Reversal of Photon-Scattering Errors in Atomic Qubits

Spontaneous photon scattering by an atomic qubit is a notable example of environment-induced error and is a fundamental limit to the fidelity of quantum operations. In the scattering process, the qubit loses its distinctive and coherent character owing to its entanglement with the photon. Using a single trapped ion, we show that by utilizing the information carried by the photon, we are able to coherently reverse this process and correct for the scattering error. We further used quantum process tomography to characterize the photon-scattering error and its correction scheme and demonstrate a correction fidelity greater than 85% whenever a photon was measured.

Figure 1

A schematic layout of our experiment. (a) Scheme of the apparatus. A single Strontium ion is trapped and laser cooled in a linear Paul trap. A weak $\pi$ polarized beam, resonant with the $S_{1/2}\mathrm{\text{−}}{P}_{1/2}$ transition, excites the ion which then spontaneously decays by emitting a photon. A photon collection apparatus consisting of a 0.31 N. A. objective lens, wave-plates, PBS and PMTs measures the polarization of photons that are emitted along the x direction. In addition, a TAC records the time interval between the rising edges of the LO and the PMT when a photon is detected. (b) A diagram of the relevant energy levels in the ${}^{88}\mathrm{Sr}^{+}$ ion. (c) Experiment sequence. The spin is prepared in some state by optical pumping and rf spin rotation; then a photon is scattered by a weak resonant beam and the spin state is measured by another spin rotation, followed by shelving and state-selective fluorescence.

Figure 2

Process tomography of the ion spin at the different stages of the experiment. The results are presented by the surface onto which all pure states are mapped and the real part of the reconstructed process matrix (the imaginary part is negligible and is not shown). (a) Without photon scattering. From the process matrix we infer qubit preparation and detection fidelity of 0.97. (b) After post-selection of single photon scattering in the $\widehat{x}$ direction, photon scattering collapses the Bloch sphere of pure states onto a pancake-shaped spheroid in the equatorial plane. The process overlap with the ideal identity operation is reduced to 50%. (c) and (d) Separating the measured result in (b), to $|V⟩$ (Rayleigh) and $|H⟩$ (Raman) photon detection events. Rayleigh scattering events leave the Bloch sphere almost intact, whereas Raman events swap between the sphere poles, while eliminating its width in the equatorial plane almost completely. Here, the loss of coherence is attributed to the random time of photon scattering events. (e) After applying a correction pulse that depends on $|H⟩$ photons detection and with a phase given by ${\varphi }_{\mathrm{TAC}}$, the resulting overlap with the identity operation increases to 83%.

Figure 3

A Ramsey experiment on the ion spin with single-photon scattering. (a) Measuring the photon polarization in the basis $|H⟩$, $|V⟩$, Probability to detect the spin in the $\mid \uparrow ⟩$ state vs the TAC phase, for Rayleigh (blue) and Raman (red) scattering events separately. The probability of measuring $\mid \uparrow ⟩$ is independent of ${\varphi }_{\mathrm{TAC}}$ for Rayleigh scattering, whereas for Raman scattering, the $\pi$ rotation around an axis that rotates in the Bloch sphere equatorial plane, produces a double fringe. (b) Measuring the photon polarization in the basis $|H^{\prime }⟩=(|H⟩+|V⟩)/\sqrt{2}$ and $|V^{\prime }⟩=(|H⟩-|V⟩)/\sqrt{2}$. Here the $\pm \pi /2$ rotation due to the photon scattering in the two polarization $|V^{\prime }⟩/|H^{\prime }⟩$ (blue/red) substitute the first Ramsey pulse. The small phase shift (0.5 rad on top the $\pi$) is due to uncompensated birefringence in our polarization analysis setup.