Precision measurement of the ground-state hyperfine constant for $^9Be^+$ in a linear Paul trap via magnetically insensitive hyperfine transitions

TL;DR

This work reports a high-precision measurement of the ground-state hyperfine constant $A$ for $^{9}$Be$^{+}$ by probing the magnetically insensitive transition $|F=2,m_{F}=0\rangle \rightarrow |F=1,m_{F}=0\rangle$ in a linear Paul trap with a Coulomb crystal. By scanning near-zero magnetic fields and fitting the observed resonances with a Breit–Rabi model incorporating higher-order Zeeman terms, the authors extract $A = -625.008840(35)\ \mathrm{MHz}$ with a relative precision of $5.6 \times 10^{-8}$, along with a parameter $k$ and zero-field offset $I_{0}$ that characterize the magnetic-field dependence. A detailed uncertainty budget isolates contributions from BeH$^{+}$ lifetime effects, magnetic-field inhomogeneities, and second-order Doppler shifts, achieving a total uncertainty of 35 Hz. The study also derives an effective nuclear Zemach radius of $4.03(5)$ fm and discusses discrepancies with some prior high-field results, highlighting avenues for further experimental and theoretical refinement in atomic-nuclear structure tests.

Abstract

Direct measurements of the ground-state magnetically insensitive hyperfine transition |F=2,mF=0>->|F=1,mF=0> of $^9Be^+$ ions have been performed using microwave-driven state transfer. The $^9Be^+$ ions are confined and laser-cooled in a linear Paul trap, forming a Coulomb crystal. The transition frequencies have been measured over a magnetic field range of $ \pm 0.5 mT $ centered at zero magnetic field, and the acquired data were fitted accounting for the high-order Zeeman effect. The hyperfine constant A is determined to be -625.008840(35) MHz, achieving a relative precision of $ 5.6 \times 10^{-8}$.

Figures (4)

• Figure 1: Experimental setup for microwave spectroscopy of $^{9}\mathrm{Be}^{+}$ ions. The upper-right inset shows a $^{9}\mathrm{Be}^{+}$ Coulomb crystal imaged by the EMICCD camera. MSG: microwave signal generator; PA: power amplifier; GL: Glan lens; PMT: photomultiplier tube detector; EMICCD: electron-multiplying intensified charge-coupled device.
• Figure 2: Energy level structure of the $^{9}\mathrm{Be}^{+}$ ion (not to scale). The diagram shows the hyperfine levels of the ground state $^{2}S_{1/2}$ and the excited state $^{2}P_{3/2}$ under a weak magnetic field. The 313 nm cooling and repumping lasers are indicated by blue and purple arrows, respectively. Microwave-driven state transfers are denoted by gray arrows. The red arrow marks the magnetically insensitive transition within the $^{2}S_{1/2}$ ground state.
• Figure 3: (a) Timing sequence of a single laser-microwave multi-pulse experiment. The five microwave transitions correspond to frequencies $f_1$, $f_2$, $f_0$, $f_2$, and $f_1$, respectively, as indicated in Fig. \ref{['fig:levels']}. (b) Observed Rabi oscillation spectrum of the transition $(F, m_{F}) = \left| 2, 2 \right\rangle \rightarrow \left| 1, 1 \right\rangle$ at its resonant frequency of $f = 1235.042\,\mathrm{MHz}$, a coil current of $I = 1\,\mathrm{A}$, and microwave signal generator output power of $P = 237\,\mathrm{mV}$. (c) Observed microwave spectrum of the transition $(F, m_{F}) = \left| 2, 2 \right\rangle \rightarrow \left| 1, 1 \right\rangle$ at a coil current of $I = 0.3\,\mathrm{A}$. The spectrum is scanned over a 36 kHz range with a step size of 100 Hz. Each data point represents the average of 120 repeated measurements.
• Figure 4: Resonance frequencies of the $^{2}S_{1/2} \left| F = 2, m_{F} = 0 \right\rangle \rightarrow \left| F = 1, m_{F} = 0 \right\rangle$ transition as a function of Helmholtz coil current. The error bars represent the uncertainties obtained from fitting the resonance curves. Current fluctuations are incorporated as horizontal error bars in the fitting process.