Robust and scalable rf spectroscopy in first-order magnetic sensitive states at second-long coherence time

TL;DR

The paper tackles the challenge of performing high-fidelity, noise-robust Ramsey spectroscopy in magnetic-sensitive, long-lived ion states to enable stringent tests of local Lorentz-invariance and precise quadrupole-moment measurements. It introduces and analyzes robust radio-frequency composite-pulse sequences, notably UR10, and compares them to GSE both theoretically and experimentally in the spin-1/2 $^2S_{1/2}$ state, before extending to the eight-level $^2F_{7/2}$ manifold. The UR10 sequence demonstrates dramatically enhanced resilience to rf-detuning and pulse-duration errors, achieving coherent Ramsey signals up to $T_D$ of several seconds and enabling the most stringent LV test in the electron-photon sector with a single Yb$^+$ ion to date; it also yields a precise quadrupole moment $\Theta = -0.0298(38)\,e a_0^2$, in agreement with optical-clock measurements. The approach scales to multi-ion crystals, offering improved sensitivity for fundamental-physics tests and high-precision metrology of metastable states, while maintaining readiness for practical implementation in larger quantum sensors.

Abstract

Trapped-ion quantum sensors have become highly sensitive tools for the search of physics beyond the Standard Model. Recently, stringent tests of local Lorentz-invariance (LLI) have been conducted with precision spectroscopy in trapped ions. We here elaborate on robust radio-frequency composite-pulse spectroscopy at second long coherence times in the magnetic sublevels of the long-lived $^{2}F_{7/2}$ state of a trapped $^{172}$Yb$^{+}$ ion which is scalable to spatially extended multi-ion systems. We compare two Ramsey-type composite rf pulse sequences, a GSE sequence and a UR10 that decouple the energy levels from magnetic field noise, enabling robust and accurate spectroscopy. Both sequences are characterized theoretically and experimentally in the spin-$1/2$\ $^{2}S_{1/2}$ electronic ground state of $^{172}$Yb$^+$ and results show that the UR10 sequence is 38 (13) times more robust against pulse duration (frequency detuning) errors than the GSE sequence. We extend our simulations to the eight-level manifold of the $^2F_{7/2}$ state, which is highly sensitive to a possible violation of LLI, and show that the UR10 sequence can be used for high-fidelity Ramsey spectroscopy in noisy environments. The UR10 sequence is implemented experimentally in the $^2F_{7/2}$ manifold and a coherent signal of up to 2.5\,s is reached. In reference we have implemented this sequence and used it to perform the most stringent test of LLI in the electron-photon sector to date with a single Yb$^{+}$ ion. Due to the high robustness of the UR10 sequence, it can be applied on larger ion crystals to improve tests of Lorentz symmetry further. We demonstrate that the sequence can also be used to extract the quadrupole moment of the meta-stable $^{2}F_{7/2}$ state, obtaining a value of $Θ\,=\,-0.0298(38)\,ea^{2}_{0}$ which is in agreement with the value deduced from clock measurements.

Figures (8)

• Figure 1: Zeeman substates within the $^{2}F_{7/2}$ manifold of $^{172}$Yb$^{+}$. With a defined quantization magnetic field $\textbf{B}$, the $^{2}F_{7/2}$ manifold splits into eight Zeeman sublevels spanning from $m_{J}\,=\,-7/2$ to $m_{J}\,=\,+7/2$. The $m_{J}^{2}$ dependent terms in the Hamiltonian lead to a non-linear energy shift $\Delta E$ caused by the quadrupole shift and a potential Lorentz violation.
• Figure 2: Bloch sphere representation of the GSE and the UR10 sequences with one repetition $N_{\text{rep}}\,=\,1$. The rf detuning $\delta\omega$ is scaled to the rf Rabi frequency $\Omega$ as $\eta_{\omega}\,=\,\delta\omega/\Omega$ and pulse duration error $\delta t$ is scaled to the $\pi$-pulse duration $t_{\pi}$ as $\eta_{t}\,=\,\delta t/t_{\pi}$. The states $\ket{\Psi_{0}}$ and $\ket{\Psi_{1}}$ are denoted in red and black, respectively. The purple circle indicates the start of the first $\pi$-pulse. Figure (a) and (c) show the state evolution of the GSE and the UR10 sequences, respectively, with $\eta_{\omega}\,=\,- 0.03$ and $\eta_{t}\,=\,0$. Figure (b) and (d) show the state evolution of the GSE and UR10 sequences, respectively, with $\eta_{\omega,t}\,=\,- 0.03$. For figure (a) and (b), the state starts off at the equator at the red dot and after a combination of three periods of free evolution and two $\pi$-pulses with specific phases, lands at the location shown with the black dot. The sequence introduces an error of $\epsilon_{\text{GSE}}\,=\,3\times10^{-3}$. For (c) and (d), the state starts off at the equator at the red dot and after a combination of eleven periods of free evolution and ten $\pi$-pulses with specific phases. A simulation with $t_{w}\,=\,100$ µs shows that the sequence introduces an error of $\epsilon_{\text{UR10}}\,=\,2\times10^{-16}$.
• Figure 3: (a) Reduced energy level diagram of $^{172}$Yb$^{+}$. Doppler cooling and repumping are carried out on the transitions near 370 nm and 935 nm, respectively. Optical pumping into either one of the $\ket{S,\pm1/2}$ states is done via a $\sigma^{\pm}$-polarized 370 nm laser beam parallel to the magnetic field (B-field). Repumpers near 1650 nm and 638 nm are used to bring the population back to the ground state after excitation to the $^{2}D_{5/2}$ or $^{2}F_{7/2}$ state on the transitions near 411 or 467 nm, respectively. (b) Laser and quantization B-field orientation. The B-field lies in the $xz$ plane at an angle of $\beta\,=\,26.8(4.0)$$^{\circ}$ with the trap axis $\mathbf{\hat{z}}$. The 467 nm (411 nm) beam points in the $-\mathbf{\hat{y}}$ ($+\mathbf{\hat{y}}$) direction with a polarization parallel (perpendicular) to the B-field. The resonant circuit including an antenna coil used for rf spectroscopy has a diameter of $d_{c}\,=\,$4.5 cm and is mounted $h_{c}\,=\,$5.5 cm above the ion. (c) Reduced experimental sequence for rf spectroscopy in the $^{2}S_{1/2}$ manifold ("$^{2}S_{1/2}$ seq.") and in the $^{2}F_{7/2}$ manifold ("$^{2}F_{7/2}$ seq."). "C" denotes Doppler-cooling via the 370 nm transition, "O" denotes optical pumping, and "rf" is the rf sequence. "E2" and "E3" represent the excitation via the electric quadrupole and electric octupole transitions, respectively. "R" is the repumping via transitions near 1650 nm and 638 nm.
• Figure 4: Example of measurements to determine (a,c) the Rabi frequency and (b,d) the rf center frequency in both the $^{2}S_{1/2}$ and $^{2}F_{7/2}$ manifold. For both measurements in the $^{2}S_{1/2}$ and $^{2}F_{7/2}$ manifolds, the ion is prepared in the $m_{J}\,=\,-1/2$ state. The population that remains in the initial state after applying a single rf pulse is measured, from which the Rabi frequency and the resonance frequency are extracted. (a) The two-level Rabi frequency in the $^{2}S_{1/2}$ state is $\Omega_{S}\,=\,2\pi\times 60.5(4)$ kHz. (b) The measured rf center frequency in the $^{2}S_{1/2}$ state is $\nu_{0}\,=\,3.5942(7)$ MHz. (c) The multi-level Rabi frequency in the $^{2}F_{7/2}$ state is $\Omega_{F}\,=\,2\pi\times 28.1(1)$ kHz. (d) The measured rf center frequency in the $^{2}F_{7/2}$ state is $\nu_{0}\,=\,3.5510(6)$ MHz. The fluctuations in (c,d) comes from slow drifts of the E3 transition frequency during the measurement.
• Figure 5: Simulation and experimental verification of the GSE and UR10 sequences in a spin-$1/2$ system at a Ramsey dark time of $T_{\text{D}}\,=\,$10 ms. Figure (a) shows the stability diagram for pulse errors $\eta_{\omega}$ and $\eta_{t}$ for the GSE sequence and figure (b) shows the UR10 sequence. The grey scale indicates the fidelity from 0 to 1. The colored lines show the region where the sequences are investigated experimentally in the $^{2}S_{1/2}$ electronic ground state of $^{172}$Yb$^{+}$. The yellow box indicates the threshold that was set in our experiment. Figure (c) and (d) show the experimental results of the GSE and the UR10 sequence, respectively. The colors of the data points correspond to the region indicated in (a) and (b), and the grey solid areas show the simulated results with a typical B-field fluctuation of $\delta B\,=\,$1 - 2 nT within a measurement time of 4 - 5 min. Each measurement point is averaged over 50 repetitions. From the simulation, the UR10 sequence tolerates 38 and 13 times larger values of $\eta_{t}$ and $\eta_{\omega}$, respectively, compared to the GSE sequence.