Figure 1

Experimental setup: A ${}^{24}$Mg${}^{+}$ ion is trapped in a linear Paul trap. Radial confinement is accomplished by an rf voltage applied to two of the four rods. A dc voltage applied to the rings provides tunable axial confinement. The ion is Doppler cooled by red-detuned laser light addressing the cycling $D_2$ transition. A single-photon camera observes the axial spatial fluorescence distribution via an imaging system.Reuse & Permissions

Figure 2

Time-averaged spatial distribution of a single laser-cooled ion trapped with secular frequency $\omega =2\pi \times 15$ kHz ($T\approx 1$ mK). The inset shows the ion image, while the plot is a histogram in the axial direction. The residuals (lower graph) of a Gaussian fit (solid line) confirm a normal distribution as expected from Brownian motion caused by the recoils of stochastic photon scattering events. The RMS width $\Delta z_{\mathrm{cam}}$ of the Gaussian and the RMS resolution $\Delta z_{\mathrm{PSF}}\approx 1$ $\mu$m of our imaging system are indicated.Reuse & Permissions

Figure 3

(a) Measurement of the spot size $\Delta z_{\mathrm{cam}}$ of the ion images with $\Delta =-2\pi \times 18.7$ MHz and $s⪅0.1$ for various trapping potentials $\omega /2\pi$. Insets: Exemplary ion images. From a fit (solid line) of Eqs. (2) and (4), we obtain an ion temperature of $T=1.02\left(3\right)$ mK and a PSF of $\Delta z_{\mathrm{PSF}}=1.13\left(3\right)$ $\mu$m RMS with statistical uncertainties. (b) The same analysis for various laser detunings $\Delta$. Results are shown as squares together with temperatures expected from Eq. (3) [dashed (gray) line]. We attribute the observed temperature excess of up to $\approx$0.2 mK to systematic micromotion; see discussion in the text. Values of $\Delta z_{\mathrm{PSF}}$ obtained for all data points agree within their statistical uncertainties. This measurement demonstrates the precision of thermal spread thermometry, providing a total accuracy of $<$0.3 mK, unchallenged by other methods in the unresolved sideband regime.Reuse & Permissions

Figure 4

Measured ion image spot size $\Delta z_{\mathrm{cam}}$ versus laser detuning for $s⪅0.1$ and fixed $\omega =2\pi \times 32$ kHz. The solid (black) line shows the result expected from Doppler cooling according to Eq. (3), together with Eqs. (2) and (4), using $s=0.1$ and $\Delta z_{\mathrm{PSF}}=1.5\mu$m. Qualitative agreement with the overall temperature behavior of Eq. (3) is found. However, quantitatively the measured spreads and thus the temperatures are about 10$\%$ higher than expected, which we attribute to rf heating due to excess micromotion, which depends on the photon scattering rate and thus decreases toward larger detunings. The long-dashed (gray) line represents Doppler cooling with an additional detuning-independent heating rate; the short-dashed (gray) line assumes a broadening of the line to $1.1\Gamma$ as caused, e.g., by magnetic fields. Neither assumption can explain the observed thermal spread. Larger error bars for larger negative detunings are due to the lower photon count rate.Reuse & Permissions