Robustness of the avian compass function described by radical pair model against biomagnetic noise

The magnetic sensing mechanism proposed to exist in avian eyes functions as a compass that detects the geomagnetic inclination aiding their migration. This mechanism is modeled by a quantum spin model known as the radical pair (RP) model. This model suggests that information about geomagnetic inclination is transmitted as biochemical signals via spin-selective recombination of the RP. However, as a delicate quantum system in a warm and noisy in vivo environment, the RP model is likely to be subject to unavoidable environmental noise. In this study, we develop a model that incorporates environmental magnetic noise, based on a Lindblad-type master equation. Our model includes the type of biological tissue and the temperature of the biological system as parameters, accounting for their effects. The results of numerical calculations indicate that the compass function of the RP model is robust against intrinsic magnetic noise arising from thermal fluctuations in the electromagnetic field within realistic biomagnetic range. However, noise from an environmental magnetic field far exceeding the biological magnetic range, on the order of hundreds to thousands of $\mu$T, significantly affects and disrupts the compass function. This work provides a foundation for quantifying environmental influences on open quantum systems in biological settings and offers a tool for connecting experimental measurements of environmental magnetic fields with quantum noise modeling.