Fundamental limits for cooling of linear quantum refrigerators

We study the asymptotic dynamics of arbitrary linear quantum open systems that are periodically driven while coupled with generic bosonic reservoirs. We obtain exact results for the heat flowing from each reservoir, and these results are valid beyond the weak-coupling or Markovian approximations. We prove the validity of the dynamical third law of thermodynamics (Nernst unattainability principle), showing that the ultimate limit for cooling is imposed by a fundamental heating mechanism that dominates at low temperatures, namely the nonresonant creation of excitation pairs in the reservoirs induced by the driving field. This quantum effect, which is missed in the weak-coupling approximation, restores the unattainability principle, the validity of which was recently challenged.

Figure 1

Minimum temperature $T_{\text{min}}$ as a function of the coupling strength ${\gamma }_0$. $T_{\text{min}}$ is numerically obtained as the lowest temperature for which ${\dot{\overline{Q}}}_{\alpha }^{\text{RP}}>\left|{\dot{\overline{Q}}}_{\alpha }^{\text{NRH}}\right|$. Results are shown for the adaptive cooling strategy considered in [2]. The system consists of a single harmonic oscillator of frequency ${\mathrm{\Omega }}_0$ in contact with two reservoirs whose spectral densities satisfy the cooling condition $I_{\alpha }\left({\mathrm{\Omega }}_0\right)\ll I_{\beta }\left({\mathrm{\Omega }}_0\right)$. Heat rates are obtained through an exact numerical evaluation (see details in Appendix pp4).

Figure 2

Typical shape of the function $p_{\alpha ,\beta }^{\left(1\right)}\left(\omega \right)$ in the weak-coupling limit.

Figure 3

Spectral densities used for the numerical evaluation of the heat rates (${\lambda }_{\alpha }={\lambda }_{\beta }=1$). They satisfy exactly the cooling condition $I_{\alpha }\left({\mathrm{\Omega }}_0\right)\ll I_{\beta }\left({\mathrm{\Omega }}_0\right)$.

Figure 4

Resonant and nonresonant contributions to the total heat rate ${\dot{\overline{Q}}}_{\alpha }$. The driving frequency is ${\omega }_d={\mathrm{\Omega }}_0-T_0$ and ${\lambda }_{\alpha }=1$.