State-selective electromagnetically induced transparency for quantum error correction in neutral atom quantum computers

We propose a way to measure the qubit state of an arbitrary subensemble of atoms in an array without significantly disturbing the quantum information in the unmeasured atoms. The idea is to first site-selectively transfer atoms out of the qubit basis so that one of the two states at a time is put into an auxiliary state. Electromagnetically induced transparency (EIT) light will then protect most states while detection light is scattered from atoms in the auxiliary state, which is made immune to the EIT protection by angular momentum selection rules and carefully chosen light polarization. The two states will be measured in turn, after which it is possible to recool and return the atoms to a qubit state. These measurements can be the basis of quantum error correction.

Figure 1

Overview of the measurement. (a) Initially, atoms in a 3D optical lattice are in random superpositions in the qubit basis (shown in blue). (b) A select group of atoms is transferred to states out of the qubit basis (shown in red, with a red beam going through them). (c) Qubit and nonqubit bases in the Cs ground state. The open (closed) circle dark blue states are transferred to the open (closed) circle light red states using a series of stimulated pulses. The red state population in the green dotted sublevel is detected, the other $F=4$ states are EIT protected, and the $F=3$ states are protected from light scattering by being far from resonance. (d) Another series of stimulated pulses exchanges the open and closed circle red states, allowing the other state population to be detected.

Figure 2

Scheme 1 EIT-protected detection on the $D2$ line. The imaging light uses ${\sigma }^{+}$ polarization and the EIT protection beam uses $\pi$ polarization. With this configuration all $F=4$ sublevels are EIT protected except $|4,4\rangle$, which can be imaged on the cycling transition $|4,4\rangle \to |5^{\prime },5^{\prime }\rangle$. For clarity, in this and all similar figures, we do not show the Zeeman shifts of hyperfine levels. EIT protection is quite insensitive to these shifts, as long as they are very small compared to ${\mathrm{\Omega }}_{\text{EIT}}$, which they are in the regime of interest. The wavelength of the imaging light should be chosen to match the imaging transition.

Figure 3

EIT protection of an idealized three-level system. (a) Non-ground-state population of the EIT-protected qubit state as a function of the detection beam detuning and the protection beam intensity. The red dashed line marks where the detection beam is resonant with the cycling transition. (b) Suppression factor for the reabsorption of scattered photons ($R$) as a function of intensity of the protection beam.

Figure 4

Scheme 1 simulation results. (a) Non-ground-state population of the nominally EIT-protected qubit state as a function of the detection beam detuning and the protection beam intensity. The red dashed line marks where the detection beam is resonant with the cycling transition. (b) Scattering rate in the upper qubit state as a function of time for different intensities. (c) Scattering rate of the population in the stretched state as a function of time. All the curves overlap completely since scattering on the cycling transition is unaffected by the EIT beam intensity. (d) Suppression factor for the reabsorption of scattered photons ($R$) as a function of the intensity of the protection beam. The result is proportional to the non-ground-state populations shown along the red dashed line in (a).

Figure 5

EIT-protection scheme 2. In this case both the imaging and EIT protection light are ${\sigma }^{+}$ polarized. The unprotected state is $|3,3\rangle$, with imaging on the open transition $|3,3\rangle \to |4^{\prime },4^{\prime }\rangle$. A set of stimulated pulses can be used to get the population out of the dark $|4,3\rangle$ and $|4,4\rangle$ states in order to continue imaging the atoms. These pulses can additionally drive sideband transitions to simultaneously cool the atoms.

Figure 6

Scheme 2 results. (a) Non-ground-state population of the EIT-protected qubit state $|3,0\rangle$ as a function of the detection beam detuning and the protection beam intensity. The red dashed line marks where the detection beam is resonant with the open transition for imaging. (b) Suppression factor for the reabsorption of scattered photons ($R$) as a function of intensity of the protection beam.

Figure 7

Scheme 2 possible error caused by a small amount of the wrong $\pi$ polarization in the protection beam. The $\pi$ polarization (denoted by the thin red arrow), which would otherwise resonantly couple $|4^{\prime },4^{\prime }\rangle$ to a bare $|4^{\prime \prime },4^{\prime \prime }\rangle$ (upper right gray dashed line), is off-resonant due to the strong ${\sigma }^{+}$ (denoted by the thick red arrow), which Rabi splits $|4^{\prime \prime },4^{\prime \prime }\rangle$ (upper right black solid lines).

Figure 8

Scheme 2 polarization error in EIT protection beam. (a) Varying fractional power is applied in the wrong $\pi$ polarization in the protection beams (with no magnetic field). The main source of error is population transfer $|3,3\rangle \to |3,2\rangle$ through a four-photon Raman transition. (b) Keeping the fractional power at $5\times {10}^{-4}$, a magnetic field of varying size is applied in order to reduce the error.

Figure 9

Selective transfer of an atom out of the qubit basis. The first step (1) in the sequence requires site specificity, while the remaining steps do not, as they are not resonant with atoms in the qubit states.

Figure 10

Level diagram for simplified EIT-protection systems discussed in Appendix pp2. The basic four levels (black solid lines) $|4,0\rangle , |4^{\prime },1^{\prime }\rangle , |5^{\prime },1^{\prime }\rangle$, and $|4^{\prime \prime },1^{\prime \prime }\rangle$ are discussed in Fig. 11. Figure 12 adds in the level $|3^{\prime \prime },1^{\prime \prime }\rangle$ (green dashed line). Finally, Fig. 13 adds in another level $|3^{\prime },1^{\prime }\rangle$ (brown dash-dotted line).

Figure 11

EIT protection of the four-level system with states $|4,0\rangle , |4^{\prime },1^{\prime }\rangle , |5^{\prime },1^{\prime }\rangle$, and $|4^{\prime \prime },1^{\prime \prime }\rangle$. (a) Compared to the results in Fig. 3, the new bright state has constant detuning from the resonant detection beam (red dashed line) at high protection beam intensities. (b) The addition of the new level causes the saturation of $R$.

Figure 12

EIT protection of the five-level system with the same states as the previous four-level system plus $|3^{\prime \prime },1^{\prime \prime }\rangle$. (a) The new bright state compared to Fig. 3 has a varying detuning in this case, with the positive slope causing it to cross 0 detuning (red dashed line) at around $800\text{W}/{\text{cm}}^2$. (b) The addition of the new level causes the peak of $R$ at the crossing point in (a).

Figure 13

EIT protection of the six-level system with the same states as the previous five-level system plus $|3^{\prime },1^{\prime }\rangle$. (a) There are two new bright states compared to Fig. 3. One of these two states has a constant detuning at high protection beam intensities, while the other has a positive slope and crosses 0 detuning (red dashed line) at around $800\text{W}/{\text{cm}}^2$. (b) The combination of these two new bright states cause the peak of $R$ at the crossing point in (a) and the saturation at high protection beam intensities.

Figure 14

EIT protection using a forbidden transition. The qubit state $|4,0\rangle$ is protected from the $\pi$-polarized imaging light by selection rules. All other $F=4$ states are EIT protected except $|4,4\rangle$.

Figure 15

Scheme 3 protection and leakage error results. (a) Suppression factor for the reabsorption of scattered photons ($R$) as a function of intensity of the protection beam, for $|4,0\rangle$ with $\pi$-polarized light [in red, corresponding to the level structure in Fig. 16] and ${\sigma }^{+}+{\sigma }^{-}$ light (in blue). In general, the protection will be some linear combination of these results depending on the polarization of the scattered light. (b) Leakage error from $|4,3\rangle$ [corresponding to Fig. 16] as a function of protection beam intensity.

Figure 16

Scheme 3 dressed states results. The non-ground-state population of the nominally EIT-protected states (a) $|4,0\rangle$ and (b) $|4,3\rangle$ as a function of the detection beam detuning and the protection beam intensity. The red dashed line marks where the detection beam is resonant with the cycling transition.