Long-Lived Circular Rydberg Qubits of Alkaline-Earth Atoms in Optical Tweezers

Coherence time and gate fidelities in Rydberg atom quantum simulators and computers are fundamentally limited by the Rydberg state lifetime. Circular Rydberg states are highly promising candidates to overcome this limitation by orders of magnitude, as they can be effectively protected from decay due to their maximum angular momentum. We report the first realization of alkaline-earth circular Rydberg atoms trapped in optical tweezers, which provide unique and novel control possibilities due to the optically active ionic core. Specifically, we demonstrate creation of very high-$n$ ($n=79$) circular states of ${}^{88}\mathrm{Sr}$. We measure lifetimes as long as 2.55 ms at room temperature, which are achieved via cavity-assisted suppression of black-body radiation. We show coherent control of a microwave qubit encoded in circular states of nearby manifolds, and characterize the qubit coherence time via Ramsey and spin-echo spectroscopy. Finally, circular-state tweezer trapping exploiting the ${\mathrm{Sr}}^{+}$ core polarizability is quantified via measurements of the trap-induced light shift on the qubit. Our work opens routes for quantum simulations with circular Rydberg states of divalent atoms, exploiting the emergent toolbox associated with the optically active core ion.

Popular Summary

Arrays of trapped neutral atoms have emerged as a promising platform for large-scale quantum computers and simulators. In those systems, Rydberg atoms—atoms in which one electron is in a highly excited state—are used to either mediate interactions between qubits or to implement the qubit itself. Their finite lifetime, therefore, sets a fundamental limit to gate fidelities or qubit coherence times. Increasing the orbital angular momentum of the Rydberg electron, however, protected the states from spontaneous decay, greatly enhancing their lifetime. We report, for the first time, the creation and control of a qubit realized in an optically trapped circular Rydberg state—the state with maximum Rydberg electron orbital angular momentum—of an alkaline-earth-like atom.

In this work, we create highly excited circular Rydberg states and characterize the coherence of the qubit. The observed lifetime of more than 2.5 ms represents the longest-lived trapped Rydberg state ever observed in a room-temperature setup. Besides being a promising candidate for overcoming the current lifetime limitations by orders of magnitude, the divalent nature of the atoms provides an optically active ionic core while the outer electron is excited to a circular Rydberg state. This allows for direct optical control, readout, and—most importantly—trapping of the atom.

Our work opens routes for quantum simulations with circular Rydberg states of divalent atoms, exploiting the emergent toolbox associated with the optically active core ion.

Figure 1

(a) Schematic drawing of the experiment. Single ${}^{88}\mathrm{Sr}$ atoms are trapped in optical tweezers (green) inside an electrode structure consisting of two transparent electrodes (red) and four circularly shaped electrodes for applying ${\sigma }^{+}$-polarized radio-frequency (rf) fields (indicated with orange arrows). Magnetic and electric control fields $E$ and $B$ are aligned with the tweezer axis. (b) Averaged fluorescence image of single atoms in the ten-tweezer array used throughout this work obtained through the transparent electrodes. (c) The atoms are excited to the $|79F,m=2⟩$ state via an off-resonant three-photon transition, from where they are transferred to the $|79\mathrm{C}⟩$ circular Rydberg state by an adiabatic rapid passage (ARP). The qubit is implemented by coherently coupling $|79\mathrm{C}⟩$ to $|77\mathrm{C}⟩$ by two microwave photons at approximately 13.9 GHz. (d) State-selective field-ionization profile of the $|79F⟩$, $|79\mathrm{C}⟩$, and $|77\mathrm{C}⟩$ states ionized by a $10\text{−}\mathrm{\mu }\mathrm{s}$ linear field ramp to $27(5)\mathrm{V}{\mathrm{cm}}^{-1}$ before (blue) and after circularization and a partial transfer to $|77\mathrm{C}⟩$ (red). The three states are well distinguishable, and by counting the events within the gray integration windows, the state populations $p_n$ are extracted. (e) Sketch of the experimental sequence showing tweezer trap depth (green), electric control field strength (red), Rydberg laser light (purple), rf field (orange), microwave for qubit control (blue), and state readout via state-selective ramped field ionization (SSFI) (brown). The employed technical components in (a) are colored accordingly. (f) Rabi oscillations on the $|79\mathrm{C}⟩\leftrightarrow |77\mathrm{C}⟩$ microwave transition. The population transfer $p_{77}/(p_{77}+p_{79})$ to find the atom in $|77\mathrm{C}⟩$ as a function of the microwave pulse length $t_{\mathrm{MW}}$ is shown. The solid blue line is a sinusoidal fit with a Gaussian decay envelope. Note that for the three areas separated by the gray dashed lines, the frequency is varied independently due to microwave power fluctuations between measurements. Error bars represent 1 standard deviation.

Figure 2

(a) Measurement of the circular-state qubit coherence via Ramsey (blue) and spin-echo (red) experiments. The population transfer $P$ into the $|77\mathrm{C}⟩$ state is plotted against the cumulative Ramsey time and echo time $t_{\mathrm{R}}+t_{\mathrm{E}}$. The microwave pulse sequences for the Ramsey and spin-echo experiments are shown schematically above the data. Solid lines are Gaussian-damped sine functions fitted to the data. The gray diamonds show the combined ion detection probability in $|79\mathrm{C}⟩$ and $|77\mathrm{C}⟩$ ($p_{77}+p_{79}$) scaled by the first data point. An exponential fit (gray dashed line) to the data reveals a population lifetime in the qubit subspace of 1.3(1) ms. (b) Maximum fringe contrast of the spin-echo signal scaled by the preparation efficiency ${\epsilon }_{\mathrm{CRS}}$, as a function of the revival time $t_{\mathrm{C}}$. The solid line shows a fit of Eq. (1) to the data. Error bars show 1 standard deviation.

Figure 3

(a) Radial cut through the calculated tweezer potential for CRS with $n=77$ (blue) and $n=100$ (red) at the trap center. The dashed lines show the contribution from the ponderomotive potential of the electron in the CRS, and the gray dotted line depicts the contribution from the ionic-core polarizability. The size of the electron wave function (shaded areas) is comparable to the tweezer waist. The total trap potential (solid lines) depends on $n$, leading to a differential light shift ${\delta }_{\mathrm{LS}}$. (b) Differential light shift ${\delta }_{\mathrm{LS}}$ for the two-photon MW transition $|n\mathrm{C}⟩\to |(n+2)\mathrm{C}⟩$ as a function of $n$. (c) Measured microwave spectrum $|79\mathrm{C}⟩\to |77\mathrm{C}⟩$ for tweezer depth $U_0=2.39\mathrm{MHz}$ (red diamonds) and $U_0=5.13\mathrm{MHz}$ (green squares) compared to the free-space resonance (blue circles). Solid lines are Gaussian fits to the data to extract the light shift. Linewidths are dominated by power broadening from different microwave powers set for the three measurements. (d) Light shift ${\delta }_{\mathrm{LS}}$ as a function of the tweezer depth $U_0$ (tweezer power $P_{\mathrm{CRS}}$) obtained from microwave spectroscopy data as in (c) (red circles) and from Ramsey oscillations as in Fig. 4 (green triangles). The blue shaded area is an ab initio calculation of the expected light shift including experimental uncertainties in the tweezer power, while the gray line is a linear fit to the data. In all panels, vertical error bars represent 1 standard deviation.

Figure 4

(a)–(c) Ramsey signal for three different values of the tweezer power $P_{\mathrm{CRS}}=(1.15,0.73,0.32)\mathrm{mW}$. For increasing power, the frequency decreases by the tweezer-induced light shift. (d) Reversible coherence time $T_2^{*}$ extracted from Ramsey measurements as a function of the tweezer power $P_{\mathrm{CRS}}$ (green circles) compared to expectations from a semiclassical dephasing model (green line). The gray dashed lines mark the correspondence to the Ramsey data in (a)–(c). (e) Spin-echo contrast obtained with the same method as in Fig. 2 but for atoms trapped in tweezers with $P_{\mathrm{CRS}}=0.32\mathrm{mW}$ (red circles). The red dashed line shows the model fitted to the nontrapped case [Fig. 2] for comparison. Error bars show 1 standard deviation.

Figure 5

Decay dynamics of $|79\mathrm{C}⟩$ (a) and $|77\mathrm{C}⟩$ (b) into neighboring circular Rydberg states. Symbols depict the populations $p_n$ in different $n$ manifolds as a function of the hold time $t$. The populations in each state are recorded via SSFI with a $24\text{−}\mathrm{\mu }\mathrm{s}$-long linear ionization ramp to $27(5)\mathrm{V}{\mathrm{cm}}^{-1}$ at various values of $t$ and are extracted by fitting multiple skewed Gaussians to the obtained ion histograms. The initial population $p_{77}$ and $p_{79}$ in (b) arises from the preparation efficiency of $|77\mathrm{C}⟩$ via an MW $\pi$ pulse from $|79\mathrm{C}⟩$. Solid lines depict a fit of a rate model to the data, which yields lifetimes for $|79\mathrm{C}⟩$ ($|77\mathrm{C}⟩$) of 2.55(10) ms [0.53(8) ms]. Insets (c) and (d) show representative ion histograms for the datasets in (a) and (b), respectively, recorded at values of $t$ marked by the dashed vertical lines. The colors indicate the principal quantum numbers labeled in (a) and apply to all subfigures.

Figure 6

(a) Calculated transition frequency $f_{m,m+1}$ between the lowest $m$ states of the $n=79$ hydrogenic manifold of ${}^{88}\mathrm{Sr}$ used in the circularization process. The gray dashed line indicates the rf frequency $f_{\mathrm{rf}}=70\mathrm{MHz}$ used in the experiment. (b) rf voltage dependence of the transfer from $|79F,m=2⟩$ to $|77\mathrm{C}⟩$. The SSFI traces are recorded with the same linear ionization ramp as in Fig. 1, after the ARP to $|79\mathrm{C}⟩$ and a subsequent MW $\pi$ pulse resonant with the $|79\mathrm{C}⟩$-to-$|77\mathrm{C}⟩$ transition.