Bose-Einstein condensation in inhomogeneous Josephson arrays

Abstract

We show that spatial Bose-Einstein condensation of non-interacting bosons occurs in dimension d < 2 over discrete structures with inhomogeneous topology and with no need of external confining potentials. Josephson junction arrays provide a physical realization of this mechanism. The topological origin of the phenomenon may open the way to the engineering of quantum devices based on Bose-Einstein condensation. The comb array, which embodies all the relevant features of this effect, is studied in detail.PACS numbers: 03.75. Fi, 85.25.Cp, The recent impressive experimental demonstration of Bose-Einstein Condensation (BEC) [1] has stimulated a new wealth of theoretical work aimed to better understanding its basic mechanisms [2] and, possibly, to exploit its consequences for the engineering of quantum devices.It is well known [3] that for an ideal gas of Bose particles BEC does not occur in dimension d ≤ 2, and an ′′ ad hoc ′′ external confining potential is needed to reach the required density of states. The same is true for free bosons living on regular periodic lattices, while the result cannot be extended to more general discrete structures lacking translational invariance.In the following we shall prove that even for d < 2 [4] non-interacting bosons may lead to Bose-Einstein condensation into a single non-degenerate state, provided one resorts to a suitable discrete non-homogeneous support structure: indeed, when the bosonic kinetic degrees of freedom do not depend on metric features only, the particles may feel a sort of effective interaction due to topology. The proposed mechanism for BEC in lower dimensional systems is then a pure effect of the structure of the ambient space and avoids as well the need of resorting to external random potentials as the ones investigated by Huang in [2]; this is a very desirable feature in view of engineering real quantum devices.In practice, the behavior of free bosons over generic discrete structures is made experimentally accessible through the realization of suitable arrays of Josephson junctions. The latter are devices that can be engineered in such a way as to realize a variety of non-homogeneous patterns. We shall show indeed that classical Josephson junction arrays arranged in a non-homogeneous geometry -not even necessarily planar -provide an example of the proposed mechanism for BEC, leading to a single state spatial condensation.Theoretical studies of Josephson junction arrays are based on the short-range Bose-Hubbard model, since the phase diagram of Josephson junction arrays may be derived [5] from an Hamiltonian describing bosons with repulsive interactions over a lattice. In d = 1 the phase diagram has been studied by analytical [6] and quantum Monte Carlo methods [7]; experimentally, Josephson junction arrays are used to study interacting bosons in one dimension. For a generic array the corresponding Hamiltonian is given bywhere A ij is the adjacency matrix: A ij = 1 if the sites i and j are nearest neighbors and A ij = 0 otherwise; a † i creates a boson at site...

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