Optical Spectroscopy of Spin Noise

Spontaneous fluctuations of the magnetization of a spin system in thermodynamic equilibrium (spin noise) manifest themselves as noise in the Faraday rotation of probe light. We show that the correlation properties of this noise over the optical spectrum can provide clear information about the composition of the spin system that is largely inaccessible for conventional linear optics. Such optical spectroscopy of spin noise, e.g., allows us to clearly distinguish between optical transitions associated with different spin subsystems, to resolve optical transitions that are unresolvable in the usual optical spectra, to unambiguously distinguish between homogeneously and inhomogeneously broadened optical bands, and to evaluate the degree of inhomogeneous broadening. These new possibilities are illustrated by theoretical calculations and by experiments on paramagnets with different degrees of inhomogeneous broadening of optical transitions [atomic vapors of ${}^{41}\mathrm{K}$ and singly charged (In,Ga)As quantum dots].

Figure 1

Modeled optical spectra of a spin system having two (a) resolved and (b) unresolved absorption lines. The top panels show the absorption (black lines) and conventional Faraday-rotation (Verdet, red/gray lines) spectra. The middle and lower panels show the calculated optical spectra of spin noise when the two transitions originate from a common spin system (correlated spin fluctuations) or from independent spin systems (uncorrelated fluctuations), respectively.

Figure 2

(a) Calculated and normalized absorption, conventional FR, and OSN spectra for spin-dependent optical transitions exhibiting different ratios of inhomogeneous to homogeneous linewidth $\epsilon ={\gamma }_{\mathrm{inh}}/{\gamma }_h=0.1$, 1, and 10. (b) Evolution of the OSN spectrum with increasing $\epsilon$. For ease of comparison, ${\gamma }_h+{\gamma }_{\mathrm{inh}}$ was fixed. (c) Enhancement factor and dip depth versus $\epsilon$. The solid lines correspond to the OSN case, and the dashed line corresponds to $\mathrm{FR}$ squared.

Figure 3

(a) The raw spin noise power spectral density from fluctuations of $4S$ electron spins in ${}^{41}\mathrm{K}$, at different detunings of the probe laser from the $D1$ optical transition (1.6106 eV). $T=110°\mathrm{C}$ ($\sim 25\%$ absorption on resonance). A 3 G transverse magnetic field shifts the spin noise to $\sim 2\mathrm{MH}z$. Essentially, no noise is observed on resonance, as is expected for a homogeneously broadened transition. (b) The OSN spectrum: the total spin noise power versus probe photon energy. The line is a fit to a single squared FR spectrum ${\theta }_h^2(E)$ with ${\gamma }_h=14.7\mu \mathrm{eV}$.

Figure 4

(a) Squares and circles show the conventional FR spectrum and the OSN spectrum from an inhomogeneously broadened ensemble of (In,Ga)As QDs that are singly charged by holes. The lines are fits using ${\theta }_h(E)$ and ${\theta }_h^2(E)$, respectively, convolved with an inhomogeneous Gaussian broadening of ${\gamma }_{\mathrm{inh}}=8.5\mathrm{meV}$ [see Eq. (4)]. (b) The total spin noise power (measured at 1.406 eV) increases dramatically as temperature drops and the homogeneous linewidth ${\gamma }_h$ of the individual QDs in the ensemble decreases. The squares and circles are two data sets taken at different locations on the sample. The line is a simulation of Eq. (6).