Errors in trapped-ion quantum gates due to spontaneous photon scattering

We analyze the error in trapped-ion, hyperfine qubit, quantum gates due to spontaneous scattering of photons from the gate laser beams. We investigate single-qubit rotations that are based on stimulated Raman transitions and two-qubit entangling phase gates that are based on spin-dependent optical dipole forces. This error is compared between different ion species currently being investigated as possible quantum-information carriers. For both gate types we show that with attainable laser powers the scattering error can be reduced to below current estimates of the fault-tolerance error threshold.

Figure 1

Relevant energy levels (not to scale) in an ion qubit, with nuclear spin $I$. The $P_{1∕2}$ and $P_{3∕2}$ excited levels are separated by an angular frequency ${\omega }_{\mathrm{f}}$. The $S_{1∕2}$ electronic ground state consists of two hyperfine levels $F=I-1∕2$ and $F=I+1∕2$. The relative energies of these two levels depends on the sign of the hyperfine constant $A_{\mathrm{hf}}$ and can vary between ion species (in this figure, $A_{\mathrm{hf}}$ is negative). The qubit is encoded in the pair of $m_F=0$ states of the two $F$ manifolds separated by an angular frequency ${\omega }_0$. Coherent manipulations of the qubit levels are performed with a pair of laser beams that are detuned by $\Delta$ from the transition to the $P_{1∕2}$ level, represented by the two straight arrows. The angular frequency difference between the two beams equals the angular frequency separation between the qubit levels ${\omega }_{\mathrm{b}}-{\omega }_{\mathrm{r}}={\omega }_0$. Some ion species have $D$ levels with energies below the $P$ manifold. Wavy arrows illustrate examples of Raman scattering events.

Figure 2

The solid line is the probability $(P_{\mathrm{SE}}=P_{\text{Raman}})$ to scatter a Raman photon (in units of $\gamma ∕{\omega }_f$) during a $\pi$ rotation vs the laser detuning in units of the excited-state fine-structure splitting. The dashed line is the probability for any type of scattering event $(P_{\mathrm{SE}}=P_{\text{total}})$ during the pulse vs detuning. The Raman scattering probability decays quadratically with $\Delta$ for $\mid \Delta \mid ⪢{\omega }_f$.

Figure 3

Laser power in each of the Raman beams vs the error in a single-ion gate ($\pi$ rotation) due to Raman scattering back into the $S_{1∕2}$ manifold [obtained using Eq. (18)]. Different lines correspond to different ion species (see legend). Here we assume Gaussian beams with $w_0=20\mu \mathrm{m}$ and a Rabi frequency ${\Omega }_R∕2\pi =0.25\mathrm{MHz}$ $({\tau }_{\pi }=1\mu \mathrm{s})$.

Figure 4

Schematic of Raman laser beam geometry assumed for the two-qubit phase gate. The gate is driven by two Raman fields, each generated by a Raman beam pair. Each pair consists of two perpendicular beams of different frequencies that intersect at the position of the ions such that the difference in their wave vector lies parallel to the trap axis. One beam of each pair is parallel to the magnetic field which sets the quantization direction. The beams’ polarizations in each pair are assumed to be linear, perpendicular to each other and to the magnetic field. Wavy arrows illustrate examples of photon scattering directions.

Figure 5

Laser power in each of the Raman beams vs the error in a two ion entangling gate due to Raman scattering back into the ${\mathcal{S}}_{1∕2}$ manifold [obtained using Eq. (27)]. Different lines correspond to different ion species (see legend). Here we assume Gaussian beams with $w_0=20\mu \mathrm{m}$, a gate time ${\tau }_{\text{gate}}=10\mu \mathrm{s}$, ${\omega }_{\text{trap}}∕2\pi =5\mathrm{MHz}$, and a single circle in phase space $(K=1)$.

Figure 6

Schematics of the trajectories traversed by the ions in phase space during the gate. The acquired phase is proportional to the encircled area. (a) Phase-space trajectory of an ideal gate. (b) Phase-space trajectory of an erroneous gate where a photon was scattered during the gate drive. The short straight arrows represent the recoil displacement. The gray-shaded area is proportional to the phase error. The area that is added to the ∣↑↓⟩ trajectory is subtracted from the ∣↓↑⟩ trajectory.