Demonstration of Two-Atom Entanglement with Ultrafast Optical Pulses

We demonstrate quantum entanglement of two trapped atomic ion qubits using a sequence of ultrafast laser pulses. Unlike previous demonstrations of entanglement mediated by the Coulomb interaction, this scheme does not require confinement to the Lamb-Dicke regime and can be less sensitive to ambient noise due to its speed. To elucidate the physics of an ultrafast phase gate, we generate a high entanglement rate using just ten pulses, each of $\sim 20\mathrm{ps}$ duration, and demonstrate an entangled Bell state with $(76\pm 1)\%$ fidelity. These results pave the way for entanglement operations within a large collection of qubits by exciting only local modes of motion.

Figure 1

(a) Experimental schematic. A single pulse from a mode-locked 355 nm laser is divided into eight subpulses by three sequential optical delay stages. The shaped pulse is then split into two paths directed through independent AOMs, used to make the interaction direction dependent. The pulses overlap in space and time at the position of the ions in a counterpropagating lin $\perp$ lin polarization configuration that produces an SDK [21]. Following gate operations, the ion qubits are measured by collecting state-dependent fluorescence from the two ions on respective PMTs when resonant lasers are applied (not shown). (b) Phase space evolution for a single SDK on a single ion. The SDK displaces the momentum of an initial motional state (in purple) by $\pm 2p_0\eta$ in phase space, correlated with the internal spin raising or lowering operator ${\sigma }_{\pm }$ in the ion. (c) Phase space evolution for two collective modes of motion under a sequence comprised of an SDK at time $t_1$, free evolution, and a second SDK at time $t_2$, in frames rotating at the respective mode frequencies. Each SDK imparts a phase depending on the magnitude of the momentum kick and the coherent state before the kick [see Eq. (3)], as well as a laser phase $2\varphi (t)[-2\varphi (t)]$ on the $|\downarrow \downarrow ⟩$ ($|\uparrow \uparrow ⟩$) states [see Eq. (4)]. In addition, the motional states acquire a phase from the free evolution [see Eq. (2)]. Because the evolution is depicted in the rotating frame, the direction of an SDK depends on the time elapsed since the previous SDK.

Figure 2

(a) The gate sequence is applied within a Ramsey experiment. The entangling gate contains five repetitions of a sequence consisting of a single SDK ($b_n=1$), followed by a wait of $M_1$ time steps ($b_n=0$), another SDK ($b_n=-1$), and a final wait of $M_2$ time steps. (b) Depiction of the trajectories followed by the COM and relative modes for a fully entangling sequence. They follow opposite circulations and enclose similar areas, and the sum leads to the gate phase. (c) We choose ${\varphi }_2$ such that we maximize $P_{\uparrow \uparrow }$, $P_{\downarrow \downarrow }$ and minimize $P_{\downarrow \uparrow }$, $P_{\uparrow \downarrow }$ (black arrow). The population in $|\uparrow \downarrow ⟩$ and $|\downarrow \uparrow ⟩$ is due mostly to SDK infidelity causing single-particle coherences. Amplitude differences of the peaks in the $P_{\downarrow \uparrow }+P_{\uparrow \downarrow }$ signal are likely caused by microwave power calibration errors. (d) The parity oscillation amplitude after choosing the value of ${\varphi }_2$ corresponding to the black arrow in (c) is proportional to $2{\rho }_{\uparrow \uparrow ,\downarrow \downarrow }$, which allows us to compute the fidelity. The best parity oscillation amplitude achieved is 0.69(1), leading to a final gate fidelity of 76(1)%. The error bars reflect the statistical uncertainty and for (d) are typically smaller than the points.

Figure 3

(a) Measured parity oscillation amplitude for various values of the gate phase ${\mathrm{\Phi }}_g$, which should be proportional to $\sin {\mathrm{\Phi }}_g$ (solid blue line). The gate phase is modified by changing the pulse schedule given by the integers $M_1$ and $M_2$ (see the main text). The insets show phase space trajectories for the COM (black solid curves) and relative (red dashed curves) modes for ${\mathrm{\Phi }}_g=\pi /1.67$ (right) and ${\mathrm{\Phi }}_g=\pi /11.9$ (left). The fidelity of the entangled state produced in each case, referenced to the ideal state ${\mathrm{\Psi }}_f({\mathrm{\Phi }}_g)$ in Eq. (6), is roughly 0.7 for all the measurements. (b) Pulse sequence for a faster sequence, where the kicks along the arms of the trajectory are switched each time, and there is trap evolution to round the corners. See Supplemental Material [24] for experimental details, including the pulse schedule. The inset shows the parity oscillation. Because there are more pulses, the gate phase is less than $\pi /2$, and there are additional infidelities introduced by the switching, the amplitude of the parity oscillation is significantly smaller than that in Fig. 2. The total gate time is about $1.95\mu \mathrm{s}$.