Noise level of a ring laser gyroscope in the femto-rad/s range

Absolute angular rotation rate measurements with sensitivity better than prad/s would be beneficial for fundamental science investigations. On this regard, large frame Earth based ring laser gyroscopes are top instrumentation as far as bandwidth, long--term operation, and sensitivity are concerned. Here, we demonstrate that the GINGERINO active--ring laser upper limiting noise is close to $2 \times 10^{-15}$ rad/s for $\sim 2 \times 10^5$ s of integration time, as estimated by the Allan deviation evaluated in a differential measurement scheme. This is more than a factor 10 better than the theoretical prediction so far accounted for ideal ring lasers shot--noise with the two beams counter--propagating inside the cavity considered as two independent propagating modes. This feature is related to the peculiarity of real ring laser system dynamics that causes phase cross--talking among the two counter--propagating modes. In this context, the independent beam model is, then, not applicable and the measured noise limit falls below the expected one.