Perpendicular laser cooling with a rotating-wall potential in a Penning trap

We investigate the impact of a rotating-wall potential on perpendicular laser cooling in a Penning ion trap. By including energy exchange with the rotating wall, we extend previous Doppler laser-cooling theory and show that low perpendicular temperatures are more readily achieved with a rotating wall than without. Detailed numerical studies determine optimal operating parameters for producing low-temperature, stable two-dimensional crystals, important for quantum information processing experiments employing Penning traps.

Figure 1

Cross-sectional sketch of the NIST Penning trap used to generate and control single-plane crystals. Electrostatic potentials applied to cylindrical electrodes generate a confining well in the direction of the trap symmetry ($\widehat{z}$) axis. The trap is immersed in a strong uniform magnetic field ($B\approx 4.5$ T) directed parallel to the $\widehat{z}$ axis. Radial confinement is due to the Lorentz force generated by ($\vec{E}\times \vec{B}$)-induced rotation through the magnetic field. The central ring electrode is segmented into eight sections and used to apply a rotating-wall potential. Doppler laser-cooling beams are directed both along the trap axis and perpendicular to the trap axis. Here PMT denotes photomultiplier tube.

Figure 2

Two-dimensional geometry for perpendicular laser cooling of a single plane array of radius $R_c$. The array rotates at the angular frequency ${\omega }_r$. The cooling laser is directed parallel to $\widehat{x}$ with wave vector $\vec{k}=k\widehat{x}$. Here $d$ denotes the offset of the perpendicular Doppler cooling laser beam from the center of the array. For $d>0$ the laser beam is directed to the side of the array that is receding from the laser beam due to the array rotation. A Gaussian beam profile $I\left(y\right)=I_{0}exp\left[\frac{-2{(y-d)}^2}{{w_y}^2}\right]$ with waist $w_y$ is assumed. For the Doppler cooling limit calculation the array is treated as a continuous medium with a 2D density $\sigma (x,y)$ as discussed in the text.

Figure 3

Equilibrium temperature $T_{\perp }$ as a function of the normalized detuning ${\mathrm{\Delta }}_d$ [Eq. (17)] of the laser from the Doppler-shifted atomic resonance and the normalized dispersion ${\mathrm{\Delta }}_w$ [Eq. (18)] in the Doppler shift across the laser-beam waist. The units of the labeled color bar and contours are millikelvin. The calculation uses Eq. (16), valid for $\left|d\right|,w_y\ll R_c$, to determine the equilibrium temperature and assumes $S_0=0.5$ and parameters for ${}^9{\mathrm{Be}}^{+}$. The single-ion Doppler laser-cooling limit for ${}^9{\mathrm{Be}}^{+}$ is 0.44 mK.

Figure 4

(a) Contours of equilibrium planar temperature $T_{\perp }$ in millikelvin plotted against laser detuning from the atomic transition frequency and offset $d$ of the laser beam from the center of the crystal. The calculation assumes ${\omega }_r/2\pi =45$ kHz, $w_y=30\mu \mathrm{m},S_0=0.5,R_c=225\mu \mathrm{m}$, and values of $m$ and ${\gamma }_0/2$ appropriate for ${}^9{\mathrm{Be}}^{+}$ ions. (b) Plot of the equilibrium net torque (in joules) imparted by the laser over the entire crystal. The same temperature contours shown in (a) are included. The torque calculation assumes ${\mathrm{\Sigma }}_0=2.77\times {10}^9{\mathrm{m}}^{-2}$.

Figure 5

Contours of planar temperature $T_{\perp }$ (in millikelvin) along with a color plot of the net torque (in joules) imparted by the perpendicular cooling laser. The calculation assumes ${\omega }_r/2\pi =45$ kHz, $w_y=60\mu \mathrm{m},S_0=0.5,R_c=225\mu \mathrm{m},{\mathrm{\Sigma }}_0=2.77\times {10}^9{\mathrm{m}}^{-2}$, and atomic parameters appropriate for ${}^9{\mathrm{Be}}^{+}$ ions.

Figure 6

Contours of planar temperature $T_{\perp }$ (in millikelvin) along with a color plot of the net torque (in joules) imparted by the perpendicular cooling laser. The calculation assumes ${\omega }_r/2\pi =200$ kHz, $w_y=30\mu \mathrm{m},S_0=0.5,R_c=225\mu \mathrm{m},{\mathrm{\Sigma }}_0=2.77\times {10}^9{\mathrm{m}}^{-2}$, and atomic parameters appropriate for ${}^9{\mathrm{Be}}^{+}$ ions. We believe the small irregularities or waviness of the $1.50-\mathrm{mK}$ contour lines are a numerical artifact.

Figure 7

In-plane equilibrium temperature $T_{\perp }$ (in millikelvin) for parameters identical to those in Fig. 4, except for the addition of a parallel laser-beam recoil heating term with $S_{\parallel }=0.2$.

Figure 8

In-plane equilibrium temperature $T_{\perp }$ (in millikelvin) for parameters identical to those in Fig. 5, except for the addition of a parallel laser-beam recoil heating term with $S_{\parallel }=0.2$.