Copyright by Kirsten Viering 2006 - Raizen Lab - The University of ...

larizations stay constant, while the quantization axis rotates according to the magnetic
field orientation. The rotation angles, as they are defined in fig. 5.6 (c), are given by
sin θ(r) =
cos θ(r) =
sin φ(r) =
cos φ(r) =
B2 x(r) + B2 y(r)
B2 x(r) + B2 y(r) + B2 z(r)
Bz(r)
B2 x(r) + B2 y(r) + B2 z(r)
Bx(r)
B2 x(r) + B2 y(r)
(5.35)
(5.36)
(5.37)
By(r)
B2 x(r) + B2 y(r)
. (5.38)
Thus we are able to determine the effective Raman-Rabi frequencies βi,0 in the
atom frame by rotating the Raman transition operator. For further calculations it is
useful to go back to the spherical tensor notation, T(k,q). k describes the rank of the
operator, q the ˆz component of the operators angular momentum.
We can define β0,0 = T(1, 0). In the absence of a magnetic field, the transition
operator T(1,0) corresponds to ∆m = 0 transitions. In the rotated frame this tensor
becomes
T(1, 0) = D 1
q0 (r) T′ (1, q), (5.39)
q
where D1 (r) are the Wigner rotation matrix elements which can be expressed using the
q0
spherical harmonics Y1q(θ(r), φ(r)),
D 1
q0 (r) = (−1)q
4π
3 Y1q(θ(r).φ(r)), (5.40)
The angles θ and φ describe the rotation of the quantization axis.
The relation between the spherical tensor operator in the lab and the rotated
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