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arXiv:quant-ph/0212120 · Physical Review A

Bipartite mode entanglement of bosonic condensates on tunneling graphs

Paolo Zanardi A

Published May 20, 2003• 10 citations
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Abstract

We study a set of L spatial bosonic modes localized on a graph $\ensuremath{\Gamma}.$ The particles are allowed to tunnel from vertex to vertex by hopping along the edges of $\ensuremath{\Gamma}.$ We analyze how, in the exact many-body eigenstates of the system, i.e., Bose-Einstein condensates over single-particle eigenfunctions, the bipartite quantum entanglement of a graph vertex with respect to the rest of the graph depends on the topology of $\ensuremath{\Gamma}.$

DOI
10.1103/physreva.67.054301

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We study a set of L spatial bosonic modes localized on a graph \ensuremathΓ.\ensuremath{\Gamma}.\ensuremathΓ. The particles are allowed to tunnel from vertex to vertex by hopping along the edges of \ensuremathΓ.\ensuremath{\Gamma}.\ensuremathΓ. We analyze how, in the exact many-body eigenstates of the system, i.e., Bose-Einstein condensates over single-particle eigenfunctions, the bipartite quantum entanglement of a graph vertex with respect to the rest of the graph depends on the topology of \ensuremathΓ.\ensuremath{\Gamma}.\ensuremathΓ.
We consider a collection of bosonic modes corresponding to the vertices of a graph Gamma.\\Gamma.Gamma. Quantum tunneling can occur only along the edges of Gamma\\GammaGamma and a local self-interaction term is present. Quantum entanglement of one vertex with respect the rest of the graph is analyzed in the ground-state of the system as a function of the tunneling amplitude tau.\\tau.tau. The topology of Gamma\\GammaGamma plays a major role in determining the tunneling amplitude tau∗\\tau^*tau∗ which leads to the maximum ground-state entanglement. Whereas in most of the cases one finds the intuitively expected result tau∗=infty\\tau^*=\\inftytau∗=infty we show that it there exists a family of graphs for which the optimal value oftau\\tautau is pushed down to a finite value. We also show that, for complete graphs, our bi-partite entanglement provides useful insights in the analysis of the cross-over between insulating and superfluid ground states
We analyze the quantum entanglement properties of bosonic particles hopping over graph structures. Mode entanglement of a graph vertex with respect to the rest of the graph is generated, starting from a product state, by turning on for a finite time a tunneling along the graph edges. The maximum achieved during the dynamical evolution by this bipartite entanglement characterizes the entangling power of a given hopping Hamiltonian. We studied this entangling power as a function of the self-interaction parameters, i.e., nonlinearities, for all the graphs up to four vertices and for two different natural choices of the initial state. The role of graph topology and self-interaction strengths in optimizing entanglement generation is extensively studied by means of exact numerical simulations and by perturbative calculations
We consider a collection of bosonic modes corresponding to the vertices of a graph Γ. Quantum tunneling can occur only along the edges of Γ and a local self-interaction term is present. Quantum entanglement of one vertex with respect to the rest of the graph (mode entanglement) is analyzed in the ground state of the system as a function of the tunneling amplitude τ. The topology of Γ plays a major role in determining the tunneling amplitude τ max that leads to the maximum value of the mode entanglement. Whereas in most of the cases one finds the intuitively expected result τ max = ∞ , we show that there exists a family of graphs for which the optimal value of τ is pushed down to a finite value. We also show that, for complete graphs, our bi-partite entanglement provides useful insights in the analysis of the crossover between insulating and superfluid ground states.
We associate a bosonic network to each complete bipartite graph. A Hamiltonian is defined with hopping terms on the edges of the graph, and a global interaction term depending on the vertex sets. For a generic graph this Hamiltonian is superintegrable, and we derive a Bethe Ansatz solution for the energy and eigenstates. These results provide the means to investigate the quantum dynamics and allow for the computation of occupancy probabilities in certain regimes. We use these results to gain an understanding of entanglement evolution in the network.
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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.