Optical Superresolution Sensing of a Trapped Ion’s Wave Packet Size

We demonstrate superresolution optical sensing of the size of the wave packet of a single trapped ion. Our method extends the well-known ground state depletion (GSD) technique to the coherent regime. Here, we use a hollow beam to strongly saturate a coherently driven dipole-forbidden transition around a subdiffraction limited area at its center and observe state dependent fluorescence. By spatially scanning this laser beam over a single trapped ${}^{40}{\mathrm{Ca}}^{+}$ ion, we are able to measure the wave packet sizes of cooled ions. Using a depletion beam waist of $4.2(1)\mu \mathrm{m}$ we reach a spatial resolution which allows us to determine a wave packet size of 39(9) nm for a near ground state cooled ion. This value matches an independently deduced value of 32(2) nm, calculated from resolved sideband spectroscopy measurements. Finally, we discuss the ultimate resolution limits of our adapted GSD imaging technique in view of applications to direct quantum wave packet imaging.

Figure 1

(a) Energy levels and transitions in ${}^{40}{\mathrm{Ca}}^{+}$: Excitation on the ${\text{S}}_{1/2}\text{−}{\text{P}}_{1/2}$ dipole transition near 397 nm is used for Doppler cooling, and we observe emitted fluorescence decay of the ${\text{P}}_{1/2}$ state for state detection. The dipole-forbidden ${\text{S}}_{1/2}\text{−}{\text{D}}_{5/2}$ transition near 729 nm is used as a vortex beam (i) to probe the spatial extent of the ion’s wave packet. Using a regular Gaussian beam (ii) we perform resolved sideband cooling and sideband spectroscopy to determine the mean phonon number. (b) Sketch of the setup: The ion is trapped in a segmented linear ion trap (yellow), the ion’s fluorescence light (blue) is collected by an $f/1.6$ objective and imaged onto an EMCCD. The vortex beam [red, (i)] shape is created from the output of a single mode fiber via a holographic diffraction grating and focused to the ion with a $f=75\mathrm{mm}$ lens ($\mathrm{NA}=0.2$). To scan the vortex beam waist over the ion wave packet, a 3D Piezo translation stage is used. A second beam [red, (ii)] near 729 nm is used for resolved sideband cooling. The coordinates shown here allow us to define all the relevant frames of references: for the trap $(\sqrt{2}{\widehat{x}}_t,\sqrt{2}{\widehat{y}}_t,{\widehat{z}}_t)=(\widehat{x}+\widehat{y},\widehat{y}-\widehat{x},\widehat{z})$ and for the beam $(\sqrt{2}{\widehat{x}}_B,{\widehat{y}}_B,\sqrt{2}{\widehat{z}}_B)=(\widehat{x}+\widehat{z},\widehat{y},\widehat{z}-\widehat{x})$, respectively.

Figure 2

Measured and simulated GSD profiles. The gray scale indicates the probability of depleting the ground state with the hollow beam pulse. Results for Doppler cooling (top row) and for sideband ground state cooling (bottom row) are displayed. Each pixel GSD value is determined as the average of 10 repetitions of a measurement sequence. The statistics follow the expected quantum projection noise. (a),(b) Large area scans at low excitation power $9\mu \mathrm{W}$, exhibiting the beam shape. (c),(d) At increased power of $250\mu \mathrm{W}$ Rabi oscillations become visible. (e),(f) Zooming in at $250\mu \mathrm{W}$ power. (g),(h) Increasing power to $1200\mu \mathrm{W}$. Here, the EPSF is sufficiently small to discriminate the size of the wave packet: for Doppler cooling, center contrast is lost, while the narrow wave packet from ground state cooling preserves contrast at center. (l),(m): Simulations of EPSF assuming a pointlike particle with Rabi oscillations corresponding for Doppler and ground state cooling, respectively. (j),(k) EPSF convoluted with spatial wave packet (dashed curve) for Doppler and ground state cooling of the axial mode, respectively. Note, that loss of contrast at the center allows us to distinguish both cases and matches with data in (e),(f).

Figure 3

One-dimensional GSD profiles by averaging the three middle rows of data (solid circles) from Figs. 2, 2, 2, 2 [(a), (b), (c), (d) here] with nonlinear fits with the full model (solid) and EPSF without wave packet convolution (dashed line). For a pulse power achieving a resolving ${\sigma }_{\mathrm{EPSF}}$ similar to the wave packet size, both models strongly differ, and the fit allows for the extraction of a value for the wave packet size: the fit for the Doppler cooled ion outputs ${\sigma }_{\mathrm{z}}=60(5)\mathrm{nm}$ (c) and the one from the axially sideband cooled ion outputs ${\sigma }_z=39(9)\mathrm{nm}$ (d).