Experimental quantum-information processing with ${{}^{43}\text{C}\text{a}}^{+}$ ions

For quantum-information processing (QIP) with trapped ions, the isotope ${{}^{43}\text{C}\text{a}}^{+}$ offers the combined advantages of a quantum memory with long coherence time, a high-fidelity readout, and the possibility of performing two-qubit gates on a quadrupole transition with a narrow-band laser. Compared to other ions used for quantum computing, ${{}^{43}\text{C}\text{a}}^{+}$ has a relatively complicated level structure. We discuss how to meet the basic requirements for QIP and demonstrate ground-state cooling, robust state initialization, and efficient readout for the hyperfine qubit with a single ${{}^{43}\text{C}\text{a}}^{+}$ ion. A microwave field and a Raman light field are used to drive qubit transitions, and the coherence times for both fields are compared. Phase errors due to interferometric instabilities in the Raman field generation do not limit the experiments on a time scale of 100 ms. We find a quantum-information storage time of many seconds for the hyperfine qubit.

Figure 1

(Color online) Energy level diagram of the valence electron of ${{}^{43}\text{C}\text{a}}^{+}$ showing the hyperfine splitting of the lowest energy levels. Laser light at 397 nm is used for Doppler cooling and detection; the lasers at 866 nm and 854 nm pump out the metastable $D$ states. A laser at 729 nm excites the ions on the transition from the $S_{1/2}\left(F=4\right)$ states to the $D_{5/2}\left(F=2,\dots ,6\right)$ states. It is used for ground-state cooling, state initialization, and state discrimination. Microwave radiation applied to an electrode close to the ions as well as a Raman light field at 397 nm can drive transitions between different levels in the hyperfine structure of the ${\text{S}}_{1/2}$-state manifold.

Figure 2

(Color online) (a) Most laser beams are sent to the ion from a plane containing also the symmetry axis of the trap (top view). With three custom-made lenses in the inverted viewports S, NE, and NW, we can tightly focus light at wavelengths 729 nm and 397 nm. The lens in S is also used to collect fluorescence light which is sent to a PMT or a camera. The quantization axis is defined by a magnetic field along SW-NE. (b) Two beams (729 nm and 866 nm) are sent in from below in a 60° angle to the trap axis (side view). The axial trapping potential is provided by two tips indicated by arrows along the $z$ axis. Unbalancing the tip voltages results in a shift of the ion crystal along the trap axis (E-W).

Figure 3

(Color online) The population in the stretched state $S_{1/2}\left(F=4,m_F=4\right)$ is plotted as a function of the duration of optical pumping. An exponential fit (solid line) reveals a time constant of $1.4\mu \text{s}$. After $10\mu \text{s}$ the population is in the desired state in 98% of the measurements. The inset shows a histogram of the success rate of 100 measurements each containing 100 experiments when two $\pi$ pulses on the quadrupole transition are applied and an additional intermediated optical pumping interval is used. This enhances the fidelity of the process to above 99.2%.

Figure 4

(Color online) Rabi oscillations on the blue axial sideband of the transition $S_{1/2}\left(F=4,m_F=4\right)\leftrightarrow D_{5/2}\left(F=6,m_F=6\right)$ after ground-state cooling. The solid line is a fit assuming a thermal state. It yields a mean occupation of the axial mode of ${\overline{n}}_{ax}=0.06$.

Figure 5

(Color online) Transfer probability measurement of an amplitude shaped laser pulse on the transition $S_{1/2}\left(m_F=1/2\right)\leftrightarrow D_{5/2}\left(m_F=3/2\right)$ of a single ${{}^{40}\text{C}\text{a}}^{+}$ ion as a function of the Rabi frequency. Data were taken for four different pulse lengths $\tau$ and frequency chirp spans ${\Delta }_c$ as given in the plot legend. The lines indicate what is theoretically expected. With enough laser power available, the transfer probability hardly changes over a wide range of Rabi frequencies.

Figure 6

(Color online) Rabi oscillations on the ${{}^{43}\text{C}\text{a}}^{+}$ hyperfine qubit mediated by a microwave field after 0, 50, and 100 ms. Each data point represent 50 individual measurements. The solid line is a weighted least-squares fit with the function $\frac{A}{2}\text{cos}\left(\pi t/{\tau }_{\pi }\right)+y_0$, resulting in $y_0=0.490\left(3\right)$, ${\tau }_{\pi }=520.83\left(3\right)\mu \text{s}$, and $A=0.974\left(11\right)$. The number of state transfers is indicated. Since the amplitude of the Rabi oscillation is still close to unity even after 200 state transfers, the microwave can serve as a reference to the Raman light field regarding power and phase stability.

Figure 7

(Color online) Rabi oscillations on the ${{}^{43}\text{C}\text{a}}^{+}$ hyperfine qubit induced by a colinear Raman light field. Each data point represents 50 individual measurements. As for the microwave excitation, we fitted a sinusoidal function to the data set from 0 to $600\mu \text{s}$, which yields a fringe amplitude $A=0.97\left(1\right)$, a $\pi$ time of ${\tau }_{\pi }=65.3\left(1\right)\mu \text{s}$, and a fringe center at $y_0=0.479\left(5\right)$. Fitting to the data points beyond 3.4 ms, a small offset phase had to be introduced and ${\tau }_{\pi }$ adjusted to $63.8\left(2\right)\mu \text{s}$, indicating a small increase of the Raman light power during the measurement. The amplitude reduced to $A=0.80\left(2\right)$, and the fringe center dropped to $y_0=0.428\left(7\right)$. A comparison with microwave excitation reveals imperfections caused by spontaneous scattering and laser amplitude fluctuations.

Figure 8

(Color online) Measurement of the qubit state $|\downarrow ⟩$-population probability after a waiting time ${\tau }_d$. Single-photon scattering events induced by residual light at 397 nm lead to a transfer of population from the $|\downarrow ⟩$ state to other Zeeman states in the ground-state manifold. The solid line is an exponential fit with a decay time constant of 410 ms.

Figure 9

(Color online) Ramsey phase experiments on the ${{}^{43}\text{C}\text{a}}^{+}$ hyperfine qubit at a magnetic field of 3.4 G with a Ramsey waiting time ${\tau }_R$ set to 100 ms. The data were taken with three different driving fields. (a) Microwave drive with ${\tau }_{\pi }=19\mu \text{s}$, (b) copropagating Raman light field with ${\tau }_{\pi }=20\mu \text{s}$, and (c) noncopropagating Raman light field where ${\tau }_{\pi }=23\mu \text{s}$. The fringe amplitudes are determined by weighted least-squares sinusoidal fits with amplitude $A$, offset phase ${\Phi }_0$, and fringe center $y_0$ as free parameters. This yields fringe amplitudes of 0.886(17), 0.879(16), and 0.922(13), respectively. Dephasing by interferometric instabilities does not limit the experiments on these time scales. For Ramsey times beyond 100 ms, we observed a further decay of the fringe amplitude.

Figure 10

(Color online) Ramsey phase experiments on the ${{}^{43}\text{C}\text{a}}^{+}$ hyperfine qubit with microwave excitation at a magnetic field of 0.5 G. (a) A scan with ${\tau }_R=50\mu \text{s}$ results an amplitude of 0.976(4) and demonstrates the ability of state initialization, manipulation, and readout. (b) For a Ramsey time ${\tau }_R=200\text{ms}$, the amplitude is 0.962(11). (c) For a Ramsey time of ${\tau }_R=1\text{s}$, a fringe amplitude of 0.847(21) was measured. Here the measurement time was about 90 min. The reduction of the amplitude is attributed to the residual sensitivity of 1.2 Hz/mG to ambient magnetic field fluctuations.