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

atoms its wavefunction Ψ(t) is typically intractable. Instead we use the density
matrix formalism which provides a convenient way of describing the evolution
of the ensemble average expectation value of the dipole operator 〈µ(t)〉 even
when Ψ(t) is unknown.
The density matrix can be defined in two steps. In the first step, we
consider the expectation value of the dipole operator for a single atom
〈µ(t)〉 = 〈Ψ(r,t)|µ|Ψ(r,t)〉. (2.57)
We then expand Ψ(r,t) in a complete set of orthonormal eigenfunctions of the
unperturbed atomic Hamiltonian H0
Ψ(r,t) =
cn(t)un(r), (2.58)
n
where all of the time dependence has been placed in the coefficients cn(t). We
then evaluate the expectation value of the dipole operator in this basis
〈µ(t)〉 =
c ∗ m(t)〈um(r)|µ|un(r)〉cn(t). (2.59)
m,n
By defining µmn ≡ 〈um(r)|µ|un(r)〉 and Rnm ≡ cnc ∗ m this can be written much
more compactly as
〈µ(t)〉 =
Rnmµmn =
(Rµ)nn = Tr(Rµ). (2.60)
m,n
n
Next we apply this approach to an atomic ensemble. To do so we define
ρnm(t) ≡ Rnm(t), (2.61)
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