Controlling the motion of an atom in an optical cavity

22 3.1. Model
of the atoms where the analytic model fails. Section 3.4 presents the steady state solution
for the analytical model. The general solution for an arbitrary number of atoms and
modes is considered as well as special cases where just one atom interacts with degenerate
modes or a single mode interact with atoms which have the same transition frequency
and decay rate. The motion of atoms in the cavity light field is the topic of the next
two sections. In section 3.5, the momentum diffusion (heating) of atoms in the cavity
light field is calculated, and in section 3.6, the velocity-dependent part of the light force
is derived. Both sections rely on the low saturation approximation, and treat the general
case as well as the case of a single mode interacting with the atoms. In section 3.7, the
limit of large atom-laser detuning is considered for a low atomic saturation. This limit is
interesting for trapping single atoms, and for laser-cooling particles which have no closed
transitions, like molecules. In section 3.8, numerical methods are described which can be
used for saturated atoms or for other situations where the analytical model fails.
3.1 Model
As a starting point, Na two-level atoms in an optical cavity supporting Nc modes of the
electromagnetic field are considered, see Fig. 3.1. The light field inside the cavity and the
internal states of the atoms are treated quantum mechanically, while the atomic motion is
treated classically. This is usually a good approximation if the atom’s temperature is well
above the recoil limit and if the cavity linewidth and the atomic linewidth are large as
compared to the recoil frequency of the atom (DLHK00). The cavity mode i is described
by a quantum field with annihilation and creation operators ai and a †
i, respectively. The
ith mode’s resonance angular frequency is denoted by ωc,i. The field of the mode i decays
with the rate κi, which is assumed to be much smaller than the inverse round trip time
of the light in the cavity. The mode i is pumped by an external field with strength ηi.
The jth atom’s raising and lowering operators are σ + j and σ − j , respectively. Its transition
angular frequency is ωa,j. The atomic dipole moments decay with a rate γj. Theith mode
is coupled to atom j with the coupling constant gij, which depends on the position rj
of the jth atom and may be a complex number. For an electric dipole transition and if
the relative difference between the ωc,i is small, the coupling constant is given by (MS99,
Eq. (14.3))
gij = g0ψi (rj) , (3.1)
where ψi is the mode function of the ith mode, see Eq. (2.7), and
g0 = µ ωc/ (2¯hɛ0V ). (3.2)
Here, µ is the dipole matrix element of the transition, V is the normalization volume, see
Eq. (2.9), and ωc is any of the ωc,i. It is assumed that the mode functions are orthonormal,
see the discussion at Eq. (2.9). An orthonormal basis can usually be found, except for
exotic cases such as unstable cavities (vdLvEvDW01).
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