Unifying paradigms of quantum refrigeration: Fundamental limits of cooling and associated work costs

In classical thermodynamics the work cost of control can typically be neglected. On the contrary, in quantum thermodynamics the cost of control constitutes a fundamental contribution to the total work cost. Here, focusing on quantum refrigeration, we investigate how the level of control determines the fundamental limits to cooling and how much work is expended in the corresponding process. We compare two extremal levels of control: first, coherent operations, where the entropy of the resource is left unchanged, and, second, incoherent operations, where only energy at maximum entropy (i.e., heat) is extracted from the resource. For minimal machines, we find that the lowest achievable temperature and associated work cost depend strongly on the type of control, in both single-cycle and asymptotic regimes. We also extend our analysis to general machines. Our work provides a unified picture of the different approaches to quantum refrigeration developed in the literature, including algorithmic cooling, autonomous quantum refrigerators, and the resource theory of quantum thermodynamics.

Figure 1

Comparison of achievable temperatures and associated work costs for scenarios 1 and 2 in the single-cycle, finite repetitions, and asymptotic regimes. The ratio $T/T_R$ is the relative cooling, $T$ being the final temperature and $T_R$ the initial one. The symbols (dots, etc) correspond to maximal cooling (i.e., achieving minimal temperature $T^{*}$) in each scenario. Here we use $T_R=1$.

Figure 2

Model for the minimal thermal machine achieving cooling and allowing for the comparison of two paradigmatic scenarios of quantum refrigeration. After initialization of the machine and target qubit with a thermal bath at room temperature $T_R$, two scenarios are proposed. In scenario 1, the free energy is provided by a hot bath. This corresponds to a low level of control, i.e., maximal entropy change. In contrast, scenario 2 describes a thermal machine requiring a high level of control (e.g., via a coherent battery), that can implement arbitrary unitary operations at zero entropy change.

Figure 3

Parametric plot of the relative temperature of the target qubit $\frac{T}{T_R}$ as a function of its work cost $\mathrm{\Delta }F$ for $E_C=0.4$ and $T_R=1$. The red solid curve corresponds to scenario 1 (incoherent operations), the blue dashed, to scenario 2 (coherent operations). When the cooling is maximal (i.e., the work cost is unrestricted), scenario 2 always outperforms scenario 1, $T_{\text{coh}}^{*}<T_{\text{inc}}^{*}$ and $\mathrm{\Delta }{F}_{\text{coh}}^{*}<\mathrm{\Delta }{F}_{\text{inc}}^{*}$. However, below a critical work cost $\mathrm{\Delta }{F}_{\text{crit}}$, scenario 1 always outperforms scenario 2.

Figure 4

Scenario 1, repeated incoherent operations. Each cycle comprises the steps of (1) the environment reset of qubit $B$ and resource input into qubit $C$ and (2) the cooling unitary operation.

Figure 5

Cooling vs the work cost for different number of repetitions of incoherent operations. Each curve is parametrized by the temperature of the hot bath, $T_H$. $E_C, E$, and $T_R$ are all set to 1.

Figure 6

Scenario 2, coherent operations, in the regime of repeated operations. Each cycle comprises the steps of (1) the environment reset of the machine and (2) cooling.

Figure 7

Scenario 2 in the regime of algorithmic cooling. Each cycle comprises the steps of (1) environment reset, (2) precooling, (3) environment reset, and (4) cooling.

Figure 8

Comparison of the work cost of using algorithmic cooling from the beginning (orange dot-dashed), as opposed to the optimal sequence of coherent operations (blue solid line), and of an autonomous refrigerator (red dashed, parametrized w.r.t. $T_H$). $E_C, E$, and $T_R$ are all set to 1.

Figure 9

Lowest achievable temperature $T^{*}$ and associated free energy change of the resource $\mathrm{\Delta }{F}^{*}$ for different cooling paradigms and machine sizes.

Figure 10

$\mathrm{\Delta }{F}_{\text{crit}}$ is plotted as a function of $T_R$ for various fixed $E_C$.

Figure 11

The internal resource versions of the coherent (solid blue) and the incoherent (dashed red) scenarios are compared. The energy gaps are fixed to $E=1$ and $E_C=\frac{1}{3}$ and the environment temperature to $T_R=1$. One sees that the incoherent scenario always outperforms the coherent one for temperatures that are reachable to both scenarios.

Figure 12

Cooling via repeated coherent operations after the first coherent operation is completed. First the machine qubits $B$ and $C$ are thermalized to the environment temperature $T_R$, following which one performs a unitary that swaps the populations of the levels $|011\rangle$ and $|100\rangle$.

Figure 13

The cycle of steps corresponding to algorithmic cooling. Steps 1 and 3 thermalize qubit $B$ to the environment. Step 2 is the precooling of qubit $C$ by a swap with $B$. Step 4 is the cooling of the target qubit via the usual coherent operation. In the case of optimizing algorithmic cooling w.r.t. the work cost (see Appendix pp9), Step 2 is replaced by a partial rather than full swap.