Precision and cost of feedback cooling

We investigate the consequences of information exchange between a system and a measurement-feedback apparatus that cools the system below the environmental temperature. A quantitative relationship between entropy pumping and information acquired about the system is derived, showing that, independent of the concrete realization of the feedback, the latter exceeds the former by a positive amount of excess information flow. This excess information flow satisfies a trade-off relation with the precision of the feedback force, which places strong constraints on both the information-theoretic cost of feedback cooling and the required magnitude of the feedback force. From these constraints, a fundamental lower bound on the energetic cost of optical feedback cooling is derived. Finally, the results are demonstrated for feedback cooling by coherent light scattering. We show that measurement precision is the major factor determining the attainable temperature. Precise measurements can also be leveraged to reduce the required feedback force, leading to significantly more energy-efficient cooling close to the fundamental bound for realistic parameter values.