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arXiv:0812.0274 · Infinite Dimensional Analysis Quantum Probability and Related Topics

BOSE–EINSTEIN CONDENSATION ON INHOMOGENEOUS AMENABLE GRAPHS

Francesco Fidaleo, Daniele Guido, Tommaso Isola

Published Jun 1, 2011• 18 citations
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Abstract

We investigate the Bose–Einstein condensation on nonhomogeneous amenable networks for the model describing arrays of Josephson junctions. The resulting topological model, whose Hamiltonian is the pure hopping one given by the opposite of the adjacency operator, also has a mathematical interest in itself. We show that for the nonhomogeneous networks like the comb graphs, particles condensate in momentum and configuration space as well. In this case different properties of the network, of geometric and probabilistic nature, such as the volume growth, the shape of the ground state, and the transience, all play a role in the condensation phenomena. The situation is quite different for homogeneous networks where just one of these parameters, e.g., the volume growth, is enough to determine the appearance of the condensation.

DOI
10.1142/s0219025711004389

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We investigate the Bose–Einstein condensation on nonhomogeneous amenable networks for the model describing arrays of Josephson junctions. The resulting topological model, whose Hamiltonian is the pure hopping one given by the opposite of the adjacency operator, also has a mathematical interest in itself. We show that for the nonhomogeneous networks like the comb graphs, particles condensate in momentum and configuration space as well. In this case different properties of the network, of geometric and probabilistic nature, such as the volume growth, the shape of the ground state, and the transience, all play a role in the condensation phenomena. The situation is quite different for homogeneous networks where just one of these parameters, e.g., the volume growth, is enough to determine the appearance of the condensation.
We study the filling of states in a pure hopping boson model on the comb lattice, a low-dimensional discrete structure where geometrical inhomogeneity induces Bose-Einstein condensation (BEC) at finite temperature. By a careful analysis of the thermodynamic limit on combs we show that, unlike the standard lattice case, BEC is characterized by a macroscopic occupation of a finite number of states with energy belonging to a small neighborhood of the ground state energy. Such a remarkable feature gives rise to an anomalous behavior in the large distance two-point correlation functions. Finally, we prove a general theorem providing the conditions for the pure hopping model to exhibit the standard behavior, i.e. to present a macroscopic occupation of the ground state only.
We argue that Josephson junction networks may be engineered to allow for the emergence of new and robust quantum coherent states. We provide a rather intuitive argument showing how the change in topology may affect the quantum properties of a bosonic particle hopping on a network. As a paradigmatic example, we analyze in detail the quantum and thermodynamical properties of non-interacting bosons hopping on a comb graph. We show how to explicitly compute the inhomogeneities in the distribution of bosons along the comb's fingers, evidencing the effects of the topology induced spatial Bose–Einstein condensation characteristic of the system. We propose an experiment enabling to detect the spatial Bose–Einstein condensation for Josephson networks built on comb graphs.
We investigate the properties of Josephson junction networks with inhomogeneous architecture. The networks are shaped as quare comb planar lattices on which Josephson junctions link superconducting islands arranged in the plane to generate the pertinent topology. Compared to the behavior of reference linear arrays, the temperature dependencies of the Josephson currents of the branches of the network exhibit relevant differences. The observed phenomena evidence new and surprising behavior of superconducting Josephson arrays as well as remarkable similarities with bosonic junction arrays.
New coherent states may be induced by pertinently engineering the topology of a network. As an example, we consider the properties of non-interacting bosons on a star network, which may be realized with a dilute atomic gas in a star-shaped deep optical lattice. The ground state is localized around the star centre and it is macroscopically occupied below the Bose–Einstein condensation temperature T c . We show that T c depends only on the number of the star arms and on the Josephson energy of the bosonic Josephson junctions and that the non-condensate fraction is simply given by the reduced temperature T / T c .
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.
The existence of a Bose-Einstein condensation in an interacting many-boson system at T = 0°K is proved under certain conditions on the particle density and the interparticle potential. Starting with the tentative assumption that the condensation exists, we study the fluctuation in the occupation number of the condensate with due regard to its interactions (1) with particles outside the condensate as well as (2) with the fluctuation itself. If the condensate fluctuation has a normalizable ground state, then the assumed existence of the condensation is tenable. For the case of the pair-Hamiltonian model satisfying the conditions for condensation, the interactions of the second category are of no importance. In the limit of infinite volume, this Hamiltonian can be diagonalized in an irreducible representation of a Bose-field operatorφ(x), whereφ(x) has nonvanishing ground state expectation value, in accordance with the usual c-number replacement of creation and destruction operators for the condensate particles. The full Hamiltonian for a system of pairwise interacting bosons is studied only in a low-density limit. Bose-Einstein condensation exists when the over-all space integral of the interparticle potential is positive. In this case the interactions of the second category play an important role in ensuring a normalizable ground state for the condensate fluctuation. There is an indication that in the limit of infinite volume the Hamiltonian cannot be diagonalized in any irreducible representation of the field operatorφ. Yet the c-number replacement of the condensate operators is legitimate as far as states of particles outside the condensate are concerned. Some speculations are made as to what may happen for systems of moderate density.