Quantum thermal machines driven by vacuum forces

We propose a quantum thermal machine composed of two nanomechanical resonators (two membranes suspended over a trench in a substrate) placed a few $\mu \mathrm{m}$ from each other. The quantum thermodynamical cycle is powered by the Casimir interaction between the resonators and the working fluid is the polariton resulting from the mixture of the flexural (out-of-plane) vibrations. With the help of piezoelectric cells, we select and sweep the polariton frequency cyclically. We calculate the performance of the proposed quantum thermal machines and show that high efficiencies are achieved thanks to (i) the strong coupling between the resonators and (ii) the large difference between the membrane stiffnesses. Our findings can be of particular importance for applications in nanomechanical technologies where a sensitive control of temperature is needed.

Figure 1

(a) Schematic representation of the quantum thermal machine. Two nanomechanical resonators (gold and graphene membranes) are clamped in a dilution chamber at temperatures $T_a$ and $T_b$. A piezoelectric cell selects the flexural (out-of-plane) mode and controls the thermodynamical cycle. (b) Diagrammatic representation of the quantum heat engine based on the lower polariton mode. Two adiabatic (${\mathcal{Q}}^{*}=1$, lighter curve) or quasiadiabatic (${\mathcal{Q}}^{*}=1.2$, darker curve) isentropic strokes (compression and expansion) and two isochoric strokes are represented. $T_a=1$ mK and $T_b=10$ mK. A quantum refrigerator is obtained by inverting the direction of the cycle.

Figure 2

Casimir energy for different values of temperature. From top to bottom, $T=\{300,150,50,10\}$ K. The solid line is the quantum ($T=0$) case.

Figure 3

(a) Spectrum of Eq. (24) near the avoided-crossing region. The golden (dashed) [respectively, gray (dot-dashed)] line represents the bare gold, ${\omega }_k^{\left(a\right)}$ [respectively, bare graphene, ${\omega }_k^{\left(b\right)}]$ flexural dispersion. The solid lines are the polariton modes and ${\mathrm{\Omega }}_1$ (${\mathrm{\Omega }}_2$) is the initial (final) mode used in the thermodynamical cycles. (b) Bare $[n_k^{\left(a\right)}$ and $n_k^{\left(b\right)}]$ and lower-polariton $\left[N_k^{\left(L\right)}\right]$ phonon number for $T_a=1$ mK and $T_b=10$ mK.

Figure 4

Efficiency and power of the QHE. (a) Efficiency for a cycle operating with two adiabatic (${\mathcal{Q}}^{*}=1$, lighter blue) and quasiadiabatic (${\mathcal{Q}}^{*}=1.005$, darker blue) isentropic strokes (compression and expansion). The performance of the machine is upper bounded by the Carnot efficiency. (b) Work extracted during the thermodynamical cycle. The circles represent the compression ratio for which the power is maximum and the shadowed area represents the region of work reversion ($W<0$). The efficiency at maximum power is lower bounded by the Curzon-Ahlborn (CA) efficiency for the adiabatic case and by the value ${\eta }_{-}$ discussed in the text for the quasiadiabatic case. The cycle amplitude is ${\mathrm{\Omega }}_2/{\mathrm{\Omega }}_1\simeq 1.32<T_2/T_1\simeq T_b/T_a$. $T_a=1.0$ mK and $T_b=10$ mK.

Figure 5

(a) Coefficient of performance (CoP) of a quantum refrigerator and (b) figure-of-merit of the refrigerator for the same parameters of Fig. 4. The performance of the machine is upper bounded by the Carnot CoP. The circles represent the compression ratio for which the power is maximum and the shadowed area represents the region of heat reversion ($Q_c<0$). The CoP at maximum power is lower bounded by the Curzon-Ahlborn CoP for the adiabatic case and by the value ${\zeta }_{-}$ discussed in the text for the quasiadiabatic case. $T_a=7.5$ mK and $T_b=10$ mK.

Figure 6

Hopfield coefficients measuring the fraction of gold (orange dashed line) and graphene (gray dot-dashed line) in the lower polariton mode $A_k$.