Absolute single-ion thermometry

We describe and experimentally implement a single-ion local thermometry technique with absolute sensitivity adaptable to all laser-cooled atomic ion species. The technique is based on the velocity-dependent spectral shape of a quasi-dark resonance tailored by two driving fields in a $J\to J$ transition such that the two fields can be derived from the same laser source, leading to a negligible relative phase shift. We validated the method and tested its performance in an experiment on a single ${}^{88}\mathrm{Sr}^{+}$ ion cooled in a surface radio-frequency trap. We first applied the technique to characterize the heating rate of the surface trap. We then measured the stationary temperature of the ion as a function of cooling laser detuning in the Doppler regime. The results agree with theoretical calculations, with an absolute error smaller than 100 $\mu \mathrm{K}$ at 500 $\mu \mathrm{K}$, in a temperature range between 0.5 and 3 mK and in the absence of adjustable parameters. This simple-to-implement and reliable method opens the way to fast absolute measurements of single-ion temperatures in future experiments dealing with heat transport in ion chains or thermodynamics at the single-ion level.

Figure 1

(a) Energy level structure of a $\mathrm{\Lambda }$ system that allows for the presence of a dark state. (b) Quasi-$\mathrm{\Lambda }$ system considered in this paper. (c) Pump and probe beam geometry with respect to the trap axes.

Figure 2

Incoherent repumping scheme. (a) Considering the level structure of the ${}^{88}\mathrm{Sr}^{+}$ ion, population trapping in the metastable level $D_{3/2}$ is avoided by exciting the $D_{3/2}\to P_{3/2}$ and $D_{5/2}\to P_{3/2}$ transitions. In this way no coherence is driven between the $S_{1/2}$ or $P_{1/2}$ levels and the other states. (b) Simplified level structure in which the $D_{3/2}$, $D_{5/2}$, and $P_{3/2}$ levels are replaced by a single $J=3/2$ level that is incoherently populated with a rate ${\mathrm{\Gamma }}_f$ and decays into the $S_{1/2}$ level with an effective rate ${\mathrm{\Gamma }}_r$.

Figure 3

Calculated fluorescence spectra in the presence of a thermal velocity distribution for a motion along the $z$ axis and for several temperatures $T_z$. The ion parameters correspond to the $5s{}^{2}S_{1/2}\to 5p{}^{2}P_{1/2}$ transition of ${}^{88}\mathrm{Sr}^{+}$. The other parameters for the calculation are ${\mathrm{\Omega }}_1=12$ MHz, ${\mathrm{\Omega }}_2=4$ MHz, ${\mathrm{\Delta }}_1=-3.7$ MHz, and $B=2\times {10}^{-4}$ T.

Figure 4

Calculated scattering rate $S_d\left(T_z\right)$ at the center of the dark resonance as a function of the temperature. The parameters used for the calculation are ${\mathrm{\Omega }}_1=12$ MHz, ${\mathrm{\Omega }}_2=4$ MHz, ${\mathrm{\Delta }}_1=-3.7$ MHz, ${\mathrm{\Delta }}_2=2$ MHz, and $B=2\times {10}^{-4}$ T.

Figure 5

Averaged experimental fluorescence spectrum (black dots). The pump detuning is ${\mathrm{\Delta }}_1=-3.7$ MHz and the ion is precooled to a sub-Doppler temperature with a cooling phase in which the probe is tuned to ${\mathrm{\Delta }}_2=1.7$ MHz. The free parameters obtained from the numerical fit (red line) are ${\mathrm{\Omega }}_2=3.78\left(1\right)$ MHz, ${\mathrm{\Omega }}_1=10.7\left(1\right)$ MHz, $B=2.06\left(1\right)\times {10}^{-4}$ T, and ion temperature $T=0.128\left(7\right)$ mK. The uncertainties do not take into account systematic shifts.

Figure 6

Averaged retrieved temperature $T_z$ as a function of the heating time (black dots). The heating rate of the trap (slope of the fit, red line) is 52(1) mK/s. Error bars are $\pm 1$ standard deviation (both statistical and systematic).

Figure 7

Averaged retrieved temperature $T_z$ as a function of the Doppler-cooling beam detuning (black dots). Measurements are obtained in the low-intensity regime $I\ll I_{\mathrm{sat}}$. The solid red line (blue dashed line) is the result of a calculation of the stationary temperature including (not including) the trap heating rate (see Fig. 6). The bottom panel of the figure shows the residuals (red dots) with the associated statistical standard deviation.