Spin-alignment noise in atomic vapor

A. A. Fomin,1 M. Yu. Petrov,1 G. G. Kozlov,1 M. M. Glazov,2, 1 I. I. Ryzhov,3, 1 M. V. Balabas,4 and V. S. Zapasskii1 1Spin Optics Laboratory, Saint Petersburg State University, 198504 St. Petersburg, Russia 2Ioffe Institute, Russian Academy of Sciences, 194021 St. Petersburg, Russia 3Department of Photonics, Saint Petersburg State University, 198504 St. Petersburg, Russia 4Department of Physics, Saint Petersburg State University, 198504 St. Petersburg, Russia (Dated: December 30, 2019)

Introduction. Properties of materials in physics are studied most frequently by measuring their response to external perturbations. In these measurements, the perturbation needed to reliably detect the response may noticeably affect characteristics of the studied system, thus hindering interpretation of the experimental data. For this reason, the signals that can be read out from the medium, in the absence of any external perturbation, in certain cases, appear to be highly valuable. The signals of this kind are usually revealed as spontaneous fluctuations of a certain physical quantity: electric current, pressure, magnetization, etc., with the information about the system contained in the spectra of these fluctuations. Investigation of these spectra is the subject of the noise spectroscopy [1]. This paper is devoted to one specific type of this spectroscopy-to the spectroscopy of polarization noise of light caused by spin fluctuations of the medium probed by this light. Of particular value is the spin noise spectroscopy (SNS) initially realized on atomic systems [2][3][4][5], where the magnetization fluctuations are detected via the noise of the Faraday rotation. Nowadays, this technique provides indispensable information on the spin dynamics also useful for the development of spintronics. During the last years, this technique has revealed a number of unique capabilities [6][7][8][9]. Specifically, the SNS made it possible to observe resonant magnetic susceptibility of nanosystem (quantum wells, quantum dots), inaccessible for the conventional EPR spectroscopy [10,11], to examine magnetization dynamics of nuclei [12,13] and to investigate nonlinear phenomena in such systems [14][15][16][17]. The SNS technique also allows one to distinguish between homogeneously and inhomogeneously broadened lines of optical transitions [18,19] or to measure homogeneous linewidth in an inhomogeneously broadened system using multiprobe noise technique [20]; intensity modulation of the probe beam made it possible to expand the frequency range of signals detected in SNS up to microwave frequencies [21,22]. With the use of tightly focused light beams, the SNS allows one to investigate noise signals with high spatial resolution and even to realize tomography of magnetic properties of bulk materials [23]. The two-beam version of SNS proposed in [24] makes it possible, in principle, to observe both temporal and spatial correlations in magnetization.
Conventional SNS is based on the detection of the spin fluctuations via fluctuations of gyrotropy of the medium. In the presence of an external magnetic field typically applied in the Voigt geometry the random fluctuations of spins precess with the Larmor frequency ω L around the field. This Larmor precession is manifested as a peak at the frequency ω L in the spin noise spectrum.
In this work, we show, both theoretically and experimentally, that for the spin larger than 1/2 the spin alignment fluctuations give rise to the noise of the linear birefringence of the medium, which is manifested in the noise of ellipticity of the light transmitted through the medium. Depending on the experimental geometry, the spin alignment fluctuations are revealed not only at the single frequency ω L , but also at the double frequency 2ω L , and produce a static contribution to the noise spectrum. Our experimental data on cesium vapors are in full agreement with the developed microscopic theory. Our findings provide the noise counterpart of the double Larmor frequency resonance in the magneto-optical susceptibility [25].
Experiment and experimental results. The experimental setup shown in Fig. 1(a) was modified as compared with the conventional arrangement used in the SNS. As a light source, we used a tunable singlefrequency Ti:sapphire laser with ring resonator providing the emission linewidth of 2-3 kHz. The magnetic field B acting upon the cell with cesium was created by a coil 20 cm in diameter that could be aligned either along (B y), or across the light beam propagation (B z), thus providing either the Faraday or the Voigt geometry, respectively. Here, y is the light propagation axis, x and z are the axes in the plane. The azimuth of the probe beam polarization could be varied by rotating linear polarizer (LP) in the beam circularly polarized with the aid of a quarter-wave plate (λ/4). After passing through the cesium cell, the probe beam hits the polarimetric detector comprised of a phase plate (λ/2 or λ/4), a polarization beamsplitter (PBS), and a differential photodetector. The half-wave plate was used when measuring the Faraday-rotation noise to balance the detector. To measure the ellipticity noise, the λ/2 plate was replaced by the λ/4 plate aligned with its axes at 45 • to polarizing directions of the PBS. In this case, the output signal of the differential photodetector vanishes when the input light beam is linearly polarized, regardless of the azimuth of its polarization plane, and becomes nonzero only for nonzero ellipticity of the input light. The output signal of the differential detector was then fed to the FFT spectrum analyzer to obtain, after multiple averaging, the power spectrum of the Faraday rotation or ellipticity noise. The light power density of the probe beam in the  cell was varied by changing the total light power and the beam cross section.
The cell with metal cesium was 50 mm long and 30 mm in diameter and did not contain any buffer gas. The measurements of the polarization noise were performed at the wavelength of the probe beam tuned in resonance with the transition 6S 1/2 (F = 4) → 6P 3/2 of the D2 line of Cs atoms (λ = 852.3 nm), with the light power density lying in the range W = 10-100 mW/cm 2 . The density of cesium vapors could be varied by changing the temperature of the cell (in the range 30-80 • C) with the aid of a heat gun.
The fluctuation spectra of the Faraday rotation contained, in the Voigt geometry, a single peak, with its frequency linearly varied with the magnetic field (ω L ∝ B). In the Faraday geometry, the Faraday rotation noise spectrum was concentrated in the vicinity of zero frequencies (see, e.g., Ref. [7]). The spectral width of the Faraday rotation noise was mainly related, in our conditions, to inhomogeneity of the field across the sample.
Key findings of our study are presented in Figs. 1(b) and 1(c) where the ellipticity noise is shown both in the Faraday and Voigt geometries. Strikingly, the ellipticity noise in the Faraday geometry, in addition to the zerofrequency peak, contains the peak at the double Larmor frequency, 2ω L . In the Voigt geometry, in addition to the zero-frequency contribution, two peaks are seen at the single (ω L ) and double (2ω L ) Larmor frequencies.
Interestingly, in this geometry, the intensity of the peak at the double Larmor frequency, 2ω L , exhibits a strong dependence on the angle θ between the incident probe beam polarization vector and the magnetic field, reaching the highest value at θ = 45 • and vanishing at θ = 0 • and 90 • , as illustrated in Fig. 2. In other words, the double frequency component of the ellipticity noise vanishes if the probe beam is polarized either along or across the external magnetic field.
The spectral width of the ω L and 2ω L components of the ellipticity noise is similar to that of the conventional spin noise contribution, suggesting that this broadening is also due to the inhomogeneity of the magnetic field. The ratio of amplitudes of the two peaks in the ellipticity noise spectra did not change substantially with light power density (Fig. 3), which showed that the peak at the double Larmor frequency was not related to effects of optical nonlinearity in cesium vapors.
From the viewpoint of symmetry, observation of the spin-precession signal (no matter, regular, or stochastic) in the Faraday geometry at the double Larmor frequency is natural. Indeed, the spins oriented along the light beam make the medium circularly anisotropic, thus giving rise to the Faraday rotation. Oscillations of spin orientation with respect to the light propagation should evidently cause appropriate oscillation of the Faraday rotation. This is what we observe in the Voigt geometry of the conventional SNS. On the other hand, the spins aligned along a certain direction across the light beam (not necessarily preferentially oriented along this direction) create a distinguished direction in the medium thus making it linearly anisotropic. Rotation of the alignment direction around the light beam will evidently cause modulation of the light beam polarization at the double frequency of the rotation. This signal should be most conveniently observed in the Faraday geometry. Figure 4 π ω L 0 (a) spin orientation (b) spin alignment t t Figure 4. Illustration of the precession effect on spin orientation (a) and spin alignment (b). The spin orientation after a half period of the precession appears to be inverted, while the spin alignment returns to its initial state.
illustrates schematically how the spin precession manifests itself in the precession of spin orientation [ Fig. 4(a)] and spin alignment [ Fig. 4(b)]. One can see that a half period of spin-orientation precession corresponds to the whole period of the alignment precession. Below we present a microscopic theory of the alignment fluctuations and discuss its manifestations in the ellipticity noise.
Model and discussion. In conventional spin noise spectroscopy, the fluctuations of the Faraday or Kerr rotation are detected. In transparent isotropic media these fluctuations are related to the fluctuations of gyrotropy of the sample and can be phenomenologically associated with the fluctuations of the asymmetric part of the dielectric susceptibility ε αβ (α, β, γ = x, y, z are the Cartesian indices): where ω is the light frequency, A(ω) describes the efficiency of conversion of the spin fluctuation δF to the Faraday rotation, κ αβγ is the Levi-Civita symbol. The analysis shows that for F > 1/2 the fluctuations of a quadratic combination of spin components, i.e., fluctuations of spin alignment, contribute to the noise of the symmetric part of the dielectric susceptibility where S(ω) is the function of light frequency and {AB} s = (AB+BA)/2 is the symmetrized product of the operators, giving rise to the birefringence noise. These fluctuations can be observed as the ellipticity noise.
Let us now calculate the autocorrelation function of the alignment fluctuations. In our experimental geometry, the quantity of interest is and its power noise spectrum given by Fourier transform over τ : [17]. The averaging in Eq. (3) is carried out over the ensemble of atoms and over t at a fixed τ . For the Faraday geometry the spin precession in magnetic field couples the correlator C zx in Eq. (3) with another correlator Calculations show that the correlators in question obey the set of kinetic equationṡ where the dot on top denotes the time derivative, ω L is the Larmor frequency and Γ is the alignment relaxation rate. The initial conditions for Eqs. (5) can be immediately found from the equilibrium density matrix and, in the high-temperature limit, k B T ω L , read The alignment noise power spectrum takes the form where we introduced the broadened δ-function, ∆(ω) = Γ/[π(ω 2 + Γ 2 )]. For the inhomogeneous magnetic field, Eq. (7) should be averaged over the values of ω L , which will give rise to an additional broadening of the peaks. Thus, the alignment fluctuations in the Faraday geometry occur at the double Larmor frequency. Indeed, while each of the spin components F z and F x precess around the magnetic field with the single frequency ω L their symmetrized product {F z F x } s rotates twice faster. It is confirmed by Eq. (3) and is in full agreement with the experimental data shown in Fig. 1(b). In agreement with the general theory and qualitative expectations, the correlator C xz vanishes for F = 1/2.
The situation appears to be richer in the Voigt geometry. In order to analyze the fluctuations of alignment we derived a set of six coupled differential equations for the correlators {F α F β } s (t + τ ){F α F β } s (t) (α, β, α , β = x, y, z) and the appropriate initial conditions. The solution in the time domain reads with C zx (0) given by Eq. (6) and θ being the angle between the probe beam polarization and the magnetic field. Accordingly, the alignment noise power spectrum in the Voigt geometry reads Here, the spectrum contains three components at Ω = 0, ±ω L and ±2ω L . The intensities of the components depend on the orientation of the probe polarization with respect to the magnetic field. At θ = 0 or π/2 only the first harmonic of the Larmor frequency is present. Indeed, in this geometry, one of the factors in the product {F z F x } s is constant, while the other factor oscillates due to the spin precession around the field. By contrast, at θ = π/4 and 3π/4, both F z and F x oscillate at the Larmor frequency as a function of time and their symmetrized product contains both static contribution Ω = 0 and the second harmonic Ω = 2ω L . Again, our model analysis is fully in line with the experimental data shown in Figs. 1(c) and 2. Noteworthy, the presence of the peak at ω L for θ = 45 • , Fig. 2(b) is related to the fact that, in our experiments, the ellipticity noise is measured at small detunings with noticeable absorption. In this case, the spin fluctuations contributing to the gyrotropy noise [Eq. (1)] observed at the single Larmor frequency manifest themselves in the ellipticity in addition to the alignment noise.
Our experiments have shown that the alignment noise vanished in the presence of a small amount of a buffer gas (∼1 Torr of Ne). The sensitivity of the alignment noise to the presence of the buffer gas is not unexpected, because spin orientation and spin alignment correspond to different components of the density matrix and can relax with different rates [26]. In our system, this could be related to the fact that the collisions between the buffer gas and Cs atoms result in the efficient relaxation of the alignment while the spin polarization is conserved [27]. Also, the effects of Doppler broadening could be different for the ellipticity and Faraday rotation noise [19].
Conclusion. We have shown that the spin noise spectra, usually detected via polarization noise of a probe laser beam, may reveal not only fluctuations of gyrotropy of the medium at Larmor frequency, but also fluctuations of linear birefringence at the double Larmor frequency. Physically, these two types of fluctuations are associated with random oscillations of spin polarization and spin alignment, respectively. According to our calculations and our experimental results, characteristics of the peak at the double Larmor frequency substantially differ from those of the conventional peak at Larmor frequency. Specifically, the intensity of the second-harmonic component is controlled by the orientation of the probe beam azimuth with respect to the magnetic field. Also, the second-harmonic peak appears to be more sensitive to the probe beam wavelength (being most pronounced inside the linewidth of the optical transition) and to conditions of spin motion (particularly, to the buffer gas pressure). At the same time, the alignment noise proves to be persistent at high light-power densities, when optical nonlinearities cannot be ignored. Here, we used cesium vapors as a model system, however, the possibilities to observe and study the alignment noise by optical means go far beyond the field of atomic physics. This technique can be applied to any system with spins larger than 1/2, including heavy-holes and excitons in semiconductors and nanosystems, spins of magnetic ions as well as the host lattice nuclei in solids.