Minimizing Dissipation via Interacting Environments: Quadratic Convergence to Landauer Bound

Abstract

We explore the fundamental limits on thermodynamic irreversibility when cooling a quantum system in the presence of a finite-size reservoir. First, we prove that for any non-interacting $n$-particle reservoir, the entropy production $Σ$ decays at most linearly with $n$. Instead, we derive a cooling protocol in which $Σ\propto 1/n^2$, which is in fact the best possible scaling. This becomes possible due to the presence of interactions in the finite-size reservoir, which must be prepared at the verge of a phase transition. Our results open the possibility of cooling with a higher energetic efficiency via interacting reservoirs.

Figures (4)

• Figure 1: Framework. a) A system (S) is cooled by external control $V(t)$ combined with the interaction with a finite-size reservoir (B) made-up of non-interacting components. b) In this case, the thermal reservoir features strong interactions between its components. We prove here that this can be exploited for more efficient cooling of the system.
• Figure 2: Thermodynamic processes for Landauer erasure. The solid lines correspond to entropy production in thermodynamic processes. RW, SSP and TL correspond to collisional processes described, respectively, in Refs. reeb2014improved, skrzypczyk2014work and taranto2024efficiently. Red curve corresponds to the process using an interacting environment described in Sec. \ref{['sec:non-inter']}. Dashed lines correspond to analytic bounds, i.e. lower-bound for non-interacting environments \ref{['eq:noninter_bound']} (blue), lower-bound for general environments \ref{['eq:rw_bound']} (red) and an upper-bound for the thermodynamic process discussed in Sec. \ref{['sec:non-inter']} [see Eq. \ref{['eq:diss_protocol']}] (grey). Parameters used: $\beta = 1$, $q = n^{-\alpha}$ with $\alpha = 3$.
• Figure 3: Heat capacity close to a phase transition. The main plot shows the rescaled heat capacity $C/n$ as a function of the inverse temperature for the Hamiltonian given by Eq. \ref{['eq:inter_spec']} with parameter $\beta = \beta_0 = 1$. We observe that for the interacting environment (red curves) phase transition occurs precisely at $\beta = \beta_0$, whereas for the non-interacting environment the rescaled heat capacity $C/n$ approaches a constant value. The inset shows that $C$ diverges as $C \propto n^2$ as a function of the number of qubits $n$, whereas $C \propto n$ for a non-interacting environment.
• Figure 4: Entropy production at criticality. The panel shows the entropy production during Landauer erasure as a function of the number of qubits $n$, realised via the optimal protocol discussed in Appendix \ref{['app:phase_transition_notsuff']}. The environment used in the protocol clearly exhibits a thermal phase transition, as demonstrated by the super-extensive scaling of heat capacity [see Eq. \ref{['eq:quadratic_heat_capacity']}]. For comparison we also plot the curves corresponding to the linear (blue) and quadratic (red) decay of entropy production. We see that the presence of a phase transition is not sufficient to observe a quadratic scaling of entropy production.