Competition between the tensor light shift and nonlinear Zeeman effect

Many precision measurements (e.g., in spectroscopy, atomic clocks, quantum-information processing, etc.) suffer from systematic errors introduced by the light shift. In our experimental configuration, however, the tensor light shift plays a positive role enabling the observation of spectral features otherwise masked by the cancellation of the transition amplitudes and creating resonances at a frequency unperturbed either by laser power or beam inhomogeneity. These phenomena occur thanks to the special relation between the nonlinear Zeeman and light shift effects. The interplay between these two perturbations is systematically studied and the cancellation of the nonlinear Zeeman effect by the tensor light shift is demonstrated.

Figure 1

Geometry of the experiment. A linearly polarized light beam propagates along the $y$ direction and interacts with a sample of Cs atoms contained in a paraffin-coated glass cell. The polarization of the light direction is tilted by an angle $\alpha$ with respect to the $z$ direction of the offset magnetic field $B_{\mathrm{off}}$. Cesium atoms additionally interact with the rf magnetic field $B_{\mathrm{rf}}$ oscillating along the $y$ axis. The polarization state of the light is measured with a balanced polarimeter consisting of a polarizing beam splitter and two photodiodes.

Figure 2

(a) Cesium 6 ${}^2$$S$${}_{1/2}\to$ 6 ${}^2$$P$${}_{1/2}$ transition ($D$${}_1$ line) energy structure with the splitting of the $F=3$ state caused by the linear (LZ) and nonlinear Zeeman (NLZ) effects. The single-ended arrow represents the frequency of the laser beam used in the experiments, the shaded circles symbolize populations of hyperfine components of the ground state, and the double-ended arrows represent the rf transitions between neighboring sublevels. (b) rf spectrum signal components generated by transitions between degenerate (linear Zeeman effect) and nondegenerate (nonlinear Zeeman effect) sublevels. Solid (dashed) lines in the spectrum represent a contribution to the signal from atoms populating sublevels with positive (negative) magnetic quantum number $m$.

Figure 3

rf spectrum of the transitions between Zeeman sublevels of the $F$ $=$ 3 state for an offset field of 143 $\mu$T. The laser light of 5.5 mW power was tuned to the $F=4\to F^{\prime }=3$ transition.

Figure 4

Dependence of the spectrum amplitudes (a) and width, $\delta \nu$, (b) on the offset field $B_{\mathrm{off}}$ for laser light power fixed at 5.6 mW. (a) The amplitudes of the component observed at lower or higher frequency (involving the $m_{\mathrm{F}}=3\leftrightarrow m_{\mathrm{F}}=2/-3\leftrightarrow -2$ transitions) are depicted with blue dots and red diamonds. Solid blue and dashed red line represent the calculated values. (b) Two straight lines represent the tensor light shift (horizontal) and NLZ shift (skew) values, while dashed blue curve depicts the calculated dependence.

Figure 5

Average frequency difference between rf resonance components in the $F=3$ manifold as a function of the laser power. The frequency of the laser beam was tuned to the $F=4\to F^{\prime }=3$ transition (∼10 GHz off the probed $F=3\to F^{\prime }=3,4$ excitations) and the signals were measured at $B_{\mathrm{off}}=143\hspace{0.5em}\mu$T. The dashed horizontal line represents the corresponding NLZ splitting value of 54.6 Hz and the thick line is a linear fit to the data.

Figure 6

Spectra of the rf transitions in the $F$ $=$ 3 ground state (a) for various offset fields around 38 $\mu$T; (b) for a magnetic field stronger than the one corresponding to the cancellation point (38.25 $\mu$T); and (c) at the cancellation point (37.5 $\mu$T). Note the fivefold magnification of the signal sizes in (b) and (c) relative to (a). The width of the narrow feature in (b) is 4 Hz, while the width of the broader pedestal in (b) and (c) is 8 Hz. All signals were recorded for a light power of 5.6 mW.

Figure 7

(a) rf spectrum demonstrating the sign reversal of the signal for a fixed bias magnetic field of 26.86 $\mu$T and with the laser power 2.52 mW (black), 2.75 mW (red), 2.82 mW (green), 3.06 mW (blue). (b) Modeling of the rf spectrum line shape for the similar (2.5, 2.75, 2.9, 3.2 mW) laser powers as in (a).

Figure 8

Dependence of the signal width on laser power with the laser polarization orthogonal to the bias magnetic field ($\alpha ={90}^{°}$). Spectra were measured for an offset magnetic field of 26.86 $\mu$T.

Figure 9

Dependence of the signal amplitude (red diamonds) and width (blue dots) on laser power. Lines represent the calculated values of the amplitude (solid) and width (dashed). The signals were recorded for $B_{\mathrm{off}}=0.286\hspace{0.5em}\mu$T.