Larmor frequency dressing by a nonharmonic transverse magnetic field

We present a theoretical and experimental study of spin precession in the presence of both a static and an orthogonal oscillating magnetic field, which is nonresonant, not harmonically related to the Larmor precession, and of arbitrary strength. Due to the intrinsic nonlinearity of the system, previous models that account only for the simple sinusoidal case cannot be applied. We suggest an alternative approach and develop a model that closely agrees with experimental data produced by an optical-pumping atomic magnetometer. We demonstrate that an appropriately designed nonharmonic field makes it possible to extract a linear response to a weak dc transverse field, despite the scalar nature of the magnetometer, which normally causes a much weaker, second-order response.

Figure 1

Schematic of experimental setup. The probe laser is tuned to the $D_2$ Cs line at 852 nm and unmodulated; the pump laser is tuned to the $D_1$ Cs line at 894 nm and is modulated synchronously with the atomic precession. Both are mixed and coupled to the sensor though polarization-maintaining fibers (PMF). The probe is attenuated down to the microwatt level before the Cs cell, the pump is stopped by an interferential filter (Int fltr) after the cell. A multi-order waveplate renders the pump polarization circular, while leaving the probe polarization linear. The Cs cell contains 23 Torr of N${}_2$ as a buffer gas, which quenches the fluorescence and slows down the Cs atom motion, making it diffusive. The cell is slightly heated to increase the Cs vapor density. The transmitted probe laser polarization is analyzed by a balanced polarimeter made of a Rochon polarizer oriented at ${45}^{\circ }$ with respect to the probe polarization axis and two photodiodes. A differential transimpedance amplifier converts the photocurrent to a voltage signal, which is then digitally acquired. That signal reproduces the Faraday rotation, associated with the time-dependent orientation of the precessing atomic spins.

Figure 2

Ratio between dressed and unperturbed precession frequency as a function of amplitude of oscillating field (expressed in dimensionless quantity $\xi =\gamma B_x/\omega$) in the case of a harmonic transverse field. According to Eq. (28) the plot reproduces the behavior of $|J_0\left(\xi \right)|$.

Figure 3

Comparison of theoretical predictions (line) and experimental points for the ratio between the dressed and unperturbed precession frequency as a function of the amplitude of a square-wave like oscillating field. According to Eq. (37) the behavior is described by the modulus of a sinc function.

Figure 4

Graphs of the function $F(\xi ,r,k,\varphi )$, see Eq. (34), for different values of $k$ and $\varphi$.

Figure 5

Comparison of theoretical predictions (line) and experimental points for different values of $\xi$ and $k$ as a function of $r$ [see Eq. (31)].

Figure 6

Comparison of theoretical predictions (line) and experimental points for $k=3$, $\xi =3$ and two values of and $r$ for the ratio between the dressed and unperturbed precession frequencies as a function of the relative phase between the two Fourier components of the oscillating field [see Eq. (31)].

Figure 7

Comparison of theoretical predictions (line) and experimental points for $k=2,r=1$ and two values of $\xi$ for the ratio between the dressed and unperturbed precession frequencies as a function of the relative phase between the two Fourier components of the oscillating field [see Eq. (31)].

Figure 8

(Upper) Comparison of theoretical predictions (line) and experimental points in presence of a small dc field superimposed on the ac field. The parameters are ${\xi }_0\equiv \gamma B_x^{\left(\mathrm{dc}\right)}/\omega =1.3\times {10}^{-2}$, $\xi =2$, $r=1$, $k=2$, and $\zeta =0.14$. The solid line is calculated using Eq. (26) while the dashed one is calculated using Eq. (22). It is clearly visible the effect of a small ${\xi }_0$. (Lower) The three second-order corrections according to Eq. (26). Notice how the term proportional to ${\xi }_0^2$ has the same value for $\varphi =0$ and $\varphi =\pi$ and thus it is not responsible of the asymmetry.