Entanglement and Tunable Spin-Spin Couplings between Trapped Ions Using Multiple Transverse Modes

We demonstrate tunable spin-spin couplings between trapped atomic ions, mediated by laser forces on multiple transverse collective modes of motion. A ${\sigma }_x{\sigma }_x$-type Ising interaction is realized between quantum bits stored in the ground hyperfine clock states of ${}^{171}\mathrm{Yb}^{+}$ ions. We demonstrate entangling gates and tailor the spin-spin couplings with two and three trapped ions. The use of closely spaced transverse modes provides a new class of interactions relevant to quantum computing and simulation with large collections of ions in a single crystal.

Figure 1

(a) Time evolution of the spin populations $P_n$ for two ions subject to a laser force tuned to bisect the transverse CM and tilt sidebands (detuned by 26.7 kHz from each mode). The ions ideally evolve from $|\downarrow \downarrow ⟩$ to the entangled state $|\downarrow \downarrow ⟩+|\uparrow \uparrow ⟩$ after a gate time of ${\tau }_g=37.5\mu \mathrm{s}$ (dashed line). (b) Time evolution of the spin populations for three ions subject to a laser force detuned 9.4 kHz blue of the transverse CM mode. The ions ideally evolve from $|\downarrow \downarrow \downarrow ⟩$ to the entangled state $|\downarrow \downarrow \downarrow ⟩+|\downarrow \uparrow \uparrow ⟩+|\uparrow \downarrow \uparrow ⟩+|\uparrow \uparrow \downarrow ⟩$ after a gate time of $\tau =53.4\mu \mathrm{s}$ (dashed line). (c) Measured parity of two (circles) and three (squares) ions after the respective gate is applied followed by a subsequent global $\pi /2$ rotation of the spins with phase $\varphi$. (For the three-ion case, an fixed global $\pi /2$ rotation is added before this to create the GHZ state.) The contrast of the sinusoidal oscillations with $N\varphi$ give the $N$–particle coherence of the state. The populations at ${\tau }_g$ in (a) and (b) along with the parity contrast in (c) imply entanglement fidelities of $F_2>95.9\%$ for $N=2$ and $F_3>84\%$ for $N=3$ ions. The lines in (a) and (b) are fits to Eq. (1) with no free parameters, and in (c) are sinusoidal fits to extract contrast.

Figure 2

(a) Time evolution of the average number of ions in $|\uparrow ⟩$ for $N=2$ ions under the bichromatic force in the far-detuned limit, showing the secular oscillation of the Ising spin-spin coupling at 4.8 kHz accompanied by small modulations from motional excitation at $\sim 56\mathrm{kHz}$. (b) Time evolution of the probability $P_0$ of all spins $|\downarrow ⟩$, with $N=3$ ions under the bichromatic force in the far-detuned limit. Here, the two couplings $J_{1,2}=J_{2,3}$ and $J_{1,3}$ are clearly visible. Solid line is a fit to the secular terms in Eq. (3) with an empirical exponential decay. (c)–(d) Measured couplings $J_{i,j}$ for two (c) and three (d) ions (points) as a function of detuning $\mu$ overlaid with theory (lines) from Eq. (4) with no free fit parameters [the overall sign in (d) is set from theory]. The normal modes positions are indicated by the dashed lines.