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 $\mathrm{SU}\mathrm{}\left(N\right)\mathrm{}$ 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 ($N^2-1\mathrm{}$)-dimensional $\mathrm{SU}\mathrm{}\left(N\right)\mathrm{}$ space contains an ($N-1\mathrm{}$)--dimensional steady-state subspace $\mathit{\Gamma }\mathrm{}\left(t\right)\mathrm{}$ whose orthonormal basis vectors ${\overrightarrow{\Gamma }}_1,\dots , {\overrightarrow{\Gamma }}_{N-1\mathrm{}}$ are given explicitly in terms of the Hamiltonian matrix elements. The motion of the system can be interpreted as a "generalized precession" of $\overrightarrow{\mathrm{S}}$ about $\mathit{\Gamma }$. Multilevel adiabatic following occurs when the angle $\chi \mathrm{}\left(t\right)\mathrm{}$ between the coherence vector $\overrightarrow{\mathrm{S}}$ and its projection onto $\mathit{\Gamma }$ is very small. The multiple dimension of $\mathit{\Gamma }$ is shown to provide a variety of paths for adiabatic inversion. The adiabatic solution is obtained by solving $N-1\mathrm{}$ simple equations for the directional cosines of $\overrightarrow{\mathrm{S}}$ on ${\overrightarrow{\Gamma }}_i$. 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.