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

component of the dipole oscillation. The dipole force can be found from the
gradient of the potential energy and is therefore a conservative field.
Fdip = −∇Udip(r) = 1
Re(α)∇I(r) (2.47)
2ǫ0c
Thus the optical dipole force comes from the dispersive interaction of the
induced dipole moment with the gradient of the driving light field [59, 64–66].
We now turn our attention to modeling α which we do following Lorentz’s
model of a classical oscillator. In this model the electron is considered to be
a classical particle bound elastically to a nucleus and possessing an oscillation
frequency ω0 which is identified with the frequency of the optical transition
of interest. Damping is included in this model through Larmor’s formula for
the power radiated by an accelerating charge [67]. This model has several
limitations - it does not model an atom with multiple transitions and does
not display any saturation behavior when strongly driven. In the far-detuned
limit neither of these limitations are of concern.
The equation of motion for a damped, driven harmonic oscillator
¨x + ΓL ˙x + ω 2 0x = − eE(t)
can be used to solve for α by noting that p = −ex = αE. The result is
α = e2
me
me
(2.48)
1
ω2 0 − ω2 , (2.49)
− iωΓL
where the damping coefficient ΓL is the classical Larmor energy damping rate
due to radiative loss
ΓL = e2 ω 2
6πǫ0mec 3.
47
(2.50)