Theory of Adiabatic Rapid Passage for Three Equally Spaced Levels

Abstract

In experiments reported elsewhere, adiabatic rapid passage (ARP) has been performed on multilevel electron-paramagnetic crystals having three equally spaced energy levels separated by $\ensuremath{\hbar}{\ensuremath{\omega}}_{0}$. All transitions among the three levels are allowed due to crystalline electric-field mixing of the pure spin states. Simultaneous ARP inversion of the two coincident transitions at ${\ensuremath{\omega}}_{0}$ produces a negative temperature simultaneously on all three transitions, at ${\ensuremath{\omega}}_{0}$ and $2{\ensuremath{\omega}}_{0}$. The present paper explains these results through an analysis of adiabatic rapid passage in a spin system having an arbitrary spin-Hamiltonian with three equally spaced energy levels. Previous treatments of ARP have been limited either to the multilevel case with pure spin states or to the two-level case with mixed states. In the present analysis, the 3\ifmmode\times\else\texttimes\fi{}3 density matrix is transformed to an equivalent matrix whose equation of motion contains a Hamiltonian which is real and constant, even with the rf perturbation applied. The transformed matrix is expanded in terms of nine orthonormal Hermitian basis matrices. Three of the expansion coefficients are directly related to state populations. The time variation of the expansion coefficients is calculated for ARP conditions. The transition probabilities of the two ${\ensuremath{\omega}}_{0}$ transitions are not required to be equal. The final values of the three relevant expansion coefficients indicate negative temperatures on all three transitions.