Faraday-rotation fluctuation spectroscopy with static and oscillating magnetic fields

By Faraday-rotation fluctuation spectroscopy, one measures the spin noise via Faraday-induced fluctuations of the polarization plane of a laser transmitting the sample. In the first part of this paper, we present a theoretical model of recent experiments on alkali-metal gas vapors and semiconductors, done in the presence of a static magnetic field. In a static field, the spin noise shows a resonance line, revealing the Larmor frequency and the spin-coherence time $T_2$ of the electrons. Second, we discuss the possibility to use an oscillating magnetic field in the Faraday setup. With an oscillating field applied, one can observe multiphoton absorption processes in the spin noise. Furthermore, an oscillating field could also help to avoid line broadening due to structural or chemical inhomogeneities in the sample, and thereby increase the precision of the spin-coherence time measurement.

Figure 1

A linear polarized laser is sent through the sample. Due to spin-orbit coupling and the Faraday effect, the laser polarization plane rotates depending on the electron spins inside the sample. While the time-averaged Faraday-rotation angle is zero, since there is no net magnetization collinear to the laser propagation direction, the Faraday-rotation fluctuations are finite.

Figure 2

Measured Faraday-rotation noise of a spin ensemble by Oestreich et al. (Ref. 29), together with a fit of a Lorentzian function as expected for a single spin, see Eq. (5).

Figure 3

Sketch of the electron spin dynamics in the Faraday setup with a static magnetic field. (a) At time $t=0$, the spin is measured. (b) Between the two measurement, the spin precesses in a static magnetic field $\mathit{B}$. (c) If the second measurement takes place after a full revolution, it reproduces the outcome of the first measurement.

Figure 4

Sketch of the spin dynamics in the Faraday setup with an oscillating field. (a) The initial measurement of the spin state. (b) In the external field the spin precesses in one direction. (c) When the field changes its sign, the spin precesses in the opposite direction. (d) After a full field-oscillation time, the spin is again in its initial state.

Figure 5

Spectrum of the spin-spin correlation function for different ratios of magnetic field strength to oscillation frequency. For a higher ratio ${\omega }_0∕\gamma$, more resonance lines appear at higher frequencies, while the signal strength decreases. Here, we chose $T_2^{-1}=0.1\gamma$.

Figure 6

Spectrum of the spin-spin correlation function for ${\omega }_0∕\gamma =5$. Upper panel: With (solid) and without (dashed) $g$-factor fluctuations. The $g$-factor fluctuations cannot lead to inhomogeneous line broadening. Lower panel: With (solid) and without (dashed) a random static magnetic field component, which describes hyperfine interaction. In contrast to the $g$-factor fluctuations, hyperfine interaction can lead to a line broadening.

Figure 7

Spin-spin correlation spectrum, as a function of the ratio ${\omega }_0∕\gamma$ and frequency $\omega$ for $T_2^{-1}=0.1\gamma$.

Figure 8

Diagrammatic representation of $S\left(t\right)$. The operators ${\widehat{s}}_x$ are placed on the upper or lower branch of the time contour at time $0$ and $t$, respectively.