Thermodynamics of a Minimal Algorithmic Cooling Refrigerator

We investigate, theoretically and experimentally, the thermodynamic performance of a minimal three-qubit heat-bath algorithmic cooling refrigerator. We analytically compute the coefficient of performance, the cooling power, and the polarization of the target qubit for an arbitrary number of cycles, taking realistic experimental imperfections into account. We determine their fundamental upper bounds in the ideal reversible limit and show that these values may be experimentally approached using a system of three qubits in a nitrogen-vacancy center in diamond.

Figure 1

Schematic illustration of the minimal three-qubit algorithmic cooling cycle: in a first (compression) step, heat is extracted from the target qubit ($t$), cooling it down while heating up the two reset qubits ($r$). In a second (refresh) step, the reset qubits are rethermalized to the bath temperature $T_h$.

Figure 2

Thermodynamic performance of the algorithmic cooling refrigerator per cycle. (a) Coefficient of performance $\zeta (n)$, Eq. (8); (b) cooling power $J(n)$, Eq. (8); and (c) polarization of the target qubit ${\epsilon }_1(n)$, Eq. (9); for various values of the damping rate $\gamma$ and of the mixing angle $\theta$. These two parameters have radically different effects: whereas the decay constant affects the asymptotic value of the polarization, the mixing angle changes the convergence rate to that value. In addition, the behavior of the cooling power mostly depends on the mixing angle, while the COP depends on both variables. The fundamental upper bounds in the reversible limit ($\gamma =0$) are shown by the blue squares. Parameters are ${\epsilon }_1(0)=0$, ${\epsilon }_2(0)={\epsilon }_3(0)=\epsilon =0.6$.

Figure 3

Comparison with the Carnot coefficient of performance. In the reversible regime ($\gamma =0$), the coefficient of performance $\zeta (n)$ (full symbols) gets close to the corresponding Carnot limit ${\zeta }_C(n)$ (empty symbols) after a few cycles. The Carnot bound is generally not reached in the presence of losses ($\gamma \ne 0$). Same parameters as in Fig. 2.

Figure 4

Experimental performance of the three-qubit algorithmic cooling refrigerator. (a) Experimental data for heat $Q(n)$ (green triangles) show excellent agreement with theory (orange diamond) with $\gamma ={10}^{-4}$ and $\theta =\pi /3.4$. (b) Cooling power $J(n)$ and COP $\zeta (n)$ also agree very well with theory [$\zeta (n)$ becomes sensitive to measurement errors for larger $n$]. Error bars correspond to the standard deviation.