Nonlinear magneto-optical rotation with modulated light in tilted magnetic fields

Larmor precession of laser-polarized atoms contained in antirelaxation-coated cells, detected via nonlinear magneto-optical rotation (NMOR), is a promising technique for a new generation of ultrasensitive atomic magnetometers. For magnetic fields directed along the light propagation direction, resonances in NMOR appear when linearly polarized light is frequency or amplitude modulated at twice the Larmor frequency. Because the frequency of these resonances depends on the magnitude but not the direction of the field, they are useful for scalar magnetometry. Additional NMOR resonances at the Larmor frequency appear when the magnetic field is tilted away from the light propagation direction in the plane defined by the light propagation and polarization vectors. These resonances, studied both experimentally and with a density matrix calculation in the present work, offer a convenient method of achieving additional information about a direction of the magnetic field.

Figure 1

Experimental setup. The magnetic field coils mounted inside the innermost shielding layer and used for creation of arbitrarily oriented magnetic field are not shown.

Figure 2

The FM NMOR in-phase and quadrature signals vs modulation frequency recorded for various angles between the magnetic field and the light-propagation direction in the $xz$ plane. A tiny residual signal observed for 90° is a result of a small misalignment in the experiment (the polarization of light is not completely parallel to the $y$ axis).

Figure 3

(Color online) The amplitude of the FM NMOR signals recorded at $2{\Omega }_L$ vs the tilt angle of the magnetic field in the $xz$ plane. The solid line is a cosine fit to the experimental points.

Figure 4

Theoretical calculation of FM NMOR signal vs modulation frequency for an $F=1\to F=0$ transition for various tilt angles of the magnetic field in the $xz$ plane.

Figure 5

(Color online) Illustration of the time dependence of FM NMOR for $\mathbf{B}$ in the $xz$ plane at 30° to the light propagation direction $\widehat{\mathbf{x}}$, using results of the calculation for an $F=1\to F^{\prime }=0$ transition. (a) Angular momentum probability surfaces depicting the dynamic part of the atomic alignment at various times during the Larmor period $\tau$. The alignment returns to its original position after time $\tau ∕2$. The double-headed arrows represent the light polarization before and after the medium. (b) Optical rotation as a function of time for the case shown in (a) (solid line), and for $\mathbf{B}$ along $\widehat{\mathbf{x}}$ (dashed line). The dominant signal has period $\tau ∕2$ (frequency $2{\Omega }_L$), corresponding to the periodicity of the polarization state shown in (a). The effect of using the modulated pump as a probe beam can be seen here as a small additional modulation with period $\tau$.

Figure 6

The FM NMOR signal vs modulation frequency ${\Omega }_m$ recorded for different tilt angles of the magnetic field in the $xy$ plane. We proved with theoretical simulations that the small signal observed at ${\Omega }_L$ for the magnetic field oriented along the $y$ axis is due to the misalignment between the magnetic field and the light polarization direction.

Figure 7

Theoretical calculation of FM NMOR signal vs modulation frequency for an $F=1\to F=0$ transition for various tilt angles of the magnetic field in the $xy$ plane.

Figure 8

The amplitude of the FM NMOR signals recorded at ${\Omega }_L$ and $2{\Omega }_L$ vs the tilt angle of the magnetic field in the $xy$ plane.

Figure 9

(Color online) As Fig. 5, but with $\mathbf{B}$ in the $xy$ plane at 17.5° to the light propagation direction $\widehat{\mathbf{x}}$. (a) The alignment axis in this case is not perpendicular to the magnetic field, so it takes a full Larmor period $\tau$ to return to its original state. (b) The optical rotation. The optical anisotropy in the $yz$ plane has components at ${\Omega }_L$ and $2{\Omega }_L$.