Implementation of an experimentally feasible controlled-phase gate on two blockaded Rydberg atoms

We investigate the implementation of a controlled-$Z$ gate on a pair of Rydberg atoms in spatially separated dipole traps where the joint excitation of both atoms into the Rydberg level is strongly suppressed (the Rydberg blockade). We follow the adiabatic gate scheme of Jaksch et al. [D. Jaksch, J. I. Cirac, P. Zoller, S. L. Rolston, R. Côté, and M. D. Lukin, Phys. Rev. Lett. 85, 2208 (2000)], where the pair of atoms is coherently excited using lasers, and apply it to the experimental setup outlined by Gaëtan et al. [A. Gaëtan, Y. Miroshnychenko, T. Wilk, A. Chotia, M. Viteau, D. Comparat, P. Pillet, A. Browaeys, and P. Grangier, Nat. Phys. 5, 115 (2009)]. We apply optimization to the experimental parameters to improve gate fidelity and consider the impact of several experimental constraints on the gate success.

Figure 1

The level scheme for a single Rydberg atom. The atom is driven to the Rydberg state via a two-photon transition, which couples the ground state $|g\rangle$ to the excited state $|r\rangle$ through the intermediate state $|i\rangle$. The effective Rabi frequencies of the transitions are ${\Omega }_R\left(t\right)$ for the red laser (which is blue-detuned by $\Delta$) and ${\Omega }_B$ for the blue laser (which is red-detuned by $\Delta +\delta$).

Figure 2

The level scheme for the two-atom system (neglecting the internal level $|i\rangle$). The transition from the joint ground state $|gg\rangle$ to the super-radiant state $|{\Psi }^{+}\rangle =\left(\right|gr\rangle +|rg\rangle )/\sqrt{2}$ is enhanced by a factor of $\sqrt{2}$, while there is no coupling to the subradiant $|{\Psi }^{-}\rangle =\left(\right|gr\rangle -|rg\rangle )/\sqrt{2}$ state. The excitation of both atoms to the Rydberg state $|rr\rangle$ is forbidden, since the interaction energy $U\left(r\right)$ has shifted the level far off-resonant with the incident lasers.

Figure 3

The decrease in infidelity of the quantum gate for the set of initial parameters given in Table 1. One sees that the infidelity decreases monotonically until saturating in a local minimum of the optimization. The final achieved infidelity was $1-F=3.8\times {10}^{-4}$.

Figure 4

The dashed (green) line is the phase accumulation of the states $|gg\rangle$ (given by ${\varphi }_{gg}$), the dotted (blue) line is that of $|ge\rangle$ (given by ${\varphi }_{ge}$), and the solid (red) line is the total entanglement phase $\varphi$, which at the final time reaches $3.0$ (note that the dashed green line and the solid red line overlap almost exactly). The parameters used are given in Table 1. The time is in units of nanoseconds, and so the duration of the gate is 500 ns.

Figure 5

(a) The solid (red) line is the population of the state $|gg\rangle$ and the dashed (green) line is the population of the state $|{\Psi }^{+}\rangle$ over the duration of the gate. (b) Similarly to (a), the solid (red) line is the population of the state $|ge\rangle$ and the dashed (green) line is the population of the state $|re\rangle$ over the duration of the gate. The populations of $|eg\rangle$ and $|er\rangle$ are respectively the same.

Figure 6

(a) The solid (red) line is the simulated probability to excite $|ge\rangle$ to $|re\rangle$, with the experimental results for the same transition shown as a dashed (green) line. (b) Similarly, the solid (red) line is the simulated probability to excite $|gg\rangle$ to $|{\Psi }^{+}\rangle$, with the experimental results for the same transition shown as a dashed (green) line.

Figure 7

The decrease in infidelity of the quantum gate for the set of initial parameters given in Table 2. The stepwise decrease of the current minimum illustrates how restarting the simplex can result in finding another (deeper) local minimum until it reaches the (supposed) global minimum. The final achieved infidelity was $1-F=1.8\times {10}^{-2}$.

Figure 8

The dashed (green) line is the phase accumulation of the states $|gg\rangle$ (given by ${\varphi }_{gg}$), the dotted (blue) line is that of $|ge\rangle$ (given by ${\varphi }_{ge}$), and the solid (red) line is the total entanglement phase $\varphi$, which at the final time reaches $-1.0$. The parameters used are given in Table 2. The time is in units of nanoseconds, and so the duration of the gate is 1.0922 $\mu$s.

Figure 9

The pulse shape for the Rabi frequency ${\Omega }_R\left(t\right)$ for the parameters in Table 2. The shape is a “degenerated flat top,” as given in Eq. (25), i.e., a Gaussian profile.

Figure 10

The solid (red) line is the population of the state $|gg\rangle$ and the dashed (green) line is the population of the super-radiant state $|{\Psi }^{+}\rangle$ over the duration of the gate.

Figure 11

The solid (red) line is the population of the state $|ge\rangle$ and the dashed (green) line is the population of the state $|re\rangle$ over the duration of the gate. The populations of $|eg\rangle$ and $|er\rangle$ are respectively the same.

Figure 12

The effect of the motional phases of the atoms on the final gate fidelity for the parameters in Table 1. Here, the effect of the motional phases is not too large, since we perform the gate over a very short time and we do not significantly excite the atoms.

Figure 13

The effect of the motional phases of the atoms on the final gate fidelity for the parameters in Table 2. Here, the effect of the motional phases is detrimental to the gate fidelity.

Figure 14

The final fidelity of the gate plotted against the average temperature of the atoms. Each (green) point is an average over 10 000 realizations of ${\vartheta }_1$ and ${\vartheta }_2$ chosen randomly in the range $[0,\pi ]$. The (red) solid line is a linear fit to the points. The (blue) dashed line shows the current temperature of 75 $\mu$K.