Finite-time thermal refrigerator in interacting Bose-Einstein Condensates

Abstract

We study a finite-time thermodynamic refrigeration cycle realized numerically in three-dimensional, weakly interacting Bose-Einstein condensates (BECs). The setup consists of three spatially separated condensates -- system, piston, and reservoir -- coupled through time-dependent potential barriers that implement compression, expansion, and contact strokes. Finite-temperature initial states are generated with the Stochastic Ginzburg-Landau equation, and the subsequent dynamics are evolved using the truncated Gross-Pitaevskii equation. To measure temperatures we use a momentum-space thermometry method that provides estimates for each condensate. We find that despite mass transfer and sound excitations, the protocol achieves successful cooling during consecutive cycles: the first cycle lowers its temperature by ~20%, and a second cycle yields additional, though reduced, cooling, reaching a final ~27% cooling from the initial state. Our results show that interacting BECs can sustain finite-time quantum thermal cycles under realistic conditions, and provide a platform for exploring different refrigeration schemes, optimized control protocols, and shortcuts to adiabaticity.

Figures (6)

• Figure 1: Representation of the thermal cycle. The panels show in color the normalized mass density $\rho /\rho _0$ in a two-dimensional slice of the condensates. From top to bottom, the figure depicts: (i) The initial state of each condensate, showing from left to right the system, the piston, and the reservoir. (ii) The compression stage of the piston. (iii) The interaction of the piston with the thermal reservoir. (iv) The subsequent expansion of the piston back to its original volume. (v) The interaction between the piston and the system, before their final decoupling. The cycle concludes after an additional integration for $20 \tau$ to reach the equilibrium.
• Figure 2: Initial PDFs of momentum for the three condensates with their respective fits using Eq. (\ref{['eq:momentum BE']}), showing good agreement between the theoretical expression and the equilibrium obtained through the preparation of the initial thermal bath. Note the three condensates (system, piston, and reservoir) have similar PDFs. The inset shows the PDF of the piston in semilog scale, to showcase the agreement also in the tail of the PDF. Results for the system and the reservoir are analogous.
• Figure 3: Mass dynamics of each condensate during the first (a) and second (b) cycle. Each panel shows the evolution of the mass in each condensate with respect to its initial mass, $M/M_0$ (where $M_0$ is the mass at the beginning of each cycle); $t_i$ is the initial time of each cycle. The dashed lines separate the (1) compression, (2) PR contact, (3) expansion, and (4) PS contact, followed by a final relaxation (see Table \ref{['tab:cycle_stages']}). The piston undergoes substantial mass transfer followed by oscillations during each contact stage, specially on the first cycle.
• Figure 4: PDFs of momentum with their corresponding best fits using Eq. (\ref{['eq:momentum BE']}) for the system (blue), piston (green), and reservoir (orange). From top left to bottom right the PDFs correspond to the end of the (a) compression stage, (b) piston-reservoir contact stage, (c) expansion stage, and (d) final state after relaxation.
• Figure 5: PDFs of momentum for the system at the beginning ($t =0$) of the first cycle (red), at its end ($t = 420 \ \tau$, in blue), and at the end of the second cycle ($t=840\ \tau$, in cyan). The dashed lines indicate the best fit using Eq. (\ref{['eq:momentum BE']}).