is negligible compared with the vibrational quantum energy i.e. This process is labeled '2' in the adjacent image. The average effect of this mechanism is cooling the ion by one vibrational energy level. When these steps are repeated a sufficient number of times is reached with a high probability. Theoretical basis The core process that provides the cooling assumes a two level system that is well localized compared to the wavelength () of the transition (Lamb-Dicke regime), such as a trapped and sufficiently cooled ion or atom. After, modeling the system as a harmonic oscillator interacting with a classical monochromatic electromagnetic field yields (in the rotating wave approximation) the Hamiltonian with and where is the number operator is the frequency spacing of the oscillator is the Rabi frequency due to the atom-light interaction is the laser detuning from 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. The absorption(emission) of photons by the atom is then governed by the off-diagonal elements, with probability of a transition between vibrational states proportional to , and for each there is a manifold, , coupled to its neighbors with strength proportional to . Three such manifolds are shown in the picture. If the transition linewidth satisfies , a sufficiently narrow laser can be tuned to a red sideband, . For an atom starting at , the predominantly probable transition will be to . 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 , and the net effect of such a cycle, on average, will be the removing of motional quanta. After some cycles, the average phonon number is , where is the ratio of the intensities of the red to blue −th sidebands. Repeating the processes many times while ensuring that spontaneous emission occurs provides cooling to . More rigorous mathematical treatment is given in Turchette et al. and Wineland et al. Specific treatment of cooling multiple ions can be found in Morigi et al. An insightful approach to the details of cooling is given in Eschner et al., and was selectively followed above. Experimental implementations For resolved sideband cooling to be effective, the process needs to start at sufficiently low . To that end, the particle is