Heating rates in a thin ion trap for microcavity experiments


We have built and characterized a novel linear ion trap. Its small horizontal electrode separation of 250 μm would previously have required microfabrication methods, while our trap was machined conventionally. The thin trap is designed to accommodate a transverse optical cavity of 0.5 mm length, a requirement for cavity-QED experiments with trapped ions in the strong coupling regime. The sandwich structure of the electrodes allows for a very accurate alignment. Employing the Doppler-recooling method, we found that intermittent laser-induced radiation pressure has a significant effect on the ion’s spectrum. This must be taken into account to correctly determine the heating rate of the trap. To this end, we have derived an analytic expression for the spectral line shape of the ion, which includes the effect of natural line broadening, heating as well as radiation pressure. We apply it to determine the accurate heating rate of the system.

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We gratefully acknowledge support from the European Commission (Marie Curie Excellence Grant MEXT-CT-2005-025703, SCALA network Contract 015714) and the EPSRC (EP/D061296/1).

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Correspondence to M. Keller.

Appendix: Line shape of an oscillating hot ion

Appendix: Line shape of an oscillating hot ion

In this Appendix, we consider the fluorescence line shape of a trapped ion exposed to intermittent excitation. The treatment of homogeneous broadening and thermal Doppler broadening is similar to the one in Ref. [24] (note that we use angular frequencies instead of scaled parameters). The most important distinguishing feature of our model is that it includes the oscillatory motion of the ion in the harmonic trapping potential resulting from the interruption of radiation pressure. For the first time, we derive an analytic expression for the spectrum of the ion under these conditions. It allows us to determine the temperature of the ion from the spectrum, without relying on the details of the recooling dynamics used in Ref. [24], which would be difficult to model in the presence of radiation pressure.

We start by considering the case of homogeneous broadening, applicable to a cold ion at rest in the trapping potential. The normalized spectrum, which is proportional to the fluorescence intensity measured in the experiment is given by a Lorentzian

$$ S_{\mathrm{cooled}}(\delta,\gamma) = \frac{\gamma}{\pi} \frac{1}{\gamma^2 + \delta^2}, $$

where δ is the detuning from resonance and γ is the half width at half maximum of the transition due to homogeneous broadening. When the ion is not at rest but undergoing oscillatory motion in the trapping potential, a periodically changing Doppler shift occurs, given by

$$ \varDelta (t) = \varDelta _0\cos\omega_z t, $$

where oscillation along the trap-axis (z-direction) with frequency ω z is assumed and the amplitude Δ 0 is given by Eq. (2).

The periodic Doppler shift leads to a modified spectrum

$$ S_\mathrm{D}(\delta,\gamma,\varDelta _0) = \frac{1}{\pi} \operatorname{Re} \biggl[ \frac{1}{\sqrt{\varDelta _0^2 - (\delta+i\gamma)^2}} \biggr]. $$

For an ion undergoing thermal motion, spectrum (10) must be averaged over a distribution of Doppler shift amplitudes Δ 0. It is convenient to express the Doppler shift through the oscillatory energy \(\epsilon=m\omega_{z}^{2} z_{0}^{2}/2\) of the ion and the recoil energy E r =h 2/2 2 [24]:

$$ \hbar \varDelta _0 = \frac{h z_0 \omega_z}{\lambda} = \sqrt{4\epsilon E_r}. $$

The probability distribution for oscillation energy ϵ of the ion is given by

$$ P(\epsilon,\bar{\epsilon}) = \frac{1}{\bar{\epsilon}} \exp \biggl( \frac{-\epsilon}{\bar{\epsilon}} \biggr). $$

The thermally broadened spectrum is then obtained as

$$ S_{\mathrm{therm}}(\delta,\gamma,\sigma) = \int\limits _0^\infty P(\epsilon,\bar{\epsilon}) S_\mathrm{D}(\delta,\gamma,\sqrt{4\epsilon E_r}/\hbar)\, d\epsilon, $$

with \(\hbar\sigma= \sqrt{2E_{r}\bar{\epsilon}}\). The integral can be evaluated and expressed in analytical form

$$ S_{\mathrm{therm}}(\delta,\gamma,\sigma) = \frac{1}{\sqrt{2\pi}\sigma } \operatorname{Re} \biggl[ \operatorname{w} \biggl( \frac{|\delta|+i\gamma}{\sqrt{2} \,\sigma} \biggr) \biggr]. $$

The function \(\operatorname{w}(z)\) is the Faddeeva function or complex error function, given by

$$ \operatorname{w}(z) = \exp\bigl(-z^2\bigr) \bigl( 1 - \mathrm{erf}(-iz) \bigr) $$

and can be readily computed [27]. Expression (14) is identical to a Voigt profile with inhomogeneous linewidth σ.

In the experimental configuration described in this paper, switching off the cooling laser results in the ion undergoing an oscillation with a fixed amplitude z 0, even before it starts heating up. This can be accommodated by using a modified distribution function, spread around an oscillation at fixed energy ϵ 0=(ħΔ 0)2/4E r .

$$ P_{\mathrm{osc}}(\epsilon,\epsilon_0) = \frac{1}{\bar{\epsilon} [2-\exp(-\epsilon_0/\bar{\epsilon}) ]} \exp \biggl(-\frac{\vert \epsilon-\epsilon_0\vert }{\bar{\epsilon}} \biggr). $$

The spectrum of a hot ion including homogeneous broadening, harmonic oscillation and thermal effects is therefore given by


The integral in (17) can be solved analytically, again using the Faddeeva function, leading to the expression given in Eq. (5) in Sect. 4.


$$z = \frac{(|\delta|+i\gamma)^2}{2\sigma^2},\quad d = \frac{\varDelta _0^2}{2\sigma^2} $$

and \(\operatorname{w}(z)\) is defined in Eq. (15). It is easy to see that for negligible oscillatory motion of the ion (Δ 0→0), the above expression reduces to (14), i.e., a Voigt-profile.

Expression (5) describes the most general fluorescence spectrum of a trapped ion. To our knowledge, it has not been derived in analytic form before. In the experiment, the homogeneous linewidth γ is determined by power broadening of the natural linewidth, σ is related to the temperature of the ion and Δ 0 is either due to secular motion in the trap or due to micromotion. Oscillation along the axis of the linear trap is initiated as a result of the sudden switch-off of the cooling laser at the beginning of the heating period. Radiation pressure of the cooling laser displaces the equilibrium position of the ion by an amount z 0 in the axial potential well. Removal of radiation pressure therefore launches oscillatory motion with amplitude z 0.

Figure 11 shows fluorescence spectra calculated from Eq. (5) for different parameters. In the figure below, the corresponding distribution of oscillation energies, which appears explicitly in Eq. (17), is plotted. While an oscillating ion is characterized by a delta-function (case (b), green line), adding thermal motion leads to a broader distribution. In case (c), red line, there is still a peak at the original oscillation energy. It is, however, approaching the pure thermal distribution of case (d).

Fig. 11

Fluorescence spectra S(δ) calculated from Eq. (5) for different sets of parameters. (a) (γ,σ,Δ 0)=2π×(20,1,0) MHz, corresponding to homogeneous broadening (Lorentzian profile); (b) (γ,σ,Δ 0)=2π×(20,1,90) MHz, corresponding to a cold oscillating ion; (c) (γ,σ,Δ 0)=2π×(20,50,90) MHz, corresponding to a hot oscillating ion; (d) (γ,σ,Δ 0)=2π×(20,50,0) MHz, corresponding to a hot ion with no oscillation (Voigt profile). The graph below shows the corresponding probability distributions P(ϵ), plotted as a function of the scaled energy \(\sqrt{4\epsilon E_{r}}\)

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Brama, E., Mortensen, A., Keller, M. et al. Heating rates in a thin ion trap for microcavity experiments. Appl. Phys. B 107, 945–954 (2012). https://doi.org/10.1007/s00340-012-5091-9

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  • Radiation Pressure
  • Fluorescence Level
  • Cooling Laser
  • Compensation Voltage
  • Stray Field