Single-Photon Atomic Cooling - Raizen Lab - The University of ...

so that the ensemble average expectation value of the dipole operator can be
expressed as
〈µ(t)〉 =
m,n
cnc ∗ mµmn = Tr[ρ(t)µ]. (2.62)
The density matrix has four elements in this case because we have
assumed a two-level system,
ρ =
ρ11 ρ12
ρ21 ρ22
. (2.63)
The diagonal elements represent the probability of finding an atom in the
respective state i.e. ρ11 is the probability of finding an atom in state |1〉. The
significance of the off-diagonal elements can be seen by calculating 〈µ(t)〉 for
our two level system. The result is
〈µ(t)〉 = ρ12µ21 + ρ11µ11 + ρ22µ22 + ρ21µ12. (2.64)
If we assume that each state |1〉 and |2〉 has definite parity and note that
the dipole operator µ has odd parity symmetry then we see immediately that
µ11 = µ22 = 0. Additionally if we assume, with no loss of generality, that
µ21 = µ12 ≡ µ then we can write the very illuminating formula
〈µ(t)〉 = µ[ρ12(t) + ρ21(t)], (2.65)
which shows that the sum of off-diagonal elements, known as the coherence
terms, is proportional to the ensemble average dipole moment.
We are now in a position to derive the optical Bloch equations. The
temporal evolution of the operator ρ is determined by the Heisenberg equation
˙ρ = 1
[H,ρ]. (2.66)
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