Unequal Coherent Ergotropy at Equal Interferometric Visibility and Energy in Quantum-Controlled Interferometers

In the quantum world, energy is not just a number stored in a battery; it is a resource with a specific texture. Scientists have long known that a quantum system can hold energy in two distinct ways. One way is simple and classical: the system sits in a high-energy state, ready to drop down and release power, much like a ball perched on a hill. The other way is purely quantum: the system exists in a delicate superposition, a state of being in two places or energy levels at once. This "superposition" is a form of coherence, a fragile alignment that allows the system to perform tasks that a classical object cannot. The total amount of work that can be extracted from such a system is called ergotropy. Crucially, researchers have realized that the total energy stored does not tell the whole story. Two batteries might hold the exact same amount of energy, yet one might be able to deliver useful work while the other cannot, simply because the energy is organized differently inside.

This distinction becomes vital when we look at how quantum systems behave in experiments designed to test the nature of reality. For decades, physicists have used interferometers—devices that split a particle's path into two and then recombine them—to study wave-particle duality. When the paths recombine, they create an interference pattern, a series of bright and dark stripes that reveal the wave-like nature of the particle. The sharpness of these stripes is called visibility. Traditionally, visibility has been the gold standard for measuring how much quantum coherence a system possesses. If the stripes are clear, the system is coherent; if they are blurry, it is not. However, a new study suggests that this visual measure is incomplete. It turns out that two quantum setups can produce identical interference patterns, looking exactly the same to an observer, yet contain vastly different amounts of usable work. The difference lies in information that the interference pattern hides.

Researchers from the University of Gdańsk and the Federal University of Paraná investigated this hidden potential by treating the path of a particle as a quantum battery. They set up two different versions of a Mach-Zehnder interferometer, a classic device with two arms. In both setups, they sent a particle prepared in a thermal state, meaning it had some random energy fluctuations typical of a warm environment. The goal was to see how much work could be extracted from the particle after it passed through the machine. The two setups differed only in the order of operations. In the first setup, known as the quantum delayed-choice experiment, the particle first passes through a fixed splitter that puts it into a superposition of both paths, and only later does a second splitter, controlled by a quantum switch, decide whether to recombine the paths or leave them separate. In the second setup, the quantum-controlled reality experiment, the order is reversed: the quantum switch decides first whether to split the path, and then a fixed splitter recombines them.

The researchers found that at the output, both machines produced identical results in the most obvious ways. The interference fringes had the exact same sharpness, or visibility. The average energy stored in the particle was the same. Even the amount of work that could be extracted from the simple, classical part of the energy was identical. Yet, when they calculated the work that could be extracted from the quantum coherence—the part that relies on the particle being in a superposition—a clear difference emerged. The second setup, where the quantum switch acted first, consistently yielded more extractable work than the first setup. This advantage was not a small fluctuation; it was a fundamental difference driven by the order in which the machine acted on the particle.

The reason for this advantage lies in a subtle form of information called path predictability. In the second setup, because the quantum switch acted first, the particle retained a slight "memory" of which path it was more likely to take, even before the final recombination. This predictability is a form of a priori knowledge about the particle's state. While the interference pattern at the end looked the same for both machines, the second machine had preserved this extra bit of structural information. The study showed that this hidden predictability could be converted into extra work. The first machine, by delaying the choice of whether to split the path, erased this predictability before the final measurement, leaving it with less usable quantum energy. The researchers demonstrated that the difference in work was directly linked to this predictability, proving that the ability to do work depends on more than just the final interference pattern.

This finding has a thermodynamic cost that balances the books. The quantum switch used to control the beam splitters acts as a detector of the particle's path. In the setup that produced more work, the switch ended up in a state that was easier to reset for the next cycle. In the setup that produced less work, the switch ended up in a more chaotic state that required more energy to clean up and return to its starting condition. This aligns with a fundamental principle of physics known as Landauer's principle, which states that erasing information costs energy. The study showed that the extra work gained in the second setup came with a lower cost to reset the controller, meaning the system was more efficient overall. The advantage was not free; it was a trade-off between the work extracted from the battery and the energy required to reset the controller, but the second setup always came out ahead or equal.

To ensure these results were robust, the researchers also considered what would happen if the particle interacted with its environment in a way that destroyed its quantum nature, a process called decoherence. They modeled a scenario where the particle's spin, an internal property, became entangled with its path due to relativistic effects, effectively scrambling the quantum information. In this harsh environment, the first setup lost all of its ability to extract work; the interference pattern vanished, and the battery became useless. The second setup, however, retained a finite amount of extractable work. Even when the interference fringes disappeared completely, the second machine could still deliver power because it relied on the path predictability that survived the scrambling. This revealed that the second setup is far more resilient, capable of functioning as a battery even when the delicate wave-like features of the particle are destroyed.

The study concludes that interference visibility is not a complete measure of the quantum resources available for work. Two systems can look identical in their interference patterns and store the same total energy, yet one can be a far superior battery because it preserves a specific type of information about the particle's path. This "a priori" information, which exists before the final measurement, acts as a thermodynamic resource that can be converted into useful work. The research highlights that the order in which quantum operations are performed matters deeply, not just for the final outcome, but for the efficiency and resilience of the energy extraction process. By showing that a system can be more robust and efficient simply by rearranging the sequence of its internal components, the work offers a new perspective on how to design quantum devices that can store and release energy in a noisy, real-world environment.