Particle creation, adiabaticity, and irreversibility in the TDHO

TL;DR

This work analyzes a time-dependent harmonic oscillator (TDHO) evolving unitarily to elucidate how particle creation, adiabaticity, and irreversibility interrelate when the system is described in different representations. It develops an exact framework based on initial, diagonal, and invariant representations connected by squeezed transformations and an explicit evolution operator $\mathcal{U}(t)$ that is valid for arbitrary $omega(t)$, including non quasi-static regions. The authors adapt quantum heat and work definitions to depend on the chosen representation, relate diagonal entropy to particle creation, and show that the largest non-invariant modes thermalize reversibly with a mode temperature $T_K$, while a macroscopic temperature $T_{macr}$ emerges in appropriate limits. Numerical and analytical results demonstrate transition dynamics and regional thermalization, with potential implications for quantum thermal machines and understanding quantum-to-classical transitions under unitary evolution.

Abstract

We present an exact description of the dynamics and the thermodynamics of a time dependent harmonic oscillator (TDHO) that follows a unitary evolution. In that context, we study the relationship between particle creation, adiabaticity and irreversibility in terms of three different representations: the initial, the diagonal and the invariant representations of the TDHO. We provide analytical results that are valid for any functional value of the frequency and along the whole evolution of the TDHO, which allows us to monitor the behavior of the thermodynamical magnitudes in regions that are not fully considered in previous works, i.e. the transition and the non quasi-static regions. We supplement the analytical results with numerical calculations and graphs. They both show how the largest modes of the non invariant representations may undergo a process of reversible thermalization, where the concept of temperature naturally arises from the unitary evolution of the oscillator with no relation to any external concept of temperature. It would allow us to monitor an unexpected classical-to-quantum transition, which might entail a violation of the third principle of classical thermodynamics. We also provide adaptations of the customary definitions of the quantum heat and work that account for the particle creation of the TDHO. They thus depend on the representation and their evolution is different for the three representations analyzed in this paper, which might have important consequences in the development of quantum thermal machines. Finally, we study the relationship between the creation of particles and the diagonal entropy, which is derived from the von Neumann entropy in the diagonalization limit. The condition of no production of entropy under unitary evolution suggests the definition of a mode temperature that would correspond to the thermal temperature is some appropriate limit.

Figures (35)

• Figure 1: Relationship between quantum and classical (thermo)-dynamics. The classical limit of the quantum theory is obtained when the action $S$, in $\hbar$ units, is very large. Classically, thermal behavior appears when a large number of particles reach a dynamical equilibrium. It is expected that quantum thermodynamics is, on the one hand, the quantum limit of equilibrium for a large number of particles and, on the other hand, it gives rise to classical thermodynamics in the classical limit.
• Figure 2: Relationship between the evolution of the TDHO in the three representations considered in this paper, where the operators $\mathcal{U}$, $U_0$, and $S_S$, are given by \ref{['Ut01']}, \ref{['aio0rels01']} and \ref{['aio0rels02']}, respectively.
• Figure 3: Frequency $\omega_1(t)$, Eq. \ref{['freq01']}. It starts from a constant value $\omega_0$ in the in region, and ends with a constant value $\omega_f$ in the out region.
• Figure 4: Time dependent probabilities of measuring the TDHO in the state $|M_0\rangle$, Eq. \ref{['Prob0_01']}, for different values of $M$, when the TDHO is initially in the vacuum state, $|0\rangle$.
• Figure 5: Time dependent probabilities of measuring the TDHO in the state $|M_0\rangle$, Eq. \ref{['Prob0_01']}, for different values of $M$, when the TDHO is initially in the number state, $|1\rangle$.