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Thermodynamic Performance of a Measurement-Driven Quantum Engine with a Two-Parameter $(p,q)$ -Deformed Harmonic Oscillator

This paper investigates a measurement-driven single-bath quantum engine utilizing a two-parameter $(p,q)$ -deformed harmonic oscillator, demonstrating that deformation can enhance measurement-induced energy input and extracted work while deriving thermodynamic quantities and identifying conditions for nonnegative heat and valid engine operation.

Original authors: Tunde Joseph Osunmusanmi, Berihu Teklu

Published 2026-09-09

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Tunde Joseph Osunmusanmi, Berihu Teklu

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

In the microscopic world where atoms and light interact, the rules of energy are written differently than in our everyday experience. For decades, scientists have studied how to build tiny engines that run on quantum mechanics, the physics governing the very small. A standard way to think about these machines is to imagine a piston moving inside a cylinder, but instead of air, the cylinder holds a single atom or a group of atoms. To make this engine run, researchers usually need two heat baths: one hot and one cold. The engine absorbs heat from the hot source, does some work, and dumps the leftover heat into the cold sink. However, a newer and more unusual idea has emerged: what if you don't need a hot bath at all? What if the energy to drive the engine comes from the act of looking at it? In quantum physics, simply measuring a system changes its state, a phenomenon known as backaction. This measurement can inject energy directly into the machine, acting as the fuel. The question researchers have been asking is whether changing the very shape of the energy levels inside the engine—making them uneven or curved rather than perfectly straight—can make this measurement-powered engine run better.

A team of researchers at Khalifa University in the United Arab Emirates has taken a deep dive into this question by designing a theoretical engine that uses a special kind of mathematical distortion to reshape its fuel source. They focused on a system called a harmonic oscillator, which is a standard model for a vibrating particle, but they modified it using two adjustable parameters. Think of this not as changing the size of the engine, but as changing the spacing between the rungs of a ladder. In a normal oscillator, the rungs are evenly spaced. In their modified version, the spacing between the rungs changes depending on how high up the ladder you are, and this change is controlled by two numbers they can tune. They then ran a series of computer simulations to see how this distorted ladder affected the engine's ability to turn the energy from a measurement into useful work.

The engine they studied operates in a cycle with four distinct steps. First, the system starts in a calm, thermal state, like a cup of coffee sitting at room temperature. The researchers then slowly change the shape of the energy ladder without letting the system lose or gain heat, a process called an adiabatic stroke. Next comes the crucial moment: they perform a measurement on the system. In the quantum world, this is not a passive observation; it is an active intervention that jolts the system and injects energy. This is the "fuel" for the engine. After this jolt, the researchers reverse the first step, slowly returning the energy ladder to its original shape. Finally, the system is allowed to settle back into its initial thermal state, releasing any excess heat. By tracking the energy flows at each step, they could calculate how much work the engine produced and how efficiently it converted the measurement energy into that work.

The researchers found that the shape of the energy ladder matters significantly. When they adjusted the two parameters to create a specific type of distortion, the engine was able to extract more work from the measurement than it could with a standard, undistorted ladder. In their simulations, the deformation allowed the system to sample energy gaps in a way that amplified the effect of the measurement. The results showed that by tuning these parameters, they could increase the amount of energy injected by the measurement and, more importantly, increase the amount of work the engine could deliver back out. In the most favorable conditions they tested, the engine converted nearly 96 percent of the measurement energy into useful work. This is a very high number, but it is important to note that this efficiency only counts the energy the engine itself produces, not the energy required to run the measurement device that provided the fuel.

However, the study also revealed important boundaries and limitations. The researchers were careful to check that their mathematical models remained physically valid. They found that if the distortion was too extreme in certain directions, the energy levels would stop behaving like a real physical system, and the engine would fail to operate. They had to filter out these impossible scenarios to find the "engine regime" where the machine actually works. In the valid regions, the deformation did not just act as a simple magnifier that made everything bigger; it changed the relative distances between energy levels, which altered how the system responded to the measurement. This suggests that the internal structure of the working material is a powerful tool for controlling performance, distinct from simply turning up the volume on the energy.

The findings suggest that engineering the spectrum of a quantum system—changing the specific arrangement of its energy levels—can be a viable strategy for improving measurement-driven engines. The team demonstrated that a two-parameter deformation offers a richer set of controls than previous models that used only one. While the study was conducted entirely through computer simulations and theoretical calculations, the results provide a clear roadmap for how such an engine might behave. The researchers emphasize that their efficiency numbers represent the performance of the working medium alone. A complete machine would also need to account for the energy cost of the measurement apparatus itself, which would lower the overall efficiency. Nevertheless, the work shows that by carefully sculpting the energy landscape of a quantum system, it is possible to enhance the conversion of measurement backaction into mechanical work, offering a new avenue for designing nanoscale machines that run on the information we gather about them.

Technical Summary: Thermodynamic Performance of a Measurement-Driven Quantum Engine with a Two-Parameter (p, q)-Deformed Harmonic Oscillator

Problem Statement
The paper addresses the thermodynamic performance of a single-bath, measurement-driven quantum engine. While standard quantum Otto cycles typically rely on adiabatic control of the oscillator frequency to alter energy level spacing, this work investigates a complementary control mechanism: spectral engineering via a two-parameter $(p, q)$ -deformed harmonic oscillator. The central question is whether the nonlinearity introduced by the deformation parameters $(p, q)$ can modify the conversion of measurement backaction into useful work, beyond a simple rescaling of the energy spectrum. The study specifically focuses on a cycle where the energetic input is provided not by a hot thermal reservoir, but by a nonselective Gaussian measurement.

Methodology
The authors formulate a four-stroke cycle for a working medium described by a $(p, q)$ -deformed harmonic oscillator:

Adiabatic Stroke I: The control vector $\lambda = (\omega, p, q)$ is changed quasistatically from an initial state $\lambda_i$ to a final state $\lambda_f$ . Level populations are preserved, but the energy spectrum changes.
Measurement Stroke II: A nonselective Gaussian measurement of the final deformed position quadrature is performed. This injects energy ( $Q_M$ ) into the system via measurement backaction. The stroke is modeled using a transition matrix derived from continuous Hermitian Kraus operators.
Adiabatic Stroke III: The control parameters are returned adiabatically to $\lambda_i$ , preserving the populations established by the measurement.
Thermalization Stroke IV: The system rethermalizes with the original single bath, rejecting heat ( $Q_T$ ).

Key Theoretical and Numerical Components:

Deformed Algebra: The working medium utilizes a $(p, q)$ -deformed algebra where the number operator eigenvalues depend nonlinearly on the excitation number and the parameters $p$ and $q$ .
Thermodynamic Quantities: The authors derive expressions for measurement heat, adiabatic work contributions, and net cycle work ( $W_{cyc}$ ) using a consistent first-law convention where work done on the system is positive.
Efficiency Definition: The study defines a "reduced working-medium efficiency" $\eta = -W_{cyc}/Q_M$ . The authors explicitly note this excludes the energetic costs of the measurement apparatus (implementation, recording, resetting).
Validity Checks: To ensure physical admissibility, the authors impose strict conditions:
- $Q_M > 0$ (Energy input from measurement).
- $W_{cyc} < 0$ (Net work output).
- $Q_T < 0$ (Heat rejection).
- $0 < \eta < 1$ .
- Spectral Ordering & Passivity: The spectrum must be ordered, and the state entering the measurement must be passive with respect to the final Hamiltonian to ensure non-negative measurement heat.
Numerical Implementation: Simulations are performed using a truncated Hilbert space ( $N=10$ ). The authors perform convergence checks against the cutoff size and verify the symmetry and normalization of the transition matrix. They scan the parameter space $0.7 \leq p_f, q_f \leq 1.3$ and $1 \leq \omega_f/\omega_i \leq 3$ .

Key Results

Deformation-Induced Enhancement: Numerical scans indicate that within the physically admissible regime (satisfying spectral ordering, passivity, and convergence checks), increasing the final deformation parameters $(p_f, q_f)$ can enhance both the measurement-induced energy input ( $Q_M$ ) and the extracted work compared to the undeformed oscillator ( $p=q=1$ ).
Efficiency: The reduced efficiency reaches a maximum of approximately $\eta \simeq 0.958$ for the specific numerical parameters used. The authors emphasize this is a reduced efficiency, not a full device-level efficiency.
Relative Performance: Relative maps comparing the deformed engine to the undeformed benchmark show positive enhancements in both work output and efficiency in the strongly deformed regions of the parameter space.
Role of Measurement Resolution: The measurement-induced heat decreases monotonically as the measurement resolution ( $\sigma$ ) increases (weaker backaction). Deformation parameters control the spectral scale sampled by the measurement, while $\sigma$ controls the strength of energy transfer.
Finite-Basis Limitations: The study acknowledges that for $p, q < 1$ , the infinite-dimensional deformed oscillator may not define a confining Gibbs state. Therefore, results in this region are interpreted as finite-basis effective-model calculations, contingent on the satisfaction of spectral validity and convergence checks.

Significance and Claims
The paper claims to demonstrate that algebraic spectral nonlinearity (via $(p, q)$ -deformation) serves as a viable control mechanism for measurement-powered cycles. The primary contribution is a consistent thermodynamic formulation that links the measurement transition matrix to the deformation-dependent energy gaps.

The authors modestly frame their findings:

The observed performance improvements are attributed to the reshaping of the spectrum relative to the specific numerical protocol used.
The results do not constitute a universal proof of superiority, as the enhancement may partly reflect a larger final spectral scale rather than a fixed-resource optimization.
The reported efficiency is strictly a "reduced working-medium efficiency" and does not account for the thermodynamic cost of the measurement apparatus.
The work identifies the conditions (passivity, unitality, spectral ordering) required for the engine to operate physically, connecting engine operation directly to the spectral properties of the deformed oscillator.

The paper concludes that while deformation can modify the conversion of measurement backaction into work, future work is required to connect these parameters to concrete physical implementations, include finite-time effects, and perform optimization under fixed-resource constraints (e.g., fixed final energy scale or fixed measurement heat).

Technical Summary: Thermodynamic Performance of a Measurement-Driven Quantum Engine with a Two-Parameter (p, q)-Deformed Harmonic Oscillator

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