Editing Resolved sideband cooling (section)

== Theoretical basis ==

The core process that provides the cooling assumes a two level system that is well localized compared to the wavelength (<math>2\pi c/\omega_0</math>) of the transition (Lamb–Dicke regime), such as a trapped and sufficiently cooled ion or atom. Modeling the system as a harmonic oscillator interacting with a classical monochromatic electromagnetic field<ref name = eschner /> yields (in the rotating wave approximation) the Hamiltonian
<math display="block">
 H = H_\text{HO} + H_\text{AL}
</math>
with
<math display="block">
 H_\text{HO} = \hbar\nu\left(n + \frac 1 2\right),
</math>
<math display="block">
 H_\text{AL} = -\hbar\Delta |e\rangle \langle e| + \hbar \frac \Omega 2 \big(|e\rangle \langle g| e^{i\mathbf k\cdot\mathbf r} + |g\rangle \langle e|e^{-i\mathbf k\cdot\mathbf r}\big),
</math>
and where
: <math>n</math> is the number operator,
: <math>\nu</math> is the frequency spacing of the oscillator,
: <math>\Omega</math> is the Rabi frequency due to the atom-light interaction,
: <math>\Delta</math> is the laser detuning from <math>\omega_0</math>,
: <math>\mathbf k</math> is the laser [[wave vector]].

That is, incidentally, the Jaynes–Cummings Hamiltonian used to describe the phenomenon of an atom coupled to a cavity in cavity QED.<ref name = wineland_98 /> The absorption(emission) of photons by the atom is then governed by the off-diagonal elements, with probability of a transition between vibrational states <math>m, n</math> proportional to <math>\big|\langle m| e^{i\mathbf k\cdot\mathbf r} |n\rangle \big|^2</math>, and for each <math>n</math> there is a manifold <math>\big\{|g, n\rangle, |e, n\rangle\big\}</math> coupled to its neighbors with strength proportional to <math>\big|\langle m| e^{i\mathbf k\cdot\mathbf r} |n\rangle\big|</math>. Three such manifolds are shown in the picture.

If the <math>\omega_0</math> transition linewidth <math>\Gamma</math> satisfies <math>\Gamma \ll \nu</math>, a sufficiently narrow laser can be tuned to a red sideband, <math>\omega_0 - q\nu, q \in \{1, 2, 3, \dots\}</math>. For an atom starting at <math>|g, n\rangle</math>, the predominantly probable transition will be to <math>|e, n - q\rangle</math>. This process is depicted by arrow "1" in the picture. In the Lamb–Dicke regime, the spontaneously emitted photon (depicted by arrow "2") will be, on average, at frequency <math>\omega_0</math>,<ref name = diedrich /> and the net effect of such a cycle, on average, will be the removing of <math>q</math> motional quanta. After some cycles, the average phonon number is <math>\bar n = R_q^{1/q}/(1 - R_q^{1/q})</math>, where <math>R_q</math> is the ratio of the intensities of the red to blue <math>q</math>-th sidebands.<ref name = turchette /> In practice, this process is normally done on the first motional sideband <math>q = 1</math> for optimal efficiency. Repeating the processes many times while ensuring that spontaneous emission occurs provides cooling to <math>\bar n \approx (\Gamma/\nu)^2 \ll 1</math>.<ref name = eschner /><ref name = wineland_98 /> More rigorous mathematical treatment is given in Turchette et al.<ref name = turchette /> and Wineland et al.<ref name = wineland_98 /> Specific treatment of cooling multiple ions can be found in [[Giovanna Morigi|Morigi]] et al.<ref name=morigi />