Addressing Two-Level Systems Variably Coupled to an Oscillating Field

We propose a simple method to spectrally resolve an array of identical two-level systems coupled to an inhomogeneous oscillating field. The addressing protocol uses a dressing field with a spatially dependent coupling to the atoms. We validate this scheme experimentally by realizing single-spin addressing of a linear chain of trapped ions that are separated by $\sim 3\mu \mathrm{m}$, dressed by a laser field that is resonant with the micromotion sideband of a narrow optical transition.

Figure 1

The system considered is an array of $N$ identical noninteracting two-level systems, subjected to an external coupling field with a Rabi frequency ${\Omega }_j$ for particle $j$. The lower panel shows a sketch of the single-spin addressing protocol, involving three steps. (i) The bare eigenstates of all particles $\uparrow$ and $\downarrow$ at the Bloch sphere poles are mapped onto the dressed eigenstates in the sphere equatorial plane (shown by thick black arrows). (ii) A driving field (shown by the thin double green arrow) which is amplitude modulated at a frequency ${\Omega }_j$ rotates the spin of particle $j$ (shown by the thin red arrow, and the red two-level system in the upper panel). (iii) The dressed states are mapped back on the bare basis at the sphere poles. All particles end up in state $\downarrow$, except particle $j$ which is in $\uparrow$.

Figure 2

Calibration of the single-spin addressing protocol. Panels (a) and (b) show the time evolution and Fourier spectrum, respectively, of the fraction of dark ions in the three-ion chain driven on the micromotion sideband, for a given axial position of the chain. (c) Calibration of the dressed state energies by Fourier transform of the Rabi nutation curve on the micromotion sideband as a function of the axial displacement. The three particular points on the map (labeled 1–3) correspond to the rf null being, respectively, on the central ion, in between two rightmost ions, and on the rightmost ion, as shown in the schematic drawing on top of the map. The ions (rf null) are represented by red (white) disks. (d) 2D scan of the phase of the amplitude-modulated driving field (with respect to the dressing beam phase) and the pulse time for a single ion. A ${\sigma }_x$ modulation is obtained for a phase $\varphi \approx 3.03\mathrm{rad}$ (vertical dashed black line).

Figure 3

Spectrum of the single-addressing protocol for a three-ion chain taken after a pulse time of $400\mu \mathrm{s}$, in a trap whose secular frequencies were $({\omega }_x,{\omega }_y,{\omega }_z)=2\pi \times (2.57,2.45,1.68)\mathrm{MHz}$. The rf frequency of the trap was 21.75 MHz.

Figure 4

Imaging the single-spin addressing protocol on a three-ion chain. The inter-ion spacing is $3\mu \mathrm{m}$. Each image below is an average of 50 pictures taken with an EMCCD camera, with an exposure time of 20 ms. In the upper image, all three ions are in the $S$ state, and therefore fluoresce. In the lower images, a single ion remained in the $S$ state while the other two were transferred to the $D$ (dark) state. Because of the long exposure time required for the EMCCD imaging, there is a significant amount of error due to the finite lifetime of the dark state, in addition to the cross talk errors. For instance, on the second image, the error is, respectively, 7(4)% and 12(5)% for the central and rightmost ions, respectively.