High-fidelity quantum gates for trapped ions under micromotion

Two- or three-dimensional Paul traps can confine a large number of ions forming a Wigner crystal, which would provide an ideal architecture for scalable quantum computation except for the micromotion, an issue that is commonly believed to be the obstacle for any high-fidelity quantum gate. Here we show that the obstacle of micromotion can be overcome with current technology, even though the magnitude of the micromotion is way outside the Lamb-Dicke region. Through exact solution of the quantum Mathieu equations, we demonstrate the principle of the gate design under micromotion using two ions in a quadrupole Paul trap as an example. The proposed micromotion quantum gates can be extended to the many-ion case and may pave a new way for scalable trapped-ion quantum computation.

Figure 1

(a) The time-dependent parameter ${\eta }_{mm}\left(t\right)$ and (b) the real and imaginary parts of function $v_{\text{c.m.}}\left(t\right)$. The blue (dark gray) curve with starting value near 1 is the real part, and the red (light gray) curve starting from 0 is the imaginary part. They have even and odd parities as a function of time and look similar to cosine and sine functions. Units of time is the trap frequency $T_z=2\pi /{\omega }_{\text{c.m.}}$. The parameters used are ion mass $m=9u$ ($u$ is the atomic mass unit), rf trap frequency ${\Omega }_T=2\pi \times 240$ MHz, and characteristic electrode size $d_0=200 \mu \text{m}$; the ac and dc voltages $V_0$ and $U_0$ are 300 and 21 V, respectively. The resulting secular trap frequencies are ${\omega }_{\text{c.m.}}=2\pi \times 0.965\text{MHz},{\omega }_r=2\pi \times 3.62\text{MHz}$ along the $z$ axis and ${\omega }_x={\omega }_y=2\pi \times 20.8\text{MHz}$ along the $x$ and $y$ axes.

Figure 2

(a) The fidelity of a two-ion conditional phase-flip gate as a function of gate time $\tau$, where the unit of time is the period of the center-of-mass motion along the $z$ direction $T_z=2\pi /{\omega }_{\text{c.m.}}$. The detuning was chosen to be $\mu =0.95{\omega }_{\text{c.m.z}}$. The blue solid line indicates the optimal results with micromotion taken into account; the red dashed line is the result for a genuine static harmonic trap without micromotion, and the gray dotted line is obtained by applying the optimal solution for a static harmonic trap to the case with micromotion, which results in poor performance. (b) The infidelity (one-fidelity) near the optimal evolution time, essentially a close-up of (a) near $\tau =20T_z$. The green dots in (b) show the time points that are an integral multiple of the micromotion period. The other parameters used are temperature of both motional degrees of freedom $k_{B}T=10\hslash {\omega }_{\text{c.m.}}$ and effective laser wave vector $\Delta k_z=8\mu {\text{m}}^{-1}$, so ${\eta }_{\text{c.m.}}\approx 0.12$ and ${\eta }_r\approx 0.09$.

Figure 3

(a) The waveform of the optimal segmented pulse calculated for the gate with duration $\tau =1.31T_z$. The thick blue (thin red) line corresponds to the case with (without) micromotion. (b) The maximal Rabi frequency $\tilde{\Omega }\equiv {max}_t\left|\Omega \left(t\right)\right|$ as a function of the gate time $\tau$. The upper blue (lower red) curve corresponds to the case with (without) micromotion.