Description of Bose-Einstein Condensates in $$\mathcal {PT}$$ -Symmetric Double Wells

Abstract

The Gross-Pitaevskii equation for a Bose-Einstein condensate in a \(\mathcal {PT}\)-symmetric double-well potential is investigated theoretically. An in- and outcoupling of atoms is modelled by an antisymmetric imaginary potential rendering the Hamiltonian non-Hermitian. Stationary states with real energies and \(\mathcal {PT}\)-symmetric wave functions are found, which proves that Bose-Einstein condensates are a good candidate for a first experimental verification of a \(\mathcal {PT}\)-symmetric quantum system. Time-resolved calculations demonstrate typical effects only observable in \(\mathcal {PT}\)-symmetric potentials, viz. an oscillation of the condensate’s probability density between these wells with an oscillation frequency critically depending on the strength of the in- and outcoupling. \(\mathcal {PT}\)-broken eigenstates with complex energy eigenvalues are also solutions of the time-independent Gross-Pitaevskii equation but are not true stationary states of its time-dependent counterpart. The comparison of a one-dimensional and a three-dimensional calculation shows that it is possible to extract highly precise quantitative results for a fully three-dimensional physical setup from a simple one-dimensional description.