Thermalization from quenching in coupled oscillators

TL;DR

The paper presents a bath-free, finite-time protocol to thermalize a quantum harmonic oscillator by coupling it to a second identical oscillator and applying a sequence of sudden quenches in frequency and coupling. Leveraging Gaussian two-mode dynamics and Ermakov equations, the thermalization condition reduces to three algebraic constraints on three tunable parameters, yielding exact analytic solutions for a dense set of discrete temperatures and enabling arbitrary-temperature approximations otherwise. A key contribution is the identification of special discrete states (SDS) where exact tuning is possible, with explicit formulas for the parameter triplets and the corresponding temperatures, including the fastest SDS case. The approach is experimentally accessible (e.g., with trapped ions) and naturally extends to heating/cooling protocols and potential multimode generalizations, providing a simple, controllable route to rapid thermal state preparation in quantum thermodynamics.

Abstract

We introduce a finite-time protocol that thermalizes a quantum harmonic oscillator, initially in its ground state, without requiring a macroscopic bath. The method uses a second oscillator as an effective environment and implements sudden quenches of the oscillator frequencies and coupling. Owing to the Gaussian nature of the dynamics, the thermalization condition reduces to three solvable equations, yielding exact analytic solutions for a dense discrete set of temperatures and numerical solutions in all other cases. Any target temperature can be approximated with arbitrary precision, with a trade-off between speed and accuracy. The simplicity of the protocol makes it a promising tool for rapid, controlled thermalization in quantum thermodynamics experiments and state preparation.

Figures (9)

• Figure 1: The family of thermal density matrices is represented by the curve $\vec{R}_\beta = \left( \coth(\beta\omega),\, 0,\ -\mathrm{cosech}(\beta\omega) \right)$. The curve lies entirely in the $X$--$Z$ plane, shown as the shaded region, where it is described by part of the hyperbola $Z=-\sqrt{X^2-1}$. The blue dot indicates the ground state, and the arrow shows the direction of increasing temperature.
• Figure 2: Contour representation of the effective 2D potential of the coupled oscillator system, for typical values of $\omega'/\omega$ and $k/\omega^2$, when the system is in (a) the uncoupled phase, and (b) the coupled phase ($0<t<\tau$).
• Figure 3: Evolution of $\rho^{(1)}_{x_1\,x'_1}$ represented in the $\vec{R}$-space, assuming a randomly chosen set of the tunable parameters $\omega'$, $k$ and $\tau$. The oscillator-1 is initially in the ground state (blue dot) and evolves, at $t=\tau$, to the green point. The red curve is the family of thermal states.
• Figure 4: (a) Protocol for preparing oscillator-1 in a thermal state. Both oscillators start in their ground states and are initially decoupled. A coupling of strength $k$ is suddenly introduced, and their frequencies are shifted to $\omega'$ for a duration $\tau$ (the active-phase). The oscillators are then decoupled and their frequencies reset to $\omega$. Tuning $k$, $\omega'$, and $\tau$ brings oscillator-1 to a thermal state at $t = \tau$. (b) The energy-frequency diagram for oscillator-1 as it undergoes the protocol.
• Figure 5: The tunable parameters $\tilde{\omega}'$ and $\tilde{k}$ as a function of the SDS (inverse-)temperatures and as given in \ref{['eq:omegapr_vs_T']} and \ref{['eq:k_vs_T']}. The dashed and dot-dashed curves correspond to $l<n$ and $l>n$ cases, respectively. Note that these relations are valid only on the SDS $T_{nl}$.