Theoretical study of Landau-Zener tunneling at the M+N level crossing

Two rules have been analytically obtained for the well-known problem of Landau-Zener tunneling at the M+N level crossing. Both rules are valid for the arbitrary crossing rate, level splitting vectors, and mixing matrix. The first rule is given by a simple expression for an arbitrary diagonal element of the S matrix. The second rule is that the S matrix has (M-1)×(M-1) and (N-1)×(N-1) triangles of zeros in its M×M and N×N submatrices, i.e., some interlevel transitions are strictly forbidden. Some other features of the off-diagonal elements are also studied, using a 2+2 level crossing as an example. Numerical and analytical results of the 2+2 crossing show difference from a 1+2 level, and we conclude that the M+N level crossing cannot be expressed as a mere composition of 1+1 level crossings except for the above two rules. Finally we discuss the possibility of a single-electron operation using these results.