Magnetically Induced Transparency-Absorption and Normal-Anomalous Dispersion Characteristics of ${}^{87}\text{Rb}$ Medium or Any J-Type Configuration Atomic Vapors Subject to a Vector Magnetic Field and a Weak Resonant Pump

TL;DR

This work develops a comprehensive analytical framework for magnetically induced transparency-absorption (MITA) and normal-anomalous dispersion (MINAD) in a weakly excited ${}^{87}\text{Rb}$ three-level (J-type) system under a vector magnetic field. By solving the optical Bloch equations in stationary, quasi-stationary, and short-pulse regimes with an eigenvector/variation-of-constants approach, it yields closed-form expressions for populations and coherences and reveals a bifurcation at zero longitudinal Zeeman splitting $\Delta_2=0$ that transitions the dynamics from single-mode to two-mode behavior. The study connects the resulting absorption and dispersion to measurable quantities through $\alpha(\omega)$ and $\Delta n(\omega)$ and demonstrates slow radio-frequency oscillations favorable for weak-field sensing alongside rapid optical-frequency oscillations enabling spectrally selective filtering and frequency-comb-like shaping. Collectively, these results provide practical guidance for implementing MITA/MINAD in precision magnetometry and photonic signal processing using atomic vapors, with explicit regimes and parameters applicable to ${}^{87}\text{Rb}$ and similar J-type configurations.

Abstract

We have developed an analytical framework for magnetically induced transparency-absorption (MITA) and normal-anomalous dispersion (MINAD) in a weakly driven ${}^{87}\text{Rb}$ vapor, or any J-type three-level system, under a vector magnetic field. By solving the Bloch equations in the stationary, quasi-stationary, and short-pulse regimes, we obtained closed-form expressions for the atomic populations and coherences and identified a bifurcation in the oscillatory dynamics at zero longitudinal Zeeman splitting. The Fourier-domain analysis reveals alternating transparency/absorption and normal/anomalous dispersion with frequency-dependent sign reversals, enabling spectrally selective filtering and group-delay effects. Slow oscillatory behavior in the radio-frequency range makes the system suitable for weak magnetic-field sensing, while fast oscillations at optical frequencies suggest applications in spectral filtering and frequency-comb-like signal shaping. The results provide a theoretical basis for experimental observation of MITA/MINAD and for optimizing atomic-vapor platforms for precision magnetometry and related photonic functionalities.

Figures (5)

• Figure 1: Schematic configuration
• Figure 2: Absorption coefficient $\omega*\mathfrak{Im}(\omega)$ with magnetically-induced absorption ($> 0$) and transparency ($\leq 0$, where $< 0$ stands for stimulated emission (gain)) periodic ranges in a frequency domain for the parameters $\Omega = 0.1$ MHz, $\Gamma = 6.06536$ MHz, $\Delta_2 = 0.7$ kHz, $L_g = 0.07$ MHz, $\rho_{11} (0) = 1/2$, $\mathfrak{Re}(\rho_{21}(0)) = 0$, $\mathfrak{Im}(\rho_{21}(0)) = 0$, $T = 20 \mu s$ (the time of Fourier transform or the time of fields' action) and in the quasi-stationary regime ($\dot{\rho}_{31} \approx \dot{\rho}_{32} \approx 0$).
• Figure 3: Dispersion $\mathfrak{Re}(\omega)$ with normal ($>0$) and anomalous ($<0$) periodic ranges in a frequency domain for the parameters $\Omega = 0.1$ MHz, $\Gamma = 6.06536$ MHz, $\Delta_2 = 0.7$ kHz, $L_g = 0.07$ MHz, $\rho_{11} (0) = 1/2$, $\mathfrak{Re}(\rho_{21}(0)) = 0$, $\mathfrak{Im}(\rho_{21}(0)) = 0$, $T = 20 \mu s$ (the time of Fourier transform or the time of fields' action) and in the quasi-stationary regime ($\dot{\rho}_{31} \approx \dot{\rho}_{32} \approx 0$).
• Figure 4: The case in Fig. \ref{['fig:AbsorpVLF']}, but in the visible region.
• Figure 5: The case in Fig. \ref{['fig:DispVLF']}, but in the visible region.