Selective Addressing of High-Rank Atomic Polarization Moments

We describe a method of selective generation and study of polarization moments of up to the highest-rank $\kappa =2F$ possible for a quantum state with total angular momentum $F$. The technique is based on nonlinear magneto-optical rotation with frequency-modulated light. Various polarization moments are distinguished by the periodicity of light-polarization rotation induced by the atoms during Larmor precession and exhibit distinct light-intensity and frequency dependences. We apply the method to study polarization moments of ${}^{87}\mathrm{R}\mathrm{b}$ atoms contained in a vapor cell with antirelaxation coating. Distinct ultranarrow (1-Hz wide) resonances, corresponding to different multipoles, appear in the magnetic-field dependence of the optical rotation. The use of the highest-multipole resonances supported by a given system has important applications in quantum and nonlinear optics and in magnetometry.

Figure 1

Surfaces for which the distance to the origin in a given direction is proportional to the probability of finding the projection $M=F$ along this direction [16]. Plotted are atomic states each consisting of population in addition to (a) quadrupole (${\rho }_0^{(2)}$), (b) hexadecapole (${\rho }_0^{(4)}$), (c) same as in (b), but rotated by $\pi /2$ around the $x$ axis, (d) the average of (b) and (c), which has a fourfold symmetry with respect to rotations around $\widehat{\mathbf{x}}$. The polarization states produced in the experiment correspond to (a) and (d) rotating around an $\widehat{\mathbf{x}}$-directed magnetic field at the Larmor frequency.

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.

Figure 3

An example of the magnetic-field dependence of the FM NMOR signals showing quadrupole resonances at $B=\pm 143.0\mu \mathrm{G}$, and the hexadecapole resonances at $\pm 71.5\mu \mathrm{G}$. Laser modulation frequency is 200 Hz, and modulation amplitude is 40 MHz peak to peak; the central frequency is tuned to the low-frequency slope of the $F=2\to F^{\prime }=1$ absorption line. Plots (a) and (b) show the in-phase component of the signal at two different light powers; plot (c) shows the quadrature component. Note the increase in the relative size of the hexadecapole signals at the higher power. The insets show zooms on hexadecapole resonances.

Figure 4

Spectral dependences of the (a) quadrupole and (b) hexadecapole signals measured at light power of $800\mu \mathrm{W}$, and (c) the low-power ($0.4\mu \mathrm{W}$) transmission spectrum (no FM). Note the different spectral dependences for the FM NMOR signals in (a) and (b), and, in particular, the absence of the signal at the $F=1\to F^{\prime }$ transitions in (b). The dashed line indicates the central laser frequency at which the measurements represented in Figs. 3, 5, 6 were taken.

Figure 5

Signal amplitude vs the input laser power. Filled circles: the quadrupole resonance; open circles: the hexadecapole resonance. The inset shows the expanded low-power region for the latter. From these data, we determine the initial exponent in the power dependence of the signal to be 1.96(6) and 3.75(37) for the quadrupole and hexadecapole cases, respectively. The expected exponents are 2 and 4.

Figure 6

Resonance width vs the input laser power. Filled circles: the quadrupole resonance; open circles: the hexadecapole resonance. The widths extrapolated to zero light power are found to be $\Delta B_q=0.848(4)\mu \mathrm{G}$ and $\Delta B_h=0.904(33)\mu \mathrm{G}$ for the quadrupole and hexadecapole cases, respectively.