Repetitive Readout and Real-Time Control of Nuclear Spin Qubits in ${}^{171}\mathrm{Yb}$ Atoms

We demonstrate high-fidelity repetitive measurements of nuclear spin qubits in an array of neutral ytterbium-171 (${}^{171}\mathrm{Yb}$) atoms. We show that the qubit state can be measured with a spin-flip probability of $0.004(4)$ for a single tweezer and $0.012(3)$ averaged over the array. This is accomplished by high cyclicity of one of the nuclear spin qubit states with an optically excited state under a magnetic field of $B=58$ G, resulting in a spin-flip probability of approximately ${10}^{-5}$ per scattered photon during fluorescence readout. The performance improves further as $\sim 1/B^2$. The state discrimination fidelity is $0.993(4)$ with a state-averaged readout survival of $0.994(3)$, limited by off-resonant scattering to dark states. We combine our measurement technique with high-contrast rotations of the nuclear spin qubit via an ac magnetic field to explore two paradigmatic scenarios, including the noncommutativity of measurements in orthogonal bases, and the quantum Zeno mechanism in which measurements “freeze” coherent evolution. Finally, we employ real-time feedforward to repetitively and deterministically prepare the qubit in the $+z$ or $-z$ direction after initializing it in a different basis and performing a measurement in the $Z$ basis. These capabilities constitute an important step towards adaptive quantum circuits with atom arrays.

Popular Summary

Many quantum science applications require the ability to measure qubits without losing them and without losing knowledge of the postmeasurement state. In neutral atom arrays, typical measurements are either destructive, in which the atom is lost with high probability, or cause the atom to populate states outside the qubit basis. This work addresses these problems by exploiting the simple ground-state structure of alkaline-earth-like ytterbium-171. A measurement technique is demonstrated to have high atom survival and a high probability that the qubits will remain in their measured state. These capabilities are then used to combine repetitive readout with qubit rotations, enabling real-time qubit control.

The measurement technique is enabled by the ability to scatter roughly ${10}^5$ photons from one qubit state before a spin flip to the other qubit state occurs. Crucially, ytterbium-171 has a relatively simple ground-state structure with no electronic angular momentum, allowing for qubit encoding directly into its nuclear-spin degree of freedom. With roughly 1000 photons being required for high-fidelity readout, this technique enables qubit readout with a spin-flip probability of 0.01. The nuclear-spin qubits are controlled with a radio-frequency magnetic field, providing a platform to combine coherent qubit manipulation and quantum nondemolition readout.

The demonstrated capabilities open the door to new opportunities in quantum computing and metrology and can be combined with use of the optical clock transition. Single-qubit control will enable the application of this measurement technique to perform midcircuit measurements.

Figure 1

Overview. (a) The experimental system consists of a glass vacuum cell with one microscope objective on either side. Atoms are illuminated with two retroreflected probe beams that each have an angle with respect to the $x$-$y$ and $y$-$z$ planes. The tweezer array lies parallel to the $y$ axis. The dc magnetic field and the tweezer electric field point in the $y$ direction. An ac magnetic field produced by the pair of coils shown points in the $z$ direction. Images recorded on the camera can be analyzed in real time to enable or disable ac pulses on the coils. (b) The relevant level structure of ${}^{171}\text{Yb}$, showing the cycling transition from the ${}^1{S}_0|m_F=-1/2⟩\equiv |0⟩$ qubit state, making it bright, while $|m_F=+1/2⟩\equiv |1⟩$ is dark during readout. (c) Energies of the four $m_F$ Zeeman states in ${}^3{P}_{1}F=3/2$ relative to free space versus the magnetic field, taken with approximately a $1$-$\text{mK}$ tweezer depth for a single array site. (d) A histogram of $12$-$\text{ms}$ fluorescence readout ($2500\text{repetitions}\times 5\text{sites}=12500\text{shots}$) of the nuclear spin qubit, where the dark peak is either zero atoms or an atom in $|1⟩$ and the bright peak is an atom in $|0⟩$. The system was initialized in $|0⟩$. The discrimination fidelity is $\mathcal{F}=0.993(4)$. Inset: averaged ($2500\text{repetitions}$) camera image of fluorescence from a five-site array with spacing of $7.8\text{μ}\mathrm{m}$.

Figure 2

Circuit legend. Simplified equivalent circuit elements. The number 0 or 1 and the open or filled circle distinguishes “qubit readout” from “atom readout”, respectively.

Figure 3

Characterizing depolarization during qubit readout. (a) Circuit diagram for studying the dependence of depolarization during qubit readout on magnetic field. A probe block of variable magnetic field is placed between two measurements performed at $58\text{G}$. The final “atom readout” pulse is used to postselect on events in which the atom survived the entire sequence. (b) The measured array average (diamonds) and single-site best (squares), SPAM corrected, and the predicted state populations (line) under real imaging conditions versus magnetic field. The error bars reflect the standard deviation of a binomial distribution. (c) Circuit diagram for directly studying the depolarization probability during qubit readout at $58\text{G}$. (d) The array-averaged histograms from each image for $17500$ shots, with the results shown together on a two-dimensional (2D) histogram. These results show that the $D\to B$ and $B\to D$ conditional depolarization probabilities during a $12$-$\text{ms}$ probe pulse are $0.025(2)$ [$0.011(3)$] and $0.025(2)$ [$0.018(4)$] averaged across the array (best site), respectively, before SPAM correction.

Figure 4

Interleaved readout and qubit rotation. (a) Circuit for qubit rotations between two qubit readout pulses. The atom readout pulse is included at the end to postselect on events in which the atom survives the entire sequence. We start with a state aligned along $|+⟩=(|0⟩+|1⟩)/\sqrt{2}$. (b) Rabi oscillations versus time, showing three groups that span the first second. No obvious contrast decay is observed. Here $P_{\text{same}}$ refers to cases where the outcome of the two qubit measurements are the same. This outcome is equally split between $|0⟩$ and $|1⟩$ due to the initial state $|+⟩$. (c) The 2D scatter plots of photon counts associated with the early-time data points with rotations of $\theta =0$, $\theta \approx \pi /2$, and $\theta =\pi$ as shown in (b). These results indicate that the $\pi$-pulse fidelity is comparable to the probability of not undergoing a spin flip during readout.

Figure 5

Repetitive readout in variable bases. (a) The $\{+Z,-Z,+X,-X\}$ measurement sequence, showing large bit string probabilities only for cases where the outcome is opposite between both the first two and last two measurements. The correlation matrix shows strong off-diagonal negative correlations for these pairs. (b) The $\{+Z,+X,-Z,-X\}$ measurement sequence, showing equal bit string probabilities for all outcomes. The correlation matrix shows no significant off-diagonal elements. The red lines show the ideal probability distributions, which are either $0$ or $1/2^2=0.25$ in (a) and $1/2^4=0.0625$ in (b). The ideal off-diagonal correlations are either $0$ or $-1$. Data in black and solid bars postselect on detecting an atom in the initial and final atom readout; data in gray and dotted bars postselect on only the initial atom readout.

Figure 6

The quantum Zeno mechanism. (a) The Zeno circuit. After initializing in $|+⟩$ and performing a qubit measurement, we repetitively interleave a rotation $R(\theta )$ and a readout $N-1$ times. We postselect on the final atom readout. (b) The probability that the measurement has the same outcome as the initial measurement, $P_{\text{same}}$, versus the measurement index for many different rotation angles $\theta$. We see monotonic decay of $P_{\text{same}}$ to 0.5 for $\theta \in [0,\pi /2]$ and damped alternations to 0.5 for $\theta \in [\pi /2,\pi ]$. The lines deviate from the ideal case only by the finite depolarization probability during readout that is fit to the $\theta =0$ case as ${\overline{P}}_{\text{depol}}=0.0127(6)$. (c) The probability that all outcomes are the same, $P_{\text{all-same}}$, versus the total rotation angle ${\theta }_{\text{tot}}=\theta \times (N-1)$ for several different numbers of repetitions $N$. The lines are $P_{\text{all-same}}=(1-{\overline{P}}_{\text{depol}}{)}^N{\text{cos}}^{2N-2}({\theta }_{\text{tot}}/(2N-2))$.

Figure 7

Repetitive real-time feedforward. (a),(c) The circuit for performing repetitive deterministic state preparation for a single qubit. After initializing in $|+⟩$ and performing an initial measurement, we repeat a loop that contains a first pulse ($\pi /2$), a first measurement, a second pulse ($0$ or $\pi$), and a second measurement. In (a), we choose the second pulse based on the first measurement outcome in order to yield the outcome of the second measurement to match the initial measurement outcome. This requires feeding forward the result obtained from an exclusive or (xor) gate between the classical bits in the initial measurement and the first measurement in the loop. The result is shown in (b), where the thick solid line shows the result averaged over $400$ shots. In (c), we choose to alternate between obtaining the opposite and the same outcome in the second measurement as that of the initial measurement, which requires alternating between exclusive not or (xnor) and xor classical logic on odd- and even-numbered loops. The result is shown in (d), where now we see full-contrast alternations. The arrows in (b) and (d) illustrate the role of the feedforward, with gray vertical lines dividing loop iterations. Loops using xor logic have light gray backgrounds and loops using xnor logic have dark gray backgrounds. Panels (e) and (f) show a single, representative trajectory for each circuit and the arrows indicate when a $\pi$ pulse is applied.

Figure 8

Comparison of the $m_F=-3/2$ and $m_F=-1/2$ imaging conditions. (a) Comparison of photon scatter during imaging. The upper histogram shows imaging using the $m_F=-3/2$ excited state with the mean number of photons collected ${\mu }_1=37.1(2)$ at $58$-$\text{G}$ magnetic field when an atom is present (see Appendix pp8), while the lower histogram shows imaging using $m_F=-1/2$ with ${\mu }_1=28.1(2)$ at $18\text{G}$. The red vertical lines show the photon collection threshold for bright-dark discrimination in each case (see Appendix pp8). (b) Measurement of atom temperature in the tweezer under the $m_F=-3/2$ (dark green circle) and $m_F=-1/2$ (light green diamond) via the release-recapture experiment. Tweezer sites are imaged and the tweezers are diabatically switched off for variable time before being turned on and imaged again. Recapture probability is calculated as that of the second image being bright, postconditioned on the first being bright as well. Optical pumping is performed before both images in the $m_F=-3/2$ case. Probabilities have been rescaled so that the mean of the four shortest-time probabilities is equal to 1. The lines show predicted probabilities obtained via Monte Carlo simulation for temperatures $4\text{μ}\mathrm{K}$ (black) up to $10\text{μ}\mathrm{K}$ (red), indicating that the temperature after imaging using $m_F=-3/2$ ($m_F=-1/2$) is approximately $5\text{μ}\mathrm{K}$ ($4\text{μ}\mathrm{K}$).

Figure 9

ac magnetic field system. (a) The schematic of the rf driver used to drive one pair of the horizontal shim coils. The driver is capable of giving a maximum output voltage around $300{\text{V}}_{\text{pp}}$ and the peak current about $0.2\text{A}$ ranges from $30\text{kHz}$ to $2\text{MHz}$. (b) The connection of the rf driver to the shim coils. The isolated output of the rf driver (green box) is connected in series with the shim coil driver (teal box). A $1$-$\text{μ}\mathrm{F}$ polypropylene film capacitor is used to bypass the rf signal through the shim coil driver.

Figure 10

Energy levels and probe beam configuration. The energy splitting between different Zeeman states in ${}^1{S}_0$ and ${}^3{P}_1$ manifolds are determined by the external magnetic field (Zeeman splitting) and tweezer depth (vector and tensor light shifts). For magnetic fields exceeding 5 G, the splittings between ${}^3{P}_1$ Zeeman states look nearly identical.

Figure 11

Off-resonant scattering-limited probe lifetime. (a) The calculated lifetime under $|m_F|=3/2$ imaging as a function of probe intensity. The dashed lines assume that the atom will be lost if it scatters to any ${}^3{P}_J$ state from ${}^3{S}_1$, while the solid lines assume that it will survive only if scattered to ${}^3{P}_1$. (b) The lifetime under $|m_F|=1/2$ imaging for the same condition, which gives a 3 times higher lifetime due to the lower off-resonant scattering rate. We set the tweezer {wavelength, power, waist ($1/e^2$ radius)} to {$760\text{nm}$, $7\text{mW}$, $670\text{nm}$} for both calculations. A probe detuning of $-1\mathrm{\Gamma }$ is used to calculate the ${}^3{P}_1$ state population.

Figure 12

Polarizabilities of the atomic states of interest. The polarizabilities of the ${}^1{S}_0$ (black); ${}^3{P}_0$ (yellow); ${}^3{P}_1$, $F=3/2$, $|m_F|=1/2$ (light green); and ${}^3{P}_1$, $F=3/2$, $|m_F|=3/2$ (dark green) are plotted as a function of the tweezer wavelength. The shaded band represents the uncertainty in the total polarizability, which arises from uncertainty in the trap waist and measured light shifts of the excited states. The magic wavelength ($759\text{nm}$, red vertical line) for the ground and clock states is also near magic for the $|m_F|=1/2$ state. We have included correction factors in the calculations for the ${}^3{P}_1$ states to match the experimentally observed differential polarizabilities.

Figure 13

State mixing of the ${}^3{P}_1$, $F=3/2$, $m_F=-3/2$ imaging state due to the vector and tensor light shifts. (a) The probabilities in the $m_F=-3/2$ (dark green) and $m_F=-1/2$ (light green) states due to the vector ($\gamma =1^{\circ }$ and $\theta =0^{\circ }$; solid) and tensor ($\theta =1^{\circ }$ and $\gamma =0^{\circ }$; dashed) light shifts as the magnitude of the applied magnetic field is varied. The gray vertical lines at $10\text{G}$ (dash-dot) and $60\text{G}$ (dotted) show field values chosen for the angle studies shown in (b). (b) We study the dependence on the angles $\theta$ and $\gamma$, which describes the misalignment of the tweezer from the axis of quantization and the degree of ellipticity of the tweezer polarization, respectively. There is a sharp rise at angles close to $0^{\circ }$ due to the sudden appearance of the off-diagonal matrix elements.

Figure 14

Depolarization probability study without postselection on atom readout. Photon counts from the first two readouts in the circuit shown in Fig. 3 used to study the depolarization probability without postselection on the final atom readout. Here, the population in the $B\to D$ depolarization channel is artificially inflated by atom loss between these two readouts, as is the population in the $D\to D$ quadrant due to failed loading of the tweezer.

Figure 15

Validation of discrimination fidelity. Using three consecutive measurement histograms $a,b,c$, with $a$ being an atom readout, postselecting data on measurement $a$ being bright, we estimate the readout fidelity by examining the behavior of the conditional probabilities ${\mathcal{F}}_{\text{read}}^{\text{B}}=Pr(B_c|B_a\wedge B_b)$ and ${\mathcal{F}}_{\text{read}}^1=Pr(D_c|B_a\wedge D_b)$ as well as the average readout fidelity ${\mathcal{F}}_{\text{read}}={\mathcal{F}}_{\text{read}}^{\text{D}}Pr(D_b)+{\mathcal{F}}_{\text{read}}^{\text{B}}\text{Pr}(B_b)$ by sweeping the thresholds ${\theta }_b$ and ${\theta }_c$ used in measurements $b$ and $c$, respectively, while holding that for measurement $a$ fixed at the vertical red line, set by the optimization of Eq. (H2). We show results for ${\theta }_c$ fixed to its optimum value with the fidelities as a function of ${\theta }_b$ only. For large ${\theta }_b$, $|0⟩$ is prepared by $b$ with a perceived near-unity probability and subsequently measured in $c$ with fidelity ${\mathcal{F}}_{\text{read}}^{\text{B}}=0.99(1)$. For small ${\theta }_b$, $|1⟩$ is perceived to be prepared with near-unity probability instead and measured with fidelity ${\mathcal{F}}_{\text{read}}^{\text{D}}=0.97(2)$. We note that ${\mathcal{F}}_{\text{read}}^{\text{D}}$ may not have fully saturated to its maximal value due to finite sampling of events occurring with vanishing probability.

Figure 16

Example probability graph for SPAM correction. The probability graph for the three-image sequence from Fig. 4 is shown, wherein each node represents a state of the system and each edge gives the probability of transitioning to another. Every possible combination of readout results is represented as the complete path from the “Start” node on the left to any of the terminal nodes with no outgoing edges on the right, and the total probability for any such path is found as the product of all the edge weights along it. Here, $|\varnothing ⟩$ is used as a shorthand for “empty tweezer or lost atom,” and we assume that the qubit state has vanishing probability of flipping during readout. Thus, a model for an experimentally measurable probability is found by summing the probabilities of all paths that satisfy a corresponding set of data analysis conditions. For example, the model for the measured $\pi$-pulse probability ${}^{\ast }{\eta }_{\pi }$ is found following Eq. (H10) as the sum over probabilities of all paths passing through a “Bright” readout 0 node and a “Dark” readout 1 node (or vice-versa) as well as a “Bright” readout 2 node, divided by that for all paths terminating in a “Bright” readout 2 node. This model is set equal to the experimentally measured value of ${}^{\ast }{\eta }_{\pi }$ and used for the simultaneous correction of all relevant parameters, as described in Appendix pp9. Generation of models from other graphs is done similarly, following Eqs. (H4)–(H8).

Figure 17

Measurement of array averaged $T_1$, $T_2^{\ast }$, and $T_2^{\text{echo}}$. (a) The circuit used to measure dark-state survival and the qubit depolarization time, $T_1$. This circuit is identical to that shown in Fig. 3 except for the addition of a holding time $\tau$ between the first two readouts. The qubit is also initialized in $|0⟩$ here in order to measure both $P_{\text{same}}$ (blue circles) and the atom survival fraction (orange triangles), which are plotted below as a function of $\tau$. The atom survival lifetime is measured to be $8.8(3)\text{s}$. The qubit depolarization time $T_1$ is estimated to be $230(50)\text{s}$, but our data extend only to $13\text{s}$. (b) The circuit used to measure the qubit dephasing time, $T_2^{\ast }$. Here, the qubit is rotated to the equator for a time $\tau$ before a $\pi /2$ pulse with phase $\phi$ relative to the first is applied and the qubit state is measured. The resulting fringes in $P_{\text{same}}$ are plotted as a function of $\phi$ in the insets, and the contrast of the fringes is plotted as a function of the hold time $\tau$ in the main plot. The decay of the contrast follows a Gaussian profile, the $1/e$ time of which we measure as $T_2^{\ast }=0.37(1)\text{s}$. (c) The same circuit except with a $\pi$ pulse in the middle of the precession time. This spin-echo sequence gives $T_2^{\text{echo}}=1.40(5)\text{s}$.