Adiabatic following in multilevel systems

The problem of achieving population inversion adiabatically in an N-level system using one or more laser fields whose detunings and/or amplitudes are continuously varied is studied analytically and numerically. The SU(N) coherence vector picture is shown to suggest unexpected inversion procedures and also to give a generalized interpretation of adiabatic following. It is shown that the (N²-1)-dimensional SU(N) space contains an (N-1)--dimensional steady-state subspace Γ(t) whose orthonormal basis vectors →Γ₁,, →Γ_N-1 are given explicitly in terms of the Hamiltonian matrix elements. The motion of the system can be interpreted as a "generalized precession" of →S about Γ. Multilevel adiabatic following occurs when the angle χ(t) between the coherence vector →S and its projection onto Γ is very small. The multiple dimension of Γ is shown to provide a variety of paths for adiabatic inversion. The adiabatic solution is obtained by solving N-1 simple equations for the directional cosines of →S on →Γᵢ. The adiabatic solution and time scale and the state taken up by the atomic variable are discussed analytically and numerically for a three-level system.