Two-qubit gates using adiabatic passage of the Stark-tuned Förster resonances in Rydberg atoms

We propose schemes of controlled-Z and controlled-not gates with ultracold neutral atoms based on deterministic phase accumulation during double adiabatic passage of the Stark-tuned Förster resonance of Rydberg states. The effect of deterministic phase accumulation during double adiabatic passage in a two-level quantum system has been analyzed in detail. Adiabatic rapid passage using nonlinearly chirped pulses with rectangle intensity profile has been discussed. Nonlinear time dependence of the energy detuning from the Förster resonance is used to achieve a high fidelity of population transfer between Rydberg states. Fidelity of two-qubit gates has been studied with an example of the $90S+96S\to 90P+95P$ Stark-tuned Förster resonance in Cs Rydberg atoms.

Figure 1

(a) Scheme of a cz gate using double adiabatic rapid passage across Stark-tuned Förster resonance. Two atoms are excited to Rydberg states. An external electric field shifts the energy levels of the Rydberg atoms so that the Förster resonance is passed adiabatically two times. Then the atoms are deexcited to the ground state. The phase shift is deterministically accumulated if both atoms are initially prepared in state $|1\rangle$. (b) Scheme of a cnot gate. Two additional $\pi /2$ pulses rotate the target qubit around the $y$ axis in the opposite directions.

Figure 2

Scheme of deterministic phase accumulation during a double adiabatic passage in a two-level quantum system. The phase shift is $\pi$ for the left-hand panel and zero for the right-hand panel. The dynamics of probability amplitudes $c_1\left(t\right), c_2\left(t\right), c_1^{\prime }\left(t\right), c_2^{\prime }\left(t\right)$ of states $|1\left(t\right),2\left(t\right)\rangle$ and of probability amplitudes ${\tilde{c}}_1\left(t\right), {\tilde{c}}_2\left(t\right), {\widetilde{c^{\prime }}}_1\left(t\right), {\widetilde{c^{\prime }}}_2\left(t\right)$ of semiclassical dressed states $|I\left(t\right),II\left(t\right)\rangle$ is shown schematically. (a), (b) Time dependencies of Rabi frequency $\mathrm{\Omega }\left(t\right)$ and of detuning $\delta \left(t\right)$. (c), (d) Numerically calculated time dependencies of the population of initial state $|1\left(t\right)\rangle$ compared with the calculations in the adiabatic approximation. (e), (f) Numerically calculated time dependencies of the phase of initial state $|1\left(t\right)\rangle$ compared with calculations in the adiabatic approximation.

Figure 3

Comparison between the schemes of double adiabatic rapid passage with linearly chirped Gaussian pulses (left-hand panel), and with rectangular shape of time-dependent Rabi frequency and nonlinear time dependence of detuning from the resonance (right-hand panel). (a), (b) Time dependence of Rabi frequency $\mathrm{\Omega }\left(t\right)$. (c), (d) Time dependence of detuning from the resonance $\delta \left(t\right)$. (e), (f) Time dependence of the population of state $|1\rangle$. (g), (h) Time dependence of the phase of state $|1\rangle$.

Figure 4

(a) Energy defect of the Förster resonance $\left|nS,\left(n+6\right)S\right.〉\to \left|nP,\left(n+5\right)P\right.〉$ in Cs Rydberg atoms. (b) Stark diagram for Cs Rydberg states with $\left|m_j\right|=1/2$. The $90S$ state is selected as zero energy level. (c) Scheme of possible transitions for various channels of the $\left|90S,96S\right.〉\to \left|90P,95P\right.〉$ Förster resonance in Cs. The $\mathit{nS}$ states with $m_j=1/2$ are initially excited.

Figure 5

(a) Time dependence of the electric field for Stark tuning of the $|90S_{1/2},96S_{1/2}\rangle \to |90P_{1/2},95P_{1/2}\rangle$ Förster resonance. (b) Time dependence of the energy defects of the interaction channels listed in Table 1 for the $|90S,96S\rangle \to |90P,95P\rangle$ Förster resonance. (c) Time dependence of the population of the collective state $|90S_{1/2},96S_{1/2}\rangle$. (d) Time dependence of phase of the collective state $|90S_{1/2},96S_{1/2}\rangle$.

Figure 6

Double adiabatic passage of the Stark-tuned Förster resonance for different interatomic distances $R$ with error correction. (a), (b), (c) Time dependencies of population of the collective state $|90S_{1/2},96S_{1/2}\rangle$ calculated for $R=24$, 25, and 26 $\mu \mathrm{m}$, respectively. (d), (e), (f) Time dependencies of phase of the collective state $|90S_{1/2},96S_{1/2}\rangle$ calculated for $d=24$, 25, and 26 $\mu \mathrm{m}$, respectively. (g), (h), (i) Calculated truth tables of a cnot gate for $R=24$, 25, and 26 $\mu \mathrm{m}$, respectively. The overlap with the ideal truth table is shown above each plot.

Figure 7

The reconstructed density matrices of the (a) ${\mathrm{\Phi }}^{+}$, (b) ${\mathrm{\Phi }}^{-}$, (c) ${\mathrm{\Psi }}^{+}$, and (d) ${\mathrm{\Psi }}^{-}$ Bell states.