Subpicosecond $X$ rotations of atomic clock states

We demonstrate subpicosecond-timescale population transfer between the pair of hyperfine ground states of atomic rubidium using a single laser-pulse. Our scheme utilizes the geometric and dynamic phases induced during Rabi oscillation through the fine-structure excited state to construct an $X$ rotation gate for the hyperfine-state qubit system. The experiment performed with a femtosecond laser and cold rubidium atoms, in a magnetooptical trap, shows over 98% maximal population transfer between the clock states.

Figure 1

Subpicosecond $X$ rotation scheme between the clock states $\left|0_{\mathrm{clock}}\right.〉=\left|S_{1/2},F=I+1/2,m_F=0\right.〉$ and $\left|1_{\mathrm{clock}}\right.〉=\left|S_{1/2},F=I-1/2,m_F=0\right.〉$, where a circularly polarized (${\sigma }^{+}$) laser pulse drives the cyclic Rabi oscillation of the ground fine-structure sublevel $\left|-\right.〉$ through the fine-structure excited state $\left|P_{1/2},m_J^{\prime }=1/2\right.〉$. While, the other ground fine-structure sublevel $\left|+\right.〉$ remains as a dark state due to the selection rule, resulting in the relative phase $\theta$ between $\left|\pm \right.〉$. Note that a $\pi$-polarized pulse cannot make this kind of qubit rotations: because the linear polarized light (the equal superposition of ${\sigma }^{\pm }$-polarized lights) drives the same Rabi oscillations to both $\left|\pm \right.〉$ (so, no relative phase).

Figure 2

$|5S_{1/2},F=2\rangle$ state probability $P\left(\mathcal{A}\right)$ vs. pulse-area $\mathcal{A}$ for ${\sigma }^{+}$ and $\pi$ resonant $D_1$ transitions, respectively, where blue circles (${\sigma }^{+}$) and red squares ($\pi$) represent the experimental data. The thick solid and dashed lines are the corresponding numerical calculations, with the thin lines showing the calculated results for the weak excitation regime. The error bars indicate the standard error of mean.

Figure 3

(a–b) Experimental results for the population of $5S_{1/2},F=2$ states after (a) ${\sigma }^{+}$ and (b) $\pi$ transitions as a function of pulse-area $\mathcal{A}$ and detuning $\mathrm{\Delta }/2\pi$. (c) Transition probability of the clock-state qubit ($m_F=0$ only), retrieved using (a), (b), and Eq. (3). (d–f) TDSE results corresponding to (a–c).

Figure 4

(a) Qubit transition probability, experimentally retrieved for only clock state contribution (green diamonds), and the corresponding calculation results (blue solid line) plotted as a function of detuning $\mathrm{\Delta }/2\pi$. (b) Measured $X$-rotation angles (blue diamonds) and full calculation (red solid line) compared to the two-level model calculation ($x$-axis). The error bars indicate the standard error of mean. The experimental data are from the complete population return points (dotted line) in Figs. 3 and 3.

Figure 5

Pauli $X$ gate fidelity $\mathcal{F}$ vs. laser-pulse duration: The population transfer ${\left|\left.〈0_{\mathrm{clock}}\right|\widehat{U}(\mathcal{A},\mathrm{\Delta })\left|1_{\mathrm{clock}}\right.〉\right|}^2$ between hyperfine clock states is optimized with pulse-area $\mathcal{A}$ and detuning $\mathrm{\Delta }$, as varying the pulse duration. At the optimal points ${\mathcal{A}}^{*}$ and ${\mathrm{\Delta }}^{*}$ for each pulse duration ($x$ axis, log scale), the numerical calculation results of the $\widehat{X}$ gate fidelity $\mathcal{F}=\overline{{\left|\left.〈{\psi }_{\mathrm{in}}\right|{\widehat{X}}^{†}\widehat{U}({\mathcal{A}}^{*},{\mathrm{\Delta }}^{*})\left|{\psi }_{\mathrm{in}}\right.〉\right|}^2}$ are shown, and the region with over 0.9999 fidelity is shaded.