Quantum Otto engine with decoupled idle levels in a non-Hermitian XY model

Abstract: We study a quantum Otto cycle in a two-qubit non-Hermitian XY model with a staggered imaginary magnetic field. The energy spectrum of this system naturally decouples into a pair of working levels that depend on the external field and a pair of idle levels that are completely independent of it, thereby providing the first concrete microscopic realization of the idle-level quantum heat engine architecture proposed by de~Oliveira and Jonathan [Phys. Rev. E 104, 044133 (2021)] in a physical spin model. Tuning the non-Hermitian parameter eta_0 drives a continuous transition from a dissipative regime with negative net work and net heat absorption from the hot reservoir into a genuine heat engine mode, while simultaneously enhancing both output work and efficiency. As eta_0 increases within the stable PT-unbroken phase, the engine efficiency rises significantly, reaching a substantial fraction of the Carnot limit. This effect originates from the compression of the idle-level gap, which redistributes the level occupation weights in the hot and cold equilibrium states and thereby modulates the absorbed heat. The numerator of the net work expression is independent of eta_0, but the denominator depends on eta_0 indirectly through hyperbolic cosine functions -- this is the mathematical root of the idle-level control mechanism. We provide a detailed analysis of the robustness of these findings against parameter variations, a critical comparison of the non-Hermitian control with the Hermitian limit, and a concrete experimental proposal for trapped-ion quantum simulators. Our results demonstrate that non-Hermiticity serves as an indispensable tool for steering both the operation mode and the performance of a quantum engine.

• The paper shows that decoupling idle levels in a non-Hermitian XY model enables independent manipulation of heat absorption and work extraction in a quantum Otto engine.
• It employs a biorthogonal thermodynamic framework to derive analytical expressions for internal energy, heat, work, and efficiency transitions via modulation of the non-Hermitian parameter.
• Numerical analysis confirms that efficiency rises from around 3.5% to 42.7% with increasing idle-level control, highlighting robust experimental feasibility and scalability.

Quantum Otto Engine with Decoupled Idle Levels in a Non-Hermitian XY Model

Model Architecture and Idle-Level Mechanism

The non-Hermitian two-qubit XY model with staggered imaginary magnetic fields investigated constitutes an explicit physical realization of the idle-level quantum heat engine architecture. This model is defined by a Hamiltonian block-diagonal in the computational basis, separating into two subspaces: working levels $$\{\ket{\uparrow\uparrow},\ket{\downarrow\downarrow}\}$$, whose eigenvalues depend exclusively on the real magnetic field $h$, and idle levels $$\{\ket{\uparrow\downarrow},\ket{\downarrow\uparrow}\}$$, governed solely by the non-Hermitian parameter $\eta_0$. The spectrum yields $E_{1,2} = \mp 2J\sqrt{h_0^2+\gamma^2}$ and $E_{3,4} = \mp 2J\sqrt{1-\eta_0^2}$, manifesting complete decoupling.

This is paradigmatic for idle-level engines in which the occupancy and energetics of a subset of levels are insulated from external work parameters. Variations in $\eta_0$ enable modulation of these idle levels independently from the working subspace, which fundamentally alters the heat absorption and entropy flow during the engine cycle.

Non-Hermitian Thermodynamic Framework

The analysis utilizes the biorthogonal equilibrium thermodynamics formalism for non-Hermitian systems with real spectra [Gardas et al., Sci. Rep. 6, 23408 (2016)]. The canonical partition function is constructed as $Z=2\cosh(\beta A)+2\cosh(\beta B)$, allowing calculation of all thermal quantities assuming an ensemble stabilized via measurement-conditioned protocols or dissipative engineering. All calculations are strictly within the $\mathcal{PT}$-unbroken phase, ensuring spectral reality and adiabatic accessibility.

Otto Cycle Implementation and Analytical Results

The quantum Otto cycle is realized via two adiabatic and two isochoric strokes, with all transitions assumed to satisfy the non-Hermitian adiabatic theorem (verified quantitatively in the appendix). Analytical expressions for internal energy, heat, and work are derived, with the net work given by:

$$W=(A_H-A_C)\left[\frac{\sinh(\beta_c A_C)}{\cosh(\beta_c A_C)+\cosh(\beta_c B)}-\frac{\sinh(\beta_h A_H)}{\cosh(\beta_h A_H)+\cosh(\beta_h B)}\right]$$

Crucially, the numerator is free of $h$0, while $h$1 enters in the denominator through hyperbolic cosines, establishing the mechanism by which idle-level compression modifies engine performance without altering the maximal work extractable from the working subspace.

Transition Dynamics: Dissipative vs Engine Regimes

Numerical analysis reveals a continuous transition as $h$2 is increased: for low $h$3, the engine operates in a dissipative regime ($h$4, $h$5), with all input energy ultimately dissipated to the cold reservoir. Above a critical value $h$6, $h$7 becomes positive, switching to genuine engine operation. The entropy production $h$8 is strictly positive throughout the dissipative regime, supporting thermodynamic consistency.

Efficiency Enhancement via Idle-Level Control

Within the heat engine regime, efficiency $h$9 increases monotonically with $$\{\ket{\uparrow\downarrow},\ket{\downarrow\uparrow}\}$$0, approaching the standard Otto limit as $$\{\ket{\uparrow\downarrow},\ket{\downarrow\uparrow}\}$$1 approaches the exceptional point ($$\{\ket{\uparrow\downarrow},\ket{\downarrow\uparrow}\}$$2), with no violation of Carnot bounds. This enhancement arises from the redistribution of level occupation differences during thermalization; as the idle gap $$\{\ket{\uparrow\downarrow},\ket{\downarrow\uparrow}\}$$3 is compressed, heat absorption is amplified while net work remains governed by $$\{\ket{\uparrow\downarrow},\ket{\downarrow\uparrow}\}$$4. This provides direct microscopic insight into the efficacy of idle-level controlled engines.

Explicit calculations show efficiency rises from $$\{\ket{\uparrow\downarrow},\ket{\downarrow\uparrow}\}$$53.5% at onset ($$\{\ket{\uparrow\downarrow},\ket{\downarrow\uparrow}\}$$6) to $$\{\ket{\uparrow\downarrow},\ket{\downarrow\uparrow}\}$$742.7% at $$\{\ket{\uparrow\downarrow},\ket{\downarrow\uparrow}\}$$8, saturating at $$\{\ket{\uparrow\downarrow},\ket{\downarrow\uparrow}\}$$9 just below the exceptional point.

Robustness and Comparative Analysis

Parameter sweeps across anisotropy $\eta_0$0 and temperature ratio $\eta_0$1 confirm the robustness of the critical regime transition and efficiency enhancement: qualitative features persist even as system scales, temperature differences, or coupling strengths vary. In the Hermitian limit ($\eta_0$2), no idle-level tunability exists, and the system remains irreversibly dissipative absent alteration of working-level parameters; this underscores the indispensable role of non-Hermiticity for achieving operational versatility and performance gains in such architectures.

Simultaneous enhancement of both net work and efficiency is demonstrated in the quasistatic regime—a feature distinct from prior non-Hermitian engine proposals, which typically target output power for fixed efficiency.

Experimental Feasibility and Future Directions

Implementation prospects in trapped-ion quantum simulators are substantiated: the required Hamiltonian engineering is available through the Mølmer–Sørensen scheme, with imaginary field realization feasible via postselection or dissipative engineering. State tomography and biorthogonal measurement schemes are compatible with current hardware and coherence timescales, making non-Hermitian idle-level quantum engines experimentally accessible.

Future work should address finite-time, nonadiabatic cycles, explore operation across exceptional points ($\eta_0$3 breaking), and incorporate resources such as entanglement and quantum correlations. This will extend connections to thermodynamic uncertainty relations and quantum information thermodynamics.

Conclusion

The two-qubit non-Hermitian XY model with staggered imaginary magnetic fields presents the first concrete microscopic realization of the idle-level quantum heat engine architecture. Idle levels decoupled from external work parameters enable robust, continuous control of engine operation mode and performance, inducing efficiency enhancement and mode switching via a non-Hermitian control knob $\eta_0$4. Analytical and numerical treatment confirm thermodynamic consistency, scalability, and experimental feasibility. Non-Hermiticity thus emerges as an essential resource for quantum thermal machine engineering, offering new avenues for experimental and theoretical exploration in quantum thermodynamics and open-system control (2606.22405).