Impact of spontaneous emission on spin-squeezed quantum sensors

The echo squeezing protocols (ESPs) are techniques that amplify the phase shift in a quantum sensor with one-axis-twist squeezing (OATS). For atomic sensors, spontaneous emission (SE) in the OATS operations is an important imperfection, which, however, is prohibitively difficult to study. SE can transfer atoms to all the Zeeman substates, making the number of relevant collective states excessively large. In this paper, we focus on a relatively simple isotope, namely Sr-88, which only has one Zeeman substate in each of the two ground states. Nevertheless, studying the effect of SE is still challenging because SE will populate all collective states, either symmetric or asymmetric, thus putting the ensemble into a mixed state. Based on analytical derivation and numerical simulations, we conclude that the GESP is more resistant to SE than the conventional echo squeezing protocol (CESP). This is another advantage of the GESP that has not been realized formerly. We also find that for the GESP employing Sr-88, the SE-induced reduction in the signal contrast is the same as that for a Ramsey protocol without squeezing and the quantum noise is suppressed by SE. This is an unexpectedly favorable result. The SE-induced suppression of quantum noise constitutes a previously unrecognized effect that challenges both earlier conclusions and intuitive expectations.