Memory-aware feedback enhances power in active information engines

We study an information engine operating in an active bath, where a Brownian particle confined in a harmonic trap undergoes feedback-driven displacement cycles. Unlike thermal environments, active baths exhibit temporally correlated fluctuations, introducing memory effects that challenge conventional feedback strategies. Extending the framework of stochastic thermodynamics to account for such memory, we analyze a feedback protocol that periodically shifts the potential minimum based on noisy measurements of the particle's position. We show that conventional feedback schemes, optimized for memoryless thermal baths, can degrade performance in active media due to the disruption of bath-particle memory by abrupt resetting. To overcome this degradation, we introduce a class of memory-preserving feedback protocols that partially retain the covariance between the particle's displacement and active noise, thereby exploiting the temporal persistence of active fluctuations. Through asymptotic analysis, we show how the feedback gain-which quantifies the strength of positional shifts-nontrivially shapes the engine's work and power profiles. In particular, we demonstrate that in active media, intermediate gains outperform full-shift resetting. Our results reveal the critical interplay between bath memory, measurement noise, and feedback gain, offering guiding principles for designing high-performance information engines in nonequilibrium environments.