Heat unexpectedly flowed from colder objects toward hotter ones when scientists placed two identical thermal processes into a quantum state where their order was no longer fixed. That counterintuitive behavior, demonstrated in a photonic experiment, became the foundation for an unusual quantum engine that simultaneously pumps heat from a colder environment to a warmer one while also delivering useful work—something impossible for an ordinary Otto cycle operating under the same conditions.
One of the oldest ideas in thermodynamics is that heat naturally moves from hotter objects to colder ones. Whether a cup of coffee cools on a table or ice melts in warm water, the direction of heat flow appears universal. That principle underlies refrigerators, engines, power plants, and countless technologies built over more than a century of thermodynamic science.
The new study explores whether that familiar rule still behaves the same way in a distinctly quantum setting, where events themselves do not necessarily happen in a fixed sequence.
Instead of changing the temperatures involved, the researchers changed something much more fundamental: the order in which a quantum system interacted with two identical thermal environments.
The result was an unexpected form of heat transfer that could briefly reverse the usual direction of energy flow under carefully defined conditions.
A quantum process without a definite order
The work centers on a concept known as indefinite causal order, a feature of quantum theory in which two operations do not occur in one fixed sequence.
In everyday experience, if event A happens before event B, that order is well defined. Quantum physics, however, allows situations where two possible orders exist in coherent superposition. One of the best-known ways to realize this idea is through the quantum switch, where a control qubit determines whether one operation occurs before another or vice versa, while itself remaining in a quantum superposition.
In this study, the researchers considered a simple quantum system—a qubit—that interacted with two identical thermalizing channels, each representing an environment at the same temperature.
Ordinarily, applying one thermalizing channel after another produces the same final thermal state regardless of which channel acts first. Nothing unusual happens because both environments are identical.
The researchers instead allowed the two possible orders to coexist simultaneously through the quantum switch.
After those interactions, they measured the control qubit. Depending on the measurement outcome, the system ended up in different thermal states.
That measurement turned out to make all the difference.
Heat sometimes flowed from cold to hot
The researchers discovered that, under certain temperature ranges, the measurement outcome could produce what they call anomalous heat flow.
If the quantum system initially started hotter than its surrounding thermal channels, it could nevertheless absorb heat from those colder channels.
Conversely, if the system initially started colder than the channels, it could release heat into the hotter surroundings.
In both situations, heat temporarily traveled opposite to the familiar thermodynamic direction.
The effect was not unrestricted. It appeared only when the temperature difference between the system and the thermal channels remained below specific limits derived by the researchers.
For a qubit, the calculations showed that heating from colder channels becomes possible when the system temperature is less than twice the channel temperature. Another mathematical condition defined the temperature range in which a colder system could become even colder despite interacting with hotter surroundings.
Outside those ranges, heat resumed its familiar direction.
The strange behavior came from quantum control
To understand why this reversal occurs, the researchers examined an equivalent theoretical description of the quantum switch.
Rather than treating the phenomenon as mysterious, they showed that the effect can be interpreted using controlled SWAP operations, quantum operations that exchange states depending on the state of a control qubit.
Their analysis revealed that the crucial ingredient is not simply quantum superposition by itself, but coherent quantum control over the ordering of interactions.
When the control qubit contains quantum coherence, the controlled operations inject thermodynamic resources into the overall process. Those additional resources allow heat exchanges that would never occur in an ordinary thermal interaction.
Importantly, the researchers also tested an alternative setup in which the control qubit merely selected between two thermalization channels without creating an indefinite order.
That version failed to produce anomalous heat flow.
The comparison showed that the unusual behavior depends specifically on the quantum switch and not simply on coherent control alone.
A photonic experiment tested the prediction
To investigate whether the theory could be reproduced experimentally, the team built a photonic simulation using single photons.
The polarization of each photon represented the thermal state of the quantum system, while different optical paths encoded the control qubit responsible for determining the order of interactions.
The experiment used a Mach–Zehnder interferometer that allowed photons to travel along two different paths simultaneously. Along one path, the two thermalizing channels acted in one sequence. Along the other path, the order was reversed.
After the two paths recombined, measurements projected the control qubit into one of two possible outcomes.
The researchers first performed a conventional experiment without indefinite causal order by fixing the control qubit in a definite state.
As expected, heat always flowed from the hotter object toward the colder one.
They then repeated the experiment with the control qubit prepared in a quantum superposition, enabling indefinite causal order.
This time the data revealed the predicted anomalous regions.
When the thermal channels were colder than the system, the measurements showed cases where the system actually gained heat from those colder channels.
When the channels were hotter than the system, measurements also identified situations where the colder system transferred heat into the hotter channels.
The measured heat changes closely followed the theoretical predictions, while separate quantum process tomography confirmed that the optical implementation of the thermalizing channels achieved an average process fidelity exceeding 99%.
Why this does not violate the second law
At first glance, heat flowing from cold to hot seems to contradict the second law of thermodynamics.
The researchers explain why that is not the case.
The anomalous heat transfer appears only after conditioning on the measurement outcome of the control qubit.
If those measurement outcomes are ignored, the average heat exchange over all outcomes is exactly the same as in the ordinary thermal process.
For example, when the system and the thermal channels begin at the same temperature, the overall net heat transfer remains zero regardless of whether indefinite causal order is used.
The unusual heat flows therefore emerge only within selected measurement outcomes rather than as a violation of thermodynamic laws governing the complete process.
The surprising heat flow became the basis for an unusual engine
Having established that heat could sometimes move in the unexpected direction, the researchers asked whether that effect could perform a useful task.
They designed a modified quantum Otto cycle, one of the standard models for quantum heat engines.
Unlike a conventional Otto engine, their design used indefinite causal order during the heat-absorption stage.
The working substance was again a qubit whose energy spacing changed during the cycle.
During one stage, the qubit interacted with two colder thermal channels through the quantum switch.
Only when the measurement produced the desired control outcome did the cycle continue. Otherwise, the system was reset and the process repeated.
This selective strategy resembles the role traditionally assigned to Maxwell’s demon, which uses information about measurements to influence thermodynamic behavior.
After the successful measurement outcome, the system absorbed heat from the colder reservoir despite already being hotter than it.
Later stages of the cycle converted part of that absorbed energy into useful work while releasing heat into a hotter reservoir.
As a result, the machine simultaneously behaved as both a refrigerator and an engine.
The experiment reproduced the complete cycle
The researchers implemented the full quantum cycle using a five-module photonic apparatus that reproduced each stage of the proposed engine.
Measurements showed that the system absorbed heat from the colder thermal channels, released heat to the hotter environment, and generated net work on an external agent.
They also evaluated the machine’s coefficient of performance (COP), which included the energetic cost of erasing the measurement information stored by the Maxwell’s demon-like controller.
The calculations accounted for the probability of obtaining the required measurement outcome because successful operation depended on that result.
The experiments matched the theoretical predictions across different operating conditions.
Interestingly, although the largest cooling effect occurred when no work was extracted, the highest overall performance appeared at a different operating point where the machine simultaneously cooled the colder source and produced useful work.
What the study adds to quantum thermodynamics
The work extends previous research on indefinite causal order beyond quantum information processing into the behavior of heat itself.
Rather than merely changing computational or communication tasks, the quantum switch altered how thermal energy moved between systems under specific conditions.
The researchers argue that unfolding the quantum switch into an equivalent theoretical picture helps explain the physical origin of the effect: the quantum control process effectively accesses thermodynamic resources associated with the coherent control system.
At the same time, the study remains a proof-of-principle demonstration.
The anomalous heat flow depends on carefully prepared quantum states, specific measurement outcomes, and a photonic simulation of thermalization channels. The proposed Otto cycle likewise operates within this controlled quantum framework and incorporates the energetic cost of information erasure associated with the measurement process.
Even with those limitations, the results illustrate that quantum processes lacking a fixed causal order can enable thermodynamic behavior unavailable in classical machines. By experimentally demonstrating both the unexpected heat flow and a quantum engine built around it, the study opens a new direction for exploring how quantum causality and thermodynamics can work together to achieve tasks that ordinary thermal systems cannot perform.
Publication details
Qing-Feng Xue et al, Anomalous Heat Flows and Quantum Otto Engine with (In)definite Causal Order, Physical Review Letters (2026). DOI: 10.1103/sx1m-pdhz. On arXiv: DOI: 10.48550/arxiv.2511.04028
