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(a) Experimental setup. TC, triplet collimator; L1, lens with focal f 1 = 500 mm; Pol, linear polarizer; L2/L3, lenses with focal f 2 = f 3 = 10 mm; N 2 , reservoir of nitrogen buffer gas; HW, half-wave plate; PBS, polarizing beam splitter; PD, photodetector; DAQ, data acquisition card; FM, flip mirror; L4/L5, lenses with focal f 4 = f 5 = 10 mm; Ch, chopper. (b) Waist measurement. Normalized transmitted power P (w 0 ,z)/P 0 versus chopper displacement for strongly focused (red) and collimated (blue) probe conditions. The dashed lines are the fit to the complementary error function. (c) Vapor cell geometry. Beam configurations across the vapor cell. The probe beam is either strongly focused (w (1) 0 = 2 μm) or collimated (w (2) 0 = 50 μm) at the center of the cell by the lens L2. The two probe conditions are obtained by removing or placing the lens L1 from the setup, respectively.
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We use paramagnetic Faraday rotation to study spin noise from unpolarized Rb vapor in a tightly-focused probe beam in the presence of N$_2$ buffer gas. We find counter-intuitive effects of diffusion on the spin noise resonance lineshape. In particular, we find that under certain conditions the spin noise resonance linewidth is smaller for a tightly...
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... order to test the described model and theoretical predictions we built the experimental setup shown in Fig. 2(a). The output beam of the laser source is fiber coupled and connected to a triplet collimator, which provides a high-quality collimated beam with a Gaussian diameter of 2w = 3.8 mm. This laser is used to probe a natural abundance Rb vapor placed within a cylindrical vapor cell, made out of Pyrex, with length l = 15 mm, diameter d cell = ...
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... flow in twisted heating wires. The temperature is monitored by a thermocouple and stabilized to 0.1 • C with an analog temperature controller. The entire system is enclosed in two μ-metal layers and one aluminium layer of magnetic shielding, while magnetic coils generate a field B y , transverse to the probe propagation direction. As shown in Fig. 2(c), we study two different probe beam shaping configurations: a strong focused case, obtained with a molded aspheric lens (L2) of focal length f 2 = 10 mm, which focuses the probe to a waist radius of w (1) 0 = 2 μm at the center of the vapor cell and a pseudocollimated case, obtained by adding a planoconvex lens (L1) with f 1 = 500 mm at ...
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... In both conditions, a second aspheric lens (L3) with f 3 = 10 mm collimates the probe beam after atomic interaction. We measure the beam waist by using a copy of the optical system outside the shielding, which consists of two aspheric lenses L4 and L5 with focal f 4 = f 5 = 10 mm, and a rotating optical chopper (as shown in the dashed region of Fig. 2(a). The probe beam is linearly polarized in the x-y plane before atomic interaction and propagates in the z axis. By applying a transverse magnetic field B y = 0.71 G, the intrinsic spin noise fluctuations oscillate at the Larmor frequency ω L = g F μ 0 B y , where g F is the Landé factor and μ 0 is the Bohr magneton. The probe beam ...
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... completeness, in Fig. 4(c), we compare the spin noise spectrum obtained with strong focusing w (1) 0 = 2 μm with two new conditions in which the probe is purely collimated with beam radius w (1) = 0.5 mm and w (2) = 1.5 mm (see figure caption), obtained by removing the lenses (L1,L2,L3) from the experimental setup described in Fig. 2 and appropriate (a) Experimental (points) and calculated (continuous lines) spin noise spectra acquired with a pseudocollimated probe at T = 100 • C for the following conditions (from top to bottom): = 50 GHz, p N 2 56.5 Torr (blue); = 75 GHz, p N 2 200 Torr (red); = 100 GHz, p N 2 500 Torr (green); = 150 GHz, p N 2 820 Torr) (black). ...
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Citations
... Noise spectra, obtained by continuous observation of a system of interest without active excitation, provide information about the dynamics of nearly undisturbed systems close to natural thermal equilibrium [1,2]. In recent years, noise spectroscopies have been developed for electronic [3], optomechanical [4], biological [5], and especially spin systems [6,7], where the technique is known as spin-noise spectroscopy [2,[8][9][10][11][12][13][14][15][16][17][18][19]. The interpretation of noise spectra benefits from weak thermal excitation, which ensures a linear response regime and allows calculation of equilibrium variances from thermodynamic principles. ...
... A complete model accounting for both spin-exchange and atomic diffusion effects is still missing from the literature; however, diffusion alone has been studied in Refs. [15,19]. From the fitted value of the spin-exchange rate, the 169 • C temperature of the vapor, and the 1.9 × 10 −14 cm 2 SE cross section [44], we infer an alkali-metal number density of 3.4 × 10 14 atoms/cm 3 . ...
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... , which are defined in Eq. (6). Higher-order corrections are presented in Subsection II B. In particular, second-order corrections to the power spectral density (PSD) of Sx are presented by Eq. (20), for which we assume Eq. (16). In addition, we conclude that Gaussian white noise leads to a decrease in the Larmor frequency [Eq. ...
We theoretically analyze an optically spin-polarized collection of atoms, which serves as a basis for atomic sensors. Assuming that the intrinsic atomic spin projection noise is negligible, we provide the closed-form autocorrelation function and the power spectral density (PSD) of the solution to a noisy version of an optically pumped Bloch equation, wherein each component of the external magnetic field is subjected to white noise. We conclude that noise in the bias B-field direction does not affect the autocorrelation function, up to first order in white noise covariance amplitudes. Moreover, the noise terms for the remaining two axes make different contributions to the magnetic noise-driven spin PSD; in particular, the contribution corresponding to noises perpendicular to the probing direction dominates at high frequencies. Some results concerning the second (and higher)-order terms are given. In particular, we anticipate a decrease in the effective Larmor frequency despite an increase in the magnetic field magnitude in the case of anisotropic transversal B-field noises. The analytic results are supported by Monte Carlo simulations employing the Euler–Maruyama method. The analytic methodology is applied to the case of a Bell–Bloom magnetometer, which reveals a non-linearity in the PSD of the magnetometer output and also a broadening effect due to magnetic field noise.
... When a linearly polarized probe light interacts with a spin ensemble and records the dynamic of the collective spin system, the measured spin noise spectrum (SNS) in Fig. 6a would be affected by various dephasing mechanisms, such as wall collision, the probe beam size, and atomic motion diffusion characteristics 31,[48][49][50] . Therefore, the SNS from an atomic vapor cell is a combination of Lorentzians (S total (Ω) → ∑Γ i χ i (γ Si , Ω) with the individual weights (Γ i , γ i ) which correspond to the overlap of the probe beam spatial profile (e.g., a Gaussian mode) with each of the spin diffusion modes. ...
Quantum noise reduction and entanglement-enhanced sensing in the acoustic frequency range is an outstanding challenge relevant for a number of applications including magnetometry and broadband noise reduction in gravitational wave detectors. Here we experimentally demonstrate quantum behavior of a macroscopic atomic spin oscillator in the acoustic frequency range. Quantum back-action of the spin measurement, ponderomotive squeezing of light, and virtual spring softening are observed at oscillation frequencies down to the sub-kHz range. Quantum noise sources characteristic of spin oscillators operating in the near-DC frequency range are identified and means for their mitigation are presented.
... Atomic vapor sensors typically show an optimum quantum-limited sensitivity with vapor number density n (and thus with particle number) due to a competition of factors: signal strength grows with increasing n, but may saturate due to density-dependent effects such as collisional line broadening [25]. Meanwhile, intrinsic noise sources including collisions and measurement backaction (MBA, disturbance of the atomic spins due to its optical readout) tend to increase with increasing n, creating the conditions for an n optimum [26]. Squeezing plays a dual role in this scenario: it reduces noise associated with the readout per se, but the associated antisqueezing can also increase MBA noise. ...
... Prior work on this topic includes theoretical study of spectroscopy of saturable media [24] and experimental spin-noise spectroscopy [25,26]. In magnetometry, Li and Novikova [27] studied a low-density magnetometer with resonant squeezed-light probing and modulated optical pumping, and observed an n-optimized signal-tonoise ratio (SNR) that improved with squeezing, with optimal sensitivity ≈250 pT= ffiffiffiffiffiffi Hz p . ...
... The fits shown in Figs. 3(b)-3(d) imply ζðνÞ ¼ 0.31ð30Þ, 1.17 (26), and 1.69(31) dB for ν ¼ 40, 500, and 1 kHz, respectively. Optical squeezing thus significantly improves the n-optimized sensitivity at intermediate and high frequencies, and does not significantly worsen the sensitivity in the low-frequency, SPN-limited regime. ...
We study the use of squeezed probe light and evasion of measurement backaction to enhance the sensitivity and measurement bandwidth of an optically pumped magnetometer (OPM) at sensitivity-optimal atom number density. By experimental observation, and in agreement with quantum noise modeling, a spin-exchange-limited OPM probed with off-resonance laser light is shown to have an optimal sensitivity determined by density-dependent quantum noise contributions. Application of squeezed probe light boosts the OPM sensitivity beyond this laser-light optimum, allowing the OPM to achieve sensitivities that it cannot reach with coherent-state probing at any density. The observed quantum sensitivity enhancement at optimal number density is enabled by measurement backaction evasion.
... The probe beam diameter is relatively close to the diffusion length and consequently lies in an intermediate regime in terms of the atomic spin noise profile [43]. However, the spin projection noise contribution is negligible as it is an order of magnitude lower than the magnetometer noise floor, which is predominantly limited by photon shot-noise [44]. ...
Magnetic field imaging is a valuable resource for signal source localization and characterization. This work reports an optically pumped magnetometer (OPM) based on the free-induction-decay (FID) protocol, that implements microfabricated cesium (Cs) vapor cell technology to visualize the magnetic field distributions resulting from various magnetic sources placed close to the cell. The slow diffusion of Cs atoms in the presence of a nitrogen (N2) buffer gas enables spatially independent measurements to be made within the same vapor cell by translating a 175 μm diameter probe beam over the sensing area. For example, the OPM was used to record temporal and spatial information to reconstruct magnetic field distributions in one and two dimensions. The optimal magnetometer sensitivity was estimated to be 0.43 pT/ $\sqrt {\mathrm {Hz}}$ H z within a Nyquist limited bandwidth of 500 Hz. Furthermore, the sensor’s dynamic range exceeds the Earth’s field of approximately 50 μT, which provides a framework for magnetic field imaging in unshielded environments.
... Atomic vapor sensors typically show an optimum quantum-limited sensitivity with vapor number density n (and thus with particle number) due to a competition of factors: signal strength grows with increasing n, but may saturate due to density-dependent effects such as collisional line-broadening [25]. Meanwhile, intrinsic noise sources including collisions and measurement back-action (MBA, disturbance of the atomic spins due to its optical readout) tend to increase with increasing n, creating the conditions for an n-optimum [26]. Squeezing plays a dual role in this scenario: it reduces noise associated with the readout per se, but the associated anti-squeezing can also increase MBA noise. ...
... Prior work on this topic includes theoretical study of spectroscopy of saturable media [24] and experimental spin-noise spectroscopy [25,26]. In magnetometry, Li et al. [27] studied a low-density magnetometer with resonant squeezed-light probing and modulated optical pumping, and observed an n-optimized signal-to-noise ratio (SNR) that improved with squeezing, with optimal sensitivity ≈ 250 pT/ √ Hz. ...
We study the use of squeezed probe light and evasion of measurement back-action to enhance the sensitivity and measurement bandwidth of an optically-pumped magnetometer (OPM) at sensitivity-optimal atom number density. By experimental observation, and in agreement with quantum noise modeling, a spin-exchange-limited OPM probed with off-resonance laser light is shown to have an optimal sensitivity determined by density-dependent quantum noise contributions. Application of squeezed probe light boosts the OPM sensitivity beyond this laser-light optimum, allowing the OPM to achieve sensitivities that it cannot reach with coherent-state probing at any density. The observed quantum sensitivity enhancement at optimal number density is enabled by measurement back-action evasion.
... Therefore, the whole cell volume could be used as an active measurement volume. Another advantage was to reduce the change of the lineshape due to atomic diffusion in the collimated probe light region, and there would be no changes in the tightly focused beam [72]. ...
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... The power spectral density is the Fourier transform of the autocovariance function of the detected optical signal [54] ...
... As just noted, the most dramatic effects occur at low buffer gas pressure, and thus in conditions for which spin destruction collisions play a minor role. Similarly, diffusion and power broadening can be made negligible by working with a large probe beam size [54]. For these reasons, we believe the effects predicted here should be observable in appropriate conditions, and that other conditions can also be modeled through straightforward adaptation of the theory given here. ...
We present a first-principles analysis of the noise spectra of alkali-metal vapors in and out of the spin-exchange-relaxation-free (SERF) regime, and we predict nonintuitive features with a potential to further improve the sensitivity of SERF media, and which must be taken into account in their use in quantum optical applications. Studying the process of spin-noise spectroscopy (SNS), we derive analytic formulas for the observable noise spectra, and for the correlation functions among different hyperfine components, which give additional insight into the spin dynamics. The analytic results indicate a variety of distortions of the spin-noise spectrum relative to simpler models, including a broad spectral background that mimics optical shot noise, interference of noise contributions from the two ground-state hyperfine levels, noise reduction at the spin-precession frequency, and “hiding” of spin-noise power that can introduce a systematic error in noise-based calibrations, e.g., for spin-squeezing experiments.
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... We compare the same noise spectrum in power density units, after subtracting white noise and shifting to zero frequency, to the analytical power spectrum S(ν) = in Fig. 4(b), where V 2 at = (ρ 2 at − ρ 2 ph ) is the atomic noise power contribution and C d (τ ) is the contribution to the spin noise time-correlation function due to diffusion of atoms across the multipass probe beam [7,56]. By using the analytical solution of C d (τ ) obtained in [56], we found good agreement against data with no free parameters and a buffer gas pressure of p N 2 = 700 Torr, beam width w = 0.2 mm, and physical l cell = 10 mm, where these quantities have been independently measured. This confirms that due to a diffusion time shorter than the intrinsic relaxation time T 2 the lineshape of the Lorentzian spin noise spectrum is broadened, giving a FWHM linewidth ν c = 1/(π T c ) of about 290 Hz, where T c is the noise correlation time including the effect of atomic diffusion across the probe beam. ...
We describe a magnetic gradiometer that operates at finite fields and uses an intense pulsed laser to polarize a 87Rb atomic ensemble and a compact vertical-cavity surface-emitting laser probe laser to detect paramagnetic Faraday rotation in a single multipass cell. We report differential magnetic sensitivity of 14fT/Hz1/2, corresponding to gradiometer sensitivity of 70fT/cm/Hz with a 0.2cm baseline, over a broad dynamic range, including Earth’s field magnitude and a common-mode rejection ratio higher than 104. We also observe a nearly quantum-noise-limited behavior of the gradiometer by comparing the experimental standard deviation of the estimated frequency difference against the Cramér-Rao lower bound in the presence of white photon shot-noise, atomic spin noise, and diffusion.
... The data shows symmetrical line shapes, which agrees with predictions from Eqs. (8) and (9). The exact line shape of the Herriott-cavity-assisted magnetometer signal is complex, especially due to the diffusion of atoms among the complicated optical patterns [2,34,35]. It has been shown that [29] we can use a combination of two Lorentzian functions to describe the atomic polarization in the frequency domain. ...
Detection dead zones and heading errors induced by light shifts are two important problems in optically pumped scalar magnetometry. We introduce an atomic orientation based single-beam magnetometry scheme to simultaneously solve these problems, using a polarization-reversing and path-bending Herriott cavity. Here, a reflection mirror is inserted into the cavity to bend the optical paths in the middle, and divide them into two separated orthogonal regions to avoid the detection dead zone. Moreover, half-wave plates are added in the center of each optical region, so that the light polarization is flipped each time it passes the wave plates and the light shift effects are spatially averaged out. This operation is demonstrated to eliminate the unnoticed heading errors induced by ac light shifts. The methods developed in this paper are robust to use, and easy to be applied in other atomic devices.
