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:
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