High-Fidelity Adaptive Qubit Detection through Repetitive Quantum Nondemolition Measurements

Using two trapped ion species ( and ) as primary and ancillary quantum systems, we implement qubit measurements based on the repetitive transfer of information and quantum nondemolition detection. The repetition provides a natural mechanism for an adaptive measurement strategy, which leads to exponentially lower error rates compared to using a fixed number of detection cycles. For a single qubit we demonstrate 99.94% measurement fidelity. We also demonstrate a technique for adaptively measuring multiple qubit states using a single ancilla, and apply the technique to spectroscopy of an optical clock transition.

Figure 1

Relevant energy levels in ${}^{27}\mathrm{Al}^{+}$ and ${}^9\mathrm{Be}^{+}$. The states $|\downarrow {⟩}_{\mathrm{Al}}$ and $|\uparrow {⟩}_{\mathrm{Al}}$ form the qubit to be measured. The $m_F=\frac{7}{2}$ Zeeman sublevel of the ${}^{3}P_1$ state forms a closed transition with $|\downarrow {⟩}_{\mathrm{Al}}$. The widely separated excited state lifetimes in ${\mathrm{Al}}^{+}$, 21 s and $300\mu \mathrm{s}$, allow for many repetitions of the detection procedure in a single experiment. The qubit states in ${\mathrm{Be}}^{+}$ are distinguished by state-dependent resonance fluorescence [17].

Figure 2

Observed (symbols) and calculated (solid line) error rates for repeated detections are plotted as a function of the threshold likelihood ratio defined as the greater of $P(\{n_j\}||\uparrow {⟩}_{\mathrm{Al}})/P(\{n_j\}||\downarrow {⟩}_{\mathrm{Al}})$ or $P(\{n_j\}||\downarrow {⟩}_{\mathrm{Al}})/P(\{n_j\}||\uparrow {⟩}_{\mathrm{Al}})$. The ${}^{1}S_0$ detections (triangles) reach a repeatability of 99.99% while ${}^{3}P_0$ detections (squares) are limited by the state lifetime to 99.94%. This is achieved for a desired likelihood ratio of ${10}^3$, requiring a mean number of detection cycles equal to 6.54. Inset: Simulation of qubit detection, using experimental histograms, comparing the case in which the number of detection cycles is fixed (filled circles) to that in which it is adaptive (open circles). The ability to estimate errors in real time significantly increases measurement efficiency.

Figure 3

Red sideband Rabi flopping on the $|\downarrow {⟩}_{\mathrm{Al}}-|{}^{3}P_1{⟩}_{\mathrm{Al}}$ transition beginning with one quantum of motion. The fluorescence signal is obtained after transferring the motional information to the qubit state of ${\mathrm{Be}}^{+}$. The three possible numbers of ground state ${\mathrm{Al}}^{+}$ ions, 2 (solid circles), 1 (empty squares), and 0 (empty triangles), are distinguished by their sideband Rabi rate. For zero ground state ions, the known signal admixture from one ground state ion due to detection errors was removed. We determine the state of excitation in the aluminum ion system after making multiple mapping sequences with pulses of duration $T1$ or $T2$.

Figure 4

Signal from ${}^{3}P_0$ spectroscopy using two ${\mathrm{Al}}^{+}$ and one ${\mathrm{Be}}^{+}$ ancilla. Obtaining this signal depends on the ability to prepare the $|\downarrow {⟩}_{\mathrm{Al}}|\downarrow {⟩}_{\mathrm{Al}}$ or $|\uparrow {⟩}_{\mathrm{Al}}|\uparrow {⟩}_{\mathrm{Al}}$ state of the ${\mathrm{Al}}^{+}$ system because transitions between $|\downarrow {⟩}_{\mathrm{Al}}|\uparrow {⟩}_{\mathrm{Al}}$ and $|\uparrow {⟩}_{\mathrm{Al}}|\downarrow {⟩}_{\mathrm{Al}}$ will not be detected. Inset: Rabi flopping on the ${}^{3}P_0$ transition. Signal contrast is limited by fluctuations in the ${}^{3}P_0$ excitation, rather than detection efficiency.