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arXiv:2512.00820 · Physics Letters A

An approach to study the adiabaticity and irreversibility in the TDHO

Salvador J. Robles-Perez, Salvador Castillo-Rivera

quant-phSubmitted Nov 30, 2025• Announced Dec 2, 2025• 0 citations• Revised Feb 19, 2026 (v2)

v1 · Nov 30, 2025 v2 · Feb 19, 2026 v3 · Sep 3, 2026

Abstract

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, facilitating the monitoring of an unexpected \emph{classical-to-quantum} transition that might entail a quantum violation of the third principle of classical thermodynamics. We adapt the standard definitions of quantum heat and work to account for the change in the populations of the energy levels in the non-quasi-static evolution of the TDHO.

DOI: 10.1016/j.physleta.2026.132089
Comments: 11 pages, 5 figures

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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, facilitating the monitoring of an unexpected \emph{classical-to-quantum} transition that might entail a quantum violation of the third principle of classical thermodynamics. We adapt the standard definitions of quantum heat and work to account for the change in the populations of the energy levels in the non-quasi-static evolution of the TDHO.

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.

We study the first law of thermodynamics utilizing a frequency modulated quantum harmonic oscillator. For an adiabatic change in frequency, the temperature of the oscillator is proportional to the frequency. Then, we present a number representing the non-adiabaticity determined by the relation between Ermakov-Lewis-Riesenfeld invariant and hamiltonian. We write the first law of thermodynamics for the oscillator's quantum state having non-adiabaticity and then find the temperature. The temperature gains a non-negative non-adiabaticity correction to the adiabatic temperature. We describe the thermalization process for the oscillator as a differential equation. The equation deals with the evolution of the exponent of the density operator. Finally, we write the first law of thermodynamics for thermalization for the harmonic oscillator.

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We study a quantum harmonic oscillator undergoing thermalization. To describe the thermalization process, we generalize the Ermakov-Lewis-Riesenfeld (ELR) invariant method for the oscillator. After imposing appropriate conditions on the thermalization process, we introduce an ansatz equation that describes the time evolution effectively. We write down the first law for thermalization in the same form as that for ordinary thermodynamics. Here, the thermalization effect appears through a change of the ELR frequency. Finally, we obtain the oscillator's energy undergoing thermalization as a function of entropy and its time derivative.