A Thermodynamic Framework for Coherently Driven Systems

TL;DR

The paper develops a thermodynamically consistent framework for coherently driven quantum systems by introducing a thermodynamic Hamiltonian $\hat{H}_{\rm TD}$ and redefining work and heat in terms of accessible output light through $P^{\rm io}$ and $J^{\rm io}$. It proves a tighter second law, requiring the output light to be more noisy than the input in the absence of extra entropy production, and demonstrates energy bookkeeping via input/output theory. The authors illustrate the framework on Kerr oscillators, a two-level system, and notably the three-level maser, where the engine can increase output coherence/noise reduction under the io-thermodynamics. They also generalize the construction to multiple channels and temperatures and offer an information-theoretic view of entropy production with displaced thermal environments. This approach provides a principled way to analyze noise, coherence, and thermodynamics in driven-dissipative quantum optical devices.

Abstract

The laws of thermodynamics are a cornerstone of physics. At the nanoscale, where fluctuations and quantum effects matter, there is no unique thermodynamic framework because thermodynamic quantities such as heat and work depend on the accessibility of the degrees of freedom. We derive a thermodynamic framework for coherently driven systems, where the output light is assumed to be accessible. The resulting second law of thermodynamics is strictly tighter than the conventional one and it demands the output light to be more noisy than the input light. We illustrate our framework across several well-established models and we show how the three-level maser can be understood as an engine that reduces the noise of a coherent drive. Our framework opens a new avenue for investigating the noise properties of driven-dissipative quantum systems.

Figures (2)

• Figure 1: Setup. A cavity with an embedded quantum system, governed by the Hamiltonian $\hat{H}'$, is driven by the input field, which contains a coherent part, given by the average of the input field $\langle\hat{b}_{\rm in}\rangle$, as well as thermal fluctuations. Compared to the input, the output field, $\mathop{\mathrm{\hat{\textit{b}}_{\text{out}}}}\nolimits$, typically has a smaller coherent part and larger fluctuations. Work ($P^{\rm io}$) and heat ($J^{\rm io}$) are given by the changes in the coherent part and the fluctuations respectively. The heat dissipated by the intracavity system is given by $J'$.
• Figure 2: Conventional and new definitions of heat and work. For (a)-(d) $\Omega/\kappa =10^4$, $\gamma/\kappa = 0.05$, $g/\kappa =0.1$, $f/\kappa =0.01$. (a) Heat current and (b) entropy production for the Kerr oscillator for different occupation numbers $n_{\rm c}$ as a function of detuning $\Delta = \Omega -\omega_{\rm d}$. Note that $P^{({\rm io})} = - J^{({\rm io})}$. Parameters: $\Omega/\kappa =10^4$, $K/\kappa = 0.05$, $f/\kappa =1$. (c) Heat currents and (d) entropy production for the two-level system for different occupation numbers as a function of detuning. (e)-(f) Heat currents and power, and entropy production for the quantum three-level maser; $\Omega/\kappa = 10^4,~g/\kappa=0.7,~f/\kappa=0.1,~\omega_2/\kappa=\omega_3/(3\kappa)=\Omega/\kappa,~\gamma_H/\kappa=0.5,~\gamma_C/\kappa = 100,~n_c=4.54,~n_C=0.007$. (e) We observe that there is a region, $k_BT_H\approx10^5$ in which the system is a heat engine, $P^\text{io}<0$, while $P>0$ (see also inset). The dashed black line in the inset indicates the zero. (f) Entropy production peaks at the heat engine regime.