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

0, ±1 and ∆mF = 0, ±1, allow excited atoms to decay into any of the states
|F = 2,mF = 0, 1, 2〉 and |F = 1,mF = 0, 1〉. Not all of these final state are
trappable so knowledge of the spontaneous decay branching ratios is important
to aid in understanding the efficiency of this cooling technique. The relative
decay rates can be calculated with the help of the Wigner-Eckart theorem [51,
72–74] which states that the matrix element of an irreducible tensor operator
T κ q between states of a general angular momentum basis is given by the product
of a constant independent of the magnetic quantum numbers (m,m ′ ,q) and
an appropriate Clebsch-Gordan coefficient [58]:
〈ξ ′ ,j ′ ,m ′ |T κ q |ξ,j,m〉 = 〈ξ′ ,j ′ ||T κ ||ξ,j〉
√ 〈j,m,κ,q|j
2j ′ + 1
′ ,m ′ 〉 (2.89)
where κ is the rank of the tensor operator and q labels its component in the
spherical basis. The quantity j represents a general angular momentum and
m is its projection along the quantization axis. The quantity indicated by the
double bars
is known as a reduced matrix element.
〈ξ ′ ,j ′ ||T κ ||ξ,j〉 (2.90)
The value of the Wigner-Eckhart theorem is that it allows the factor-
ization of matrix elements into two terms, one of which, the reduced matrix
element, depends only on the physical observable of interest and the other,
the Clebsch-Gordan coefficient, depends only on the orientation of the physi-
cal observables with respect to the quantization axis. This is extremely useful
because if one is able to find the value of one matrix element, then this the-
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