Resolved sideband cooling

As of the writing of this article, the scheme behind what we refer to as resolved sideband cooling today is attributed,[1][2] to D.J. Wineland and H. Dehmelt, in their article ‘‘Proposed ${\displaystyle 10^{14}\delta \nu /\nu }$ 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.

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.

${\displaystyle \omega =\omega _{0}-\nu }$,

where ${\displaystyle \omega _{0}}$ is the internal atomic transition frequency and ${\displaystyle \nu }$ is the harmonic oscillation frequency of the atom. In this case the atom undergoes the transition

${\displaystyle \vert g,n\rangle \rightarrow \vert e,n-1\rangle }$,

where ${\displaystyle \vert a,m\rangle }$ 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.

${\displaystyle \vert e,n-1\rangle \rightarrow \vert g,n-1\rangle .}$

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 ${\displaystyle \vert g,0\rangle }$ is reached with a high probability.[8]

The core process that provides the cooling assumes a two level system that is well localized compared to the wavelength (${\displaystyle 2\pi c/\omega _{0}}$) of the transition (Lamb-Dicke regime), such as a trapped and sufficiently cooled ion or atom. After,[2] modeling the system as a harmonic oscillator interacting with a classical monochromatic electromagnetic field yields (in the rotating wave approximation) the Hamiltonian

${\displaystyle H=H_{HO}+H_{AL}}$

with

${\displaystyle H_{HO}=\hbar \nu \left(n+{\frac {1}{2}}\right)}$

${\displaystyle H_{AL}=-\hbar \Delta \left|e\right\rangle \left\langle e\right|+\hbar {\frac {\Omega }{2}}\left(\left|e\right\rangle \left\langle g\right|e^{i\mathbf {k} \cdot \mathbf {r} }+\left|g\right\rangle \left\langle e\right|e^{-i\mathbf {k} \cdot \mathbf {r} }\right)}$

and where

${\displaystyle n}$ is the number operator

${\displaystyle \nu }$ is the frequency spacing of the oscillator

${\displaystyle \Omega }$ is the Rabi frequency due to the atom-light interaction

${\displaystyle \Delta }$ is the laser detuning from ${\displaystyle \omega _{0}}$

${\displaystyle \mathbf {k} }$ 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.[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 ${\displaystyle m,n}$ proportional to ${\displaystyle \left|\left\langle m\right|e^{i\mathbf {k} \cdot \mathbf {r} }\left|n\right\rangle \right|^{2}}$, and for each ${\displaystyle n}$ there is a manifold, ${\displaystyle \{\left|g,n\right\rangle ,\left|e,n\right\rangle \}}$, coupled to its neighbors with strength proportional to ${\displaystyle \left|\left\langle m\right|e^{i\mathbf {k} \cdot \mathbf {r} }\left|n\right\rangle \right|}$. Three such manifolds are shown in the picture.

If the ${\displaystyle \omega _{0}}$ transition linewidth, ${\displaystyle \Gamma \ll \nu }$, a sufficiently narrow laser can be tuned to a red sideband, ${\displaystyle \omega _{0}-q\nu ,q\in \{1,2,3,..\}}$. For an atom starting at ${\displaystyle \left|g,n\right\rangle }$, the predominantly probable transition will be to ${\displaystyle \left|e,n-q\right\rangle }$. 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 ${\displaystyle \omega _{0}}$,[6] and the net effect of such a cycle, on average, will be the removing of ${\displaystyle q}$ motional quanta. After some cycles, the average phonon number is ${\displaystyle {\bar {n}}={\frac {R_{q}^{1/q}}{1-R_{q}^{1/q}}}}$, where ${\displaystyle R_{q}}$ is the ratio of the intensities of the red to blue ${\displaystyle q}$^−th sidebands.[10] Repeating the processes many times while ensuring that spontaneous emission occurs provides cooling to ${\displaystyle {\bar {n}}\approx (\Gamma /\nu )^{2}\ll 1}$.[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] An insightful approach to the details of cooling is given in Eschner et al.,[2] and was selectively followed above.

For resolved sideband cooling to be effective, the process needs to start at sufficiently low ${\displaystyle {\bar {n}}}$. 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. Examples of that are the cooling of Ca⁺
ions and the Raman sideband cooling of Cs atoms.

Example: cooling of Ca⁺
ions

Relevant Ca⁺
structure and light: blue - Doppler cooling; red - sideband cooling path; yellow - spontaneous decay; green - spin polarization ${\displaystyle \sigma ^{-}}$ pulses

The energy levels relevant to the cooling scheme for Ca⁺
ions are the S_1/2, P_1/2, P_3/2, D_3/2, and D_5/2, which are additionally split by a static magnetic field to their Zeeman manifolds. Doppler cooling is applied on the dipole S_1/2 - P_1/2 transition (397 nm), however, there is about 6% probability of spontaneous decay to the long-lived D_3/2 state, so that state is simultaneously pumped out (at 866 nm) to improve Doppler cooling. Sideband cooling is performed on the narrow quadrupole transition S_1/2 - D_5/2 (729 nm), however, the long-lived D_5/2 state needs to be pumped out to the short lived P_3/2 state (at 854 nm) to recycle the ion to the ground S_1/2 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:

• the ion is Doppler cooled with 397 nm and 866 nm light, with 854 nm light on as well
• the ion is spin polarized to the S_1/2(m=-1/2) state by applying a ${\displaystyle \sigma ^{-}}$ 397 nm light for the last few moments of the Doppler cooling process
• sideband cooling loops are applied at the first red sideband of the D_5/2(m=-5/2) 729 nm transition
• to ensure the population ends up in the S_1/2(m=-1/2) state, another ${\displaystyle \sigma ^{-}}$ 397 nm pulse is applied
• manipulation is carried out and analysis is carried out by applying 729 nm light at the frequency of interest
• detection is carried out with 397 nm and 866 nm light: discrimination between dark (D) and bright (S) state is based on a pre-determined threshold value of fluorescence counts

Variations of this scheme relaxing the requirements or improving the results are being investigated/used by several ion-trapping groups.

• Laser cooling
• Amplitude modulation