Optimization of Yb${}^{+}$ fluorescence and hyperfine-qubit detection

Fluorescence of single, trapped ${}^{171}\mathrm{Yb}$${}^{+}$ ions has been experimentally studied as a function of laser polarization, power, and detuning and as a function of magnetic field strength. The suppression of efficient fluorescence by coherent population trapping and the counteracting effect of the magnetic field are found to agree with theoretical predictions. For comparison, a fluorescence study has also been made of the isotope ${}^{174}\mathrm{Yb}$${}^{+}$ for which coherent population trapping is absent on the main fluorescence and laser cooling transition. Finally, state-sensitive fluorescence detection of the ${}^{171}\mathrm{Yb}$${}^{+}$ hyperfine qubit is studied, including the role of coherent population trapping in the optimization of detection parameters. A qubit detection fidelity of greater than $97\%$ is achieved.

Figure 1

Basic Yb${}^{+}$ atomic structure involved in fluorescence detection and cooling, including branching ratios $\alpha$ and $\beta$. Hyperfine states for ${}^{171}\mathrm{Yb}$${}^{+}$ (nuclear spin $I=1/2$) are also shown. Additional low-lying ${}^2{D}_{5/2}$ and ${}^2{F}_{7/2}$ states have been omitted for clarity.

Figure 2

Zeeman resolved structure of primary ${}^2{S}_{1/2}$-${}^2{P}_{1/2}$ transition for (a) ${}^{171}\mathrm{Yb}$${}^{+}$ and (b) ${}^{174}\mathrm{Yb}$${}^{+}$. Zeeman splittings due to magnetic field $B$ are indicated in terms of ${\delta }_B={\mu }_{B}B/\hslash$. Laser detuning $\Delta$ is defined relative to the $B=0$ transitions as shown. The polarization-resolved branching ratios for ${}^{171}\mathrm{Yb}$${}^{+}$ are all 1/3 while those for ${}^{174}\mathrm{Yb}$${}^{+}$ are 1/3 and 2/3 for $\pi$ and $\sigma$ transitions respectively. The UV hyperfine repump transition and the optical pumping transition used for qubit initialization to ${}^2{S}_{1/2}|F=0,m_F=0\rangle$ are also shown for the case of ${}^{171}\mathrm{Yb}$${}^{+}$.

Figure 3

Schematic of essential experimental details for Doppler laser-cooling and fluorescence detection of trapped Yb${}^{+}$ ions. The primary 369.5-nm transition is driven by a frequency-doubled 739-nm diode laser system, frequency narrowed and long-term stabilized to molecular iodine. About 200 $\mu$W of the fiber-coupled UV power is directed through two free-space resonant EOMs for UV hyperfine repump at 7.37 GHz (second sideband) and hyperfine-qubit initialization at 2.105 GHz. An 80-MHz AOM directs $~60$ $\mu$W to the trap while the zero order is used as a power drift monitor. Before being focused into the trap, the 369.5-nm beam has its vertical polarization rotated as shown by angle ${\theta }_E=2{\theta }_{\lambda /2}$ as it passes through a half-wave plate oriented at ${\theta }_{\lambda /2}$ from the vertical. The 369.5-nm laser is focused into the trap along the axis defined to be $\mathrm{x̂}$. The 935.2-nm repump transition is driven by a stabilized diode laser which is fiber-coupled into a broadband fiber modulator for hyperfine repumping and passes through a half-wave plate before being focused into the vacuum chamber. A single magnetic coil provides a magnetic bias field with direction defined by spherical coordinates ${\theta }_B$ and ${\varphi }_B$. The field is nominally directed along the $\mathrm{ŷ}$ axis. The inset shows the orientation of the linear Paul trap within the vacuum chamber.

Figure 4

Calibration of magnetic field at trapped ion’s location using microwave spectroscopy of ${}^{171}\mathrm{Yb}$${}^{+}$. Shown are measured transitions from ground hyperfine state $|F=0,m_F=0\rangle$ to $|F=1,m_F=-1\rangle$ (triangles), $|F=1,m_F=0\rangle$ (circles), and $|F=1,m_F=+1\rangle$ (squares) as a function of bias coil current $I_{\mathrm{coil}}$. Vertical axis is referenced to inferred resonance frequency ${\nu }_0=12\hspace{0.5em}642.8$ MHz at zero magnetic field. Solid lines are a fit to Breit-Rabi theory with three magnetic-field parameters, (i) current-dependent Zeeman shift 3.991(2) MHz/A corresponding to $\mathit{dB}/\mathit{dI}=2.851(1)$ G/A, (ii) offset current $I_o=-0.344(1)$ A, and (iii) minimum Zeeman splitting 0.952(7) MHz. The last two parameters characterize a constant background field of about 1 G. The fit lines match the data to $\pm 10$ kHz.

Figure 5

Scan of UV laser frequency to locate resonance of a single trapped ${}^{171}\mathrm{Yb}$${}^{+}$ ion. The frequency-doubled laser is scanned with the fundamental locked to iodine by tuning lock offset frequency $\nu$ relative to resonance ${\nu }_{0,171}$. Data shown are taken at ${\delta }_B/2\pi =8.19$ MHz, $s_0=1.590(4)$, and ${\theta }_{\mathit{BE}}=57.5^{°}$. The line is a fit to a Lorentzian with exponential suppression above resonance. Residuals are shown in the top panel. Fit parameters are a peak amplitude coefficient, lock offset on resonance ${\nu }_{0,171}=10\hspace{0.5em}193.00(5)$ MHz, and linewidth 24.6(2) MHz, where errors include sensitivity to initial guesses. The horizontal axis is recalculated using the fit value of ${\nu }_{0,171}$.

Figure 6

(a) Fluorescence counts for ${}^{171}\mathrm{Yb}$${}^{+}$ versus 369.5-nm laser-polarization angle ${\theta }_E$ at four different magnetic bias coil currents {$-0.343$,0.805,1.695,2.505}, A corresponding to Zeeman shifts ${\delta }_B/2\pi =\{0.95,4.67,8.17,11.39\}$ MHz. Laser powers at 369.5 nm are {1.4, 6.5, 10.1, 15.6} $\mu$W respectively. Detuning is $-8.0$ MHz for all data. Counts are recorded in 1-s intervals, averaged three times. Error bars, which are smaller than data markers, are the quadrature sum of Poissonian error and $0.4^{°}$ uncertainty in ${\theta }_E$ transferred to the vertical axis through slope interpolation. Lines show a global weighted fit to all currents with four parameters to account for dependence of magnetic field orientation on coil current and residual background field, three amplitude coefficients with independent values for 0.8 A and $-0.34$ A plots, and a single saturation power 1.262(8) $\mu$W for all data. The fit to 206 points, which excludes select data (see text), gives a reduced ${\chi }^2$ of 0.97. (b) Location ${\theta }_{E,\min }$ of the sharp minimum as a function of magnetic bias coil current. Inset shows a magnified scale in the range 0.8–2.5 A used in subsequent data. The line shows the prediction obtained from the global fit in (a).

Figure 7

${}^{171}\mathrm{Yb}$${}^{+}$ fluorescence versus UV laser power and Zeeman shift ${\delta }_B$. The data are taken with UV detuning $-8.0$ MHz and laser-polarization angle ${\theta }_E=41.8^{°}$, which is near-optimal linear polarization for all data. The lines are a single weighted global fit to all data. Error bars are suppressed for clarity. The reduced ${\chi }^2$ is 1.05 for 1496 data points. Fit parameters are saturation power 1.226(1) $\mu$W, and the amplitude coefficient is 3645(2). See text and Fig. 8 for further quantitative assessment of fit.

Figure 8

(a) Contour plot of ${}^{171}\mathrm{Yb}$${}^{+}$ fluorescence counts as a function of power and magnetic field (same data set as Fig. 7). Contour scale is shown at top. The solid line is the approximate expression for power at peak fluorescence [Eq. (5)]. Arrows and horizontal dotted line indicate location of cross sections shown in (b) at three different Zeeman splittings. (b) Counts in 10 ms as a function of laser power at three Zeeman splittings ${\delta }_B/2\pi =\{11.41,8.19,4.68\}$ MHz corresponding to coil currents {2.50,1.69,0.80} A. Errors, determined from repeated measurements, are the quadrature sum of Poissonian and $2\%$ fractional error before averaging. Lines are a single weighted global fit as shown in Fig. 7. Residuals shown for each Zeeman splitting. Drop lines indicate expected peak count rate and location according to Eqs. (5) and (6) and the values of the global fit parameters.

Figure 9

(a) Scan of UV laser frequency to locate resonance of single trapped ${}^{174}\mathrm{Yb}$${}^{+}$ ion. The frequency-doubled laser is scanned with the fundamental locked to iodine by tuning lock offset-frequency $\nu$. The parameters for the scan are ${\delta }_B/2\pi =8.19$ MHz, ${\theta }_{\mathit{BE}}=2.8^{°}$, and saturation parameter $s_0=0.63$. The solid line is a fit to a Lorentzian with exponential suppression above resonance. The fit yields a resonance location ${\nu }_{0,174}=8975.8(1)$ MHz and a linewidth of 24.6(1) MHz, including all broadening terms. The horizontal axis is offset and scaled using the fit value of ${\nu }_{0,174}$ to indicate UV detuning. (b) Scan of UV laser frequency to locate resonance of ${}^{174}\mathrm{Yb}$${}^{+}$ in the presence of sympathetic cooling by simultaneously trapped ${}^{172}\mathrm{Yb}$${}^{+}$ ion. The horizontal scale for the scan has been shifted by 2 MHz to account for a measured ac Stark shift from the ${}^{172}\mathrm{Yb}$${}^{+}$ cooling laser. Other parameters are similar to (a). Fit resonance position is slightly shifted by 0.39(5) MHz relative to resonance in (a), attributed to systematic error in ac Stark shift determination. The fit linewidth is 25.5(2) MHz. The fit includes a baseline $-7.6(6)$ to account for drifts in background ${}^{172}\mathrm{Yb}$${}^{+}$ fluorescence.

Figure 10

${}^{174}\mathrm{Yb}$${}^{+}$ resonance extracted from laser power scans at ${\delta }_B/2\pi =8.19$ MHz and ${\theta }_{\mathit{BE}}=2.8^{°}$. (a) Counts in 10 ms, averaged 10 times, are shown as a function of 369.5-nm laser power at the range of detunings indicated. Detunings are determined from resonance location as shown in Fig. 9. Errors, determined from repeated measurements, are the quadrature sum of Poissonian and $2\%$ fractional error before averaging. Each line is a separate weighted fit to a scan at a given detuning. Fit function is a general saturation form (see text) with two fit parameters, amplitude coefficient and saturation-related factor $\mathit{ds}/\mathit{dp}$. Reduced ${\chi }^2$ for various fits is generally good, on average 1.0(2), except for two points near resonance. (b) Plot of fit values $\mathit{ds}/\mathit{dp}$ obtained as a function of laser detuning. The line is an unweighted fit to Eq. (10) with fixed zero baseline and fixed resonance position. Two points near the peak are excluded (see text). Residuals shown above main plot. The natural linewidth from the fit is $\gamma /2\pi =19.6(1)$ MHz and saturation power is $p_{\mathrm{sat}}=1.25(1)$ $\mu$W. If two points at peak are included, these values change to 20.1(2) MHz and 1.28(1) $\mu$W, respectively.

Figure 11

${}^{174}\mathrm{Yb}$${}^{+}$ fluorescence as a function of 369.5-nm laser power and linear polarization angle ${\theta }_E$. Zeeman shift is ${\delta }_B/2\pi =11.4$ MHz ($I_{\mathrm{coil}}=2.5$ A). Laser detuning is fixed at $\Delta /2\pi =-8.4$ MHz. Data consist of fifty-four 68-point scans of laser power at half-wave plate positions over a ${240}^{°}$ range. Fluorescence photons are counted by a PMT in a 10-ms integration time, averaged 10 times. Error bars suppressed for clarity. The lines are a global weighted fit to theory in Eq. (9) with two free parameters, an amplitude coefficient 3660(2) related to photon collection efficiency, and saturation power $p_{\mathrm{sat}}=1.247(2)$ $\mu$W. Other parameters are fixed, including magnetic field angles as per calibration in Fig. 6. The reduced ${\chi }^2$ of 1.39 for 3672 total points, while not acceptable at a reasonable confidence level, is sufficiently low to give reasonable parameter errors. See text and Fig. 12 for detailed quantitative assessment of this data and fit.

Figure 12

(a) Contour plot of ${}^{174}\mathrm{Yb}$${}^{+}$ polarization data from Fig. 11. Lines and arrows indicate location of data cross sections in (b) at three laser powers (i) 1.1 $\mu$W, (ii) 10.5 $\mu$W, and (iii) 66.6 $\mu$W corresponding to saturation parameter $s_0$ of 0.9, 8.4, and 53.4 respectively. (b) Cross sections of fluorescence-count data as a function of laser-polarization angle ${\theta }_E$ at three laser powers mentioned above. The count rates are nomarlized to the maximum value of the fit model, occuring at ${\theta }_E=97.2^{°}$, corresponding to ${\theta }_{\mathit{BE}}=0^{°}$. Solid lines are global fits from Fig. 11. (c) Alternative polarization analysis technique using separate weighted fits of each laser power scan in Fig. 11 to a generalized saturation form to extract amplitude coefficient and saturation-related parameter $\mathit{ds}/\mathit{dp}$. Fit values of $\mathit{ds}/\mathit{dp}$ are plotted as a function of polarization angle ${\theta }_E$. Error bars are the statistical errors from the fits. Solid line is an unweighted fit to ${}^{174}\mathrm{Yb}$${}^{+}$ theory in Eq. (9) with one free parameter $p_{\mathrm{sat}}=1.247(3)$ $\mu$W. Fit residuals, showing reasonably random scatter, are included above the main plot.

Figure 13

${}^{174}\mathrm{Yb}$${}^{+}$ fluorescence versus UV laser power at three magnetic fields, and comparison to ${}^{171}\mathrm{Yb}$${}^{+}$ behavior. Fluorescence counts are collected in 10-ms intervals, averaged 10 times. The three Zeeman shifts ${\delta }_B/2\pi =\{4.68,8.19,11.41\}$ MHz as indicated correspond to bias coil currents {0.80,1.69,2.50} A. The three ${}^{174}\mathrm{Yb}$${}^{+}$ scans are taken at near-optimal linear polarization ${\theta }_E=97.8^{°}({\theta }_{\mathit{BE}}=2.8^{°})$ and with laser detuning $-8.4$ MHz. The three ${}^{171}\mathrm{Yb}$${}^{+}$ scans, reproduced from Fig. 8, are taken at near-optimal linear polarization ${\theta }_E=41.8^{°}({\theta }_{\mathit{BE}}=57.5^{°})$ and at UV detuning of $-8.0$ MHz. Solid lines for ${}^{171}\mathrm{Yb}$${}^{+}$ are global fits also reproduced from Fig. 8. Solid lines for ${}^{174}\mathrm{Yb}$${}^{+}$ are a global fit to a total of five power scans (340 points) at the three magnetic field values. The fit gives a reduced ${\chi }^2$ of 1.07. The two fit parameters are amplitude coefficient 3638(5) and saturation power $p_{\mathrm{sat}}=1.264(4)$ $\mu$W. Fit residuals for ${}^{174}\mathrm{Yb}$${}^{+}$ are shown in the top panels.

Figure 14

Histograms of photon counts for a ${}^{171}\mathrm{Yb}$${}^{+}$ ion initially prepared in the “bright” $|\uparrow \rangle$ state or “dark” $|\downarrow \rangle$ state at two different detection laser powers, [(a) and (b)] 2.23 $\mu$W and [(c) and (d)] 8.54 $\mu$W, corresponding to a saturation parameter $s_0$ of 2.2 and 8.6 respectively. [(a) and (b)] For near-optimal power and best detection fidelity; [(c) and (d)] at a power larger than optimal. Other parameters are Zeeman splitting ${\delta }_B/2\pi =8.19$ MHz, laser detuning $-4.1$ MHz, laser polarization ${\theta }_{\mathit{BE}}=57.4^{°}$, detection time 0.4 ms, and photon collection efficiency $\eta =2.9\times {10}^{-3}$. Bars are experimental data. Circles are a theoretical prediction with no free parameters that incorporates separate measurements of $|\uparrow \rangle$ preparation fidelity, and bright-state leakage rate and background count rate as functions of detection laser power.

Figure 15

${}^{171}\mathrm{Yb}$${}^{+}$ hyperfine-qubit detection error. (a) Measured detection error of “bright” $|\uparrow \rangle$ state (solid circles) and “dark” $|\downarrow \rangle$ state (open circles) obtained from 400 repeated experiments. Parameters are the same as in Fig. 14. Discriminator is set to optimal value $n_o=1$ ($|\uparrow \rangle$ is nominally identified with $n>n_o$ and $|\downarrow \rangle$ with $n⩽n_o$). The solid lines are a theoretical prediction with no free parameters that incorporates measured $|\uparrow \rangle$ preparation fidelity and measurements of bright-state leakage rate and background count rate as functions of detection laser power. The dashed line is ideal prediction for $|\downarrow \rangle$ detection ignoring background counts. The dash-dot line is ideal prediction for $|\uparrow \rangle$ detection using calculated bright-state leakage rate from Eq. (11). (b) The solid squares are average counts during 0.4 ms detection of $|\uparrow \rangle$ shown in (a). The superimposed line is fluorescence theory, including effect of measured bright state leakage rate. For comparison, open squares show steady-state fluorescence counts with hyperfine repump activated, similar to those in Fig. 8. The count values have been scaled to match 0.4-ms detection time and power axis has been scaled by factor 1.28 to account for loss of power to repump sidebands. Value of $p_{\mathrm{sat}}$ for curves shown is 0.99 $\mu$W.