Controlling the motion of an atom in an optical cavity

42 3.5. The momentum diffusion constant
Because a general expression for Z −1 is complicated, it is more appropriate to calculate
Z −1 for a specific situation. A special case is investigated in section 3.5.1.
Now, consider the second part of the momentum diffusion matrix, Dse,j, which is
due to spontaneous emission of the atom. This part can be calculated by determining
the correlations of the force operator Fse,j, which describes the interaction of the atomic
dipole with the modes of the light field which are not supported by the cavity. This part
of momentum diffusion has been calculated in (HGHR98, appendix B). The result is
Dse,j =¯h 2 k 2 γj
σ + j σ−
j P, (3.79)
where P is a 3 × 3-matrix with a trace of 1, which depends on the polarization of the light
field. For circularly polarized light propagating in the z-direction, P is given by
⎛
3 0 0
10
⎜ 3
P = ⎝ 0 0 10
2
0 0 5 .
⎞
⎟
⎠ (3.80)
The total diffusion matrix, Dj, is the sum of the two contributions Eq. (3.78) and
(3.79):
Dj = Dcav,j + Dse,j. (3.81)
In some contexts, only the total heating rate of the atom by momentum diffusion is
important. In this case, a scalar diffusion coefficient, Dj, can be calculated as the trace of
the diffusion matrix, Dj. This scalar gives the total heating rate without specifying the
’direction’ of the heating.
3.5.1 Single mode
As in section 3.4.1, it is assumed that the cavity supports only a single mode which has
a resonance frequency close to the atomic resonance frequency, and that all atoms have
the same decay rate γ and the same laser-atom detuning ∆a. In this case, the inverse
of the matrix Z is given in Eq. (3.28). The results for the expectation values needed in
Eq. (3.77) are stated in Eqs. (3.35) and (3.37). After using Eq. (3.40) and a bit of algebra,
one arrives at the result
Dcav,j =¯h 2 γ|η| 2 (∇jg1j) ⊗
∇jg ∗
1j
(∆cγ +∆aκ) 2
1
1+S2
1+
4∆a|g1j| 2
γ (∆cγ +∆aκ)
1
1+S2
. (3.82)
In a coordinate system in which one axis points in the direction of ∇g1j, this diffusion
matrix is particularly simple: It is a diagonal matrix with only one non-zero element,
Dcav,j, which can be calculated by taking the trace of Dcav,j, i.e. by adding all diagonal
42