Experimental Characterization of a Spin Quantum Heat Engine

Developments in the thermodynamics of small quantum systems envisage nonclassical thermal machines. In this scenario, energy fluctuations play a relevant role in the description of irreversibility. We experimentally implement a quantum heat engine based on a spin-$1/2$ system and nuclear magnetic resonance techniques. Irreversibility at a microscope scale is fully characterized by the assessment of energy fluctuations associated with the work and heat flows. We also investigate the efficiency lag related to the entropy production at finite time. The implemented heat engine operates in a regime where both thermal and quantum fluctuations (associated with transitions among the instantaneous energy eigenstates) are relevant to its description. Performing a quantum Otto cycle at maximum power, the proof-of-concept quantum heat engine is able to reach an efficiency for work extraction ($\eta \approx 42\%$) very close to its thermodynamic limit ($\eta =44\%$).

Figure 1

Quantum heat engine schematics. (a) Thermodynamic cycle employing a spin $1/2$ as the working medium. (b) Simplified pulse sequence of the experimental protocol. ${}^1\mathrm{H}$ and ${}^{13}\mathrm{C}$ nuclear spins are initially prepared in thermal states corresponding to hot and cold spin temperatures, respectively. Blue (red) circles represent $x$ ($y$) rotations by the displayed angle produced by transverse rf pulses. Orange connections stand for free evolutions under the scalar interaction (${\mathcal{H}}_J$) during the time displayed above the symbol. The unitary driving for the energy gap expansion (compression) protocol is implemented by a time-modulated rf field resonant with the ${}^{13}\mathrm{C}$ nuclear spin. The hydrogen nucleus is used to deliver the heat at the proper part of the cycle, working as a heat bus. (c) Required temperatures for work extraction at finite-time operation mode. The engine extracts work only if the hot ($T_2$) and cold ($T_1$) source temperatures correspond to a point below the curve defined by the energy level transition probability $\xi$.

Figure 2

Extracted work probability distribution of the quantum engine with Hamiltonian driving time lengths (a) $\tau =100\mu \mathrm{s}$ and (b) $\tau =500\mu \mathrm{s}$. Cold and hot source temperatures are set at $k_B{T}_1=(6.6\pm 0.1)\mathrm{peV}$ and $k_B{T}_2^B=(40.5\pm 3.7)\mathrm{peV}$, respectively. The experimental data (points) are well fitted by a sum of nine Lorentzian peaks (the solid line) centered approximately at 0, $\pm h({\nu }_2-{\nu }_1)$, $\pm {\nu }_1$, $\pm {\nu }_2$, and $\pm h({\nu }_2+{\nu }_1)$ (dashed columns), in agreement with the theoretical expectation (see Fig. S3 in the Supplemental Material [63]). The error bars are smaller than the symbol size and are not shown.

Figure 3

Spin quantum engine figures of merit: (a) average extracted work, (b) efficiency, (c) efficiency lag due to entropy production [according Eq. (5); the minimum lag is ${\eta }_{\text{Carnot}}-{\eta }_{\text{Otto}}$], and (d) extracted power as a function of the driving protocol time length ($\tau$). Points represent experimental data. The dashed lines are based on theoretical predictions and numerical simulations. In all experiments, the spin temperature of the cold source is set at $k_B{T}_1=(6.6\pm 0.1)\mathrm{peV}$. Data in blue and red correspond to implementations with the hot source spin temperatures set at $k_B{T}_2^A=(21.5\pm 0.4)\mathrm{peV}$ and $k_B{T}_2^B=(40.5\pm 3.7)\mathrm{peV}$, respectively.