wigner functions of the forbidden harmonics show squeezed-like Gaussian distributions. Furthermore, the second-order correlation function $g^{(2)}(0)$ for these fluctuation-induced harmonics is found to be $\ge 3$ (indicating superbunching), in contrast to the near-Poissonian statistics ( $g^{(2)}(0) \approx 1$ ) of classically allowed harmonics.

Significance and Claims
The paper establishes quantum states of light as a new degree of freedom for controlling nonlinear optical processes in solids. The authors claim that this mechanism offers a route to "all-optical symmetry engineering," allowing for the manipulation of harmonic selection rules and the generation of attosecond pulses in solids without the need for single-cycle drivers.

The work highlights a fundamental difference between atomic and solid-state HHG: in solids, the crystal symmetry imposes specific constraints that quantum fluctuations can selectively relax (breaking dynamical symmetry while preserving spatial crystal symmetry). This opens avenues for:

Symmetry Engineering: Tailoring harmonic spectra by tuning the quantum state of the driver.
Attosecond Science: Generating shorter, more isolated attosecond pulses in condensed matter systems.
Quantum Spectroscopy: Using the quantum statistics of the emitted light (e.g., $g^{(2)}(0)$ and Wigner functions) to probe the symmetry properties and electronic dynamics of the material.

The authors conclude that this approach leverages the larger damage threshold of materials under quantum light drivers and the increased bandwidth from symmetry relaxation to advance solid-state attosecond science.