High-order magnetic resonance in the presence of strong longitudinal radio-frequency magnetic field based on atomic alignment states

A longitudinal radio-frequency (rf) magnetic field resonating at the Larmor frequency along the $\widehat{z}$ direction in the presence of a static transverse magnetic field s investigated. Optical-radio-frequency double resonance in ${}^{85}\mathrm{Rb}$ characterizes the strength of an external magnetic field through the precession frequency of ${}^{85}\mathrm{Rb}$ atoms. The measurements are carried out in a paraffin-coated ${}^{85}\mathrm{Rb}$ cell with a spin alignment configuration. An intriguing discovery of magnetic resonance peaks is observed at subharmonics of the Larmor frequency, accompanied by the emergence of nonlinear effects in the vicinity of the resonance. Notably, when the Larmor frequency reaches zero, the optical rotation signals induced by the rf magnetic field demonstrate a superposition of high-order harmonics of the rf frequency. Based on the simplified density matrix formalism and perturbation treatment, analytical expressions for the optical rotation are consistent with experimental outcomes. These findings hold the potential of extending our understanding on unveiling the dynamics of population and coherence among Zeeman sublevels, thereby advancing our knowledge in the field of alignment magnetometry.

Figure 1

The general diagram depicts the interaction of two linearly polarized lights with the $F_g=2,3\to F_e=2$ transitions. The probe light (red line) and the repump light (blue line) create alignment in the $F_g=3$ states, with the quantization axis along the $\widehat{z}$ direction. The ground Zeeman sublevels are separated by an energy equivalent to the Larmor frequency ${\mathrm{\Omega }}_L$. The magnetic Rabi frequency between these Zeeman sublevels is denoted as ${\mathrm{\Omega }}_{\mathrm{rf}}$ (green line). For clarity, we only depict the allowed spontaneous decay for the $|F_e=2,m_F=0\rangle$ state, as denoted by the red dashed line.

Figure 2

Schematic of the experimental setup. A triaxial coil constructed by Lee-Whiting coils and saddle coils, is designed to provide a static or oscillated magnetic field. The static magnetic field direction is along the $\widehat{z}$ direction, which generated by a signal generator. A driving rf field along the $\widehat{z}$ direction is produced by a signal generator. The repump and probe beams, emitted from a distributed feedback laser and separated by about $5\mathrm{mm}$, propagate along the $\widehat{y}$ direction with the initial polarization along the $\widehat{z}$ direction. Optical rotation signal of probe beam induced by atoms is detected and analyzed by a polarimeter and lock-in amplifier. A polarimeter contains a pair of photodiodes $\mathrm{PD}+$ and $\mathrm{PD}-$, a mirror, HWP, and PBS. LP is linearly polarized plate; HWP is half-wave plate; PD is photodiode; PBS is polarized beam splitter.

Figure 3

The distributions of the magnetic resonance curves are centered at ${\mathrm{\Omega }}_L/m$, where $m$ can be numerically simulated by Eq. (9). The transverse frequency ${\omega }_{x0}=2\pi \times 5\mathrm{Hz}, A_0=0.2$, the Larmor frequency ${\mathrm{\Omega }}_L=2\pi \times 10\mathrm{kHz}$. The rf field with amplitude of ${\omega }_1=0.8{\mathrm{\Omega }}_L$ and frequency of $\omega$ varying from $2\pi \times 90\mathrm{Hz}$ to $2\pi \times 10.1\mathrm{kHz}$.

Figure 4

Magnetic resonance peaks were observed at subharmonics of the Larmor frequency with the same gain of photodiode amplifier under two different transverse frequencies ${\omega }_{x0}=2\pi \times 5\mathrm{Hz}$ and $2\pi \times 20\mathrm{Hz}$. (a)–(d) are magnetic resonance curves centered at frequency of ${\mathrm{\Omega }}_L/6, {\mathrm{\Omega }}_L/3, {\mathrm{\Omega }}_L/2$, and ${\mathrm{\Omega }}_L$. The nonlinear effect near the resonance frequency shown in blue triangle lines and green inverted triangle lines in (c) and (d) can be interpreted by Eqs. (13) and (A2b).

Figure 5

Magnetic resonance curves at center of ${\mathrm{\Omega }}_L=2\pi \times 10\mathrm{kHz}$ simulated by perturbation theory. (a) shows the real part of ${\rho }_{12}^{\mathrm{ms}}, {\rho }_{12}^{\mathrm{mt}}$, and ${\rho }_{12}^{\mathrm{m}}$. (b) is imaginary part of ${\rho }_{12}^{\mathrm{ms}}, {\rho }_{12}^{\mathrm{mt}}$, and ${\rho }_{12}^{\mathrm{m}}$. The deformations of Lorentzian magnetic resonance curves come from ${\rho }_{12}^{\left(3\right)}$.

Figure 6

Measurements of optical rotation signal at different modulation parameters of $\beta$. The static magnetic field in $\widehat{x}$ direction is about $21.4\mathrm{pT}$, which in $\widehat{y}$ and $\widehat{z}$ axes are zero. The relaxation rate $R_{\mathrm{rel}}=2\pi \times 10\mathrm{Hz}$. The triangles are experimental data and black lines are numerical fitting curves.