Editing Resolved sideband cooling (section)

== Conceptual description ==

[[File: Sideband cooling level diagram.png|thumb|right|An atom undergoing resolved sideband cooling. Driven transitions are shown with straight arrows and spontaneous transitions with wiggly arrows. After each driven transition, the atom reaches an excited state with one less motional quanta than the state it came from. For example, the atom starts in the ground ''n''&nbsp;=&nbsp;3 state and is driven to the excited ''n''&nbsp;=&nbsp;2 state. The motional quantum number n does not change in spontaneous transitions.]]

'''Resolved sideband cooling''' is a laser-cooling technique that can be used to cool strongly trapped atoms to the quantum [[ground state]] of their motion. The atoms are usually precooled using the [[Doppler cooling|Doppler laser cooling]]. Subsequently, the resolved [[sideband]] cooling is used to cool the atoms beyond the [[Doppler cooling limit]].

A cold trapped atom can be treated to a good approximation as a [[quantum-mechanical]] [[harmonic oscillator]]. If the spontaneous decay rate is much smaller than the vibrational frequency of the atom in the trap, the [[energy level]]s of the system will be an evenly spaced frequency ladder, with adjacent levels spaced by an energy <math>\hbar \nu</math>. Each level is denoted by a motional quantum number ''n'', which describes the amount of motional energy present at that level. These motional quanta can be understood in the same way as for the [[quantum harmonic oscillator]]. A ladder of levels will be available for each internal state of the atom. For example, in the figure at right both the ground (''g'') and excited (''e'') states have their own ladder of vibrational levels.

Suppose a two-level atom whose ground state is denoted by ''g'' and [[excited state]] by ''e''. Efficient laser cooling occurs when the frequency of the laser beam is tuned to the red sideband i.e.
<math display="block">
 \omega = \omega_0 - \nu,
</math>
where <math>\omega_0</math> is the internal atomic transition frequency corresponding to at transition between ''g'' and ''e'', and <math>\nu</math> is the harmonic-oscillation frequency of the atom. In this case the atom undergoes the transition
<math display="block">
 |g, n\rangle \to |e, n - 1\rangle,
</math>
where <math>|a, m\rangle</math> represents the state of an ion whose internal atomic state is ''a'', and the motional state is ''m''.

If the recoil energy of the atom is negligible compared with the vibrational quantum energy, subsequent [[spontaneous emission]] occurs predominantly at the [[carrier frequency]]. This means that the vibrational quantum number remains constant. This transition is
<math display="block">
 |e, n - 1\rangle \to |g, n - 1\rangle.
</math>

The overall effect of one of these cycles is to reduce the vibrational quantum number of the atom by one. To cool to the ground state, this cycle is repeated many times until <math>|g, n = 0\rangle</math> is reached with a high probability.<ref name = mech />