Resolved-sideband laser cooling in a Penning trap

We report the laser cooling of a single $^{40}\text{Ca}^+$ ion in a Penning trap to the motional ground state in one dimension. Cooling is performed in the strong binding limit on the 729-nm electric quadrupole $S_{1/2}\leftrightarrow D_{5/2}$ transition, broadened by a quench laser coupling the $D_{5/2}$ and $P_{3/2}$ levels. We find the final ground state occupation to be $98\pm1\%$. We measure the heating rate of the trap to be very low with $\dot{\bar{n}}\approx 0.3\pm0.2\textrm{s}^{-1}$ for trap frequencies from $150-400\textrm{kHz}$, consistent with the large ion-electrode distance.

Cold, trapped ions are one of the leading systems with which to study quantum processes that require excellent environmental isolation, including quantum computation [1], quantum simulation [2], frequency standards [3] and measurements of fundamental constants [4].
Penning traps [5,6] provide an alternative to the more common radiofrequency (RF) trap and offer an excellent degree of precision and isolation.Penning traps can achieve reasonable trapping frequencies at large (∼ 1 cm) electrode distances, reducing the ion heating rate and other environmental noise; they provide near-perfect frequency-selective state preparation and readout due to the large Zeeman splitting; and they can be used without any oscillating fields, making them suitable for trapping 2-and 3-D Coulomb crystals [7] and ions in delicate states that can be perturbed by RF fields (e.g.Rydberg ions [8]).These types of experiment often require the ion to be confined to the motional ground state, which is generally achieved via resolved sideband cooling.The sideband cooling technique was perfected in RF traps many years ago [9,10], but due to the increased technical complexities associated with the Penning trap, it has yet to be realised in this system.
In this Letter we demonstrate the application of resolved sideband cooling to the Penning trap, cooling the axial motion of a single calcium ion to its quantum ground state with 99% probability.We measure the heating rate of the trap, which we find to be amongst the lowest reported to date.The low heating rate is consistent with the large dimensions of the trap, which has a characteristic dimension of d 0 = 1.32 cm, with the distance to the nearest electrode d = 1.08 cm.Finally, we demonstrate the coherence of our cooled ion by observing its Rabi dynamics.
We recently reported work on resolved sideband spectroscopy and thermometry of a single ion in a Penning trap [11] and the experiments described here use a modified version of the same apparatus.We trap a single 40 Ca + ion in a Penning trap consisting of a stack of concentric cylindrical electrodes, held in a 1.84 T axial field provided by a superconducting solenoid magnet.Doppler cooling in the axial and radial directions is performed on the 397-nm dipole-allowed S1 /2 ↔ P1 /2 transition, with a set of four laser frequencies around 866 nm applied along the trap axis to repump population trapped in the metastable D3 /2 states.At high magnetic fields, j-state mixing [12] provides a small but significant branching ratio of 4.2×10 −7 B 2 /T 2 to the D5 /2 manifold, and a further set of four laser frequencies around 854 nm are required to prevent population trapping in these states.A laser at 729 nm, addressing the electric quadrupole S1 /2 ↔ D5 /2 transition, is used for resolved sideband spectroscopy via the electron shelving technique.Details of the trap geometry, Doppler cooling laser systems and spectroscopy scheme can be found in [11] and [13].
We have made two major changes to the experiment to enable us to perform sideband cooling.The first is an increase in the power of the 729-nm laser through use of a tapered amplifier.This increases the power at the ion from approximately 4 mW to 40 mW, allowing us to reach high enough Rabi frequencies (up to Ω 0 ∼ 2π × 50 kHz) to sideband cool on this transition.We do not require this boost to increase the resonant absorption rate, but to ensure that the cooling remains robust against small frequency drifts of several kHz during the course of the experiment, which are due to mechanical instabilities of the trap within the magnetic field.The effective width of spectral features increases with the Rabi frequency, making small detunings from the centre of the feature less significant.
The second change is to introduce a weak, oscillating quadrupolar 'axialisation' field [14] to couple the magnetron and modified cyclotron radial trap modes.The work in the previous paper [11] was performed without any oscillating fields, using an intensity gradient across the radial cooling beam to cool the otherwise-unstable magnetron motion [15].However, for higher axial frequencies, the required radial intensity gradient increases, and with our current optical system we were unable to reliably cool ions above axial frequencies of ∼ 200 kHz.The axialisation technique, which had been used by the group in earlier experiments [16,17], works at all trap frequencies, but had previously been limited by a small misalignment of the axialisation field with respect to the DC trapping potential.Having rectified this through the use of a more versatile axialisation drive supply we can now efficiently cool at all axial frequencies up to the stability limit of the trap, and achieve much lower phonon numbers after Doppler cooling.
The ion is initially Doppler cooled, before one of the two 397-nm lasers is switched off to optically pump population into the S1 /2 (m j = − 1  2 ) sub-level.Sideband cooling is performed by applying the 729-nm laser to the first red axial sideband of the S1 /2 (m j = − 1 2 ) ↔ D5 /2 (m j = − 3 2 ) transition, while using a weak 854-nm quench laser to increase the scattering rate by emptying the D5 /2 level via P3 /2 .Provided the saturation parameter of the quench laser is small, the P3 /2 level can be adiabatically eliminated and the system behaves like a virtual two level system with a controllable linewidth set by the properties of the quench laser and upper level [18].The second 397-nm laser and 866-nm repump lasers are applied throughout the sequence to ensure that population decaying on the P3 /2 ↔ S1 /2 transition is optically pumped into the correct (m j = − 1 2 ) ground state sub-level.After sideband cooling, electron shelving spectroscopy is performed as described in [11].

RESULTS
The ion is trapped with an axial frequency of 389 kHz and Doppler cooled for 5 ms.We measure an axial temperature of 0.54 mK, 1.4 times the Doppler cooling limit, corresponding to an average phonon number of n = 29 ± 1. Figure 1 shows a typical axial sideband spectrum after this step.We do not measure a radial temperature directly during these experiments, but previous work suggests this is several times higher than the axial temperature [11].Note that for this trap frequency, the 729-nm axial Lamb-Dicke parameter is η 729 = 0.155 and the ion remains somewhat outside the Lamb-Dicke regime (η 2 729 (2n+1) ∼ 1.4) after Doppler cooling, though not far enough to prevent us from cooling efficiently on the first red sideband.
Figure 2 shows the first red and blue sidebands after 21 ms of sideband cooling.The blue sideband is fitted to a Rabi sinc profile on a constant background.Because the blue sideband Rabi dynamics in n = 0 are identical to the red sideband dynamics in n = 1, and n 1, the red sideband is correctly described by the sum of a uniform background and the blue profile scaled by n.The fit shows the average phonon number after sideband cooling to be n = 0.014 ± 0.009.
The heating rate of the trap is measured by cooling to the ground state and inserting a delay period before spectroscopy is performed, during which no cooling is applied.The change in the ratio of the red and blue sideband heights then allows the change in phonon number during this delay to be determined.We perform two heating rate measurements, with delays of 50 ms and 100 ms, the results of which are summarised in Figure 3.The heating rate of ṅ = 2.5±0.3 s −1 is very low, as would be expected from such a large trap.
In Figure 4 we compare our trap heating rates to those of a variety of other traps, plotted against the distance to the nearest electrode.Here we have taken the usual approach of plotting the phonon heating rate in terms of an inferred noise spectral density, S E (ω) = 4mhω ṅ/e 2 , scaled by the trap frequency as ωS E (ω).Note that we have not included any cryogenically cooled traps, nor those that have undergone any in-situ surface treatment.For ease of comparison, we have also only considered traps with 3-dimensional electrode structures.Our heating rate, shown in the lower right corner, is comparable to those presented in [19] and amongst the lowest reported to date in any type of trap.
There has been a great deal of research over the last decade into the underlying mechanism of heating in ion traps, and in particular the 'anomalous' heating due to the proximity of electrode surfaces [20].It has become increasingly apparent in recent years that the dominant heating mechanism varies between traps and that this behaviour cannot necessarily be encapsulated in a single power-law, such as the oft-quoted 1/d 4 electrode distance scaling [21].With this proviso, we note that the data in Figure 4 clearly display a steep power law scaling, of approximately 1/d 3 .
A recent measurement of the heating rate of twodimensional crystals in a Penning trap with d = 2 cm [22] provided an indirect constraint on the single ion anomalous heating rate of ṅ < 5 s −1 at a trap frequency of 795 kHz, equivalent to a rate of ṅ = 21 s −1 at 389 kHz.Our result provides a constraint that is nearly an order of magnitude lower, despite the lower ion-electrode distance of d = 1.08 cm.However, if the scaling suggested by the data in Figure 4 was valid for all d, an even lower heating rate might be expected in our trap.It is not yet clear whether this increased heating is due to the presence of a scale-independent process such as soft collisions with background gas, or due to technical noise and EM pickup on the trap electrodes, and we are conducting further experiments to ascertain the source.
We now present some basic demonstrations of the coherence of our qubit and probe laser.We probe the ion on the carrier transition and observe Rabi oscillations after Doppler cooling only (Figure 5(a)) and after sideband cooling to the motional ground state (Figure 5(b)).After Doppler cooling, the ion is in an approximately thermal distribution with n = 35 ± 5, determined by fitting to a full spectrum as in Figure 1.The Rabi oscillations rapidly lose contrast due to the large number of contributing frequencies, and the corresponding fit provides FIG.4: The reported heating rates of a variety of linear, ring and needle traps, shown as a frequency-scaled, noise spectral density plotted against the distance to the nearest electrode.The result reported in this Letter is shown in the lower right corner.No uncertainty data is available for the results marked with "X'.[Data sources: DesLauriers [23], Diedrich [9], Roos [10], Schulz [24], Stick [25], Poulsen [19], Benhelm [26], Blakestad [27], DeVoe [28], Monroe [29], Tamm [30]] an independent measure of n = 31 ± 2.
After ground state cooling, with n ∼ 0.01, the Rabi oscillations are well defined, with a coherence time of τ = 0.7±0.1 ms, approximately consistent with our spectroscopically measured linewidth of ∆ν = 0.6 ± 0.4 kHz [11].Note that the fit to the first half of the data gives a Rabi frequency of Ω 0 = 2π × 41 kHz and the second half Ω 0 = 2π × 39 kHz, with a noisy discontinuity between the two.This is due to uncontrolled intensity noise on the 729-nm probe laser, whose power is boosted by an injection locked slave laser followed by a tapered amplifier  and is thus affected by mechanical drifts in these systems.We are currently developing a power-noise-eating feedback system for this laser and are also improving the beam pointing stability into the trap.
Figure 6 shows the Rabi dynamics as the probe laser is detuned from the carrier transition.The probe time is 100 µs and the Rabi-Lorentzian fit gives a pulse area on resonance of 3.25π.
We have demonstrated the application of resolved sideband laser cooling to an ion in a Penning trap, achieving occupancy of the quantum ground state with 99% probability.We have measured the heating rate of our ion trap and found it to be exceptionally low.We believe these results provide compelling evidence that Penning trapped ions are an excellent choice of system for a range of precision experiments in quantum information [31], quantum thermodynamics [32] and fundamental measurement [33].

FIG. 1 :
FIG.1: Typical axial motional sideband spectrum after Doppler cooling.The solid line is a fit to a comb of Lorentzians with heights driven by the Rabi dynamics of a thermally distributed population, giving n = 29 ± 1, equivalent to a temperature of 0.54 ± 0.02 mK.

FIG. 2 :
FIG.2:(a) First red and (b) first blue sidebands after 21 ms of cooling on the first red sideband.The ratio of the sideband heights above the background shows that the average phonon number is n = 0.014 ± 0.009.

FIG. 3 :
FIG. 3:The heating rate is determined by measuring the phonon number after a variable delay between cooling and spectroscopy.The fitted line gives a heating rate of ṅ = 2.5 ± 0.3 s −1 .

FIG. 5 : 2 )
FIG. 5: (a) Rabi oscillation on the carrier of the S1 /2 (mj = + 1 2 ) ↔ D5 /2 (mj = − 1 2 ) transition after Doppler cooling.The fit is to the summed Rabi dynamics of a thermal distribution of Fock states and gives Ω0 ∼ 2π × 27 kHz.The oscillations rapidly lose contrast as n ∼ 30.(b) Rabi oscillation on the carrier after ground state cooling.The Rabi frequency is intially Ω0 ∼ 2π × 41 kHz but laser intensity noise causes an abrupt shift at 70 µs and the fit to the remainder of the plot gives Ω0 ∼ 2π × 39 kHz.The overall visibility decays exponentially with a coherence time of τ = 0.7 ± 0.1 ms.

FIG. 6 :
FIG.6: High-resolution scan of the Rabi dynamics of a 100 µs probe pulse on the carrier after sideband cooling to the ground-state.The solid line is a fit to the theory from which we deduce a pulse area on resonance of 3.25π