A generalized echo squeezing protocol with near-Heisenberg limit sensitivity and strong robustness against excess noise and variation in squeezing parameter

We present a generalized echo squeezing protocol (GESP) as a generalization of the Schr\"odinger cat state protocol (SCSP) with the value of the squeezing parameter being an arbitrary number rather than pi/2. We show analytically that over a broad range of the squeezing parameter the sensitivity reaches the Heisenberg limit (HL) within a factor of root-2. For a large number of particles, N, this plateau interval is almost the whole range from zero to pi/2, and the sensitivity is independent of the parity of N. Therefore, it is possible to operate a sensor over a wide interval of the squeezing parameter without changing the sensitivity. This is to be contrasted with the conventional echo squeezing protocol (CESP) which only works for a very small interval. In contrast to the CESP, the sensitivity of the GESP is close to the quantum Cram\'er-Rao bound over the whole range of the squeezing parameter. The enhancement in sensitivity for the GESP is due to a combination of two parameters: the phase magnification factor (PMF) and the noise amplification factor (NAF). As the value of the squeezing parameter increases, both PMF and NAF increase, keeping the ratio of PMF/NAF constant, yielding an enhancement of sensitivity at the HL within a factor of root-2. Thus, the robustness of the GESP against excess noise easily exceeds that of the CESP for a broad range of values of the squeezing parameter. As such, in the context of an experimental study, it should be possible to achieve a net enhancement in sensitivity higher than that for the CESP, under typical conditions where the excess noise exceeds the unsqueezed quantum projection noise. Finally, we consider the fragility of the GESP against collisions with background particles, and show how a balance between the fragility and the robustness against excess noise would in practice determine the optimal choice of parameters for the GESP.