Nonlinear magneto-optical rotation with frequency-modulated light in the geophysical field range

Recent work investigating resonant nonlinear magneto-optical rotation (NMOR) related to long-lived $({\tau }_{\mathrm{rel}}\sim 1\mathrm{s})$ ground-state atomic coherences has demonstrated potential magnetometric sensitivities exceeding ${10}^{-11}\mathrm{G}∕\sqrt{\mathrm{Hz}}$ for small $(\lesssim 1\mathrm{\mu }\mathrm{G}$) magnetic fields. In the present work, NMOR using frequency-modulated light (FM NMOR) is studied in the regime where the longitudinal magnetic field is in the geophysical range$(\sim 500\mathrm{mG})$, of particular interest for many applications. In this regime a splitting of the FM NMOR resonance due to the nonlinear Zeeman effect is observed. At sufficiently high light intensities, there is also a splitting of the FM NMOR resonances due to ac Stark shifts induced by the optical field, as well as evidence of alignment-to-orientation conversion type processes. The consequences of these effects for FM-NMOR-based atomic magnetometry in the geophysical field range are considered.

Figure 1

Level diagram (not to scale) for the $F=2\to F^{\prime }=1$ transition investigated in the present work. Dashed bars represent the unperturbed energies of the ground-state Zeeman sublevels; solid bars represent the energies shifted by a relatively strong magnetic field. Linearly polarized light represented here as a superposition of left- and right-circularly polarized light (transitions indicated by double-sided arrows) interacts with three separate coherent superpositions of ground-state Zeeman sublevels, each with $\mathrm{\Delta }{M}_F=2$. Due to the nonlinear Zeeman effect, the energy difference between the $M_F=-2,0$ sublevels and the $M_F=0,2$ sublevels differs by $\mp {\delta }^{\mathrm{NLZ}}$ from the energy difference between the $M_F=-1,1$ sublevels [see Eqs. (1, 2) and surrounding discussion]. This yields three separate FM NMOR resonances at sufficiently high fields (where ${\delta }^{\mathrm{NLZ}}\gtrsim {\gamma }_{\mathrm{rel}}$), whose positions as a function of the modulation frequency ${\Omega }_m$ are shown in the inset. Here we ignore coherences and the Zeeman effect in the excited state, which do not play a prominent role in the effects discussed in the present work.

Figure 2

Simplified diagram of the FM NMOR setup. The balanced polarimeter incorporating the polarizer, analyzer, and photodiodes PD1 and PD2 detects signals due to time-dependent optical rotation of linearly polarized, frequency-modulated light. Not shown are a set of four-layer magnetic shields and a set of coils to control the dc magnetic field as well as some gradients.

Figure 3

Upper traces: Optical rotation signals for $B=405\mathrm{mG}$ and light power of $95\mathrm{\mu }\mathrm{W}$. Solid (hollow) triangles show the in-phase (quadrature) component. Overlying the data is a fit to three dispersive (absorptive) Lorentzians. Lower traces: fitted in-phase components of the three different $\mathrm{\Delta }{M}_F=2$ resonances, plotted separately.

Figure 4

Schematic representation (not to scale) of the differential ac Stark effect for an $F=2\to F^{\prime }=1$ transition. Linearly polarized light, tuned to the low-frequency side of the atomic resonance $(\Delta <0)$, interacts with three separate coherent superpositions of ground-state Zeeman sublevels with $\mathrm{\Delta }{M}_F=2$. Dashed bars indicate the energies of the Zeeman sublevels shifted by the magnetic field (Fig. 1). The solid bars indicate the energies of the Zeeman sublevels shifted by the ac Stark effect due to the optical electric field of sufficiently intense light. Because the transitions have different dipole moments (indicated by the thickness of the transition arrows and the numerical values next to them), the Zeeman sublevels are shifted by different amounts. Thus the ac Stark effect can lead to relative shifts by $\pm ({\delta }_{\pm 2}-{\delta }_0)$ of the quantum-beat frequencies for the coherences between the states $\mid F=2,M_F=0⟩$ and $\mid F=2,M_F=\pm 2⟩$, respectively. The differential shifts due to the ac Stark effect can either add to or cancel the shifts due to the nonlinear Zeeman effect, depending on the sign of the optical detuning $\Delta$.

Figure 5

Density matrix calculations of in-phase (solid line) and quadrature (dashed line) optical rotation signals as a function of modulation frequency for an $F=2\to F^{\prime }=1$ transition. Parameters in units of the natural linewidth ${\gamma }_0$ are transit width ${\gamma }_t={10}^{-6}$, modulation depth $=1$, average laser detuning $=-1$, Rabi frequency (reduced dipole matrix element times optical electric field) $=3\times {10}^{-3}$, Larmor frequency ${\Omega }_L=5\times {10}^{-2}$, and nonlinear Zeeman parameter (shift per Larmor frequency squared) $={10}^{-3}$.

Figure 6

Top panel: Total splitting of the FM NMOR resonances resulting from both the ac Stark shift and the nonlinear Zeeman effect as a function of light power for different magnetic fields. Overlying the data are linear fits. Bottom panel: nonlinear Zeeman shift determined by extrapolating to zero light power the total splitting $\delta$ in the top panel. Error bars on each point are determined by the scatter of the data in the top panel. The solid line is the theoretical prediction, given by ${\delta }^{\mathrm{NLZ}}\approx ({\mu }_{B}B{)}^2∕{\Delta }_{\mathrm{HF}}$.

Figure 7

Width (top panel, defined in the text) and amplitude (bottom panel) of the central resonance of the in-phase component for several different magnetic fields. Points of equal magnetic field are connected to guide the eye.

Figure 8

Ratio of the in-phase to quadrature amplitudes for the central resonance at several different magnetic fields and modulation depths. Overlying the data for a modulation depth of $\approx 40\mathrm{MHz}$ is a smooth curve to guide the eye.