Thermodynamics of Coherence-Selective Quantum Reset Protocols

1. Problem Statement

Open quantum systems are often modeled using repeated interaction protocols where a system interacts with an environment (bath) over discrete time intervals, followed by a "reset" of the environment. Previous work established two limiting protocols:

• Repeated Interaction (RI): The environment is reset, and all system-environment coherences are erased ($\eta=0$).
• Evolving Correlations (EC): The environment is reset, but system-environment coherences are preserved ($\eta=1$).

The authors identify a gap in understanding the thermodynamic consequences of intermediate protocols where a tunable fraction of coherence is retained. Specifically, they ask:

• How does the retention of system-environment memory (coherence) affect the exact stroboscopic fixed point?
• What is the thermodynamic cost (heat dissipation and entropy production) of selectively preserving this coherence?
• Is the protocol that maximizes memory retention the same as the one that maximizes heat dissipation?

2. Methodology

The authors develop an exact theory within the single-particle density matrix (SPDM) formalism for quadratic open quantum systems. This approach avoids Markovian approximations or weak-coupling assumptions.

• Model: A single fermionic level coupled to a semi-infinite tight-binding chain (structured bath). The Hamiltonian is quadratic, ensuring the dynamics close exactly at the SPDM level.
• Coherence-Selective Reset Rule: They introduce a one-parameter family of reset channels labeled by $\eta \in [0, 1]$.
  • $\eta = 0$: Complete coherence erasure (RI limit).
  • $\eta = 1$: Complete coherence preservation (EC limit).
  • $0 < \eta < 1$: Partial retention of the system-environment coherence block $\rho_{SE}$, while the environment block $\rho_{EE}$ is reset to a reference state $C_0$.
• Mathematical Framework:
  • The dynamics are formulated as an exact affine map acting on the vector of SPDM entries: $W_{n+1} = A_\eta(\tau)W_n + B_\eta(\tau)$.
  • The authors solve for the unique stroboscopic fixed point ($W_{fp}$) where the state repeats after every cycle.
  • Thermodynamics: They define the reset heat current ($J_Q$) and entropy production rate ($\sigma$) based on the energy change of the environment block during the non-unitary reset step, attributed to a "super-environment."
• Observables:
  • Retained coherence spectrum ($S_c^*$).
  • Reset heat current ($J_Q^*$).
  • Coherence efficiency ratio ($R^* = \text{Coherence} / \text{Heat}$).
  • Analysis is performed for both zero and non-zero chemical potentials ($\mu$).

3. Key Contributions

1. Exact Interpolation: The paper provides the first exact analytical and numerical framework for continuously interpolating between coherence erasure and preservation in a structured bath, moving beyond the binary RI/EC dichotomy.
2. Non-Monotonic Thermodynamics: A central theoretical discovery is that the relationship between memory retention and thermodynamic cost is non-monotonic.
3. Decoupling of Optimization Goals: The authors demonstrate that the operating points for maximizing coherence, maximizing heat dissipation, and maximizing coherence-per-cost are distinct and do not coincide.
4. Role of Bath Structure: The study rigorously quantifies how the spectral density of the bath (in-band vs. out-of-band levels) scales the magnitude of these effects without altering the fundamental tradeoff geometry.
5. Chemical Potential Analysis: They show that a non-zero chemical potential acts as a "filling-biased deformation" of the fixed-point geometry rather than introducing a new independent particle-transport tradeoff.

4. Key Results

• Coherence vs. Heat Tradeoff:

  • Retained Coherence ($C_{SE}^*$): Increases monotonically with the retention parameter $\eta$. The maximum memory is stored at $\eta \to 1$ (EC limit).
  • Reset Heat Current ($J_Q^*$): Is non-monotonic with respect to $\eta$. It starts low at $\eta=0$, rises to a maximum at an intermediate value of $\eta$, and then decreases as $\eta \to 1$.
  • Conclusion: The protocol that stores the most coherence is not the one that dissipates the most heat. The heat-optimal protocol requires a balance where coherence is preserved enough to drive energy exchange but erased enough to maintain a strong reset action.

• Operating Diagrams:

  • Coherence-Optimal: Driven toward the coherence-preserving endpoint ($\eta \approx 1$).
  • Coherence-per-Cost Optimal: Also driven toward the coherence-preserving endpoint, indicating that pure decoherence is thermodynamically inefficient.
  • Heat-Optimal: Located at an intermediate $\eta$ that shifts significantly with the reset interval $\tau$.

• Spectral Dependence:

  • In-band levels (resonant with bath modes) exhibit significantly higher coherence and heat currents than out-of-band levels.
  • However, the qualitative shape of the tradeoff (the "arch" in the coherence-heat plane) remains identical regardless of spectral placement.

• Non-Zero Chemical Potential ($\mu \neq 0$):

  • The fixed-point particle current vanishes ($J_N^* = 0$) because the system block is unchanged by the reset and total particle number is conserved during the unitary step.
  • A non-zero $\mu$ deforms the coherence-heat landscape by shifting the Fermi profile, effectively acting as a control knob that biases the optimal operating points but does not create a new independent transport channel.

5. Significance

• Memory Engineering: The work establishes coherence-selective resetting as a distinct control principle. It provides a blueprint for engineering quantum memory in open systems, showing that one can tune the "memory vs. cost" tradeoff precisely.
• Thermodynamic Optimization: It challenges the assumption that maximizing a single resource (like coherence) automatically optimizes thermodynamic efficiency. The results suggest that for applications requiring high memory retention with low dissipation, one should operate near the EC limit, whereas for applications requiring maximum energy throughput, an intermediate protocol is superior.
• Benchmarking: The exact solvability of the model provides a rigorous benchmark for testing approximate methods (like master equations) in the study of non-Markovian dynamics and quantum thermodynamics.
• Fundamental Physics: The study clarifies the organizing principle behind the Zeno and anti-Zeno effects in reset protocols, identifying the erasure vs. retention of system-environment coherence as the key factor governing asymptotic behavior and thermodynamic cost.

In summary, the paper reveals that partial coherence preservation creates a unique thermodynamic regime where memory storage and heat dissipation are optimized at different operating points, offering a new paradigm for controlling open quantum systems.