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We apply a many-body Wentzel-Kramers-Brillouin (WKB) approach to determinethe leading quantum corrections to the semiclassical dynamics of the Josephsonmodel, describing interacting bosons able to tunnel between two localizedstates. The semiclassical dynamics is known to divide between regularoscillations and self-trapped oscillations where the sign of the imbalanceremains fixed. In both cases, the WKB wave functions are matched to Airyfunctions, yielding a modified Bohr-Sommerfeld quantization condition. At thecritical energy dividing normal and self-trapped oscillations, the WKB wavefunctions should instead be matched to parabolic cylinder functions, leading toa quantization formula that is not just the Bohr-Sommerfeld formula, andrecovering the known logarithmic quantum break times at this energy. This workthus provides another illustration of the usefulness of the WKB approach incertain many-body problems.

WKB approach and quantum corrections to classical dynamics in the Josephson problem