We demonstrate minimization of ion micromotion in a linear Paul trap with the use of a high finesse cavity. The excess ion micromotion projected along the optical cavity axis or along the laser propagation direction manifests itself as sideband peaks around the carrier in the ion-cavity emission spectrum. By minimizing the sideband height in the emission spectrum, we are able to reduce the micromotion amplitude along two directions to approximately the spread of the ground state wave function. This method is useful for cavity QED experiments as it describes the possibility of efficient 3-D micromotion compensation despite optical access limitations imposed by the cavity mirrors. We also show that, in principle, sub-nanometer micromotion compensation is achievable with our current system.
© 2013 Optical Society of America
Trapped ions have become an increasingly important technology for a wide range of applications including precision metrology [1, 2] and quantum information processing (QIP) [3–6]. Due to the levels of precision demanded in these applications, it is important that the internal and motional degrees of freedom are well controlled. For instance, the quantum gate proposed by Cirac and Zoller requires the ion to be in motional ground-state for high fidelity operation [7, 8].
In an ideal radio frequency (RF) trap, a cold ion is fixed at the zero of the RF electric field and no excess motion should be present. However, in practice, the presence of stray DC fields or a phase difference between the RF electrodes can induce excess micromotion. In the frame of the ion, this micromotion is equivalent to a modulation of the cooling laser and leads to sideband generation in the emission spectrum. Adverse effects include trap heating, reduction in the laser Rabi rate, and imperfect Raman-type state transfer [3, 9–12]. Second order Doppler shifts due to excess micromotion are also a significant limitation to the attainable accuracy of atomic clocks. Thus, the detection and compensation of excess micromotion is an important requirement for many applications.
A variety of techniques for the minimization of micromotion have been discussed in the literature [11, 13–16]. A typical fluorescence technique  uses resonant lasers to probe on an atomic transition. The emission spectrum of the ion will contain sidebands along the main peak (carrier) if the micromotion is present. Through this method the micromotion can be reduced by increasing the ratio of the carrier height relative to the sidebands. However, this requires three driving lasers oriented in a geometry that allows the minimization of micromotion in all dimensions.
While these techniques are widely used in ion trap experiments, their implementation can be hindered by limited optical access or cannot readily quantify the degree to which the micro-motion is compensated. This is particularly problematic in cavity QED experiments, where one axis is blocked by the cavity mirrors. In addition, fluorescence techniques often require the RF drive frequency to be much larger than the linewidth of the optical transition used. This is not always easy to satisfy, particularly for heavier ions, which require much higher amplitudes of the RF drive for the same drive frequency compared to a light ion. Moreover, the fluorescence probing for micromotion detection is usually done on the cooling transitions, which have rather large linewidths.
In this article, we present a method to minimize excess ion micromotion by using a high finesse cavity as a spectrum analyzer for light scattered into the cavity from a probe beam. The moderate single atom cooperativity of the cavity enhances the amount of light scattered into the cavity  and, in the presence of excess micromotion, frequency sidebands at the RF drive frequency in the cavity emission spectrum appear . The heights of the sideband peaks allow us to directly measure the amplitude of the micromotion along two orthogonal directions and micromotion compensation is achieved by minimizing the sidebands. Our approach is applicable as long as the RF drive frequency (Ω) is much greater than the cavity linewidth (κ), a condition easily fulfilled for most cavity QED experiments implemented with high finesse cavities. This provides a distinct advantage over free space methods as the linewidth of the optical transition can be substantially larger than the cavity linewidth. In our case, the optical transition linewidth is ≈ 20MHz while the cavity linewidth (κ/2π) is ≈ 200kHz.
2. The Model
We consider a set up in which an intra-cavity ion is probed transversely to the cavity as depicted in Fig. 1. When the detuning, Δ, of the probe from the atomic resonance is large relative to the linewidth, we can adiabatically eliminate the excited state. This results in an effective Hamiltonian which, in the interaction picture, is given by
To determine the limits of this approach to micromotion detection, we consider the terms involving the first order sidebands for the case when the ion is located at the cavity antinode, ϕ = π/2. In this case the number of photons, Ns, collected at the micromotion sideband in an integration time τ is where Nc = Ic0τ is the number of photons collected at resonance. Background counts at the RF sideband come from both off resonant scattering into the cavity and dark counts from the counting module. If the micromotion features are well resolved, the background will be approximately constant around the sideband and dominated by dark counts from the counting module. Taking the mean counts of the background to be Nb and assuming Poissonian statistics ( ), the signal to noise ratio, SNR, will be then given byEq. (8). While the higher order sidebands can be used for detecting the micromotion along y when the ion is located at a cavity antinode, the effect is greatly reduced due to the dependence on higher order terms in βx & βy.
Experimentally, we probe the ion-cavity emission orthogonally to the cavity axis using a red-detuned 493nm laser with the cavity tuned to resonance with the micromotion sideband. Standard micromotion compensation techniques using additional bias voltages on trap electrodes are then used to minimize the cavity emission at this sideband. We note that the detection process can result in heating of the ion. This heating results in a decrease of the effective cooperativity of the cavity which, in turn, affects the minimum detectable micromotion. However the detection window can be interspersed with cooling pulses in order to minimize the effects of such heating.
3. The Experiment
The experimental setup is illustrated in Fig. 1 in which a high finesse cavity is aligned with its optical axis transverse to a linear Paul trap [19–21]. Details of the ion trap have been reported elsewhere . Briefly, a 5.3MHz RF potential with an amplitude of 125V is applied via a step-up transformer to two diagonally opposing electrodes. A small DC voltage applied to the other two bias electrodes ensures a splitting of the transverse trapping frequencies and rotates the principle axes of the trap with respect to the propagation direction of the cooling lasers. Axial confinement is provided by two axial electrodes separated by 2.4mm and held at 33V. Using this configuration, we achieve trapping frequencies of 2π × (1.2, 1.1, 0.40)MHz for a single 138Ba+ ion. In addition, microcomotion compensation is achieved via additional DC voltages applied on the bias electrodes and two aluminum plates used to shield the mirror surfaces from the barium oven.
The relevant lasers and level structure for 138Ba+ are shown in Fig. 2. Doppler cooling is achieved by driving the 6S1/2 → 6P1/2 transitions at 493nm and repumping on the 5D3/2 → 6P1/2 transitions at 650nm. The 493nm cooling laser (D1) and the 650nm repumping laser (D2) are both red-detuned by ≈ 15MHz for optimum cooling. Both lasers are combined into a single fiber and sent into the trap along the z direction defined by a 3 Gauss magnetic field. The D1 and D2 beams are both linearly polarized perpendicular to the magnetic field to avoid unwanted dark states in the cooling cycle. The probe laser (Rp) is red-detuned by 110MHz from the S1/2 ↔ P1/2 transition and sent into the trap along the x direction. Rp is linearly polarized along the magnetic field direction and drives the cavity-induced Raman transition as illustrated in Fig. 2.
The dual coated high finesse cavity is approximately 5mm long with a finesse of 85000 at 493nm and 75000 at 650nm. Cavity QED parameters relevant to the 493nm probing transition are (g, κ, γ) = 2π × (2.13, 0.172, 10.05)MHz where g is the cavity coupling strength for the S1/2 ↔ P1/2π–transition and γ is the total dipole decay rate of the P1/2 level. The cavity mirrors are identical with an approximately equal loss of 20ppm in both transmission and absorption at 493nm, inferred from the ∼ 24% resonant transmission and the corresponding cavity finesse. Thus, approximately 24% of the cavity photons will be collected from each side of the cavity.
The cavity length is stabilized via the Pound-Drever-Hall technique  to the sideband of a low linewidth 650nm laser . The sideband is generated by a wideband electro-optic modulator (EOM). Changing the EOM drive frequency allows us to tune the cavity resonance relative to the fixed frequency of the 650nm locking laser. This laser is approximately 500GHz detuned from the 5D3/2 → 6P1/2 repump transition and thus has no influence on the system. Additionally, the EOM drive frequency is chosen such that any cavity resonance near to 650nm is at least a few GHz away from the repump transition. This ensures there is no cavity dynamics associated with the repumping process. In order to ensure a well defined Δc, the 493nm probe laser is referenced to the the 650nm locking laser via a transfer cavity.
The cavity output is first passed through a dichroic mirror to separate the 493nm output from the transmission of the 650nm locking laser. Further filtering is done using a bandpass filter with a specified transmission of 97% at 493nm and attenuation of 85dB at 650nm. The light is then coupled via a single mode fiber to a single photon counting module (SPCM). From the transmission of the cavity at 493nm (24%), the fibre coupling efficiency (70%), and the quantum efficiency of the SPCM at 493nm (45%) we estimate an overall detection efficiency of intra-cavity photons of approximately 7.5(2)%.
The cavity itself sits on an attocube nanopositioner which provides vertical adjustment of the cavity relative to the ion trap. However, the vertical motion also results in a displacement of the ion along the cavity axis, with the vertical to axial displacement having a ratio of 30 : 1. Thus a vertical movement of ≈ 3.6μm of the cavity allows us to move the ion from a cavity node to antinode. The small vertical displacement does not significantly alter either the output coupling to the SPCM or the degree of micromotion compensation. In addition, the vertical displacement is much less than the cavity mode waist (≈ 40μm) and thus does not significantly alter the transverse position of the ion relative to the cavity mode. By maximizing (minimizing) the scattering into the cavity we can locate the ion at the antinode (node) of the cavity to an accuracy of about ±10nm limited by the step size of the nanopositioner.
As noted earlier, heating of the ion during probing results in an increase in the minimum detectable micromotion. To avoid this, each 200μs is preceded by a 1ms pulse of Doppler cooling in each experimental cycle. With the ion maximally coupled to the cavity this results in the collection of ∼ 3 – 5 photon counts near to the cavity resonance with a background of ∼0.1 counts. This is repeated 1000 times to give a total integration time of 0.2s.
Typical emission spectra are shown in Fig. 3 and Fig. 4. In Fig. 3(a) the ion is located at the antinode of the cavity. First and second order micromotion sidebands at ±Ω and ±2Ω are clearly visible. After compensating the micromotion along the probe direction the first order sidebands are eliminated as shown in Fig. 3(b). The second order sidebands still persist due to higher order terms as given in Eq. (5). From this equation, the second order sidebands are proportional to , which give rise to a sensitivity of the cavity emission to micromotion along the cavity axis. By compensating the micromotion along this axis these second order sidebands can also be elimnated as shown in Fig. 3(c). However, since this effect is higher order, use of the second sideband is much less sensitive to the micromotion amplitude along this direction and greater sensitivity is gained by shifting the ion to the node. This is evident by the spectrum in Fig. 3(d) taken after the ion is moved to the node. Residual micromotion along the cavity axis is still apparent from the presence of the first order sidebands allowing further micromotion compensation along that direction. We also note that the spectrum in this case contains three additional peaks near to resonance. The peak at resonance is due to a residual offset from the cavity node. The other two peaks are motional sidebands due to the secular motion of the ion. These peaks are unresolved in the previous figures due to the presence of the carrier.
The linewidth of the transmission profiles in Fig. 3 and Fig. 4 is in principle given by the cavity linewidth (κ/π = 350kHz). However the emission profiles are broadened to approximately 600kHz due to a small birefringence of the cavity which splits the orthogonally polarized modes by 240kHz. These polarization modes happen to align with the vertical and horizontal axes, respectively, to within a few degrees. Due to our limited optical access we can only probe at an angle of 45° to the vertical. Thus the probe couples equally to both modes of the cavity. For the same reason, the motional sidebands are not clearly resolved.
Due to our trap geometry, the motion induced by the RF drive is inherently 2-D , hence the micromotion along the trap symmetry axis is negligible. With the micromotion fully compensated we obtain the spectrum shown in Fig. 4 which is taken with the ion located at the antinode. The inset shows the data near to the RF sideband frequency which is statistically flat with no clear signature of a sideband present consistent with a signal to noise ratio of one. The data within the inset has a mean of 100 counts with a standard deviation of 10 and the maximum counts on the carrier is 5000. Thus, from Eq. (8) we infer a minimum micromotion amplitude along the probe direction of 7.0(2)nm which is approximately the spread of the ground state wave function along the transverse trap axes. Recently, micromotion compensation to the level of 1nm has been reported in a 30s integration time . Within the same integration time we would expect to improve our compensation to approximately 2.0nm.
With our present system there are a number of factors that limit the achievable compensation. Thermal motion of the ions reduces the effective cooperativity of the cavity  in our case by a factor of ∼ 0.6. This could be improved with better cooling or tighter confinement of the ion. Due to the small cavity birefringence, the effective scattering into the cavity is reduced by a factor of ∼ 0.6. Together, these two factors reduce the total signal by a factor of 2.8 and hence the SNR by 1.7. Finally, the two equally dominating factors that limit the background counts are dark counts from the SPCM (∼ 250/s) and residual counts from the 650nm locking beam (∼ 250/s). An additional filter would eliminate the counts from the locking beam and SPCMs with 15 counts/s are available. At this point, the off-resonance scattering from the carrier transition located at Ω away becomes a contributing factor (∼ 26/s). In principle, the RF drive frequency Ω can be increased to a level such that the contribution from the off-resonance scattering becomes negligible. Thus our background could be reduced by a factor of ∼ 30, improving the SNR by a factor of 2.4. Altogether, a factor of 4 improvement in the micromotion compensation is therefore possible with our current system making sub-nanometer compensation possible.
The ratio of cooling to detection time used in our experiment is not optimal. We have measured the heating in the trap to be approximately 4 and 1.3 phonon in the axial and radial directions respectively, over the 200μs of probing period. The corresponding reduction in the single atom cooperativity is only 2% on average, and thus probing period of 200us could possibly be increased to provide a longer integration time for the same cycle time of the experiment. However, even if the cooling pulses could be eliminated altogether, the resulting factor of 6 increase in the integration time only equates to a reduction in the minimum detectable micromotion amplitude by a factor of 1.5.
As noted earlier, the micromotion in our trap is inherently 2-D therefore our current setup is sufficient to fully compensate the excess ion micromotion. Nonetheless, as a generalization to any trap geometry, the micromotion minimization can be done easily by having an additional probing laser aligned to the ẑ axis.
In summary, we have presented a method to minimize excess ion micromotion which is well suited to cavity QED experiments equipped with high finesse cavities. Its applicability in the situation where only 2-D optical access is available also makes it a potentially useful technique in future micro-fabricated surface ion trap implemented with high finesse cavity . We have also shown that sub nanometer micromotion compensation is readily achievable by this approach. Such levels of compensation are important for precision metrology  and the study of atom-ion collisions [27, 28], in which micromotion is a significant limiting factor.
We thank Markus Baden, Kyle Arnold and Andrew Bah for help with preparing the manuscript. This research was supported by the National Research Foundation and the Ministry of Education of Singapore.
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