Possibility of magic trapping of a three-level system for Rydberg blockade implementation

The Rydberg blockade mechanism has shown noteworthy promise for scalable quantum computation with neutral atoms. Both qubit states and gate-mediating Rydberg state belong to the same optically trapped atom. The trapping fields, while being essential, induce detrimental decoherence. Here we theoretically demonstrate that this Stark-induced decoherence may be completely removed using powerful concepts of magic optical traps. We analyze magic trapping of a prototype three-level system: a Rydberg state along with two qubit states: hyperfine states attached to a $J=1/2$ ground state. Our numerical results show that, while such a magic trap for alkali metals would require prohibitively large magnetic fields, the group IIIB metals such as Al are suitable candidates.

Figure 1

Stark shifts affecting the Rydberg blockade three-level system (not to scale). Qubit levels $|0\rangle$ and $|1\rangle$ are hyperfine levels attached to the electronic ground state, while $|R\rangle$ is a Rydberg state. The unperturbed levels (solid) acquire a shift from the trapping laser. (a) For an arbitrary trapping laser, each state is shifted (dashed) by a different amount according to the local laser intensity. (b) With magic trapping conditions, all levels experience the same shift (dotted), and differential phase accumulation due to the trap vanishes.

Figure 2

Relation of angles to unit vectors in Eq. (2). (a) For linear polarization, $\widehat{k}$, $\widehat{\varepsilon }$, and $\widehat{B}$ are the laser wave vector, laser polarization, and quantization axis, respectively. ${\theta }_p$ is the angle between the polarization and the quantization axis, defined by the magnetic field. (b) For circular polarization, the relevant angle is ${\theta }_k$, the angle between the wave vector and the magnetic field. (${\theta }_p$ is no longer well defined as $\widehat{\varepsilon }$ is time dependent.)

Figure 3

Polarizabilities of the $5s$ state (dashed), the Rydberg state (solid), and the ratio $B/B_m$ (lower frame) for the hyperfine transition $|F=2,M_F=1\rangle$ to $|F=1,M_F=-1\rangle$ in ${}^{87}$Rb. Since ${\alpha }_{5s}^a\ll {\alpha }_{5s}^S$, the magic $\omega$ simply occurs where ${\alpha }_{\mathrm{Ryd}}={\alpha }_{5s_{1/2}}^S$ at approximately $\omega =0.1062$ a.u. ($\lambda =429$ nm). $B/B_m$ is obtained from Eq. (4). Near the circled magic $\omega$, $B/B_m$ diverges, so magic trapping is impossible.

Figure 4

Left-hand side (solid) and right-hand side (dashed) of Eq. (8). The range for magic trapping of the Rydberg transition is given by Eq. (7) and lies between the resonance at $\omega =0.1155$ a.u. ($\lambda =394$ nm) and the dotted line at $\omega =0.121$ a.u. ($\lambda =377$ nm). The magic frequency and magic angle for the qubit transition are set by the curves’ intersection, just under $\omega =0.121$ a.u. This combination of ${\omega }_{\mathrm{magic}}$ and ${\theta }_{\mathrm{magic}}$ will allow Stark and Zeeman insensitive trapping for the three-level system in Al.