Statistics and systematics of electron EDM searches with BaF

TL;DR

The NL-$e$EDM experiment investigates the electron EDM using BaF molecules in a spin-precession scheme that links the phase evolution to $d_e$ and to potential systematic biases via a molecular enhancement factor $W_d$. The current 34-hour data set yields a constraint $d_e = 2(3) \times 10^{-25}$ $e$ cm while simultaneously bounding key bias parameters such as the external field $E$ and laser intensities. The approach self-consistently determines experimental parameters by modeling the spin-precession signal, enabling robust limits on systematics. Looking ahead, phase-2 upgrades with a cryogenic buffer-gas beam, laser cooling, hexapole focusing, and improved fluorescence detection aim to boost molecular flux and spin-precession time, achieving orders-of-magnitude gains in statistical sensitivity and potentially competitive eEDM bounds within the coming years.

Abstract

The NL-$e$EDM experiment searches for a non-zero electric dipole moment of the electron $d_e$ ($e$EDM) in the ground state of barium monofluoride (BaF). A beam of BaF from a supersonic expansion source is probed with the spin precession method presented in \cite{Boeschoten2024}. This method permits the extraction of an $e$EDM value as well as values for parameters causing a possible systematic bias leading to a false $e$EDM. The currently achievable sensitivity is limited by statistics collected in a period of 34 hours and yields an $d_e$ of $2(3) \times 10^{-25}$ $e\,$cm. Furthermore, from the same dataset sufficiently strong limits on parameters which can induce a false $e$EDM are extracted. These are mainly the electric field \textbf{E} and the intensity of the lasers fields in the fiducial volume of the experiment. We summarize the steps required to upgrade of the experiment to reach a competitive level on $d_e$, e.g. an intense laser-cooled beam from a cryogenic buffer gas source and the light collection efficiency of fluorescence.

Figures (8)

• Figure 1: Overview of the modular experimental setup to scale. (A) Supersonic expansion source creating BaF molecules with a velocity of about 600m/s and a narrow velocity spread. The source is operated at repetition rates from 5--30 Hz. (B) Optical probing region for determination of the number of molecules and preparation of the molecules in a specific hyperfine state of the $X ^2\Sigma^+$ ground state. These molecules enter (C) the region in which the spin-precession measurement is performed, consisting of a five layer magnetically shielded volume with magnetic fields of order nT and electric fields of several kV/cm. The environmental fields of order 50 $\mu$T are reduced to less than 5 $\mu$T with rectangular field coils around the region. (D) The spin precession is read out by probing the population distribution in the two hyperfine states of the ground state.
• Figure 2: The superposition in the $X^2\Sigma^+,v=0,N=0,F=1$ state is created by a two-photon transition via the electronically excited state $A^2\Pi_{1/2},v=0,J=1/2$. The coupling is achieved by two laser fields with Rabi frequencies $\Omega_S$ and $\Omega_P$ at a typical detuning $\Delta=1$ GHz from the $X^2\Sigma^+ - A^2\Pi_{1/2}$ resonance. The detuning $\delta=\omega_{PS}-\omega_{\rm{HFS}}(E)$ is several kHz from two-photon resonance, where $\omega_{\rm{HFS}}(E)=\omega_{\rm{HFS}}^0+\omega_{\text{tensor}}(E)$ and $\omega_{PS}=\omega_P-\omega_S$. (b) The $X^2\Sigma^+,v=0,N=0$ sublevels of the ground state in electric and magnetic fields. The hyperfine splitting in absence of external fields $\omega_{\text{HFS}}^0$ is around $65.85~\text{MHz}$. The tensor Stark shift of the $m_F=\pm1$ levels, $\omega_{\text{tensor}}(E)$, is around $15~\text{kHz}$ at an electric field of 2 kV/cm. The tensor Stark shift of the $m_F=0$ level is approximately twice that of $\omega_{\text{tensor}}$ with opposite sign. (c) The timing sequence of the laser-light pulses with effective Rabi frequency $\Omega_{PS}$, where typical pulse lengths are $t = 80~\mu$s and the pulse separation period is $T=1$ ms. Energy levels and timings are not to scale.
• Figure 3: An example of the spin-precession signal $P_0(\delta)$ while keeping the parameters constant ($B = 4.40$ nT, $E = 0$ kV/cm, $t=80 ~\mu$s and $T= 2$ ms). The green line is according to the model of Eq. \ref{['eq:SpinPression']}. The hyperfine structure splitting in the ground state is $\omega^0_{\rm{HFS}} = 65.84854(3)$ MHz. The central fringes of the spectrum are most sensitive to a possible $e$EDM.
• Figure 4: The magnetic field inside of the shielding is derived from spin-precession signals when the magnetic shield was exposed to a magnetic field change of $\pm50 ~\mu\mathrm{T}$. The field change inside the shielding was suppressed by $1.3(8) \times 10^{-6}$ in agreement with numerical simulations in COMSOL.
• Figure 5: Pulses with precise timing and phase coherence for the spin-precession methods are generated from light of one laser at 860 nm (TOPTICA TApro). The light is split by a beam splitter into two beams. Each beam is passed through an AOM which is driven by RF pulses generated DDS function generators referenced to a GPS controlled Rb-clock. The pulsed 1st order beams of each AOM are aligned into optical fibers E and F.