Ytterbium Nuclear-Spin Qubits in an Optical Tweezer Array (2024)

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  Figure 1

  ${}^{171}\mathrm{Yb}$ optical tweezer arrays. (a)${}^{171}\mathrm{Yb}$ atoms are trapped in an array of 100 optical tweezer sites. (b)The nuclear spin $I=1/2$ is an environmentally well-isolated two-level system. The level diagram shows the imaging, cooling, and enhanced loading transitions. Light scattered from the 399nm ${{}^{1}S}_0\to {}^1{P}_1$ transition is collected with the high-NA objective. While imaging, we apply cooling beams red detuned by ${\mathrm{\Delta }}_C/(2\pi )=-2.04(5)\mathrm{MHz}$ ($11{\mathrm{\Gamma }}_g$) from the light-shifted ${{}^{1}S}_0\to {}^3{P}_1$ $|F^{\prime }=3/2,m_{F^{\prime }}=\pm 1/2⟩$ transition. The inset shows driven Rabi oscillations of the nuclear spin with submicrosecond $\pi$ times. (c)Histograms of collected photons over all tweezer sites for 120ms exposures under two different loading schemes—enhanced loading (black line) and standard (zero-field, red-detuned light) loading (gray line). The dashed red line is the threshold that we use to define detection of an atom in the enhanced loading case. (d)Average of 500 images of the tweezer array. (e)Single-shot image with near-deterministic tweezer array loading.

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  Figure 2

  Near-deterministic loading. (a)Loading efficiency ($P_{\text{load}}$) dependence on applied magnetic field and loading beams detuning (${\mathrm{\Delta }}_L$) from the free-space resonance. The star denotes the parameters chosen for daily operation: magnetic field of 4.9G and ${\mathrm{\Delta }}_L/2\pi =+9.76\mathrm{MHz}$. The intensity of the loading beams changes as a function of ${\mathrm{\Delta }}_L$, in the starred region corresponding to $515I_{\mathrm{sat}}$, where $I_{\mathrm{sat}}=0.138\mathrm{mW}/{\mathrm{cm}}^2$ is the saturation intensity of the ${{}^{1}S}_0\leftrightarrow {{}^{3}P}_1$ transition. The resonance frequencies of the transitions to the $m_{F^{\prime }}$ states are plotted as a function of the magnetic field. (b)Levels involved in the loading scheme for parameter space denoted with the star in (a). Accounting for tweezer light and magnetic shifts, the loading beams are blue detuned from $m_{F^{\prime }}=+1/2$ by ${\delta }_L=+2\pi \times 2.8\mathrm{MHz}$ ($15.6\mathrm{\Gamma }$). (c)Probability of obtaining a given fill fraction for a single-shot image. (d)Optimal average loading efficiency ($P_{\text{load}}$) as a function of the tweezer array size. The error bars correspond to standard deviations of the binomial distributions given by the measured probabilities.

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  Figure 3

  Nuclear-spin control. (a)The Raman level diagram corresponding to the weak-drive regime. Two copropagating beams drive Raman transitions between the nuclear-spin states. The two beams have variable relative frequency, one with $\pi$ polarization and the other with polarization normal to the atom plane (both ${\sigma }_{+}$ and ${\sigma }_{-}$ polarizations). The rotations are performed through two excited states $|F^{\prime }=3/2,m_{F^{\prime }}=\pm 1/2⟩$ but in the weak-driving regime one of these Raman transitions (dashed lines) is far off resonance compared to ${\mathrm{\Omega }}_X$ and effectively does not contribute. The detuning from the $F^{\prime }=1/2$ states is much larger than ${\mathrm{\Delta }}_X$ and these states do not significantly affect the dynamics. (b)As we sweep the detuning between the Raman beams, the resonances corresponding to the two Raman pathways become visible. (c)Fixing the frequency to the left resonance, we drive Rabi oscillations at the kilohertz scale. The plotted probabilities of detecting the $|1⟩$ state are normalized by the measured atom survival probability for the experiments in the absence of blow-away, which is 96.5(2)% for both (b) and (c); see Appendixpp3. (d)In the strong-driving regime, a single $X$ beam with linear polarization ${\mathit{\epsilon }}_x$ tilted out of the atom plane drives two Raman transitions between the spin states. A second beam, $Z$, splits the nuclear-spin states and drives oscillations around the $Z$ axis. A small magnetic field $\mathbf{B}$ splits the nuclear-spin states by 1.25kHz. (e)Level diagram for $X$ and $Z$ beams. The beam detunings are ${\mathrm{\Delta }}_X/(2\pi )\simeq -180\mathrm{MHz}$ and ${\mathrm{\Delta }}_Z/(2\pi )=-164\mathrm{MHz}$, while the excited state splittings are much smaller, ${\mathrm{\Delta }}_e/(2\pi )\simeq 2\mathrm{MHz}$. (f)The top panel shows $X$ Rabi oscillations at 1.77MHz. In the bottom panel, we measure $Z$ oscillations at 0.77MHz with a Ramsey-type sequence, where the $Z$ beam is turned on for variable time between two $X(\pi /2)$ pulses. The survival probability for these experiments without blow-away is 96.3(1)%. The accompanying Bloch spheres show example trajectories for the two types of experiments. (g)Clifford randomized benchmarking using Clifford gates compiled of $X(\pi /2)$ and $Z(\pi /2)$. The target output state is randomized between $|0⟩$ and $|1⟩$. The black circles show the measured probabilities of obtaining the target state at a given Clifford gate depth. At each depth, we run 40 different sets of randomized experiments. The error bars are given by the standard deviation over the 40 sets of experiments. The extracted average gate infidelity per Clifford gate is $5.2(5)\times {10}^{-3}$. The red circles show the simulated success probabilities using the estimated scattering error rates of the $X$ and $Z$ beams and shot-to-shot fractional intensity noise of 1%, with 3% fractional intensity noise from experiment to experiment (Appendixpp4).

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  Figure 4

  Qubit coherence. (a)Ramsey experiment. Population of state $|1⟩$ oscillates at the frequency given by the splitting of two ground states. The fit (teal) is a cosine of a single frequency and phase, with an envelope given by the lifetime and Gaussian dephasing. The teal region in the lower plot corresponds to the envelope of the fit shown in the callouts. The $T_2^{*}$ extracted from the fit to the oscillations is 3.7(4)s. The measurement of atom lifetime in tweezers is also plotted (gray), with $1/e$ time of 6.42(3)s. The error bars correspond to standard deviations of the binomial distributions given by the measured probabilities. (b)Spin-echo experiment. Inset: population of state $|1⟩$ oscillates with the duration of $Z(t)$ gate applied before the final $X(\pi /2)$ pulse. The contrast of the recorded fringe (orange) decreases with dark time $T$ due to the finite lifetime in tweezers (gray), of $7.13{(}_{-5}^{+6})\mathrm{s}$, and decoherence described by a Gaussian decay. The $T_2$ inferred from the fit is 7.9(4)s. The orange error bars are given by the square root of the covariance matrix diagonal entry corresponding to the fit contrast parameter. (c)Depolarization time. $T_1$ dependence on applied magnetic field (purple) and tweezer depth (green). $T_1$ is extended approximately exponentially with increasing bias field, and is invariant with changing tweezer depth. The error bars in (c) are similarly given by the square root of the covariance matrix diagonal entry corresponding to the spin depolarization time parameter of the fits.

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  Figure 5

  Raman-sideband cooling to the 3D motional ground state. (a)Level diagram for the two-photon transitions employed in cooling. An atom in the $|m_F=+1/2,n⟩$ state absorbs a photon from the $\pi$-polarized beam (RB2-4) and emits into the ${\sigma }^{+}$-polarized beam (RB1). With the relative frequency difference of the two beams tuned a trap frequency away from the carrier, the atom can change its motional state to $n-1$ (red sideband) or $n+1$ (blue sideband), depending on the sign of the difference. The Raman detuning from the intermediate excited state is ${\mathrm{\Delta }}_{\mathrm{RB}}/2\pi =-183\mathrm{MHz}$. (b)Optical pumping (OP) scheme for energy dissipation during cooling. In the Lamb-Dicke regime, the scattering from the pumping beam is unlikely to alter the atom’s motional state but pumps into the opposite spin state, where the cooling can begin anew. We operate with optical pumping ${\delta }_{\mathrm{OP}}/2\pi =-2\mathrm{MHz}$ red detuned from resonance. (c)Raman beams (RB1-4), optical pumping, and magnetic field ($B$) geometry. RB1 paired with RB2-4 addresses motional states along the $i$, $j$, and $k$ axes, respectively. (d)Continuous sideband cooling. RB1 and OP beams are continuously illuminating the atoms, while the remaining RB2-4 are turned on and off iteratively. (e)Pulsed sideband cooling. A pulse of RB1 and one of RB2-4 drives an approximate $\pi$ rotation on the relevant sideband, followed by a step of optical pumping. (f)Sideband thermometry along the $i$, $j$, and $k$ axes. Spectra are acquired before sideband cooling (gray squares), after 3D optimized cooling (black open circles), and after cooling optimized for radial (orange points) and axial (red points) directions. Each panel shows the red sideband (left) and blue sideband (right) for the corresponding cooling direction. Axial spectra additionally include the carrier (middle). Mean phonon occupation number ($\overline{n}$) after sideband cooling is quoted for each axis. The error bars correspond to standard deviations of the binomial distributions given by the measured probabilities.

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  Figure 6

  Overview of the experimental sequence in the science cell. The unit of time is milliseconds, and duty cycle is typically less than 1s. The blue MOT uses ${{}^{1}S}_0\leftrightarrow {{}^{1}P}_1$ transition, while the green MOT utilizes ${{}^{1}S}_0\leftrightarrow {{}^{3}P}_1$ narrow-line transition. The compressed MOT (CMOT) increases the average atom number loaded into a tweezer. With the near-deterministic loading scheme, we are left with a single atom in the tweezer $>90\%$ of the time, on average. The green cooling beam is on during blue imaging, as well as during the cooling before and after imaging.

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  Figure 7

  Optics around the objective lens. The 399 and 556nm laser beams for the MOTs are focused to the back focal plane of the objective lens and reflected by a mirror, small enough not to degrade the quality of either the tweezers or the image.

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  Figure 8

  Procedure employed to optimize tweezer balance. (a)Each tweezer’s intensity distribution is measured with light picked off before the objective lens and focused onto a camera. The intensity measurement is multiplied by an experimentally determined weighted mask and used to determine the errors for each tweezer. From this 2D array of errors, row and column errors are extracted and proportionally fed back to AOD1 and AOD2, respectively. (b)Example of the spectroscopy result for a uniform mask ($M_{ij}=1$). This information is further utilized to generate a weighted mask. The frequency variable is the red detuning from free-space resonance. (c)Initial (gray) and final (black) distributions of the ${{}^{1}S}_0\leftrightarrow {{}^{3}P}_1$, $F^{\prime }=3/2$, $m_{F^{\prime }}=1/2$ resonance for the 100 individual tweezers. The red line shows the median of the initial distribution, which is the chosen target value for the balancing.

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  Figure 9

  Infidelity and loss probability for an image of duration 120ms, with variable power in the 399nm imaging beam. $I_{\mathrm{sat}}^b=63\mathrm{mW}/{\mathrm{cm}}^2$ is the saturation intensity of the transition at 399nm. The red curve is an exponential fit to the infidelity data, and the green curve is a quadratic fit to the loss data. Both shaded regions are $1\sigma$ confidence intervals.

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  Figure 10

  Nuclear-spin preparation, manipulation, and detection. (a)We prepare the spin state in $|0⟩$ by optical pumping (OP) through ${}^3{P}_1$ $|F^{\prime }=1/2,m_{F^{\prime }}=+1/2⟩$ and detect the state destructively by driving the cycling transition $|0⟩\leftrightarrow |F^{\prime }=3/2,m_{F^{\prime }}=+3/2⟩$ until atoms in the $|0⟩$ state are expelled from their tweezers. We split the $|F^{\prime }=3/2,m_{F^{\prime }}=+1/2⟩$ and $|F^{\prime }=3/2,m_{F^{\prime }}=+3/2⟩$ states by ${\mathrm{\Delta }}_b/(2\pi )=35\mathrm{MHz}$ during blow-away detection to minimize off-resonant pumping through the $|F^{\prime }=3/2,m_{F^{\prime }}=+1/2⟩$ level by the blow-away beam. (b)$X$ rotations in the high-Rabi regime. The two Raman pathways couple the states $|0⟩$ and $|1⟩$ through two different excited states $|2⟩$ and $|3⟩$. The splitting between the nuclear-spin states is ${\mathrm{\Delta }}_N$, determined by both the external magnetic field and light shifts from the drive beam. The detuning from the excited states is ${\mathrm{\Delta }}_X/(2\pi )\simeq -180\mathrm{MHz}$, much larger than the ${\mathrm{\Delta }}_e/(2\pi )=2.65\mathrm{MHz}$ splitting between the $|F^{\prime }=3/2,m_{F^{\prime }}=\pm 1/2⟩$ levels. The Rabi coupling in the different arms is set by the dipole coupling matrix elements and the polarization angle of the drive beam. (c)Level diagram showing the ${\sigma }^{+}$-polarized $Z$ beam coupling to the $|F^{\prime }=3/2,m_{F^{\prime }}=+1/2⟩$ and $|F^{\prime }=3/2,m_{F^{\prime }}=+3/2⟩$ states. In the case of ${\mathrm{\Delta }}_Z\gg {\mathrm{\Delta }}_e$, the larger coupling matrix element on the stretched transition to $|F^{\prime }=3/2,m_{F^{\prime }}=+3/2⟩$ gives a larger light shift on the $|0⟩$ spin level. This splits the nuclear-spin states and allows us to perform fast rotations about the $Z$ axis.

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  Figure 11

  Estimating the scattering rate of the $Z$ beam. The spins are prepared in the $|1⟩$ state and the $Z$ beam is turned on for the time shown. Scattering causes the spin to relax to a population set by the branching ratios from the two excited levels that the $Z$ beam scatters from: $|F^{\prime }=3/2,m_{F^{\prime }}=+1/2⟩$ and $|F^{\prime }=3/2,m_{F^{\prime }}=+3/2⟩$. The decay timescale is 0.43(4)ms, consistent with the scattering rate estimated from the observed differential light shift produced by this beam. The error bars correspond to standard deviations of the binomial distributions given by the measured probabilities.

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  Figure 12

  By first preparing the atoms in $|0⟩$, waiting a variable delay, and then blowing away that spin state, the depolarization time constant $T_1$ can be measured. The survival probability is plotted as a function of delay time for three different magnetic field conditions: $B\approx 0\mathrm{G}$ (blue), 0.7G (green), and 2G (red). The lines represent the fit applied to the data. The fits yield values for $T_1$ of 17.0(1.5) s for the blue points, 26.7(4.8) s for the green points, and 119(29) s for the red points. Error bars represent $1\sigma$ uncertainty of the binomial distributions given by the measured probabilities.

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  Figure 13

  Clock pulse motional dephasing. The calculated infidelity of the population transfer to the clock state $|e⟩$ is plotted as a function of trap frequency and mean motional occupation number $\overline{n}$. The Rabi frequency is fixed at ${\mathrm{\Omega }}_c/(2\pi )=200\mathrm{kHz}$. The trap is assumed to be magic, with equal trap frequencies in the ground and clock states.

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