Measurements with prediction and retrodiction on the collective spin of 10^{11} atoms beat the standard quantum limit

Quantum probes using $N$ uncorrelated particles give a limit on the measurement sensitivity referred to as the standard quantum limit (SQL). The SQL, however, can be overcome by exploiting quantum entangled states, such as spin squeezed states. We report generation of a quantum state, that surpasses the SQL for probing of the collective spin of $10^{11}$ $\text{Rb}$ atoms contained in a vapor cell. The state is prepared and verified by sequences of stroboscopic quantum non-demolition (QND) measurements, and we apply the theory of past quantum states to obtain the spin state information from the outcomes of both earlier and later QND measurements. In this way, we obtain a conditional noise reduction of 5.6 dB, and a metrologically-relevant squeezing of $4.5\pm0.40~\text{dB}$. The past quantum state yields tighter information on the spin component than we can obtain by a conventional QND measurement. Our squeezing results are obtained with 1000 times more atoms than in any previous experiments with a corresponding record $4.6\times10^{-13} rad^2$ variance of the angular fluctuations of a squeezed collective spin.