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Effects of Non-idealities and Quantization of the Center of Mass Motion on Symmetric and Asymmetric Collective States in a Collective State Atomic Interferometer

We investigate the behavior of an ensemble of $N$ non-interacting, identical atoms, excited by a laser. In general, the $i$-th atom sees a Rabi frequency $Ω_i$, an initial position dependent laser phase $ϕ_i$, and a motion induced Doppler shift of $δ_i$. When $Ω_i$ or $δ_i$ is distinct for each atom, the system evolves into a superposition of $2^N$ intercoupled states, of which there are $N+1$ symmetric and $(2^N-(N+1))$ asymmetric collective states. For a collective state atomic interferometer (COSAIN) we recently proposed, it is important to understand the behavior of all the collective states under various conditions. In this paper, we show how to formulate the properties of these states under various non-idealities, and use this formulation to understand the dynamics thereof. We also consider the effect of treating the center of mass degree of freedom of the atoms quantum mechanically on the description of the collective states, illustrating that it is indeed possible to construct a generalized collective state, as needed for the COSAIN, when each atom is assumed to be in a localized wave packet. The analysis presented in this paper is important for understanding the dynamics of the COSAIN, and will help advance the analysis and optimization of spin squeezing in the presence of practically unavoidable non-idealities as well as in the domain where the center of mass motion of the atoms is quantized.