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We discuss the formation of crystalline electron clusters in semiconductorquantum dots and of crystalline patterns of neutral bosons in harmonic traps.In a first example, we use calculations for two electrons in an ellipticquantum dot to show that the electrons can localize and form a molecular dimer.The calculated singlet-triplet splitting (J) as a function of the magneticfield (B) agrees with cotunneling measurements, with its behavior reflectingthe effective dissociation of the dimer for large B. Knowledge of the dot shapeand of J(B) allows determination of the degree of entanglement. In a secondexample, we study strongly repelling neutral bosons in two-dimensional harmonictraps. Going beyond the Gross-Pitaevskii (GP) mean-field approximation, we showthat bosons can localize and form polygonal-ring-like crystalline patterns. Thetotal energy of the crystalline phase saturates in contrast to the GP solution,and its spatial extent becomes smaller than that of the GP condensate.

Electron and boson clusters in confined geometries: Symmetry breaking in quantum dots and harmonic traps