⚛️ quantum physics

Work extraction from long-lived quantum coherence of a three-level system

This paper demonstrates that long-lived quantum coherence in a three-level system with degenerate or nearly degenerate excited states can be converted into population asymmetry via energy-preserving unitary operations to maximize single-shot work extraction, offering a thermodynamic advantage for future quantum thermal machines.

Original author: Wenjing Chen, Si-Wei Han, Ziye Wang, Xiaoshan Feng, Jun Feng

Published 2026-11-03

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CC BY 4.0

Original author: Wenjing Chen, Si-Wei Han, Ziye Wang, Xiaoshan Feng, Jun Feng

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 author. For technical accuracy, refer to the original paper. Read full disclaimer

In the microscopic world where atoms and light interact, the rules of energy are governed by a strange and delicate phenomenon known as quantum coherence. To understand this, imagine a system where an atom can exist in two excited states at the same time, not as a mixture of one or the other, but as a single, unified state that holds a specific relationship between the two. This relationship, or coherence, is usually fragile; the moment the atom bumps into its surroundings, this delicate link usually snaps, and the system settles into a predictable, ordinary state. For decades, scientists have wondered if this fleeting quantum link could be more than just a curiosity. Could it be treated as a fuel, a resource that could be burned to perform useful work, much like steam drives a turbine? This question sits at the heart of quantum thermodynamics, a field trying to understand how the laws of heat and energy apply to the smallest machines imaginable.

A team of researchers at Xi'an Jiaotong University has now shown that under very specific conditions, this quantum fuel can be generated and sustained, and then harvested to extract energy. They focused on a simple model: a three-level atom, which has one ground state and two excited states that are perfectly matched in energy. When this atom is placed in contact with a warm environment, or a thermal bath, the environment usually destroys any quantum connections. However, the researchers found that if the two excited states are perfectly identical and their internal "antennas" (called transition dipoles) are aligned in the same direction, the environment cannot tell the difference between the two paths the atom might take to return to its ground state. This confusion creates a protective effect, allowing a steady, long-lasting quantum coherence to form and persist, even while the atom is constantly exchanging energy with its surroundings.

The researchers demonstrated that this persistent coherence is not just a static feature but a usable resource. They designed a process to convert this invisible quantum link into a tangible imbalance in the atom's energy levels, which can then be used to generate work. In their first proposed method, they start with an atom that has settled into this special, coherent state. They apply a precise operation that rotates the atom's internal state without changing its total energy. This rotation transforms the quantum coherence into a population asymmetry, meaning one of the two excited states becomes more crowded with energy than the other. Once this imbalance is created, the researchers can manipulate the energy levels of the atom while it interacts with a heat bath, effectively running a microscopic engine cycle. By carefully raising and lowering the energy of the excited states, they can extract work from the system. Because the initial state still retains some coherence after the first cycle, this process can be repeated, slowly draining the remaining quantum resource until the atom finally settles into a standard, non-coherent state where no more work can be extracted.

Recognizing that repeating a process many times is inefficient, the team also developed a refined, single-shot protocol. This optimized approach is designed to extract the maximum possible amount of work in one continuous, smooth cycle. In this version, the researchers again use a rotation to turn the quantum coherence into a population imbalance. However, instead of isolating the atom to shift its energy levels, they keep the atom in contact with the thermal bath while they slowly adjust the energy levels. This allows the system to stay in a state of equilibrium throughout the process, ensuring that no energy is wasted. By the end of this single cycle, the atom has been converted into a standard thermal state, and the entire amount of free energy that was stored in the initial quantum coherence has been successfully converted into useful work.

The study confirms that quantum coherence acts as a genuine thermodynamic resource, capable of boosting the amount of work a system can produce beyond what is possible with classical energy distributions alone. The researchers found that the amount of extractable work depends heavily on the temperature of the environment and the energy gap between the atom's levels; lower temperatures and specific energy gaps allow for greater extraction. While the current work is theoretical, relying on mathematical models and simulations rather than physical experiments, it provides a clear blueprint for how future quantum machines might be built. These findings suggest that by engineering systems where quantum coherence can survive and be manipulated, we could design highly efficient quantum thermal machines or batteries that operate on principles fundamentally different from anything found in the macroscopic world. The ability to turn the subtle, invisible connections of the quantum world into practical energy represents a significant step toward harnessing the full potential of quantum thermodynamics.

Technical Summary: Work Extraction from Long-Lived Quantum Coherence of a Three-Level System

Problem Statement
Quantum thermodynamics investigates how thermodynamic laws emerge from quantum mechanics and how quantum resources, specifically quantum coherence, influence microscopic energy conversion. While quantum coherence is typically destroyed by environmental interactions in open systems, specific multilevel configurations—such as a three-level V-type system with degenerate excited states and aligned transition dipoles—can sustain "noise-induced" or "long-lived" coherence. This phenomenon arises from Fano interference between indistinguishable transition pathways. A critical, yet under-explored, question is whether this steady-state coherence can be rigorously quantified as a thermodynamic resource and converted into useful work. Most existing studies focus on the dynamical role of such coherence rather than its resource nature (consumption and interconvertibility) within a quantification framework.

Methodology
The authors analyze a three-level V-type open quantum system coupled to a thermal bath via dipole-monopole interaction. The study proceeds in two main stages:

Dynamical Analysis: The system is modeled using a Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation. The authors derive the conditions for long-lived coherence, focusing on two scenarios:
- Exactly Degenerate System: Two excited states $|1\rangle$ and $|2\rangle$ share the same energy $\omega$ . When transition dipoles are perfectly aligned, cross-damping terms in the dissipator prevent the decay of off-diagonal density matrix elements, leading to a non-thermal asymptotic steady state with residual coherence.
- Nearly Degenerate System: A small energy splitting $\delta$ is introduced. The authors show that while a true steady-state coherence does not exist, metastable coherence persists over a long timescale ( $\tau_S \ll t \ll 1/\delta$ ).
Thermodynamic Protocols: To extract work, the authors propose two protocols utilizing energy-preserving unitary operations to convert quantum coherence into population asymmetry, which is then exploited via controlled level shifts and thermalization:
- Protocol 1 (Repeated Cycles): Starting from the coherent asymptotic state of the degenerate system, an energy-preserving unitary rotation converts coherence into population asymmetry between the excited states. The system then undergoes a cycle of adiabatic level shifting, thermalization with an ancillary bath, and level lowering. This process extracts work but leaves residual coherence, requiring repeated iterations to fully deplete the resource.
- Protocol 2 (Single-Shot Quasistatic Cycle): An optimized, single-cycle protocol designed to extract the Maximum Extractable Work (MEW). This protocol involves a quasistatic process where the system remains in contact with the thermal bath during level shifts. It utilizes a specific unitary rotation followed by simultaneous, fine-tuned energy adjustments of both excited levels to drive the system directly to a Gibbs state.

Key Results

Coherence as a Resource: The study confirms that for aligned dipoles in a degenerate V-system, the steady state retains non-zero quantum coherence ( $C_{\ell_1} \neq 0$ ) even without external driving. This coherence is determined by the bath temperature and transition energy.
Free Energy Difference (FED): The authors establish that the Maximum Extractable Work (MEW) is bounded by the Free Energy Difference (FED) between the system's state and the Gibbs state. They demonstrate that quantum coherence reduces the von Neumann entropy of the system, thereby increasing the FED and the potential work output compared to classical populations alone.
Protocol Performance:
- Protocol 1 successfully extracts work by incrementally consuming coherence over multiple cycles. However, the work extracted in a single round is strictly less than the total FED, indicating that multiple rounds are necessary to approach the theoretical limit.
- Protocol 2 achieves the theoretical upper bound. By operating in the quasistatic limit and maintaining thermal contact during energy shifts, this single-shot protocol fully converts the available FED (including the contribution from coherence) into work, saturating the MEW bound.
Parameter Dependence: The extracted work and FED are shown to decrease monotonically with increasing inverse temperature ( $\beta$ ) and energy level spacing ( $\omega$ ). Lower temperatures and smaller energy gaps generally favor higher work extraction from coherence.

Significance and Claims
The paper claims to provide a rigorous quantification framework for the thermodynamic utility of noise-induced quantum coherence. By demonstrating that long-lived coherence in a V-type system can be systematically converted into practical work, the authors highlight the thermodynamic advantages of such systems.

The primary significance lies in the proposal of Protocol 2, which offers a route to circumvent the time-window restrictions often found in finite-time coherence extraction (e.g., in two-level systems). By utilizing steady-state coherence, the protocol provides a robust and reproducible resource for work extraction. The authors suggest these findings could influence the future design of coherence-driven quantum thermal machines and quantum devices, such as quantum batteries, by utilizing three-level systems as fundamental building blocks. The work bridges the gap between the dynamical generation of coherence and its operational use as a thermodynamic fuel.

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