🔬 optics

Even-harmonic generation through nonequilibrium steady-state symmetry breaking

This paper demonstrates that in a boundary-driven Su-Schrieffer-Heeger chain, dissipative dynamics can break the inversion symmetry of the nonequilibrium steady state to generate even harmonics in high-harmonic generation, thereby establishing a direct link between the emitted even-harmonic intensity and the transport current without modifying the underlying Hamiltonian.

Original authors: Milad Jangjan, Arman Kashef, Siamak Pooyan, Stefan Scheel, Dieter Bauer

Published 2026-07-31

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Milad Jangjan, Arman Kashef, Siamak Pooyan, Stefan Scheel, Dieter Bauer

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

The Dance of Light and Electrons

Imagine a world where light doesn't just bounce off things but actually breaks them apart into a rainbow of new colors. This is the realm of High-Harmonic Generation (HHG), a process where a super-strong laser hits a material and forces its electrons to dance so wildly that they scream out light at frequencies much higher than the laser itself. Think of it like plucking a guitar string so hard it doesn't just make a note, but suddenly starts singing in a squeaky, high-pitched whistle. Scientists love this because the "whistle" tells them everything about the material's hidden structure.

Usually, there's a strict rule to this musical performance. If the material is perfectly symmetrical—like a mirror image of itself on the left and right—it can only sing "odd" notes (like the 1st, 3rd, or 5th harmonics). The "even" notes (2nd, 4th, 6th) are forbidden by the laws of symmetry. For decades, if scientists saw an even note, they assumed the material itself was broken or asymmetrical. But what if the material was perfectly symmetrical, yet the music still changed? What if the secret wasn't in the shape of the room, but in the flow of traffic inside it? This is the puzzle that a team of physicists at the University of Rostock set out to solve.

The One-Way Street in a Symmetrical Room

In their new study, the researchers explored a fascinating scenario using a theoretical model called the Su-Schrieffer-Heeger (SSH) chain. Imagine a long, perfectly symmetrical row of stepping stones (atoms) where electrons can hop from one to the next. In a normal, quiet room, this chain is perfectly balanced; the electrons hop left and right with equal ease, and the system has no preferred direction. If you shine a laser on this symmetrical chain, it should only produce those forbidden "odd" harmonics.

However, the researchers decided to turn this quiet room into a busy highway. They attached a "source" reservoir (a pump) to one end of the chain and a "sink" reservoir (a drain) to the other. This setup forces electrons to constantly flow through the chain, creating a steady DC current. It's like opening a valve at one end of a perfectly symmetrical pipe and a drain at the other; the water rushes through, creating a one-way flow.

Here is the magic: The researchers found that even though the chain itself (the "Hamiltonian") remained perfectly symmetrical, the flow of electrons broke the symmetry of the system's state. The electrons weren't just sitting still or hopping randomly anymore; they were rushing in a specific direction. This "nonequilibrium steady state" meant the system was no longer symmetric in how it behaved, even if the building blocks were.

Breaking the Rules of Music

The team simulated this scenario using advanced computer models based on the Lindblad master equation, a mathematical tool used to describe how open systems (systems that exchange energy or particles with their surroundings) behave. They shone a laser pulse on this current-carrying chain and listened to the light it emitted.

The results were clear and surprising. When the current was flowing, the system started singing the "forbidden" even harmonics. The intensity of these new even notes was directly linked to how strong the current was. In their simulations, with a dissipation strength (the rate of pumping and draining) of 0.001, the even harmonics appeared prominently. In contrast, when the flow was almost stopped (a dissipation strength of 10⁻¹²), the system went back to singing only the odd harmonics, obeying the old rules.

The researchers showed that this wasn't because the atoms moved out of place or the chain was physically broken. Instead, the mere existence of the current created a "symmetry breaking" in the quantum state of the electrons. It's as if the traffic jam itself changed the rules of the road, allowing cars to turn in directions they were previously forbidden to.

The Waiting Game

The study also revealed a crucial detail about when this happens. The system doesn't switch to this new state instantly. The researchers found that the electrons need time to settle into their flowing rhythm. If they turned on the laser too quickly, before the current had fully established itself, the even harmonics didn't appear.

They calculated that the time it takes for the system to reach this steady state depends on how strong the "pump" and "drain" are. With a very weak pump (small γ), the system takes a very long time to settle. In their simulations, they showed that if they waited for a relaxation time of 10⁶ (a million time steps), the even harmonics were barely visible. But if they waited longer, up to 10⁷ time steps, the even harmonics became strong and clear. This proves that the effect is tied to the establishment of a true, stable flow of electricity, not just a fleeting moment.

Why It Matters

This work suggests a powerful new way to look at materials. Previously, scientists thought that if they saw even harmonics, the material itself must be asymmetrical. This paper shows that you can have a perfectly symmetrical material that acts asymmetrical simply because it's carrying a current.

The authors propose that this mechanism could be used as a new kind of "ultrafast nonlinear spectroscopy." By listening for these even harmonics, scientists might be able to detect and measure tiny electrical currents in quantum materials without needing to touch them or break them apart. It turns the emission of light into a sensitive probe for the invisible flow of electricity.

While this study was a simulation on a theoretical one-dimensional chain, it opens the door to exploring how similar effects might happen in real, complex materials. The researchers suggest that future work could look at how this plays out in materials with more complex interactions or in higher dimensions, potentially revealing new ways to control light and electricity in the quantum world. For now, they have shown that in the quantum realm, the direction of the flow can change the song just as much as the shape of the instrument.

Technical Summary: Even-Harmonic Generation through Nonequilibrium Steady-State Symmetry Breaking

Problem Statement
In inversion-symmetric quantum systems driven by linearly polarized light, high-harmonic generation (HHG) is strictly constrained by symmetry to emit only odd harmonics. The observation of even harmonics is traditionally interpreted as a signature of inversion-symmetry breaking, typically achieved through structural asymmetry, static bias fields, or asymmetric driving. However, a fundamental question remains regarding how nonequilibrium transport currents modify ultrafast nonlinear optical responses in centrosymmetric systems where the underlying Hamiltonian retains inversion symmetry. While recent studies suggest that finite currents can induce asymmetric electronic distributions, the specific mechanism by which dissipative boundary driving in open quantum systems breaks the selection rules for HHG without altering the lattice structure has remained largely unexplored.

Methodology
The authors investigate this phenomenon using a one-dimensional Su-Schrieffer-Heeger (SSH) chain coupled to source and sink reservoirs, forming an open quantum system. The theoretical framework relies on the Lindblad master equation to describe the dissipative dynamics.

Model: The system consists of a dimerized lattice with alternating hopping amplitudes ( $v$ and $w$ ). The chain is coupled to a gain reservoir (source) at the last site and a loss reservoir (sink) at the first site with rates $\gamma_g$ and $\gamma_l$ , respectively.
Dynamics: The evolution of the many-body density matrix $\rho$ is reduced to the dynamics of the single-particle correlation matrix $C_{mn}(t) = \langle c^\dagger_m c_n \rangle$ . The authors derive a closed equation of motion for $C(t)$ based on the Lindblad formalism, which includes coherent unitary evolution driven by the Hamiltonian and dissipative terms accounting for particle exchange with the reservoirs.
Steady State: The nonequilibrium steady state (NESS) is obtained analytically by solving the resulting Sylvester (Lyapunov) equation for the correlation matrix, $MC_{ss} + C_{ss}N = -\Gamma^+$ . This approach allows for the exact determination of the steady-state density matrix without time integration.
Driving and HHG Calculation: A spatiotemporal laser field is applied to the system, localized in the bulk to avoid direct driving of the reservoir-coupled sites. The HHG spectrum is calculated from the Fourier transform of the time-dependent expectation value of the current operator, $\langle \hat{J}(t) \rangle$ .
Validation: To address experimental realism, the authors also simulate the time-dependent Lindblad evolution starting from an initial state to observe the relaxation process towards the steady state, comparing the transient dynamics with the exact steady-state solution via fidelity analysis.

Key Contributions

Symmetry Breaking Mechanism: The paper demonstrates that the selection rule for HHG in open quantum systems is governed by the symmetry of the full Liouvillian superoperator and the resulting steady-state density matrix, rather than solely by the symmetry of the Hamiltonian.
Current-Induced Even Harmonics: It is shown that dissipative boundary driving generates a finite DC current in an inversion-symmetric SSH chain. This current-carrying steady state breaks the inversion symmetry of the density matrix ( $P\rho_{ss}P^{-1} \neq \rho_{ss}$ ), thereby lifting the suppression of even harmonics.
Direct Correlation: The study establishes a direct link between the magnitude of the transport current and the intensity of the generated even harmonics. The emergence of even harmonics is not due to structural lattice distortion but is a direct consequence of the nonequilibrium steady-state symmetry breaking.

Results

Spectral Evolution: In the limit of vanishing dissipation ( $\gamma \to 0$ ), the system behaves as a closed chain, and the HHG spectrum is dominated by odd harmonics, consistent with conventional selection rules. As the dissipation strength $\gamma$ increases, even-order harmonics emerge and their intensity grows, modifying the low-order spectrum.
Current Dependence: The steady-state current increases with the reservoir coupling strength. The appearance of even harmonics coincides with the regime where the system supports a significant nonequilibrium current.
Relaxation Dynamics: The analysis of the Liouvillian gap reveals that in the weak-dissipation regime, the relaxation time to the steady state diverges. Consequently, the emergence of even harmonics depends critically on the available relaxation time ( $\tau$ ). If the system is probed before fully relaxing to the steady state, the memory of the initial inversion-symmetric state suppresses even harmonics. Fidelity calculations confirm that the crossover to even-harmonic generation is governed by the timescale of Liouvillian relaxation.
Particle Distribution: The steady-state particle density transitions from a vacuum-like state at very low $\gamma$ to a half-filled state at high $\gamma$ , with the regime of significant even-harmonic generation corresponding to the establishment of a robust current-carrying state.

Significance
The paper claims to uncover a mechanism for even-harmonic generation that relies solely on nonequilibrium steady-state symmetry breaking induced by dissipative transport, without requiring any modification to the underlying Hamiltonian or structural inversion-symmetry breaking of the lattice. This finding redefines the understanding of harmonic selection rules in open quantum systems, identifying the symmetry of the nonequilibrium density matrix as the determining factor.

The authors posit that their results provide a route to probing transport currents through ultrafast nonlinear spectroscopy in centrosymmetric quantum systems. By establishing HHG as a sensitive probe of current-carrying nonequilibrium steady states, the work suggests new opportunities for exploring the interplay between transport, dissipation, and symmetry breaking in driven open quantum materials. The study remains modest in scope, focusing on noninteracting one-dimensional chains, and identifies the extension to interacting systems, higher dimensions, and non-Markovian descriptions (e.g., via Keldysh Green's functions) as necessary future directions.