When does a pair of Fermions act like a Boson?

Entanglement and Composite Bosons

Under what circumstances can a pair of fermions be treated as an elementary boson? Many authors have done detailed studies of this question, as it applies, for example, to atomic Bose-Einstein condensates, excitons, and Cooper pairs in superconductors. In a 2005 paper, C. K. Law presented evidence that the question can be answered in general in terms of entanglement: two fermions can be treated as an elementary boson if they are sufficiently entangled. Consider, for example, a single hydrogen atom in a harmonic trap. Within the atom, the proton and electron are strongly entangled with respect to their position variables; for example, wherever the proton might be found—it could be anywhere in the trap—the electron is sure to be nearby. Law suggests that this entanglement is the essential property underlying the (approximate) bosonic behavior of the composite particle, allowing, for example, a collection of many hydrogen atoms to form a Bose-Einstein condensate.