An approach to study the adiabaticity and irreversibility in the TDHO

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.

💡 Research Summary

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The paper investigates the interplay between parametric amplification (particle creation), adiabaticity, and irreversibility in a time‑dependent quantum harmonic oscillator (TDHO) that evolves unitarily. Using the Lewis‑Riesenfeld invariant and the Ermakov equation, the authors obtain exact solutions for the TDHO wavefunctions for an arbitrary frequency profile ω(t). They express the time‑evolved state initially prepared in a number state |N⟩ as a superposition of the instantaneous energy eigenstates |M, ω(t)⟩, with transition amplitudes given by Legendre functions of a complex argument. The corresponding transition probabilities P_M(N; t) are derived analytically and shown to deviate from the Kronecker delta when the frequency varies rapidly (non‑quasi‑static regime).

By constructing the density matrix in the instantaneous eigenbasis, the authors separate diagonal (population) and off‑diagonal (coherence) contributions. For high‑lying modes they approximate the Legendre functions and, using Stirling’s formula, demonstrate that the diagonal part takes a thermal‑like form
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