🔬 materials science

Quantum heat current in Terahertz-driven phonon systems

This paper investigates the ultrafast quantum thermodynamics of THz-driven optical phonons within an open quantum system framework, demonstrating that intense laser pulses can induce significant non-Markovian deviations in heat current, thereby offering a new pathway to control dissipation in solid-state systems.

Original authors: Yulong Qiao, Richard. Matthias Geilhufe

Published 2026-08-28

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CC BY 4.0

Original authors: Yulong Qiao, Richard. Matthias Geilhufe

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 a solid crystal not as a rigid, static block, but as a vast, intricate lattice of atoms constantly vibrating. These vibrations, known as phonons, are the fundamental way heat and sound travel through materials. For decades, scientists have understood how these vibrations behave when they are calm or when they are gently warmed, relying on rules that assume the environment around them forgets any interaction almost instantly. However, a new frontier has opened with the invention of incredibly fast, powerful laser pulses. These pulses, operating at terahertz frequencies, can shake the atoms of a crystal with such intensity and speed that they force the material into states it never reaches in nature. The question facing researchers is whether our old, simplified rules still apply when we push matter this hard, or if the atoms begin to remember their past interactions in ways that change how heat flows.

A team of physicists at Chalmers University of Technology has taken a deep dive into this question by treating the vibrating atoms not just as classical objects, but as quantum systems. In the quantum world, particles do not just sit still; they possess a minimum amount of jittery motion even at absolute zero, known as zero-point motion. The researchers focused on a specific type of vibration found in materials like strontium titanate, which acts like a soft, squishy spring that can be tuned by temperature. They built a detailed theoretical model to simulate what happens when a terahertz laser pulse hits these atoms. Instead of assuming the surrounding environment acts like a simple, forgetful sponge that absorbs energy immediately, they modeled it as a complex system with a "memory." This means the environment can hold onto energy for a moment and then give it back to the vibrating atoms, a phenomenon known as non-Markovian behavior.

The core of their investigation involved calculating the flow of heat between the laser-driven atoms and their thermal environment. In everyday terms, they tracked how much energy the atoms absorbed from the laser and how much they dumped back into the surrounding lattice. Their calculations revealed a striking difference depending on how long the laser pulse lasted. When the researchers used a longer pulse, lasting about 5 picoseconds, the atoms behaved in a predictable, standard way: they absorbed energy and then slowly released it into the environment, with the heat flowing in one direction until everything settled down. This matched the traditional, simplified view of physics where the environment has no memory.

However, the story changed completely when they shortened the laser pulse to just 1 picosecond. In this ultrafast scenario, the atoms did not simply release their energy and move on. Instead, the heat flow reversed direction. After the initial burst of energy, the atoms began to pull energy back from the environment, causing the vibration to revive and oscillate in a way that defied the standard rules. This backflow of energy is a clear signature of non-Markovian dynamics, proving that the environment was indeed remembering the interaction and returning the energy to the system. The researchers found that this effect was driven by the quantum nature of the vibrations; even at very low temperatures where classical heat would vanish, the quantum "jitter" of the atoms ensured that this complex exchange of energy continued.

The study demonstrates that ultrafast laser pulses act as a sensitive probe, capable of revealing hidden complexities in how solids handle heat. By simply changing the duration of the laser pulse from 5 picoseconds down to 1 picosecond, the researchers could switch the material's behavior from a simple, forgetful state to a complex, memory-retaining one. This suggests that the standard assumption of a "memoryless" environment is often an oversimplification that fails when materials are driven by intense, rapid forces. The findings imply that to truly understand and control quantum materials, scientists must account for these memory effects, which can be manipulated by tuning the speed of the laser driving them.

Looking ahead, the authors suggest that these theoretical predictions could be tested in real-world experiments using pump-probe techniques. In such an experiment, a terahertz pulse would excite the material, and an ultrafast X-ray beam would take snapshots of the atomic motion with femtosecond precision. If the researchers observe the atoms suddenly re-accelerating or reviving their motion after the laser pulse has ended, it would be direct evidence of this energy backflow. Such an observation would not only confirm the presence of non-Markovian dynamics but also provide a new way to map the specific interactions between atoms and their environment, offering a deeper toolkit for designing future quantum materials.

Technical Summary: Quantum Heat Current in Terahertz-Driven Phonon Systems

Problem Statement
The manipulation of quantum materials via high-intensity, ultrafast terahertz (THz) laser pulses has enabled the resonant driving of optical phonon modes, leading to phenomena such as transient ferroelectricity and dynamical multiferroicity. While classical frameworks have successfully described these phonon dynamics phenomenologically, they fail at low temperatures or in materials with light atoms where quantum nuclear effects (e.g., zero-point motion) become dominant. Furthermore, standard classical stochastic models often rely on the Markovian approximation, assuming a memoryless bath with a constant decay rate. However, in realistic solid-state systems, phonon modes can couple strongly to slow degrees of freedom (e.g., structural phase transition soft modes), rendering the effective bath non-Markovian (possessing memory effects). There is a lack of a microscopic quantum description that accounts for both quantum fluctuations and non-Markovian dissipation to understand how heat is exchanged between a driven phonon and its thermal environment.

Methodology
The authors construct a microscopic quantum model treating the driven optical phonon as an open quantum system coupled to a thermal bath.

Hamiltonian: The system is described by a Caldeira–Leggett-type Hamiltonian. The phonon is modeled as a harmonic oscillator with frequency $\omega_0$ driven by a time-dependent Gaussian laser force $F(t)$ . The bath is represented as a collection of harmonic oscillators coupled linearly to the phonon coordinate, including a counter-term to preserve the physical phonon frequency.
Dynamics: The evolution is governed by a generalized quantum Langevin equation in the Heisenberg picture. This equation incorporates a friction kernel $\gamma(t)$ and a quantum noise operator $\hat{\xi}(t)$ , derived from the bath's spectral density $J(\omega)$ .
Heat Current Formulation: The authors derive a general expression for the quantum heat current $J(t)$ flowing from the bath to the phonon. This formulation does not rely on the weak-coupling approximation and is exact for the defined model. The heat current is expressed as the sum of a dissipative term (governed by the friction kernel and velocity correlations) and a fluctuation term (governed by the noise correlation function).
Numerical Analysis: The study employs a Lorentzian spectral density for the bath to model a structured environment. The authors analyze the system's response to laser pulses of varying durations ( $\tau$ ), comparing "long" pulses (narrow frequency spectrum) against "short" pulses (broad frequency spectrum) to probe the frequency dependence of the bath's response function.

Key Contributions

Derivation of Quantum Heat Current: The paper provides an exact derivation of the quantum heat current for a driven phonon system coupled to a structured bath, explicitly separating dissipative and fluctuation contributions within a non-Markovian framework.
Identification of Non-Markovian Signatures: The work demonstrates that the non-Markovian nature of the bath (memory effects) is not merely a theoretical nuance but has observable consequences in the time-domain dynamics of the phonon displacement and the heat current.
Role of Ultrafast Driving: The authors establish that the duration of the driving pulse is a critical parameter. While long pulses effectively average out the bath's frequency structure (appearing Markovian), ultrafast short pulses possess broad frequency spectra that can resolve the non-Lorentzian structure of the bath's spectral density.

Results

Phonon Dynamics: For long pulses ( $\tau = 5.0$ ps), the phonon displacement $\langle \hat{Q}(t) \rangle$ exhibits monotonic decay, mimicking standard Markovian relaxation. In contrast, short pulses ( $\tau = 1.0$ ps) induce non-Markovian dynamics characterized by rapid decay followed by "revivals" or quantum beats in the phonon amplitude.
Heat Current Behavior: Correspondingly, the heat current $J(t)$ for long pulses is unidirectional (energy flows from phonon to bath). For short pulses, the heat current changes sign, indicating a transient backflow of energy from the bath to the phonon. This backflow is a direct signature of non-Markovian memory effects.
Quantification of Non-Markovianity: The authors introduce a metric $M_J$ , defined as the ratio of backflow energy to total exchanged energy. $M_J$ is near zero in the Markovian limit but becomes significantly positive in the presence of strong non-Markovian coupling.
Quantum vs. Classical: The study highlights that while classical heat currents vanish as temperature approaches zero, the quantum heat current components (dissipation and fluctuation) saturate at finite values determined by zero-point motion, even though their net sum vanishes in equilibrium.

Significance and Claims
The paper claims that ultrafast THz pulses serve as a sensitive probe for non-Markovian dissipation in driven solid-state systems. By systematically varying the pulse duration, one can transition from observing effective Markovian behavior to resolving the underlying non-Markovian physics of the phonon-bath interaction. The authors suggest that these effects could be experimentally accessed via pump-probe experiments (e.g., using ultrafast X-ray diffraction) on materials like SrTiO $_3$ , where the observation of anomalous re-acceleration or revivals in lattice motion would provide direct information about the bath spectral density. The work provides a theoretical pathway to control and characterize non-Markovian quantum thermodynamics in quantum materials.

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