Resolved sideband cooling

Aside from the curiosity of having a particle at zero point energy, such preparation of a particle in a definite state with high probability (initialization) is an essential part of state manipulation experiments in quantum optics and quantum computing.

laser fluorescence spectroscopy on Tl+ mono-ion oscillator III (sideband cooling)".

[3] The clarification is important, as at the time of the latter article, the term also designated what we call today Doppler cooling,[2] which was experimentally realized with atomic ion clouds in 1978 by W. Neuhauser[4] and independently by D. J. Wineland.

[5] An experiment that demonstrates resolved sideband cooling unequivocally in its contemporary meaning is that of Diedrich et al.[6] Similarly unequivocal realization with non-Rydberg neutral atoms was demonstrated in 1998 by S. E. Hamann et al.[7] via Raman cooling.

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 levels of the system will be an evenly spaced frequency ladder, with adjacent levels spaced by an energy

These motional quanta can be understood in the same way as for the quantum harmonic oscillator.

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.

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.

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

[8] 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.

Modeling the system as a harmonic oscillator interacting with a classical monochromatic electromagnetic field[2] yields (in the rotating wave approximation) the Hamiltonian

[9] The absorption(emission) of photons by the atom is then governed by the off-diagonal elements, with probability of a transition between vibrational states

, a sufficiently narrow laser can be tuned to a red sideband,

In the Lamb–Dicke regime, the spontaneously emitted photon (depicted by arrow "2") will be, on average, at frequency

Repeating the processes many times while ensuring that spontaneous emission occurs provides cooling to

[2][9] More rigorous mathematical treatment is given in Turchette et al.[10] and Wineland et al.[9] Specific treatment of cooling multiple ions can be found in Morigi et al.[11] For resolved sideband cooling to be effective, the process needs to start at sufficiently low

To that end, the particle is usually first cooled to the Doppler limit, then some sideband cooling cycles are applied, and finally, a measurement is taken or state manipulation is carried out.

A more or less direct application of this scheme was demonstrated by Diedrich et al.[6] with the caveat that the narrow quadrupole transition used for cooling connects the ground state to a long-lived state, and the latter had to be pumped out to achieve optimal cooling efficiency.

It is not uncommon, however, that additional steps are needed in the process, due to the atomic structure of the cooled species.

The energy levels relevant to the cooling scheme for Ca+ ions are the S1/2, P1/2, P3/2, D3/2, and D5/2, which are additionally split by a static magnetic field to their Zeeman manifolds.

Doppler cooling is applied on the dipole S1/2 - P1/2 transition (397 nm), however, there is about 6% probability of spontaneous decay to the long-lived D3/2 state, so that state is simultaneously pumped out (at 866 nm) to improve Doppler cooling.

Sideband cooling is performed on the narrow quadrupole transition S1/2 - D5/2 (729 nm), however, the long-lived D5/2 state needs to be pumped out to the short lived P3/2 state (at 854 nm) to recycle the ion to the ground S1/2 state and maintain cooling performance.

One possible implementation was carried out by Leibfried et al.[12] and a similar one is detailed by Roos.

[13] For each data point in the 729 nm absorption spectrum, a few hundred iterations of the following are executed: Variations of this scheme relaxing the requirements or improving the results are being investigated/used by several ion-trapping groups.

In the Cs cooling experiment carried out by Hamann et al.,[7] trapping is provided by an isotropic optical lattice in a magnetic field, which also provides Raman coupling to the red sideband of the Zeeman manifolds.