Modified One-Axis-Twist Squeezing for Deterministic Orientation Control of Schr\"odinger Cat States with Unknown Atom-Number Parity

The Schr\"odinger cat state protocol (SCSP) makes use of a Schr\"odinger cat (SC) state generated with one-axis-twist squeezing (OATS) to enhance the sensitivity. In principle, the SCSP can magnify the phase shift by a factor N, the number of atoms under interrogation, accompanied by a quantum noise amplification by a factor of root-N, enabling an atomic sensor to reach the Heisenberg limit. However, the current SCSP can reach this benchmark only if the parity of N is known, which is almost impossible for a large ensemble. The reason is the orientation of the SC state generated with the conventional OATS depends on the parity of N. In this paper, we propose a modified OATS (MOATS) operation that generates the SC state aligned in a certain direction regardless of the parity of N.