Violating the thermodynamic uncertainty relation in the three-level maser

Nanoscale heat engines are subject to large fluctuations which affect their precision. The thermodynamic uncertainty relation (TUR) provides a trade-off between output power, fluctuations, and entropic cost. This trade-off may be overcome by systems exhibiting quantum coherence. This Letter provides a study of the TUR in a prototypical quantum heat engine, the Scovil–Schulz-DuBois maser. Comparison with a classical reference system allows us to determine the effect of quantum coherence on the performance of the heat engine. We identify analytically regions where coherence suppresses fluctuations, implying a quantum advantage, as well as regions where fluctuations are enhanced by coherence. This quantum effect cannot be anticipated from the off-diagonal elements of the density matrix. Because the fluctuations are not encoded in the steady state alone, TUR violations are a consequence of coherence that goes beyond steady-state coherence. While the system violates the conventional TUR, it adheres to a recent formulation of a quantum TUR. We further show that parameters where the engine operates close to the conventional limit are prevalent and TUR violations in the quantum model are not uncommon.

Figure 1

Illustration of the three-level maser with energies ${\omega }_{\alpha }$ for $\alpha \in \{x,u,l\}$. Bath $\alpha$ with population $n_{\alpha }$ induces a transition with rate ${\gamma }_{\alpha }$. The external ac field connects the levels $u$ and $l$ with strength $\epsilon$ and frequency ${\omega }_d=\mathrm{\Delta }+{\omega }_u-{\omega }_l$. In one cycle of operation, the system is excited from bath $l$ to an excited state $x$, from which it relaxes to the upper laser level $u$ by emitting heat to bath $u$. The cycle is closed by a photon that is emitted into the driving field, producing work.

Figure 2

Probe of thermodynamic uncertainty for the SSDB maser $\mathcal{Q}$ (solid blue line) and the classical reference system ${\mathcal{Q}}^{\mathrm{cl}}$ (dashed orange line) along (a) driving strength $\epsilon$, (b) bath population $n_u$, and (c) detuning $\mathrm{\Delta }$. The fixed parameters are ${\gamma }_u=2$, ${\gamma }_l=0.1$, and $n_l=5$, while the parameters $n_u=0.027$, $\epsilon =0.15$, and $\mathrm{\Delta }=0$ are varied according to the respective panels. The classical TUR limit is indicated by the dotted black line, and the quantum TUR bound $\mathcal{B}$ is indicated by the dash-dotted green line.

Figure 3

Histograms of sampled values of (a) $\mathcal{Q}$ and (b) ${\mathcal{Q}}^{\mathrm{cl}}$ from Monte Carlo exploration of the parameter space. Insets show the subset of the sampled data in the vicinity of the classical TUR limit. The parameters are sampled from the uniform distributions ${\gamma }_{\alpha }\in [{10}^{-4};5]$, $n_{\alpha }\in [{10}^{-4};10]$, $\epsilon \in [{10}^{-4};1]$, and $\mathrm{\Delta }\in [0;1]$, and the ${10}^8$ data points are arranged in bins with a width of 0.01.

Figure 4

Heat maps of (a) $\mathcal{Q}$ for the SSDB maser, (b) the off-diagonal matrix element $|{\rho }_{ul}|$ and its imaginary component (inset), and (c) ${\mathcal{Q}}^{\mathrm{cl}}$ for the classical reference system. The maser parameters are ${\gamma }_u=2$, $n_u=0.027$, ${\gamma }_l=0.1$, and $n_l=5$. While $\mathcal{Q}$ shows a behavior similar to $\mathrm{Im}\left[{\rho }_{ul}\right]$, ${\mathcal{Q}}^{\mathrm{cl}}$ shows behavior similar to $|{\rho }_{ul}|$.