An atom in a near-resonant light field of frequency experiences both a conservative force (due to stimulated photon exchange) and a dissipative force (due to spontaneous photon emission)[82,50]. The conservative force is the gradient of a spatially-dependent potential which can be viewed as the time-averaged induced dipole interaction energy (proportional to the real part of the classical polarizability) in the electric field, or equally well as the `light shift' (that is, energy level shift due to the ac Stark effect) of the atomic ground state [2,85].

We assume that we apply a monochromatic light field of detuning
to an
alkali atom (with the
transition resonant at
),
in the *far-detuned* regime
( greater
than the excited state hyperfine splitting, but much less than )
and the *low saturation* regime
(
, where the Rabi flopping rate
is defined ^{8.1} by
, the dipole matrix element multiplied by
the electric field amplitude).
The dipole potential has both a scalar and a magnetic part:

where is the spontaneous decay rate, is the spatially-dependent saturation parameter, and the potential is taken to be much less than the ground state hyperfine splitting. Only the magnetic part is affected by , which is defined as the projection of the total angular momentum on the direction of the local effective magnetic field . The constant is the nuclear Landé g-factor appropriate for the of the ground state. The scalar potential is identical to the case of a two-level atom, apart from the strength factor which is for detuning from the D1 line, for the D2 [85].

It is important to be precise with the definition of the
saturation parameter.
We write

The effective magnetic field has a strength and direction
given by the circularly polarized component of the electric field[85],
which can be written thus:

The fact that has its sign controlled by the sign of the detuning allows both attractive (red-detuned) and repulsive (blue-detuned) potentials to be created. The potential scales as but the spontaneous emission rate scales as ; from this follows the well-known result that, if coherence time is an important factor, it is best to be far off-resonance and use high intensities in order to achieve the desired trap depth [85,2].

For simplicity, in this chapter we will restrict our further analysis and simulations to , although our initial calculations suggest that the effect of the magnetic part of our potential when trapping in other states will not pose major problems (assuming the spin axis adiabatically follows the field direction), and can even be used to our advantage by increasing the depth and the transverse oscillation frequency in the case . Also, we will consider the effect of only a single resonance (choosing D2 because it has a larger than D1), which is a valid approximation when the detunings from this resonance are much less than the alkali atom fine structure splitting . Even when it becomes advantageous to use a large detuning of the order of , it is possible to cancel the effect of the other resonance by a simple shift in the two detunings (as we will see at the end of Section 8.3.1).

If we now have two light fields of differing frequency, the
atomic potentials add[82,156], as long as
we assume that the timescale of atomic motion is much slower than the
beating period (that is, the inverse
of the frequency difference).
In our case, atomic motion occurs at Hz and our light field
frequency
difference is Hz, so this assumption is valid.
Choosing equal but opposite detunings
about the D2 line, the trapping
potential for is