Raman sideband cooling of a ${}^{138}\mathrm{Ba}^{+}$ ion using a Zeeman interval

Motional ground state cooling and internal state preparation are important elements for quantum logic spectroscopy (QLS), a class of quantum information processing. Since QLS does not require the high gate fidelities usually associated with quantum computation and quantum simulation, it is possible to make simplifying choices in ion species and quantum protocols at the expense of some fidelity. Here, we report sideband cooling and motional state detection protocols for ${}^{138}\mathrm{Ba}^{+}$ of sufficient fidelity for QLS without an extremely narrow-band laser or the use of a species with hyperfine structure. We use the two $S_{1/2}$ Zeeman sublevels of ${}^{138}\mathrm{Ba}^{+}$ to Raman sideband cool a single ion to the motional ground state. Because of the small Zeeman splitting, continuous near-resonant Raman sideband cooling of ${}^{138}\mathrm{Ba}^{+}$ requires only the Doppler cooling lasers and two additional acousto-optic modulators. Observing the near-resonant Raman optical pumping fluorescence, we extract relevant experimental parameters and demonstrate a final average motional quantum number $\overline{n}\ll 1$. We additionally employ a second, far-off-resonant laser driving Raman $\pi$ pulses between the two Zeeman sublevels to provide motional state detection for QLS and to confirm the sideband cooling efficiency, measuring a final $\overline{n}=0.15\left(6\right)$.

Figure 1

(a) Relevant transitions in ${}^{138}\mathrm{Ba}^{+}$. All laser sources except the 455 nm free-running laser diode are ECDL systems locked by wavemeter. Each laser source is shuttered via AOM and mechanical shutter with only the former used when fast timing is required. (b) Apparatus and beam geometry. Note that the magnetic field is parallel to the ${\sigma }^{+}$-polarized Raman pump beam and perpendicular to the $\pi$-polarized Raman probe beam and that $\mathrm{\Delta }\vec{k}$ lies along the trap $z$ axis. Inset: an intensified CCD image of a single fluorescing ${}^{138}\mathrm{Ba}^{+}$ in the ion trap.

Figure 2

AOM setup for the near- and far-off-resonant 493 nm beams. An amplified photodiode (PD) and servo circuit control the laser power to the double-pass Raman AOMs. All AOM frequencies are $\sim 80$ MHz. A set of flip mirrors (not shown) couples light from before the Doppler $\left(-1x\right)$ AOM to the polarization-maintaining single-mode fiber (PMF) to the far-off-resonant $\left(+1x\right)$ AOM allowing alignment of the far-off-resonant beam path using resonant light.

Figure 3

Stages of ${}^{138}\mathrm{Ba}^{+}$ cooling. Solid lines show applied laser drives. Horizontal dashed lines show the red and blue motional sidebands of the driven Raman transition when relevant and the virtual state of the Raman transition (black), detuned by $\approx 80$ MHz. (a) Doppler cooling lasts 20 ms. (b) Near-resonant Raman sideband cooling lasts either 1 or 10 ms. The PMT is gated with the near-resonant Raman beams.

Figure 4

Near-resonant Raman 493 nm fluorescence spectrum with red and blue motional sidebands resolved. The red circles (blue diamonds, $y$-axis offset for display) are data corresponding to PMT exposure for the first 1 (10) ms of near-resonant Raman exposure. Solid lines are Fano profile fits. Note that the RSB amplitude after 10 ms of cooling is small, indicating that $\overline{n}\ll 1$. Error bars are given by the standard deviation of the mean of each data point, and are comparable to the marker size.

Figure 5

Stages of ${}^{138}\mathrm{Ba}^{+}$ far-off-resonant Raman state detection that occur after near-resonant Raman cooling steps (a) and (b) in Fig. 3. Solid arrows show applied laser drives, and dashed arrows indicate dominant spontaneous emission channels with corresponding branching fractions. Horizontal dashed lines show the red and blue motional sidebands of the driven Raman transition when relevant and the virtual state of the Raman transition (black), detuned by $\approx 80$ MHz ($\approx 59$ GHz) for the near-resonant (far-off-resonant) Raman beams. (c) Optically pump to the $m_j=+\frac{1}{2}$ state. (d) Far-off-resonant Raman $\pi$ pulse, conditional on motional state. (e) Protect population in $m_j=-\frac{1}{2}$ by pumping to $D_{3/2}$. (f) Shelve any population remaining in $S_{1/2}$ to $D_{5/2}$. (g) Determination of $D_{3/2}$ versus $D_{5/2}$ ion state by observing fluorescence with the PMT. (h) Deshelve any population in $D_{5/2}$ and additional Doppler cooling (not shown).

Figure 6

Far-off-resonant Raman excitation Rabi oscillations on (a) the carrier and (b) the BSB. Multiple Rabi oscillations are observed on the carrier with ${\mathrm{\Omega }}_{\text{eff,carrier}}=2\pi \times 50.92\left(3\right)$ kHz with a coherence time of 130(5) $\mu \mathrm{s}$. An efficient $\pi$ pulse on the BSB cannot be performed due to the coherence time of 134(11) $\mu \mathrm{s}$ being comparable to the $\pi$ pulse time with ${\mathrm{\Omega }}_{\text{eff,BSB}}=2\pi \times 6.02\left(8\right)$ kHz. Data points in each data set were taken randomly and interleaved.

Figure 7

Far-off-resonant Raman frequency spectrum showing (a) the full spectrum and (b) zoom-in of the motional sidebands. A $\pi$ pulse is used on the carrier. The longer sideband $\pi$ pulse is used on the RSB and BSB as well as the background points that help the fit establish the background shelving level. Near equal blue and red sideband amplitudes are observed if no near-resonant Raman cooling is performed [red diamonds, $y$ axis offset in (a) for display]. The blue circles correspond to 10 ms of near-resonant Raman cooling on the red sideband with an observed $\overline{n}=0.15\left(6\right)$. Data points in each data set were taken randomly and interleaved. Error bars are given by binomial statistics. The slight shift in carrier frequency between the traces is due to magnetic field drift changing the Zeeman splitting of the $S_{1/2}$ state.

Figure 8

Near-resonant Raman frequency scan around the RSB while measuring the RSB/BSB amplitude ratio (related to ion temperature) with 10 ms of sideband cooling. The minimum amplitude ratio is reached while sitting on the peak of the red sideband measured by the near-resonant Raman fluorescence. Data points were taken randomly and interleaved. Error bars are given by the propagation of error using binomial statistics for the red and blue sideband amplitudes.

Figure 9

Determination of optimal optical pumping time for the step shown in Fig. 5. Far-off-resonant $\pi$ pulse off (red diamonds). Far-off-resonant $\pi$ pulse on (blue circles). We see that the minimum optical pumping time for minimum (maximum) bright fraction with the far-off-resonant $\pi$ pulse off (on) is $\approx 10 \mu \mathrm{s}$. Data points in each data set were taken randomly and interleaved. Error bars are given by binomial statistics.

Figure 10

Determination of optimal protect pulse time for the step shown in Fig. 5. The detection contrast (black squares) is calculated as the difference between the experiment without the far-off-resonant Raman $\pi$ pulse (red diamonds) and with the far-off-resonant $\pi$ pulse (blue circles). The detection contrast is maximum at a protect pulse time of $\approx 10 \mu \mathrm{s}$ as expected from the minimum optical pumping time determined in Appendix pp1. Data points in each data set were taken randomly and interleaved. Error bars are given by binomial statistics.