Entanglement enhances cooling in microscopic quantum refrigerators

Small self-contained quantum thermal machines function without external source of work or control but using only incoherent interactions with thermal baths. Here we investigate the role of entanglement in a small self-contained quantum refrigerator. We first show that entanglement is detrimental as far as efficiency is concerned—fridges operating at efficiencies close to the Carnot limit do not feature any entanglement. Moving away from the Carnot regime, we show that entanglement can enhance cooling and energy transport. Hence, a truly quantum refrigerator can outperform a classical one. Furthermore, the amount of entanglement alone quantifies the enhancement in cooling.

Figure 1

Schematic diagram of the quantum refrigerator. The fridge consists of three qubits (inside the yellow circle), each in weak thermal contact (wiggly lines) with a bath at a different temperature. The qubits interact via the weak interaction Hamiltonian $H_{\mathrm{int}}$, which couples the two degenerate levels $|010\rangle$ and $|101\rangle$, depicted by the arrows. The lower qubit [purple (dark gray)] is the object to be cooled. At equilibrium, it reaches a temperature $T_S<T_C$. The other two qubits [red and blue (medium gray)] are the machine qubits, connected to heat baths at temperatures $T_R$ and $T_H$.

Figure 2

Entanglement can enhance cooling. (a) Density plot depicting the relative advantage $\zeta$ in cooling a qubit at $T_C=1K$, as a function of the available bath temperatures ($T_R$ and $T_H$) for optimal refrigerators. Entanglement provides an advantage ($\zeta >1$) for all points above the red dashed line. In the dark blue region, below the red dashed line, the optimal fridge is separable on all bipartitions ($\zeta =1$). For $T_R$ close to $T_C$ and sufficiently high $T_H$, entanglement is a widely present feature of the optimal quantum refrigerator. (b) Slices through the density plot for three different values of $T_H$, showing in more detail the relative advantage in cooling. The parameters used here are $p_C={10}^{-5}$, $E_1=1J$, and parameter bounds $p_R$,$p_H$,$g\le {10}^{-4}$.

Figure 3

Cooling advantage is determined by the amount of entanglement. Plot of relative cooling enhancement $\zeta$ against amount of entanglement on the bipartition $R|CH$ (measured by the concurrence $C$), as evaluated from Fig. 2. Since all points lie on a single curve, it follows that $\zeta$ is determined solely by $C$ and does not depend on the temperatures of the baths $T_R$ and $T_H$. The fact that the behavior is monotonic strongly suggests a functional relationship. For convenience the data plotted correspond to the three horizontal slices of Fig. 2. Taking random sample points from Fig. 2 leads to a similar result.