Breaking the Carnot limit without violating the second law: A thermodynamic analysis of off-resonant quantum light generation

The Carnot limit, formulated in 1824, represents the maximal efficiency of a classical heat engine. In this work we present a thermodynamical analysis of a light amplifier based on a three-level atom coupled off-resonantly to a single quantized cavity mode and to two heat reservoirs with positive temperatures. Based on standard work and heat flow equilibrium, we show that for a cavity blue-detuned with respect to the atomic resonance, the system can surpass the Carnot limit. Nevertheless, the second law of thermodynamics is still obeyed, as the total entropy always increases. By analyzing a semiclassical version of the model, we derive a formula for the critical frequency for which the Carnot limit is broken and a formula for the amplifier efficiency which agrees with its quantum counterpart. In the semiclassical regime, however, the second law is not satisfied and hence it does not offer a physically acceptable description of the system. Finally, we show that breaking the Carnot limit occurs also in a blue-detuned quantum amplifier with output coupling, which represents a realistic model of a laser or maser.

Figure 1

Three-level system interacting with two heat reservoirs (hot and cold) and a quantized cavity mode.

Figure 2

(a) Quasi-steady-state efficiency (solid green) and Carnot efficiency (dashed blue). (b) Total entropy production. (c) Atomic (dashed red), field (dotted blue), and atomic-field (solid green) entropies. (d) Average photon number. (e) Quasi-semiclassical energy, $\tilde{E}$. Parameters: ${\omega }_f=50\lambda =1.05({\omega }_1-{\omega }_2)$, $\frac{{\omega }_1-{\omega }_2}{\lambda }={10}^3$, $\frac{\lambda }{{\Gamma }_{1\left(2\right)}}={10}^3$, $n_1=5$, and $n_2=4.3$.

Figure 3

(a) Steady-state efficiency (solid green) and Carnot efficiency (dashed blue). (b) Entropy production without the (positive) contribution of the reservoir coupled to the cavity mirrors. (c) Average photon number of the damped amplifier. (d) Average photon number of the detuned JCM (without damping and amplification). Parameters: ${\omega }_f=50\lambda =1.05({\omega }_1-{\omega }_2)$, $\frac{{\omega }_1-{\omega }_2}{\lambda }={10}^3$, $\frac{\lambda }{{\Gamma }_{1\left(2\right)}}={10}^3$, $n_1=5$, $n_2=4.3$, and $\frac{{\Gamma }_{1\left(2\right)}}{{\Gamma }_f}={10}^3$.