Picosecond ion-qubit manipulation and spin-phonon entanglement with resonant laser pulses

Ultrafast spin-phonon entanglement based on spin-dependent momentum kicks (SDKs) provides an approach to realize fast entangling gates with intrinsic robustness and scalability for trapped ion quantum computing. Such SDKs so far have been implemented on a nanosecond timescale by off-resonant Raman transitions where each laser pulse is split into a sequence of perturbation pulses with carefully designed temporal patterns. Here we report an experimental realization of ultrafast qubit manipulation and spin-phonon entanglement in picoseconds using SDKs from single resonant laser pulses on the magnetic-field-insensitive hyperfine qubit states. This experiment demonstrates a convenient approach to ultrafast SDKs on noise-insensitive ion-spin qubits, with improvement in its speed by more than an order of magnitude. It removes the need to engineer the pattern of a sequence of perturbation pulses and is less vulnerable to noise, simplifying the approach to large-scale trapped-ion quantum computing based on fast quantum gates with SDKs.

Figure 1

Experimental scheme. (a) Relevant energy levels of a ${}^{171}\mathrm{Yb}^{+}$ ion. The two hyperfine qubit states $|0\rangle$ and $|1\rangle$ are coupled with equal amplitude to an excited state $|e\rangle$ by a resonant ultrafast laser pulse, whose bandwidth is much broader than the qubit frequency ${\omega }_{01}$. (b) A single laser pulse (indicated by the arrow) drives the Rabi oscillation between the superposition state $|+\rangle \equiv \left(\right|0\rangle +|1\rangle )/\sqrt{2}$ and the excited state $|e\rangle$. In particular, when the area of the pulse is set to $\theta =2\pi$, we get a ${\sigma }_x$ gate or a $\pi$ pulse on the hyperfine qubit. (c) If we apply two ultrafast pulses, each with an area of $\pi$, from opposite directions (indicated by the two vertical arrows), we get a spin-dependent momentum kick (SDK). By setting the separation $t_s$ between the two pulses to be a multiple of $2\pi /{\omega }_{01}$, while $|-\rangle$ is still unchanged, $|+\rangle$ now acquires a minus sign together with a momentum kick from the wave-vector difference between the counterpropagating laser beams. In addition, there is an optical phase, $e^{i\mathrm{\Delta }\varphi }$, from the path difference of the two beams, which needs to be canceled in the experimental sequence.

Figure 2

Ultrafast single-qubit operation in 2 ps. By adjusting the intensity $I$ of a single laser pulse, we get Rabi oscillation between $|+\rangle$ and $|e\rangle$ versus the pulse area $\theta \propto \sqrt{I}$. When $\theta =2\pi$, we get $U=-|+\rangle \langle +|+|-\rangle \langle -|=-{\sigma }_x$ as a single-qubit $\pi$ pulse. (a) By initializing the qubit in $|0\rangle$ (a dark state under the detection laser), we scan the bright-state population versus the pulse area as the blue dots. Each data point is averaged over 100 samples with error bars representing one standard deviation. The green dashed line, the red dotted line, and the purple dash-dotted line are the theoretical evolutions for populations in $|0\rangle , |1\rangle$, and $|e\rangle$, respectively. Together, we compute the theoretical evolution of the bright state population as the blue solid curve, which agrees well with the experimental result. (b) By tuning the laser intensity to the fitted peak in panel (a), we obtain the desired ultrafast single-qubit $\pi$ pulse. Starting from $|0\rangle$, we further measure the population in the bright state versus the pulse number: ideally, after an even number of pulses the qubit shall remain in the dark state, while an odd number of pulses flip the qubit into the bright state. Experimentally, we fit the slope of the two sets of data as $(1.46\pm 0.05)\%$ and $-(0.72\pm 0.03)\%$, which gives an average gate error of $1.1\%$. The asymmetry between these two errors can come from the leakage to the other $S$ or $P$ levels due to, e.g., the imperfect laser polarization or pulse length, which are more likely to result in a bright state.

Figure 3

Ultrafast spin-phonon entanglement by SDK. (a) The qubit and the phonon states are initialized by Doppler cooling followed by optical pumping [11]. Then we apply two SDKs with a separation of $\tau$ and finally we measure the qubit state. (b) Schematic evolution in the phase space of the phonon mode. The qubit is initialized in $|0\rangle$ (purple or light gray circles), which is a superposition of $|+\rangle$ (red, outer trajectories) and $|-\rangle$ (blue, inner trajectories). An SDK leads to a displacement for $|+\rangle$ while keeping $|-\rangle$ unchanged and thus creates spin-phonon entanglement. Then the phonon state rotates at the angular frequency ${\omega }_m$ (arcs in the clockwise direction) and in the meantime the spin states $|+\rangle$ and $|-\rangle$ evolve into each other at the qubit frequency ${\omega }_{01}$ (not shown). The two left subplots: If we apply the second SDK at $\tau =2\pi k/{\omega }_{01}$, the two momentum kicks will act on the same trajectory. When $\tau$ is around half a trap period, the two trajectories meet again and hence the spin and the phonon states disentangle; while if $\tau$ is around a full trap period, the two trajectories are further separated to give stronger entanglement. The two right subplots: If we apply the second SDK at $\tau =2\pi (k+1/2)/{\omega }_{01}$, the two momentum kicks will act on the two trajectories separately. This time, we get the largest spin-phonon entanglement when $\tau$ is around half a trap period, and zero for a full trap period. (c) Measured evolution of the bright state population. When $\tau =2\pi (k+1/2)/{\omega }_{01}$ (blue squares), we get oscillation vs $\tau$ as the spin and photon states entangle and disentangle. On the other hand, if we choose $\tau =2\pi k/{\omega }_{01}$ (red dots), the uncontrolled optical phase accumulates, and thus the average value over 100 samples stays around 0.5 independent of the separation time $\tau$.

Figure 4

We initialize the qubit in $|+\rangle$ by a $\pi /2$ microwave pulse, wait for time $T$ where we insert two SDKs separated by $t_s=2\pi (k+1/2)/{\omega }_{01}$, and then apply another $\pi /2$ microwave pulse and measure the final qubit state. For a fixed $T=50 \mu \mathrm{s}$ and $\tau =0.803 \mu \mathrm{s}$, we scan the detuning $\delta \omega$ between the frequency of the microwave and the qubit. A contrast of 0.78 is extracted from the Ramsey fringes.

Figure 5

Experimental setup. The mode-locked 739-nm pulsed laser first goes through our pulse selection system which consists of two Glan-Taylor polarizers (GTP) and an electro-optic modulator (EOM). The EOM is driven by an amplifier which can convert the TTL signal to a 175-V Vpp pulse signal. A second harmonic generator system (SHGS) then converts the laser wavelength to 369.5 nm. An acousto-optic modulator (AOM) is used to further suppress the stray light, whose zeroth-order light is blocked by a beam dump (BD) while the first-order light is coupled to a hollow fiber and is split into two beams by a polarization beam splitter (PBS). A right-angle prism (RAP) serves as a delay stage to adjust the relative distance of the two beams. The quantization axis of the ion is set along the laser beams by a magnetic field so that we can get a pure circular polarization.

Figure 6

Experimental calibration of the SHGS under different input power of the 739-nm laser. (a) The output 370-nm laser power vs the input laser power. (b) The frequency doubling efficiency. With the EOM switched on (off), we get about 1.5-W (about 30 mW) laser power from the pulse selection system in Fig. 5, which in turn becomes more than 200 mW (about 5 $\mu \mathrm{W}$) after the SHGS due to the nonlinear frequency doubling effect. This gives us an extinction ratio above 40 000:1.