A Thermodynamic Framework for Coherently Driven Systems

Technical Summary: Thermodynamic Framework for Coherently Driven Systems

Problem Statement
The extension of thermodynamic laws to the nanoscale and quantum regime faces a fundamental obstacle in defining "work." In macroscopic systems, work is associated with accessible, controllable degrees of freedom, while heat corresponds to inaccessible microscopic fluctuations. At the nanoscale, where all degrees of freedom are microscopic, this distinction becomes ambiguous. Specifically, for coherently driven quantum systems (e.g., an optical cavity driven by coherent light), the conventional definition of work relies on a time-dependent Hamiltonian. Under this framework, all energy leaking from the cavity is treated as heat, even if the output light retains coherence and could theoretically be utilized to drive another system. This leads to a conceptual inconsistency where useful energy is misclassified as waste heat, potentially resulting in violations of the second law of thermodynamics in non-secular master equation descriptions. While previous suggestions proposed treating the coherent part of the output as work, a thermodynamically consistent framework ensuring non-negative entropy production for these definitions had not been established.

Methodology
The authors develop a new thermodynamic framework for a quantum system embedded in a cavity, driven by a coherent field and coupled to a bosonic bath. The system is described by a quantum master equation (Eq. 4) that includes a time-dependent drive Hamiltonian and dissipators for the cavity and the internal system.

1. Thermodynamic Hamiltonian: The authors utilize a thermodynamic Hamiltonian ($\hat{H}_{TD}$) rather than the dynamical Hamiltonian ($\hat{H}(t)$) to define internal energy. This ensures consistency with the Markovian master equation and the specific frequency ($\omega_d$) of the drive populating the cavity.
2. Decomposition of Energy Flow: The core methodological innovation is the decomposition of the total energy exchange between the cavity and its environment into two distinct components based on the input and output field operators ($\hat{b}_{in}$ and $\hat{b}_{out}$):
   • Coherent Part (Work): Defined by the change in the squared modulus of the field expectation values ($|\langle \hat{b}_{out} \rangle|^2 - |\langle \hat{b}_{in} \rangle|^2$).
   • Fluctuating Part (Heat): Defined by the change in the field variances (covariances), representing the noise or incoherent fluctuations.
3. Mathematical Transformation: To prove thermodynamic consistency, the authors perform a transformation on the master equation. They rewrite the dissipator $L_c$ and the Hamiltonian $\hat{H}$ into shifted forms ($L_s$ and $\hat{H}_s$) that incorporate the mean field values. This transformation leaves the master equation invariant but allows the heat and work currents to be derived in a manner analogous to the conventional framework, ensuring the validity of Spohn's inequality.
4. Information-Theoretic Derivation: The framework is further justified by interpreting entropy production as the loss of information when describing the environment. By defining the environment's effective state as a displaced thermal state (accounting for the coherent drive) rather than a standard Gibbs state, the authors derive the new second law from the non-negativity of quantum relative entropy.

Key Contributions and Results

• New Definitions of Work and Heat: The paper establishes that if the output light is accessible, work ($P_{io}$) should be defined by the change in the coherent amplitude of the field, and heat ($J_{io}$) by the change in field fluctuations (noise).
  • $P_{io} = -\omega_d (|\langle \hat{b}_{out} \rangle|^2 - |\langle \hat{b}_{in} \rangle|^2)$
  • $J_{io} = -\omega_d (\langle\langle \hat{b}_{out}^\dagger \hat{b}_{out} \rangle\rangle - \langle\langle \hat{b}_{in}^\dagger \hat{b}_{in} \rangle\rangle)$
• Strictly Tighter Second Law: The authors prove that the entropy production rate ($\dot{\Sigma}_{io}$) in this new framework is strictly tighter than or equal to the conventional entropy production rate ($\dot{\Sigma}$): $\dot{\Sigma} \geq \dot{\Sigma}_{io} \geq 0$. This implies that having access to the output light reduces the apparent irreversibility of the process.
• Noise Requirement: A direct consequence of the new second law is that, in the absence of other entropy sources, the noise in the output light must be larger than the noise in the input light.
• Model Applications: The framework is applied to three specific models:
  1. Empty Cavity: In the new framework, both work and heat vanish in the steady state, correctly reflecting that the output is identical to the input (up to a phase) and thus no energy is dissipated or extracted.
  2. Kerr Oscillator: The new framework shows that entropy production is less sensitive to thermal occupation numbers compared to the conventional definition, as it isolates irreversibility to the loss of coherence due to nonlinear effects.
  3. Three-Level Maser: The conventional framework predicts that the system cannot operate as a heat engine ($P > 0$) due to the energetic cost of maintaining coherence. The new framework resolves this, identifying regions where the system acts as a heat engine ($P_{io} < 0$) by reducing the noise in the output light, effectively using the heat current between baths to increase output coherence.

Significance
The paper claims to solve a critical problem in quantum thermodynamics by providing a consistent framework for coherently driven systems where the output is accessible. By distinguishing between coherent work and fluctuating heat, the authors demonstrate that the conventional approach underestimates the efficiency and overestimates the irreversibility of such systems. The framework is presented as a tool for investigating the noise properties of driven-dissipative quantum systems, with specific relevance to optical devices like circulators and amplifiers, and nonreciprocal devices. The authors note that their formalism currently assumes the presence of a phase reference to quantify the coherent part of the output; quantifying work without a perfect phase reference remains an open problem. Additionally, the framework opens avenues for investigating fluctuation theorems and thermodynamic uncertainty relations in this context.