Effects of non-idealities and quantization of the center of mass motion on symmetric and asymmetric collective states in a collective state atomic interferometer

Abstract

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.

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“…Thus, when transformed to the collective state basis, there are also 2 N collective states. For identical Rabi frequencies and resonant frequencies, however, only the generalized symmetric states [17], of which there are only (N + 1), are relevant, and the rest of the (2 N − N − 1) states become decoupled. The case where inhomogeneity of the Rabi frequencies and different Doppler shifts experienced by different atoms are taken into account is presented at the end of this section.…”

“…The case where inhomogeneity of the Rabi frequencies and different Doppler shifts experienced by different atoms are taken into account is presented at the end of this section. We also note that if different atoms see different phase factors from the excitation fields, these factors can be absorbed into the definition of the generalized symmetric states [17]. The simplified symmetric states, known as the conventional Dicke states [16], represent the case where it is assumed that the mean separation between the atoms is much less than the wavelength corresponding to the two level transition (which, for the co-propagating off resonant Raman excitation, is ∼ (k 1 − k 2 ) −1 ).…”

“…The simplified symmetric states, known as the conventional Dicke states [16], represent the case where it is assumed that the mean separation between the atoms is much less than the wavelength corresponding to the two level transition (which, for the co-propagating off resonant Raman excitation, is ∼ (k 1 − k 2 ) −1 ). While this constraint is not necessary for the concept proposed here [17], it is easier to describe the process initially under this constraint. The observables computed remain correct when this constraint is not met.…”

“…is the Rabi frequency between collective states [16,17]. The states are separated by δ in energy and couple at different rates.…”

Investigation of coupling of a laser diode to photonic crystal fiber via hyperbolic microlens on the fiber tip by ABCD matrix formalism

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