resolved sideband cooling

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 $10^\{14\}\backslash delta\backslash nu/\backslash nu$ laser fluorescence spectroscopy on {{chem|Tl|+}} 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.

File: Sideband cooling level diagram.png

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 will be an evenly spaced frequency ladder, with adjacent levels spaced by an energy $\backslash hbar\; \backslash nu$. Each level is denoted by a motional quantum number n, which describes the amount of motional energy present at that level. These motional quanta can be understood in the same way as for the quantum harmonic oscillator. A ladder of levels will be available for each internal state of the atom. 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.

\omega = \omega_0 - \nu,

where $\backslash omega\_0$ is the internal atomic transition frequency corresponding to at transition between g and e, and $\backslash nu$ is the harmonic-oscillation frequency of the atom. In this case the atom undergoes the transition

|g, n\rangle \to |e, n - 1\rangle,

where $|a,\; m\backslash rangle$ 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. This transition is

|e, n - 1\rangle \to |g, n - 1\rangle.

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 $|g,\; n\; =\; 0\backslash rangle$ is reached with a high probability.

The core process that provides the cooling assumes a two level system that is well localized compared to the wavelength ($2\backslash pi\; c/\backslash omega\_0$) 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 yields (in the rotating wave approximation) the Hamiltonian

H = H_\text{HO} + H_\text{AL}

with

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

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),

and where

: $n$ is the number operator,

: $\backslash nu$ is the frequency spacing of the oscillator,

: $\backslash Omega$ is the Rabi frequency due to the atom-light interaction,

: $\backslash Delta$ is the laser detuning from $\backslash omega\_0$,

: $\backslash mathbf\; k$ 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. The absorption(emission) of photons by the atom is then governed by the off-diagonal elements, with probability of a transition between vibrational states $m,\; n$ proportional to $\backslash big|\backslash langle\; m|\; e^\{i\backslash mathbf\; k\backslash cdot\backslash mathbf\; r\}\; |n\backslash rangle\; \backslash big|^2$, and for each $n$ there is a manifold $\backslash big\backslash \{|g,\; n\backslash rangle,\; |e,\; n\backslash rangle\backslash big\backslash \}$ coupled to its neighbors with strength proportional to $\backslash big|\backslash langle\; m|\; e^\{i\backslash mathbf\; k\backslash cdot\backslash mathbf\; r\}\; |n\backslash rangle\backslash big|$. Three such manifolds are shown in the picture.

If the $\backslash omega\_0$ transition linewidth $\backslash Gamma$ satisfies $\backslash Gamma\; \backslash ll\; \backslash nu$, a sufficiently narrow laser can be tuned to a red sideband, $\backslash omega\_0\; -\; q\backslash nu,\; q\; \backslash in\; \backslash \{1,\; 2,\; 3,\; \backslash dots\backslash \}$. For an atom starting at $|g,\; n\backslash rangle$, the predominantly probable transition will be to $|e,\; n\; -\; q\backslash rangle$. 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 $\backslash omega\_0$, and the net effect of such a cycle, on average, will be the removing of $q$ motional quanta. After some cycles, the average phonon number is $\backslash bar\; n\; =\; R\_q^\{1/q\}/(1\; -\; R\_q^\{1/q\})$, where $R\_q$ is the ratio of the intensities of the red to blue $q$-th sidebands. In practice, this process is normally done on the first motional sideband $q\; =\; 1$ for optimal efficiency. Repeating the processes many times while ensuring that spontaneous emission occurs provides cooling to $\backslash bar\; n\; \backslash approx\; (\backslash Gamma/\backslash nu)^2\; \backslash ll\; 1$. 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.

For resolved sideband cooling to be effective, the process needs to start at sufficiently low $\backslash bar\; n$. 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. 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 {{chem|Ca|+}} ions and the Raman sideband cooling of {{chem|Cs}} atoms.

File:Internal structure of Ca 40 ion with zeeman splitting.png

The energy levels relevant to the cooling scheme for {{chem|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. 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:

• 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 $\backslash sigma^-$ 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 $\backslash sigma^-$ 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.

A Raman transition replaces the one-photon transition used in the sideband above by a two-photon process via a virtual level. In the {{chem|Cs}} 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:

• preparation of cold sample of $10^6$ {{chem|Cs}} atoms is carried out in optical molasses, in a magneto-optic trap
• atoms are allowed to occupy a 2D, near resonance lattice
• the lattice is changed adiabatically to a far off resonance lattice, which leaves the sample sufficiently well cooled for sideband cooling to be effective (Lamb-Dicke regime)
• a magnetic field is turned on to tune the Raman coupling to the red motional sideband
• relaxation between the hyperfine states is provided by a pump/repump laser pair
• after some time, pumping is intensified to transfer the population to a specific hyperfine state
• lattice is turned off and time of flight techniques are employed to perform Stern-Gerlach analysis

• Laser cooling
• Amplitude modulation

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