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arXiv:1512.08817 · Physical Review A

Higher-order spin-noise spectroscopy of atomic spins in fluctuating external fields

Fuxiang Li, Scott A. Crooker, Nikolai A. Sinitsyn

Published Mar 9, 2016• 21 citations
arXivHTMLPDF

Abstract

We discuss the effect of external noisy magnetic fields on mesoscopic spin fluctuations that can be probed in semiconductors and atomic vapors by means of optical spin-noise spectroscopy. We show that conventional arguments of the law of large numbers do not apply to spin correlations induced by external fields, namely, the magnitude of the 4th-order spin cumulant grows as $\ensuremath{\sim}{N}^{2}$ with the number $N$ of observed spins, i.e., it is not suppressed in comparison to the 2nd-order cumulant. This allows us to design a simple experiment to measure the 4th-order cumulant of spin fluctuations in an atomic system near thermodynamic equilibrium and develop a quantitative theory that explains all observations.

DOI
10.1103/physreva.93.033814

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We discuss the effect of external noisy magnetic fields on mesoscopic spin fluctuations that can be probed in semiconductors and atomic vapors by means of optical spin-noise spectroscopy. We show that conventional arguments of the law of large numbers do not apply to spin correlations induced by external fields, namely, the magnitude of the 4th-order spin cumulant grows as \ensuremath∼N2\ensuremath{\sim}{N}^{2}\ensuremath∼N2 with the number NNN of observed spins, i.e., it is not suppressed in comparison to the 2nd-order cumulant. This allows us to design a simple experiment to measure the 4th-order cumulant of spin fluctuations in an atomic system near thermodynamic equilibrium and develop a quantitative theory that explains all observations.
Abstract We theoretically and numerically investigate the spin fluctuations induced in a thermal atomic ensemble by an external fluctuating uniaxial magnetic field, in the context of a standard spin noise spectroscopy (SNS) experiment. We show that additional spin noise is excited, which dramatically depends on the magnetic noise variance and bandwidth, as well as on the power of the probe light and its polarization direction. We develop an analytical perturbative model proving that this spin noise first emerges from the residual optical pumping in the medium, which is then converted into spin fluctuations by the magnetic noise and eventually detected using SNS. The system studied is a spin-1 system, which thus shows both Faraday rotation and ellipticity noises induced by the random magnetic fluctuations. The analytical model gives results in perfect agreement with the numerical simulations, with potential applications in future experimental characterization of stray field properties and their influence on spin dynamics.
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 41K and singly charged (In,Ga)As quantum dots].
We report noninvasive detection of spin coherence in a collection of Raman-driven cold atoms using dispersive Faraday rotation fluctuation measurements, which opens possibilities of probing spin correlations in quantum gases and other similar systems. We demonstrate five orders of magnitude enhancement of the measured signal strength as compared to traditional spin noise spectroscopy with thermal atoms in equilibrium. Our observations are in good agreement with the comprehensive theoretical modeling of the driven atoms at various temperatures. The extracted spin relaxation rate of cold rubidium atoms with atom number density $\ensuremath{\sim}10{\phantom{\rule{0.16em}{0ex}}}^{9}/{\mathrm{cm}}^{3}$ is of the order of $3\phantom{\rule{0.16em}{0ex}}\ifmmode\times\else\texttimes\fi{}\phantom{\rule{0.16em}{0ex}}10{\phantom{\rule{0.16em}{0ex}}}^{3}\phantom{\rule{4pt}{0ex}}{\mathrm{s}}^{\ensuremath{-}1}$ at 150 $\ensuremath{\mu}\mathrm{K}$, two orders of magnitude less than $3\phantom{\rule{0.16em}{0ex}}\ifmmode\times\else\texttimes\fi{}\phantom{\rule{0.16em}{0ex}}10{\phantom{\rule{0.16em}{0ex}}}^{5}\phantom{\rule{4pt}{0ex}}{\mathrm{s}}^{\ensuremath{-}1}$ of that of a thermal atomic vapor with atom number density $\ensuremath{\sim}10{\phantom{\rule{0.16em}{0ex}}}^{12}/{\mathrm{cm}}^{3}$ at 373 K.
We describe a relatively non-invasive optical spectroscopy technique, known as spin noise spectroscopy (SNS), to probe spontaneous magnetization fluctuations in an alkali atomic vapor at thermal equilibrium. The atomic magnetization at a non-zero temperature fluctuates about its equilibrium value, and such magnetization (spin) noise imparts fluctuation in the measured Faraday rotation of a far-detuned, linearly polarized probe laser beam passing through the vapor. This Faraday rotation fluctuation reveals the intrinsic spin dynamics of the electrons in the alkali vapor. We present a systematic experimental study of the SNS realized in thermal rubidium vapor and illustrate how one can measure various magnetic and chemical properties of the sample with minimal perturbation. We also demonstrate the applicability of this spectroscopy technique in measuring the spin populations and their dynamics in different hyperfine and magnetic levels by optically pumping the atomic system.
Spectroscopy of atomic spin noise allows the determination of many sample properties such as energy-level structures and transverse-spin relaxation rates without the need to drive the system away from equilibrium with applied fields. Analysis of noise measurements can also reveal the processes that limit the sensitivity of a measured signal. Standard methods for spectroscopic noise measurements in atomic vapors rely on far-off-resonant optical probing. We present a method for investigating spin noise with a probe beam frequency relatively close to resonance and show how the back action of the probe on the atomic state can be subtracted from the recorded noise spectra. We verify this method experimentally by showing noise data that scale with the square root of the number of atoms, after the probe back action has been properly subtracted.
We explore the applications of spin noise spectroscopy (SNS) for detection of the spin properties of atomic ensembles in and out of equilibrium. In SNS, a linearly polarized far-detuned probe beam on passing through an ensemble of atomic spins acquires the information of the spin correlations of the system which is extracted using its time-resolved Faraday-rotation noise. We measure various atomic, magnetic and sub-atomic properties as well as perform precision magnetometry using SNS in rubidium atomic vapor in thermal equilibrium. Thereafter, we manipulate the relative spin populations between different ground state hyperfine levels of rubidium by controlled optical pumping which drives the system out of equilibrium. We then apply SNS to probe such spin imbalance non-perturbatively. We further use this driven atomic vapor to demonstrate that SNS can have better resolution than typical absorption spectroscopy in detecting spectral lines in the presence of various spectral broadening mechanisms.