Efficiency of a Quantum Otto Heat Engine Operating under a Reservoir at Effective Negative Temperatures

We perform an experiment in which a quantum heat engine works under two reservoirs, one at a positive spin temperature and the other at an effective negative spin temperature, i.e., when the spin system presents population inversion. We show that the efficiency of this engine can be greater than that when both reservoirs are at positive temperatures. We also demonstrate the counterintuitive result that the Otto efficiency can be beaten only when the quantum engine is operating in the finite-time mode.

Figure 1

Circuit to QOHE implementation protocol. The blue (red) circles represent transverse rf pulses in the $x$ ($y$) direction that produce rotations by the angles displayed into the circle. The $\theta$ and $\varphi$ angles were adjusted to produce the desired populations (see the main text).

Figure 2

(a) Transition probability $\xi$ versus excited state population $p_{\text{hot}}^{+}$ of the ${}^{13}\mathrm{C}$ nuclear spin. Red and blue regions separate the regimes where efficiency is improved (blue) as compared with the conventional QOHE (red). (b) Transition probability $\xi$ versus expansion or compression (assumed the same) time $\tau$. Note that, due to the protocol adopted here, this transition probability is upper limited by $\xi =1/2$.

Figure 3

Efficiency ($\eta$) against the population ($p_{\text{hot}}^{+}$) of the excited state. Dashed lines are the theoretical results while the dots are the experimental measurements. (a) $\eta$ vs $p_{\text{hot}}^{+}$ for several finite operation times, keeping the frequency ratio fixed (${\nu }_{\text{cold}}=2.0\mathrm{kHz}$ and ${\nu }_{\text{hot}}=3.6\mathrm{kHz}$). The intersection of the curves happens at the transition point between the regimes $\eta <{\eta }_{\text{Otto}}$ and $\eta \ge {\eta }_{\text{Otto}}$. (b) $\eta$ vs $p_{\text{hot}}^{+}$ for different frequency ratios ${\nu }_{\text{cold}}/{\nu }_{\text{hot}}$, with ${\nu }_{\text{cold}}=2\mathrm{kHz}$. The driving time is fixed in $\tau =200\mu \mathrm{s}$.