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arxiv: 2605.24483 · v1 · pith:BXQEWF5Rnew · submitted 2026-05-23 · 🪐 quant-ph

Quantum Otto machine with q-deformed P\"oschl-Teller oscillator

Collins O. Edet , Norshamsuri Ali , Rosdisham Endut , O. Abah This is my paper

Pith reviewed 2026-06-30 13:23 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum Otto cycleq-deformed potentialPöschl-Teller oscillatorheat engine efficiencyrefrigerator performancethermodynamic performanceparameter spacequantum thermal machine
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The pith

q-deformation of the modified Pöschl-Teller potential partitions the Otto cycle into separate regions of optimal heat-engine and refrigerator performance.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines a quantum Otto cycle whose working substance is a q-deformed version of the modified Pöschl-Teller oscillator. It derives closed-form energy levels and wave functions for this potential and inserts them into the standard thermodynamic expressions for the cycle. The central result is that varying the deformation parameter q and the potential parameter Δ partitions the (q, Δ) plane into regions where the device functions best as a heat engine or as a refrigerator. A sympathetic reader cares because the deformation offers a tunable knob that can be used to optimize either efficiency or cooling power without changing the cycle protocol itself.

Core claim

The q-deformed modified Pöschl-Teller potential modifies the energy spectrum of the working substance, which in turn produces separate performance domains in the (q, Δ) parameter space. Low values of Δ combined with high values of q yield the highest heat-engine efficiencies, while high Δ and low q improve the coefficient of performance when the cycle operates as a refrigerator. The efficiency reaches its maximum in the low-Δ, high-q corner of the space.

What carries the argument

The q-deformed modified Pöschl-Teller potential, whose energy eigenvalues depend explicitly on both the deformation parameter q and the potential strength parameter Δ; these eigenvalues enter the heat and work calculations for the Otto cycle and thereby control the thermodynamic figures of merit.

If this is right

  • Heat engine efficiency is maximized when Δ is small and q is large.
  • Refrigerator coefficient of performance improves when Δ is large and q is small.
  • The working substance can be switched between engine and refrigerator regimes by tuning q and Δ alone.
  • Distinct performance regions appear in the two-dimensional parameter plane.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the same deformation can be realized in other potentials, similar tunable regions may appear in other quantum thermal machines.
  • Experimental implementation would require a physical system whose spectrum matches the q-deformed Pöschl-Teller form over a controllable range of q and Δ.
  • The separation of regimes suggests that parameter tuning could replace cycle redesign in some quantum heat devices.

Load-bearing premise

The derived analytical energy spectrum and wave functions of the q-deformed potential can be substituted directly into the thermodynamic formulas for the Otto cycle without additional quantum or validity corrections.

What would settle it

A numerical diagonalization of the q-deformed Hamiltonian that produces energy levels differing from the reported analytic spectrum would falsify the performance maps.

Figures

Figures reproduced from arXiv: 2605.24483 by Collins O. Edet, Norshamsuri Ali, O. Abah, Rosdisham Endut.

Figure 1
Figure 1. Figure 1: The q-deformed modified P¨oschl–Teller potential as a function of x for different: (a) α = 0.5 (dotted red line), α = 1 (dot blue line), and α = 2.0 (solid black line) (b) q = 0.5 (dotted red line), q = 0.7 (dot blue line), q = 0.9 (dot green line)and q = 1.0 (solid black line) (standard P¨osch-Teller potential). c ∆ = 0.9 (dotted red line), ∆ = 1.8 (dotted blue line), ∆ = 2.7 (dotted green), and ∆ = 3.6 (… view at source ↗
Figure 2
Figure 2. Figure 2: Energy E q n of the q-deformed P¨oschl–Teller potential as a function of (a) q (b) α and (c) ∆, for different energy level n. Here, n = 1 (dotted red line), n = 2 (dot blue line), n = 3(dot green line) and n = 4 (solid black line). Other parameters used: q = 1, ∆= 2, and α= 1.5 (a) and α= 0.5(b). p = α 2h¯ 2 2µ , and σ = 1 2 − q∆2− 1 4 α2q2 + 1 4 . In the ther￾mal equilibrium state, the density matrix is g… view at source ↗
Figure 3
Figure 3. Figure 3: The schematic of the Entropy-temperature (S-T) [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Quantum Otto cycle as a heat engine. (a) Heat exchange with the hot reservoir, Q q h (b) Heat exchange with the cold reservoir, Q q c (c) Total work output, Wq (d) Efficiency η q , as function of the depth of the potential ∆(3.7 < ∆ < 5) and the deformation parameter q(0.8 < q < 0.9). Parameters used: αc = 0.5, αh = 1.118, Th = 5, and Tc = 1. Qc are achieved in the regime of large q and small ∆, whereas th… view at source ↗
Figure 5
Figure 5. Figure 5: Quantum Otto refrigerator cycle (a) Heat exchange with the hot reservoir, Q q h (b) Heat exchange with the cold reservoir, Q q c (c) Total work output, Wq (d) Coefficient of performance COPq as function of the potential well depth ∆(0.9 < ∆ < 1) and q(0.80 < q < 1). Parameters used: αc = 0.5, αh = 1.118, Th = 5, and Tc = 1. modified P¨oschl-Teller potential model to demonstrate that the quantum thermal dev… view at source ↗
read the original abstract

We study the impact of the potential parameters of the q-deformed modified P\"oschl-Teller potential on the thermodynamic performance of a quantum Otto cycle, where the $q$-deformed modified P\"oschl-Teller potential serves as the working substance. Analytical expressions for the energy spectrum and wave functions are derived, enabling a systematic investigation of heat exchange, work output, efficiency, and coefficient of performance. We show that $q$-deformation modifies the energy spectrum and creates distinct performance regions in the ($q$, $\Delta$) parameter space. Low ($\Delta$) and high ($q$) favour optimal heat engine efficiency, whereas high ($\Delta$) and low ($q$) improve refrigerator performance. The heat engine efficiency peaks in the low-($\Delta$), high-($q$) regime. These results highlight the q-deformed modified P\"oschl-Teller potential as a versatile and tunable platform for exploring potential parameter-driven effects in quantum thermal machines.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript examines a quantum Otto cycle with the q-deformed modified Pöschl-Teller oscillator as the working substance. It claims to derive analytical expressions for the energy spectrum and wave functions of the deformed potential, then inserts these into the thermodynamic expressions for heat, work, efficiency, and COP to map performance across the (q, Δ) parameter space, reporting that low Δ with high q optimizes heat-engine efficiency while high Δ with low q optimizes refrigerator performance, with efficiency peaking in the low-Δ high-q regime.

Significance. If the spectrum derivation holds and the direct insertion into the Otto-cycle formulas is valid, the work supplies an explicit, tunable example of how a deformation parameter can create distinct performance regimes in a quantum thermal machine, offering a concrete platform for exploring parameter-driven optimization in quantum thermodynamics.

major comments (2)
  1. [Energy spectrum section] § on energy spectrum derivation (the section presenting the analytical E_n(q, Δ) and wave functions): the manuscript states the final expressions but supplies no intermediate derivation steps from the q-deformed Schrödinger equation, no reduction check to the q=1 Pöschl-Teller case, and no domain restrictions on q; because every subsequent heat, work, and efficiency formula is obtained by direct substitution of these E_n, the absence of verification makes the reported (q, Δ) performance regions unverifiable.
  2. [Performance analysis section] § on Otto-cycle performance (the section mapping efficiency and COP in (q, Δ) space): the claimed optimal regions and efficiency peak are presented without error propagation from the spectrum, without numerical cross-checks against exact diagonalization for sample (q, Δ) values, and without discussion of adiabaticity assumptions; this directly affects the load-bearing claim that low-Δ high-q is optimal for the engine.
minor comments (2)
  1. Notation for the q-numbers appearing in the spectrum formula should be defined explicitly (standard [n]_q or a custom replacement) to avoid ambiguity when readers attempt to reproduce the expressions.
  2. Figure captions for the performance-region plots should state the fixed values of temperature ratio, cycle times, and any other parameters held constant while varying q and Δ.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The points raised identify areas where additional detail will improve clarity and verifiability. We address each major comment below and will incorporate the suggested revisions.

read point-by-point responses

  1. Referee: [Energy spectrum section] § on energy spectrum derivation (the section presenting the analytical E_n(q, Δ) and wave functions): the manuscript states the final expressions but supplies no intermediate derivation steps from the q-deformed Schrödinger equation, no reduction check to the q=1 Pöschl-Teller case, and no domain restrictions on q; because every subsequent heat, work, and efficiency formula is obtained by direct substitution of these E_n, the absence of verification makes the reported (q, Δ) performance regions unverifiable.

    Authors: We agree that the intermediate steps were omitted. The energy spectrum was obtained by solving the q-deformed Schrödinger equation via a change of variables and series solution, but these steps were not shown. In the revised manuscript we will insert the full derivation, explicitly recover the standard Pöschl-Teller spectrum at q=1, and state the domain q>0, Δ>0 required for normalizability. This will allow direct verification of all subsequent thermodynamic expressions. revision: yes

  2. Referee: [Performance analysis section] § on Otto-cycle performance (the section mapping efficiency and COP in (q, Δ) space): the claimed optimal regions and efficiency peak are presented without error propagation from the spectrum, without numerical cross-checks against exact diagonalization for sample (q, Δ) values, and without discussion of adiabaticity assumptions; this directly affects the load-bearing claim that low-Δ high-q is optimal for the engine.

    Authors: We acknowledge the value of explicit verification. We will add (i) a comparison of the analytical E_n(q,Δ) against numerical diagonalization for several representative (q,Δ) pairs, (ii) a brief discussion of the adiabaticity condition underlying the Otto cycle, and (iii) propagation of the spectrum uncertainty into the efficiency and COP expressions. These additions will substantiate the reported optimal regimes without altering the analytical conclusions. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation inserts independently derived spectrum into cycle formulas

full rationale

The paper states it derives analytical expressions for the q-deformed energy spectrum and wave functions, then substitutes those expressions into the standard Otto-cycle thermodynamic quantities. No equation is defined in terms of its own output, no fitted parameter is relabeled as a prediction, and no load-bearing premise rests on a self-citation chain. The reported (q, Δ) performance regions are direct consequences of the spectrum formulas rather than a renaming or self-referential construction. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger is limited to the parameters and assumptions explicitly named there.

free parameters (2)
  • q
    Deformation parameter scanned across values to produce performance maps.
  • Δ
    Potential parameter scanned across values to produce performance maps.
axioms (1)
  • domain assumption The q-deformed modified Pöschl-Teller potential admits an analytically solvable energy spectrum that can be used directly in thermodynamic cycle calculations.
    Required to obtain closed-form heat and work expressions.

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