Precision Measurement and Compensation of Optical Stark Shifts for an Ion-Trap Quantum Processor

Using optical Ramsey interferometry, we precisely measure the laser-induced ac-Stark shift on the $S_{1/2}–D_{5/2}$ “quantum bit” transition near 729 nm in a single trapped ${}^{40}\mathrm{C}\mathrm{a}^{+}$ ion. We cancel this shift using an additional laser field. This technique is of particular importance for the implementation of quantum information processing with cold trapped ions. As a simple application we measure the atomic phase evolution during a $n\times 2\pi$ rotation of the quantum bit.

Figure 1

(a) Level scheme of ${\mathrm{C}\mathrm{a}}^{+}$ ion. (b) Ramsey method to detect the ac-Stark effect. The $\pi /2$ pulses are resonant; the Stark pulse is detuned. See text for more details.

Figure 2

(a) Ramsey pattern: The ac-Stark shift is determined as the oscillation frequency of $P_D(\tau )$; here ${\delta }_{\mathrm{a}\mathrm{c}}/2\pi =13.9(0.2)\mathrm{k}\mathrm{H}\mathrm{z}$. The phase offset is due to a small detuning of the Ramsey pulses. (b) Compensation of the ac-Stark effect: The ion is illuminated by the Stark pulse and an additional off-resonant compensation laser field which causes an equal ac-Stark shift, but with opposite sign. Note that the frequency of both Ramsey pulses is detuned from the $|S⟩$ to $|D⟩$ transition frequency. Detuning and temporal separation of the pulses are chosen such that their relative phase is $\pi /2$; hence, $P_D$ is close to 0.5. This improves the sensitivity to ac-Stark shifts. Data are taken alternating, with $\tau =0\mu \mathrm{s}$ (solid) and $200\mu \mathrm{s}$ (open); see text for details. We estimate a residual ${\delta }_{\mathrm{a}\mathrm{c}}/2\pi =0.25(3)\mathrm{k}\mathrm{H}\mathrm{z}$.

Figure 3

The measured ac-Stark shift data [see Fig. 2a] are normalized according to the measured laser power $I(\Delta )/I(0)$ which varies by about 50% over the whole tuning range of $\Delta$. This normalized data (squares) and calculated (line) Stark shift ${\delta }_{\mathrm{a}\mathrm{c}}$ [Eq. (1)] are plotted versus the detuning ${\Delta }_{\mathrm{a}\mathrm{c}}$ of the Stark pulse from the $|S⟩–|D⟩$ resonance. The divergences are due to the $(m=-\frac{1}{2})\leftrightarrow (m^{\prime }=-\frac{5}{2},-\frac{1}{2},+\frac{3}{2})$ resonances (from left to right). Two data points at large detunings are not shown. They read ${\delta }_{\mathrm{a}\mathrm{c}}/2\pi =5.88$ and 8.49 kHz for detunings ${\Delta }_{\mathrm{a}\mathrm{c}}/2\pi =40$ and 60 MHz, respectively, and are equally well described by the theoretical curve.

Figure 4

(a) Resonant Rabi oscillations on the blue sideband of the $|S⟩–|D⟩$ transition. The period of the population oscillation is $131(1)\mu \mathrm{s}$, as found from the fit to the data. (b) Ramsey $\pi /2$ pulses on the $|S⟩–|D⟩$ carrier transition enclose the Rabi flopping on the sideband. The phase of the $|S⟩$ state is revealed to oscillate with a period of $257(2)\mu \mathrm{s}$. The ratio of both periods is 1.96(3) and agrees well with the expected value of 2.