Implementation of the quantum interferometry concept to spin-1 atomic Bose-Einstein condensates is analyzed by employing a polar state evolved in time. In order to identify the best interferometric configurations, the quantum Fisher information (QFI) is maximized. Three optimal configurations are identified, among which one was not reported in the literature yet, although it gives the highest value of the QFI in experimentally achievable short time dynamics. Details of the most optimal configurations are investigated based on the error-propagation formula which includes the interaction-based readout protocol to reduce the destructive effect of detection noise. In order to obtain Heisenberg scaling accessible by present day experimental techniques, an efficient measurement and a method for the inversion of dynamics were developed, as necessary for the protocol's implementation. A Niezgoda et al ( ) where D N J J N J J J N J , , 0 2 1 1 2 , 5 N J J 0 2 ( ) H J J J N J J J J H J 8 New J. Phys. 21 (2019) 093037 A Niezgoda et al J J J J 4 2 , 4 2 I z z z z L =ˆ, was already realized experimentally [15] and the measurement of J z 2 was performed. Here, we just demonstrate the origin and rightness of such a choice pointing out its sensitivity to the detection noise. Indeed, one can show numerically that J J J J 4 2 44 II z † † J g g a a , 4 9 z l l l , 0 0 = -ˆˆˆˆ( ) † † which are symmetric when l=S and anti-symmetric for l=A. The above spin operators have cyclic commutation relations, e.g. J J J , 2 i x l y l z l , , , = [ˆˆ]ˆ. The Hamiltonian that we used in the previous subsections expressed in terms of the symmetric and anti-symmetric operators reads: c J J J . 5 0 in which the coefficient D 0 (N, J, 0) has a closed form expression D N J J N J J J N J J N J , , 0 2 1 1 2 e 2 1 . B . 9 N J J 0 2 1 J J N 1 4 † † † a a J J J a J J A a 0 1 2 1 1 2 0 1 2 0 . C.11 J J J
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