Controlling the motion of an atom in an optical cavity

14 2.2. Classical calculation
atoms. Also, because the mirrors have a large reflectance, the field distribution of the
near-resonant mode will in a very good approximation be given by the mode supported
by a cavity which is made of perfectly reflecting mirrors with the same geometry. Under
these circumstances, the electric field distribution inside the cavity which polarizes the
atoms can be approximated by the mode of a cavity made of perfectly reflective mirrors.
As a last assumption, the frequency of the light field should be far larger than the round
trip frequency of the light inside the mode, which in turn should be much larger than the
frequencies attributed to other mechanisms influencing the light field, e.g. the motion of
the atoms or the decay rate of the atomic dipole moment. Then,
E (r,t)=A (t) e −iωlt ψ (r)+Escat (r,t) , (2.12)
where ψ (r) is the mode out of the ψn (r) in Eq. (2.7) which is close to resonance, and
Escat (r,t) is the part of the electric field scattered by the atoms which is orthogonal to
themode. TheamplitudeA (t) is under the above assumptions slowly varying in time,
i.e.
d2A (t)
dt2
dA
(t)
≪ ωl
dt ≪ ω2 l |A (t)|. (2.13)
As the polarization of the atoms is also induced by the light field, it can be written as
P (r,t)=P0 (r,t) e −iωlt
(2.14)
with
d2P0 (r,t)
dt2
dP0
(r,t)
≪ ωl
dt ≪ ω2 l |P0 (r,t)|. (2.15)
Using this, time derivatives of P0 (r,t) and the second time derivative of A (t) canbe
neglected, and Eq. (2.11) can be recast as
dA (t)
= i
dt
ω2 l − ω2 c
A (t)+i
2ωl
ωl
d
2ɛ0V
3 rψ ∗ (r) P0 (r,t)
= i∆c A (t)+i ωl
d
2ɛ0V
3 rψ ∗ (r) P0 (r,t) , (2.16)
where ∆c = ωl − ωc is the detuning of the laser from the cavity resonance, and ∆c ≪ ωl
was used to get to the second equation.
Now, the polarization, P0, is left to be calculated. As was explained above, P0 is
induced by the electric field E (r,t), which can be approximated by the electric field in
the mode, E (r,t) − Escat (r,t), at the position of the other atoms. For two-level atoms
which are exposed to monochromatic light with an intensity well below the saturation
intensity at the position of the atoms, the polarization is given by (MS99, Eq. (5.26))
P0 (r,t)=iA (t) µ2
¯h
1
δ (r − rj) ψ (rj) . (2.17)
γ − i∆a j
14