Analysis of correlations between atoms in terms of bivectors under coherent population trapping conditions

Description of states of quantum systems in terms of bivectors (in the general case, polyvectors) is proposed as an alternative to vectors in a Hilbert space. This approach is developed for slow atoms under coherent population trapping conditions (in a dark state). The field of local two-dimensional dark subspaces is naturally associated with a local bivector field. An analysis in terms of bivectors is performed to recover the state of an atom or an atomic ensemble by using the condition for its being in some unknown dark state. This approach exposes natural correlations between atoms. The corresponding many-particle states belong to the class of states with positive partial transpose.