Magneto-optic spectroscopy with linearly polarized modulated light: Theory and experiment

We investigate the polarization modulation between two linear orthogonal states of the laser beam that synchronously pumps time-dependent atomic alignment in caesium atoms exposed to a static magnetic field. Because of the atomic alignment symmetry two independent groups of resonances can be distinguished in the transmitted light: when modulation frequency matches either the Larmor frequency or its second harmonics, ${\omega }_L$ and $2{\omega }_L$, respectively. We report on our experiments, and discuss a model that perfectly reproduces the observed spectra. We have observed that the amplitudes and linewidths of resonances at ${\omega }_L$ and $2{\omega }_L$ show magnetic-field direction dependence. This peculiar behavior makes our approach interesting for future application to atomic magnetometry, in view of a dead-zone free high-sensitivity magnetometer.

Figure 1

The interaction geometry. The laser beam propagating along the $z$ axis passes through the caesium cell kept at room temperature. The transmitted signal is recorded and analyzed in standard way. A magnetic field whose strength is constant, and the direction can be varied, is applied to the atoms, The Larmor frequency ${\omega }_L$ is typically about $2\pi \times 2$ kHz. The beam polarization is switched between two orthogonal states via a polarization modulator: the horizontal (${\epsilon }_x$) and the vertical (${\epsilon }_y$) linear polarization, at a rate $\Omega$ which is swept across the resonances. In the lower part is sketched the time dependence of the polarization vector $\epsilon$ components.

Figure 2

In-phase and quadrature lock-in signals at the first ($q=1$) and third ($q=3$) harmonic of $\Omega$ normalized taking into account the appropriate scaling factors: black and red-dashed lines represent experiment and model results, respectively. The magnetic-field direction is defined by $\theta ={45}^{\circ }$ and $\phi ={120}^{\circ }$.

Figure 3

Amplitude of the in-phase (absorptive) part of the lock-in resonance at $\Omega =2{\omega }_L$ as a function of the detection harmonic for odd $q$ value, i.e., $q=1,3,5,7,9,11,13$. The dashed line is obtained dividing $A_1$ by $q^2$. Inset: Relative amplitude of the resonances at $\Omega =2{\omega }_L$ (black) and ${\omega }_L$ (gray); the relative amplitude defined as $D_q/A_q$ for each resonance is expected to be 1.

Figure 4

Linewidth ($\Gamma /2\pi$) of resonances at ${\omega }_L/q$ and $2{\omega }_L/q$ as a function of $q$. The dots are the average values obtained by fit of the experimental resonances. The lines are calculated accounting for the $1/q$ dependence of the ${\Gamma }_i$, where the initial values, i.e., ${\Gamma }_1/2\pi =9.55$ Hz and ${\Gamma }_2/2\pi =8.46$ Hz at $q=1$, are calculated for the set of experimental parameters by applying Eqs. (11).

Figure 5

In-phase (absorptive) and quadrature (dispersive) lock-in signal at the second ($q=2$) and fourth ($q=4$) harmonic of the modulation frequency, as a function of the dimensionless ratio between the modulation frequency $\Omega$ and the Larmor frequency ${\omega }_L$. The spectra are normalized with the appropriate scaling factors. The angles $\theta$ and $\varphi$ are ${45}^{\circ }$ and ${50}^{\circ }$. Black and dashed-red lines represent experiment and model results, respectively.

Figure 6

Relative amplitude of the $2{\omega }_L$ group of resonances as a function of the detection harmonic for even $q$ value, i.e., $q=2,4,6,8,10$. We have studied the subharmonic resonances up to $n=9$.

Figure 7

The linewidths ${\Gamma }_1$ and ${\Gamma }_2$ of the resonance at $2{\omega }_L$ (upper plot) and ${\omega }_L$ (lower plot): dots are obtained by fitting the experimental resonances, solid lines are the average experimental values, i.e., $9.3\left(7\right)$ Hz and $8\left(1\right)$ Hz, while the dotted lines are the theoretical predictions for $\theta ={45}^{\circ }$, i.e. $9.55$ and $8.46$ Hz, respectively.

Figure 8

Experimental (dots) and theoretical (line) values for the normalized amplitudes ($\propto C_1$ and $C_2$) of $2{\omega }_L$ and ${\omega }_L$ resonance alternate in a complementary way when the direction of the magnetic field is varied. We remark that in the $\theta ={90}^{\circ }$ inset, the experimental amplitude minima do not reach exactly zero, even though strongly suppressed; the resonances are clearly visible. This is not in contrast with the model predictions when all orders in $\eta$ are taken into account, as we have verified by numerical simulation.

Figure 9

Linewidth of resonance at ${\omega }_L$ (${\Gamma }_2/2\pi$), and $2{\omega }_L$ (${\Gamma }_1/2\pi$) as a function of $\theta$: dots are experiments and line is Eqs. (11).