Analysis of correlations between atoms in terms of bivectors under coherent population trapping conditions
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Analysis of Correlations between Atoms in Terms of Bivectors under Coherent Population Trapping Conditions L. V. Il’ichev Institute of Automation and Electrometry, Siberian Division, Russian Academy of Sciences, Novosibirsk, 630090 Russia e-mail: [email protected] Received July 25, 2005
Abstract—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. PACS numbers: 32.80.Qk, 42.50.–p DOI: 10.1134/S1063776106040066
1. INTRODUCTION Investigation of correlations in many-body quantum systems is a subject of topical interest largely motivated by the rapid development of quantum information theory and its applications. The concept of an entangled state is a central and the most intriguing one in quantum mechanics and quantum informatics. Entanglement essentially means quantum-mechanical correlation between fragments of an integral system. Identification and quantification of mixed-state entanglement require a nontrivial analysis [1]. The problem is further complicated by particle indistinguishability. No unified criterion for entanglement in a system of identical particles has been found to this day [2–6]. However, such a criterion may be unnecessary in view of the conditional nature of entanglement [5, 7], which depends on the particular experimental setting and the observer’s capabilities. Analyses of entanglement in a degenerate gas of noninteracting electrons were presented in [8–10]. In that context, natural correlation means that no special procedure is required to create entanglement. Since the fermionic nature of particles is essential, the results obtained cannot be applied to bosons. The existence of natural entanglement in a Bose–Einstein condensate (BEC) was demonstrated in [11]. However, this occupation-number entanglement between regions in the BEC volume does not involve any particle spin characteristics. This example highlights the operational nature of entanglement. This paper deals with spin correlations in an ensemble of identical bosonic atoms with a coherence length larger than the mean interatomic distance. This setting
is best suited for exposing the quantum consequences of particle indistinguishability, since the statistical operator of the system cannot be factored into a product of the statistical operators of individual atoms. Another condition for spin correlation is coherent population trapping (CPT), which can be caused by resonant interaction betwee
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