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arXiv:2608.26690

Adiabatic Otto-like quantum thermodynamical cycle in the non-quasi-static regime

Salvador J. Robles-Pérez, Salvador Castillo-Rivera

quant-phcond-mat.stat-mechphysics.atom-phSubmitted Aug 27, 2026• Announced Aug 28, 2026• 0 citations
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Abstract

We show a finite-time Otto-like quantum thermodynamic cycle that preserves the adiabatic population structure of a time-dependent harmonic oscillator in the non-quasi-static regime. In the conventional energy representation, finite-rate driving induces non-adiabatic population redistribution and leaves residual excitations after the Hamiltonian has returned to its initial value. We show that this difficulty can be avoided by formulating the dynamics in the Lewis-Riesenfeld invariant representation, without modifying the physical Hamiltonian through auxiliary counterdiabatic driving. For a parametric Mathieu protocol, quantum inertia produces a mismatch between the spatial width of the working mode and its transient dressed energy scale. We propose an experimental implementation of this scheme in a trapped-ion Paul trap using stimulated Raman interactions, with independent control of the laser detuning and beam intersection angle. This provides a finite-time implementation in which the invariant population structure is preserved while the physical trap frequency evolves non-quasi-statically. Our results establish a clear distinction between adiabatic operation and quasi-static driving, providing a route toward finite-time quantum thermal cycles that retain the adiabatic energy structure without requiring the quasi-static limit.

Comments
10 pages, 5 figures

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We show a finite-time Otto-like quantum thermodynamic cycle that preserves the adiabatic population structure of a time-dependent harmonic oscillator in the non-quasi-static regime. In the conventional energy representation, finite-rate driving induces non-adiabatic population redistribution and leaves residual excitations after the Hamiltonian has returned to its initial value. We show that this difficulty can be avoided by formulating the dynamics in the Lewis-Riesenfeld invariant representation, without modifying the physical Hamiltonian through auxiliary counterdiabatic driving. For a parametric Mathieu protocol, quantum inertia produces a mismatch between the spatial width of the working mode and its transient dressed energy scale. We propose an experimental implementation of this scheme in a trapped-ion Paul trap using stimulated Raman interactions, with independent control of the laser detuning and beam intersection angle. This provides a finite-time implementation in which the invariant population structure is preserved while the physical trap frequency evolves non-quasi-statically. Our results establish a clear distinction between adiabatic operation and quasi-static driving, providing a route toward finite-time quantum thermal cycles that retain the adiabatic energy structure without requiring the quasi-static limit.
This work studies the relationship between parametric amplification (or particle creation), adiabaticity and irreversibility in the non-quasi-static regime of a time-dependent quantum harmonic oscillator (TDHO) that evolves unitarily. We provide analytical results for the evolution of the TDHO valid for any functional value of the frequency, which enables us to monitor the behavior of the thermodynamical magnitudes in the non-quasi-static regime. In the latter, the largest modes of the energy eigenstates commonly undergo a process of spontaneous thermalization, where the concept of temperature naturally arises from the unitary evolution of the oscillator, i.e. without relation to any external source of temperature or thermal bath. As the evolution is unitary, this thermalization process can be reversible. We extend the standard definitions of quantum heat and work to account for the change in the populations of the energy levels and relate it to the diagonal entropy to obtain an effective temperature from the associated second principle of thermodynamics. The qualitative behavior of this temperature is similar to the temperature of the largest modes and is only significant in the non quasi-static region, returning to the original value T→0T\rightarrow 0T→0 afterwards. This effect is briefly analyzed in the case of a periodically driven quantum system, where the energy eigenstate mixing and the temperature grow and return to the original value periodically.
The quantum Otto cycle serves as a bridge between the macroscopic world of heat engines and the quantum regime of thermal devices composed from a single element. We compile recent studies of the quantum Otto cycle with a harmonic oscillator as a working medium. This model has the advantage that it is analytically trackable. In addition, an experimental realization has been achieved, employing a single ion in a harmonic trap. The review is embedded in the field of quantum thermodynamics and quantum open systems. The basic principles of the theory are explained by a specific example illuminating the basic definitions of work and heat. The relation between quantum observables and the state of the system is emphasized. The dynamical description of the cycle is based on a completely positive map formulated as a propagator for each stroke of the engine. Explicit solutions for these propagators are described on a vector space of quantum thermodynamical observables. These solutions which employ different assumptions and techniques are compared. The tradeoff between power and efficiency is the focal point of finite-time-thermodynamics. The dynamical model enables the study of finite time cycles limiting time on the adiabatic and the thermalization times. Explicit finite time solutions are found which are frictionless (meaning that no coherence is generated), and are also known as shortcuts to adiabaticity.The transition from frictionless to sudden adiabats is characterized by a non-hermitian degeneracy in the propagator. In addition, the influence of noise on the control is illustrated. These results are used to close the cycles either as engines or as refrigerators. The properties of the limit cycle are described. Methods to optimize the power by controlling the thermalization time are also introduced. At high temperatures, the Novikov–Curzon–Ahlborn efficiency at maximum power is obtained. The sudden limit of the engine which allows finite power at zero cycle time is shown. The refrigerator cycle is described within the frictionless limit, with emphasis on the cooling rate when the cold bath temperature approaches zero.
We investigate the performance of a quantum Otto refrigerator operating in finite time and exploiting local counterdiabatic techniques. We evaluate its coefficient of performance and cooling power when the working medium consists of a quantum harmonic oscillator with a time-dependent frequency. We find that the quantum refrigerator outperforms its conventional counterpart, except for very short cycle times, even when the driving cost of the local counterdiabatic driving is included. We moreover derive upper bounds on the performance of the thermal machine based on quantum speed limits and show that they are tighter than the second law of thermodynamics.
The assumption that quasi-static transformations do not quantitatively alter the equilibrium expectation of observables is at the heart of thermodynamics and, in the quantum realm, its validity may be confirmed by the application of adiabatic perturbation theory. Yet, this scenario does not straightforwardly apply to Bosonic systems whose excitation energy is slowly driven through the zero. Here, we prove that the universal slow dynamics of such systems is always non-adiabatic and the quantum corrections to the equilibrium observables become rate independent for any dynamical protocol in the slow drive limit. These findings overturn the common expectation for quasi-static processes as they demonstrate that a system as simple and general as the quantum harmonic oscillator, does not allow for a slow-drive limit, but it always displays sudden quench dynamics.
Quasiperiodic systems combine sharply different spectral and localisation regimes with a dense hierarchy of dynamical time scales. We show that a finite-rate quantum Otto cycle can use this structure as a thermodynamic probe, revealing which parts of the spectrum become dynamically accessible and thermodynamically relevant during a driven transformation. For a generalised Aubry-André-Harper chain, the resulting response is not determined by the static quasiperiodic regime alone: extended, near-critical, and crossing protocols do not exhibit universal ordering of work fluctuations. Instead, the statistics reflect transition pathways that are simultaneously dynamically coupled, thermally occupied, and energetically significant. The same spectral accessibility separates thermodynamic recovery from microscopic adiabatic recovery. Positive mean work can reappear while the population transfer dynamics remain strongly nonadiabatic because finite-rate population transfer is redistributed toward energetically less costly channels rather than suppressed. The full work distribution reveals a further separation, with a finite rate window in which positive-work cycles form a majority although the mean work remains negative. This behaviour persists in finite-size analysis. A thermally weighted spectral susceptibility connects these effects to a distributed hierarchy of dynamically active spectral regions rather than to a single minimum-gap bottleneck. Finite thermal contacts preserve this picture while introducing a second time scale and intercycle memory. Quantum Otto cycles can therefore provide a fluctuation-resolved thermodynamic probe of quasiperiodic spectral accessibility, complementary to direct spectroscopic and eigenstate-resolved diagnostics.
Abstract The assumption that quasi-static transformations do not quantitatively alter the equilibrium expectation of observables is at the heart of thermodynamics and, in the quantum realm, its validity may be confirmed by the application of adiabatic perturbation theory. Yet, this scenario does not straightforwardly apply to Bosonic systems whose excitation energy is slowly driven through the zero. Here, we prove that the universal slow dynamics of such systems is always non-adiabatic and the quantum corrections to the equilibrium observables become rate independent for any dynamical protocol in the slow drive limit. These findings overturn the common expectation for quasi-static processes as they demonstrate that a system as simple and general as the quantum harmonic oscillator, does not allow for a slow-drive limit, but it always displays sudden quench dynamics.