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We investigate the exact nature of the superfluid-to-Mott-insulator crossoverfor interacting bosons on an optical lattice in a one-dimensional, harmonictrap by high-precision density-matrix renormalization-group calculations. Theresults reveal an intermediate regime characterized by a cascade of microscopicsteps. These arise as a consequence of individual boson "squeezing" events anddisplay an even-odd alternation dependent on the trap symmetry. We discuss theexperimental observation of this behavior, which is generic in an externaltrapping potential.

Quantized squeezing and even-odd asymmetry of trapped bosons