Sideband cooling is a well-established technique for preparing trapped ions in the ground state of their motion in the trap.  However, this technique has mainly been used in radiofrequency traps. Its application in Penning traps has been slower, due to the properties of the Penning trap.  In particular, the strong magnetic field gives rise to very large Zeeman shifts and this complicates the atomic level structure, leading to a requirement for many laser frequencies.  Also, the relatively low trapping frequencies available in the Penning trap can lead to a requirement for cooling on multiple sidebands rather than a single one.

Ions that are trapped in a Penning trap and cooled to the ground state of motion have potential applications in areas such as quantum information processing and quantum simulation as well as sympathetic cooling and quantum logic applied to precision measurements with protons and antiprotons in traps.

Content of Sideband Cooling

Doppler Cooling

Preparing the ion using Doppler cooling

A single ion in a trap is an example of a quantum mechanical simple harmonic oscillator.  The energy levels therefore have a spacing of ħω, where ω is the trap oscillation frequency (typically a few hundred kHz for the axial motion in our trap).  After Doppler cooling, the ion typically has a motional quantum number (n) of the order of 25, corresponding to a temperature of ~0.5 mK for calcium ions in our trap.  A typical spectrum after Doppler cooling, obtained by probing the ion for a fixed period at different frequencies of the 729 nm qubit laser, can be seen in Figure 1.

[1] a
Figure 1. Typical axial motional sideband spectrum after Doppler cooling, with a 10-μs probe pulse length. The solid line is a fit to the Rabi dynamics of a thermally distributed population, giving n=24(1), equivalent to a temperature of 0.45(2) mK [1].


Sideband Cooling of the axial motion

Sideband cooling of the axial motion

The aim of sideband cooling is to “walk” the ion down the ladder of energy states until it is in the ground state (n=0). This is achieved by exciting the ion with laser radiation tuned to the first red sideband of a narrow transition, such that after each excitation (and subsequent decay back to the ground state) the value of n reduces by 1 on average. In our trap, the laser wavelength for Ca+ is 729 nm.

In Figure 1 (in Doppler Cooling section), the first red and blue sidebands (at ± 400 kHz from the central carrier frequency) are roughly equal in amplitude. As n approaches zero, these sidebands become smaller and asymmetrical, because when the ion is in the ground state there is no possibility of excitation on the red sideband. The spectrum shown in Figure 2 is obtained after 10 ms of sideband cooling, and the strong asymmetry of the sidebands indicates that the ion is in the ground state for 98% of the time. The heating rate we have measured is very low compared to RF traps because the electrodes are relatively far away from the ion.

[1] b
Figure 2. (a) First red and (b) first blue sidebands after sideband cooling, with a trap frequency of 389 kHz and a probe time of 100 μs. The solid line is a fit to the Rabi dynamics with a constant background, which gives an average value of <n>=0.02(1) [1].


Sideband cooling of a two-ion Coulomb crystal

Sideband cooling of a two-ion Coulomb crystal

We have also carried out sideband cooling of small Coulomb crystals in the Penning trap.  For two ions, the stable configuration at low trapping potential has each ion on the trap axis but displaced by equal and opposite amounts form the trap centre. Sideband cooling of the ions is complicated because the Lamb-Dicke parameter for the system (related to the ratio of the size of the ground state wavefunction to the wavelength of the light used for probing) is unusually large.  At some values of n, the amplitude of the first sideband becomes very small and this impedes the sideband cooling process.  In order to overcome this, and to address the two axial modes at the same time, we have to excite the ion on a variety of sidebands in order to bring both modes of motion to the ground state.  This becomes a very complex process, but it is still possible to achieve close to ground state cooling in this system (see Figure 3).

Figure 3. Spectrum of the excitation probability of one ion in a sideband-cooled two-ion axial crystal at an axial frequency of 162 kHz, showing a fit to the carrier and the first-order sidebands of both motions. The final quantum numbers for the centre of mass and breathing modes are 0.30(4) and 0.07(3) respectively [2].


Sideband cooling of the radial motion

Sideband cooling of the radial motion

Cooling the radial modes of ions in a Penning trap is complicated by the form of the motion, which can be described as an epicyclic superposition of two orbital motions. Both of the radial modes are associated with negative potential energy (compared with an ion at trap centre), and one of them is associated with a negative total energy. This means that energy must be added to this mode in order to reduce the amplitude of motion.  Standard techniques are available to overcome this problem in order to Doppler cool the radial motion of an ion.  We have now also achieved sideband cooling of the radial motion.  This is difficult not only because two modes have to be cooled simultaneously (as with a two-ion crystal) but also because the starting point for sideband cooling is typically at extremely high values of n for one of the radial motions.  We have developed ways to reduce the initial quantum numbers so that sideband cooling can be successfully applied, and Figure 4 shows the resulting spectrum with both modes close to their ground state [3].

Figure 4. Continuous spectrum after 68 ms of sideband cooling with a probe time of 280 ms. Fit parameters using a two-mode thermal model: n+ = 0.35(5), n- = 1.7(2).


[1] Goodwin J F et al. Resolved-sideband laser cooling in a Penning trapPhys. Rev. Lett. 2016;116(14): 143002. doi: 10.1103/PhysRevLett.116.143002. 

[2] Stutter G et al. Sideband cooling of small ion Coulomb crystals in a Penning trapJournal of Modern Optics. 2018;65(5-6): 549-559. doi: 10.1080/09500340.2017.1376719. 

[3] Hrmo P et al. Sideband cooling of the radial modes of motion of a single ion in a Penning trap. Phys. Rev. A. 2019;100: 043414. doi: 10.1103/PhysRevA.100.043414.



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