formula_8, where the operators formula_9 and formula_10 are the bosonic creation and annihilation operators and formula_11 is the angular frequency of the mode. On the other hand, the two-level atom is equivalent to a spin-half whose state can be described using a three-dimensional Bloch vector. (It should be understood that "two-level atom" here is not an actual atom "with" spin, but rather a generic two-level quantum system whose Hilbert space is isomorphic "to" a spin-half.) The atom is coupled to the field through its polarization operator formula_12. The operators formula_13 and formula_14 are the raising and lowering operators of the atom. The operator formula_15 is the atomic inversion operator, and formula_16 is the atomic transition frequency.