Resolved sideband cooling

Resolved sideband cooling is a laser cooling technique allowing cooling of tightly bound atoms and ions beyond the Doppler cooling limit, potentially to their motional ground state. 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.
Historical notes.
As of the writing of this article, the scheme behind what we refer to as "resolved sideband cooling" today is attributed, to D.J. Wineland and H. Dehmelt, in their article ‘‘Proposed formula_1 laser fluorescence spectroscopy on mono-ion oscillator III (sideband cooling).’’ The clarification is important as at the time of the latter article, the term also designated what we call today Doppler cooling, which was experimentally realized with atomic ion clouds in 1978 by W. Neuhauser and independently by D.J. Wineland. An experiment that demonstrates resolved sideband cooling unequivocally in its contemporary meaning is that of Diedrich et al. Similarly unequivocal realization with non-Rydberg neutral atoms was demonstrated in 1998 by S. E. Hamann et al. via Raman cooling.
Conceptual description.
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 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 levels of the system can be resolved as consisting of internal levels each corresponding to a ladder of vibrational states.
Suppose a two-level atom whose ground state is shown 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.
formula_2,
where formula_3 is the internal atomic transition frequency and formula_4 is the harmonic oscillation frequency of the atom. In this case the atom undergoes the transition
formula_5,
where formula_6 represents the state of an ion whose internal atomic state is "a" and the motional state is "m". This process is labeled '1' in the adjacent image.
Subsequent spontaneous emission occurs predominantly at the carrier frequency if the recoil energy of the atom is negligible compared with the vibrational quantum energy i.e.
formula_7
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 formula_8 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 (formula_9) 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
formula_10
with
formula_11
formula_12
and where
formula_13 is the number operator
formula_4 is the frequency spacing of the oscillator
formula_15 is the Rabi frequency due to the atom-light interaction
formula_16 is the laser detuning from formula_17
formula_18 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 formula_19 proportional to formula_20, and for each formula_13 there is a manifold, formula_22, coupled to its neighbors with strength proportional to formula_23. Three such manifolds are shown in the picture.
If the formula_17 transition linewidth formula_25 satisfies formula_26, a sufficiently narrow laser can be tuned to a red sideband, formula_27. For an atom starting at formula_28, the predominantly probable transition will be to formula_29. 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 formula_17, and the net effect of such a cycle, on average, will be the removing of formula_31 motional quanta. After some cycles, the average phonon number is formula_32, where formula_33 is the ratio of the intensities of the red to blue formula_31−th sidebands. Repeating the processes many times while ensuring that spontaneous emission occurs provides cooling to formula_35. 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 formula_36. 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. 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. Examples of that are the cooling of ions and the Raman sideband cooling of atoms.
Example: cooling of ions.
The energy levels relevant to the cooling scheme for 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. and a similar one is detailed by Roos. 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.
Example: Raman sideband cooling of atoms.
A Raman transition replaces the one-photon transition used in the sideband above by a two-photon process via a virtual level. In the cooling experiment carried out by Hamann et al., 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. The process followed in is: