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arXiv:0709.3817 · Journal of Physics B Atomic Molecular and Optical Physics

Laser cooling in the Penning trap: an analytical model for cooling rates in the presence of an axializing field

Richard Hendricks, E. S. Phillips, D. M. Segal, Richard Claude Thompson

Published Jan 22, 2008• 14 citations
arXivHTMLPDF

Abstract

Ions stored in Penning traps may have useful applications in the field of quantum information processing. There are, however, difficulties associated with the laser cooling of one of the radial motions of ions in these traps, namely the magnetron motion. The application of a small radio-frequency quadrupolar electric potential resonant with the sum of the two radial motional frequencies has been shown to couple these motions and to lead to more efficient laser cooling. We present an analytical model that enables us to determine laser cooling rates in the presence of such an ‘axializing’ field. It is found that this field leads to an averaging of the laser cooling rates for the two motions and hence improves the overall laser cooling efficiency. The model also predicts shifts in the motional frequencies due to the axializing field that are in qualitative agreement with those measured in recent experiments. It is possible to determine laser cooling rates experimentally by studying the phase response of the cooled ions to a near resonant excitation field. Using the model developed in this paper, we study the expected phase response when an axializing field is present.

DOI
10.1088/0953-4075/41/3/035301

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Ions stored in Penning traps may have useful applications in the field of quantum information processing. There are, however, difficulties associated with the laser cooling of one of the radial motions of ions in these traps, namely the magnetron motion. The application of a small radio-frequency quadrupolar electric potential resonant with the sum of the two radial motional frequencies has been shown to couple these motions and to lead to more efficient laser cooling. We present an analytical model that enables us to determine laser cooling rates in the presence of such an ‘axializing’ field. It is found that this field leads to an averaging of the laser cooling rates for the two motions and hence improves the overall laser cooling efficiency. The model also predicts shifts in the motional frequencies due to the axializing field that are in qualitative agreement with those measured in recent experiments. It is possible to determine laser cooling rates experimentally by studying the phase response of the cooled ions to a near resonant excitation field. Using the model developed in this paper, we study the expected phase response when an axializing field is present.
We report the experimental characterization of axialization---a method of reducing the magnetron motion of a small number of ions stored in a Penning trap. This is an important step in the investigation of the suitability of Penning traps for quantum-information processing. The magnetron motion was coupled to the laser-cooled modified cyclotron motion by the application of a near-resonant oscillating quadrupole potential (the ``axialization drive''). Measurement of cooling rates of the radial motions of the ions showed an order-of-magnitude increase in the damping rate of the magnetron motion with the axialization drive applied. The experimental results are in good qualitative agreement with a recent theoretical study. In particular, a classical avoided crossing was observed in the motional frequencies as the axialization drive frequency was swept through the optimum value, proving that axialization is indeed a resonant effect.
We report the first demonstration of the axialization of laser cooled ions in a Penning trap. Axialization involves the application of a small radial quadrupole drive which couples the cyclotron and magnetron motions. This enhances the laser cooling, allowing tighter confinement of the ions to the central axis of the trap than is otherwise possible. Using an intensified charge-coupled device (ICCD) camera we have imaged the axialization process for the first time. For a single ion, we recorded a large decrease of the magnetron amplitude corresponding to a reduction in ion temperature of approximately 2 orders of magnitude to an upper limit of order 10 mK. We have discovered dynamics specific to the laser cooled system which depend critically on the axial drive frequency and amplitude.
We present a model for laser cooling of a single ion in the Penning trap. The model solves the equations of motion in the presence of the damping caused by the interaction with the laser beam, and predicts for the first time the dependence of the cooling rates for the two radial degrees of freedom in the trap as a function of the laser detuning, laser beam offset and saturation parameter. The conditions derived for both radial degrees of freedom to be cooled simultaneously agree with those found in earlier studies. The results indicate that under conditions where both motions are cooled simultaneously, the cooling rates are both significantly smaller than the maximum rate for either motion considered in isolation. Furthermore, the magnetron cooling rate is typically much lower than the cyclotron cooling rate, as has been found in experiments. The model indicates how the cooling rate for a single ion may be optimized.
We report a study of the axialization and laser cooling of single ions and small clouds of ions in a Penning trap. A weak radiofrequency (rf) signal applied to a segmented ring electrode couples the magnetron motion to the cyclotron motion, which results in improved laser cooling of the magnetron motion. This allows us to approach the trapping conditions of a Paul trap, but without any micromotion. Using an ICCD camera we show that the motion of a single ion can be confined to dimensions of the order of 20 µm or less. We have measured increased magnetron cooling rates for clouds of a few ions, using an rf-photon correlation technique. For certain laser cooling conditions, the magnetron motion of the centre of mass of the cloud grows and stabilizes at a large value. This results in the ions orbiting the centre of the trap together in a small cloud, as confirmed by photon–photon correlation measurements.
Efficient Three-Dimensional Sub-Doppler Cooling of 40^{40}40Ca+^++ in a Penning Trap
We demonstrate efficient sub-Doppler laser cooling of the three eigenmodes of a 40^{40}40Ca+^++ ion confined in a compact Penning trap operating with a magnetic field of 0.91 T. Using the same set of laser beams as required for the initial Doppler laser cooling operation, we detune the laser frequencies to produce a narrow two-photon dark resonance. The process achieves a 1/e cooling time constant of 108(8) μμμs, ultimately reducing the mean thermal axial mode occupation from 72(23) to 1.5(3) in 800 μμμs as measured by resonantly probing an electric quadrupole transition near 729 nm. A parametric drive is applied to the trap electrodes which coherently exchanges the axial mode occupation with that of each radial mode, allowing for three-dimensional sub-Doppler cooling using only the axially-propagating laser beams. This sub-Doppler cooling is achieved for an axial oscillation frequency of $ω_z = 2π~\times~$221 kHz, which places the motion well outside of the Lamb Dicke confinement regime at the Doppler laser cooling limit. Our measured cooling rate and final mode occupation are in good agreement with a semiclassical model which combines a Lindblad master equation solution for ion-photon interactions with classical harmonic oscillator motion of the trapped ion.
We present a phase-space picture of laser cooling of the radial motion of ions in a Penning trap. This picture enables a particularly simple derivation of the condition for simultaneous cooling of all degrees of freedom of a single ion to be obtained. It also allows a physically intuitive approach to be taken to the cooling process. Using this approach, we discuss different aspects of the cooling of a single ion in a Penning trap, including the formation of ``trapped'' states where the steady-state motion amplitude is nonzero for either the magnetron or modified cyclotron motion. We use an analytical approach where the approximation of small-amplitude motions can be made, and we use numerical calculations for larger-amplitude motions. The best procedure to use for effective laser cooling is derived. A similar approach is then used to treat the laser cooling of two ions in the Penning trap. We show that the use of the phase-space approach allows new insights into many aspects of the laser-cooling process to be gained, and we indicate how these insights may be applied in experimental investigations.