# Anomalous high-density spin noise in a strongly interacting atomic vapor

J. Delpy, N. Fayard, Fabien Bretenaker, F. Goldfarb

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J. Delpy, N. Fayard, Fabien Bretenaker, F. Goldfarb. Anomalous high-density spin noise in a strongly interacting atomic vapor. Physical Review Research, 2025, 7 (1), pp.013298. 10.1103/PhysRevResearch.7.013298. hal-05376186

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Submitted on 21 Nov 2025

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Anomalous high-density spin noise in a strongly interacting atomic vapor

J. Delpy, N. Fayard, F. Bretenaker, and F. Goldfarb

Université Paris-Saclay, CNRS, Ecole Normale Supérieure Paris-Saclay, CentraleSupélec, LuMIn, F-91405 Orsay, France

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(Received 25 November 2024; accepted 12 February 2025; published 21 March 2025)

Spin noise spectroscopy (SNS) has become a mainstream approach to probe the dynamics of a spin ensemble in and out of equilibrium. Current models describing spin noise in interacting samples are based on an effective single-particle dynamics in a bath. Here, we report the observation of a strong interaction regime which significantly affects the spin dynamics. Performing SNS in a dense rubidium vapor, we observe anomalous distortions of the usual spin noise spectra, which we attribute to resonant dipole-dipole interaction within the ensemble. As the density of the vapor increases, we observe a dramatic broadening of the usual resonances and the emergence of an unexpected extra low-frequency noise component. We use a simple microscopic two-body numerical model to reproduce and discuss these observations. Our results suggest that the spectra cannot be described by the usual models of single-atom dynamics and arise from the evolution of interacting pairs of atoms. This work opens the way to the study of many-body spin noise or higher-order correlators in atomic vapors using SNS.

DOI: 10.1103/PhysRevResearch.7.013298

I. INTRODUCTION

The optical measurement of spontaneous angular-momentum fluctuations in particle ensembles, called spin noise spectroscopy (SNS) [1,2], is a powerful approach to the characterization of spin dynamics. First introduced in dilute atomic vapors [3–7], it was extended to condensed phases such as semiconductors [8], quantum wells [9], or quantum dots [10–12]. This method quickly gave new insights into the relaxation and decoherence channels appearing in such systems. In atomic ensembles, SNS reveals ground-state energy structures both in noninvasive [2,13,14] and invasive [15–18] regimes. The latter has gained considerable interest in the last decade, since it allows us to characterize the interaction of the system with its environment. Indeed, driving a system out of equilibrium allows us to obtain information beyond the fluctuation-dissipation theorem [19,20]. For instance, the coupling induced by a radio-frequency field between Zeeman sublevels was characterized using SNS [15]. Noisy magnetic fields were also shown to give rise to nonthermal spin noise in weakly pumped atomic ensembles [21] and even non-Gaussian spin dynamics, as revealed by higher-order SNS [22].

However, the use of SNS to probe interactions within an ensemble has been mainly restricted to an extensive characterizations of spin-exchange (SE) collisions in single-species systems [3,23,24] and two-species systems [25,26]. Spin-exchange originates from the Pauli exclusion principle and is thus a short-range interaction, with a cross section $\sigma \propto a_0^2$, where $a_0$ is the Bohr radius. As a consequence, such an

impact-like interaction is so short-lived that the spin dynamics remain essentially that of independent particles. Therefore, spin noise spectra can be inferred from mean-field descriptions of the evolution of single-particle operators [27–29] in which the SE simply materializes as a change in the spin decoherence rate [3]. Only in the spin-exchange relaxation-free (SERF) conditions has this picture recently been proven incomplete. In this regime, arising spin correlations lead to changes in the spin noise (SN) lineshapes, which can be described by mean-field hyperfine spin equations of evolution [24,25,30]. Moreover, entanglement induced by SE collisions was studied theoretically [31] but lacks experimental evidence. Therefore, the capability of SNS to probe collective effects beyond the single-particle description needs to be further investigated in the search for many-body regimes or higher-order spin correlations in vapors.

In this article, we investigate the spin noise in a system in which stronger, long-range interactions are expected to occur. We perform SNS in a high-density rubidium cell close to the $D_2$ optical transition, where resonant dipole-dipole coupling between different atoms can kick in. This long-range interaction that we observe in SNS is well known to produce a collective optical response in vapors at high densities [32–35]. We thus study the potential manifestation of such a collective regime in the spin noise spectrum of the vapor. A following fundamental question we consider is whether these collective effects simply lead to a renormalization of a single-particle spectrum, or if they result in new spectral features. This paper investigates these points in the following way: Sec. II describes the experimental SNS apparatus and reports unexpected, density-dependent changes at high densities. Section III introduces a microscopic model to simulate the spin fluctuations of a system of two atoms coupled by resonant dipole-dipole interactions, including atomic motion. Finally, Sec. IV compares simulations and experiments and discusses how the anomalous changes in the experimental spin noise

<footnote>Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.</footnote>
<figure><img src="image_3.png" /><figcaption>FIG. 1. (a) Experimental setup used for SNS experiments. DL: diode laser, GT: Glan-Thompson polarizer, MS: magnetic shielding, HWP: half-wave plate, WP: Wollaston prism, BPD: balanced photodetection. Spin-noise spectra obtained for a cell temperature (b) T = 90°C and (c) T = 175°C. The shaded area in panel (c) shows the low-frequency noise discussed in the text.</figcaption></figure>

spectra can emerge from the coupling of the atoms. Finally, we propose a possible explanation to the low-frequency noise and discuss its implications in regard to a correlated spin noise regime.

## II. EXPERIMENTAL OBSERVATIONS OF COLLECTIVE SPECTRA

A schematics of the apparatus is shown in Fig. 1(a). An external cavity diode laser is tuned at 780 nm with a power of a few mW to probe spin noise in the vicinity of the $D_2$ line of rubidium. We use a Glan-Thomson polarizer to ensure a good linear polarization of the beam. It is then focused into the vapor cell to reach a diameter of about $\phi \simeq 300$ µm. A 1-mm-thick vapor cell is embedded in a metallic oven, on which are fixed two 15 W heating resistors fed with DC currents. The temperature is fine-tuned using a homemade temperature controller. The typical achievable temperatures range from ambient temperature to around 180°C. We use the empirical Killian formula [36] to infer the atomic density from the cell temperature, measured with an N-type thermocouple. We verified a good agreement (within 10%) between the calculated density and the one inferred from transmission measurements in the temperature range [50–100°C]. At higher temperatures, obtaining a transmission model is complicated due to multiple reflections in the cell [37] and we only assume that the calculated density remains precise in this regime. We probe the spin noise in the vapor by measuring the stochastic Faraday-like rotations of the probe polarization $\delta\theta(t) \propto \langle S_z \rangle(t)$, where $S_z$ is the projection of the collective spin on the laser propagation axis z [1,2,4]. A half-wave (HW) plate followed by a Wollaston prism (WP) splits the beam at the output of the cell in two well-balanced paths of orthogonal polarization, which are then sent on two balanced photodiodes (model Thorlabs PDB450A). The difference of the photocurrents is then proportional to the tiny Faraday rotation angle: $\Delta I = I_+ - I_- \propto \delta\theta(t)$. We feed this differential photocurrent into an Agilent

EXA N9010A electrical spectrum analyzer (ESA) to get the SN spectrum of the atomic vapor, i.e., the power spectral density PSD($S_z$). Following the usual SNS setups [1,2], a pair of coils is used to create a transverse magnetic field aligned along the x axis, forcing the spontaneous spin coherences to oscillate at the Larmor frequency, typically a few MHz. Consequently, the spin noise resonances appear in a region of the spectrum which is free from low-frequency technical noises other than the laser shot noise and the ESA electronic noise background. This point is detailed in the Sec. I of the Supplemental Material (SM) [38].

A typical example of a SN spectrum obtained at relatively low density $n \simeq 10^{12}$ at $cm^{-3}$ ($T = 90$°C) can be seen in Fig. 1(b). The probe power is 2 mW and the magnetic field is $B \simeq 19$ G. In the following, we define the detuning as $\Delta = \omega_p - \omega_0$ with $\omega_p$ the probe laser frequency and $\omega_0$ the $5^2S_{1/2}$, $F = 3 \rightarrow 5^2P_{3/2}$ (center-of-mass) transition frequency of the $^{85}$Rb isotope. This spectrum was obtained for $\Delta/2\pi = 1.5$ GHz. This corresponds to a probe laser tuned halfway between the $F = 2 \rightarrow F'$ and $F = 3 \rightarrow F'$ hyperfine transitions between the $5^2S_{1/2}$ ground state and the $5^2P_{3/2}$ excited state of $^{85}$Rb, meaning that all ground-state hyperfine levels contribute to the spin noise signal. This detuning is much larger than the Doppler broadening of the ground-state hyperfine transitions and thus the absorption is negligible. It is also much greater than the excited-state hyperfine splitting of both species, meaning that the rank-2 (tensorial) polarizability component is weak and dominated by the scalar and vectorial polarizability effects. Therefore, linear birefringence and dichroism are negligible and the measured spin noise spectra arise solely from circular birefringence effects. The laser shot noise floor has been subtracted from the data. The spin noise spectrum then shows one SN resonance for each isotope, the width of which is mainly determined by the transit rate of the atoms through the laser beam in the ballistic regime: $\gamma_l/2\pi \simeq \bar{v}/(2\pi\phi) = 250$ kHz. Here $\bar{v} = \sqrt{3k_bT/m}$ is the average thermal velocity in the vapor, with $m$ the rubidium atomic mass. We note, as discussed in Sec. II of the Supplemental Material [38], that a small correction to the linewidth is due to the weak nonlinear Zeeman splitting at the magnetic field $B = 19$ G that we used [3,18], and thus slightly broadens the SN peaks. The simple spectrum of Fig. 1(b) is very well understood and arises from the stochastic precession of independent spins in a transverse magnetic field under relaxation processes such as transit decoherence or SE collisions.

To probe higher-density regimes, we perform SNS in a 1-mm-thick cell, thin enough to obtain low optical depth even close to resonance and thus reduce light absorption. Consequently, we are able to work at large atomic densities (up to around $4 \times 10^{14}$ at $cm^{-3}$ at $T = 180$°C) while keeping the detuning as low as $\Delta/2\pi = 1.5$ GHz. This would not be possible using typical cm-long spectroscopy cells because of the complete extinction of the beam. At such densities in our experiment, absorption is not negligible but the output transmission (typically 25%) is still substantial. In this configuration, we report drastic changes in the spin noise lineshapes when the density of the vapor increases. A SN spectrum recorded in similar conditions to that of Fig. 1(b), except for a higher temperature of $T = 175$°C, is shown in Fig. 1(c). It shows two important features: (i) the large broadening of both
spin-resonance lines, up to widths of the order of 1 MHz, and (ii) the emergence of a broad low-frequency noise component observed by SNS. We verified for consistency that these line-shapes do not change significantly when the heater is switched off; thus they cannot be explained by spurious misaligned magnetic fields induced by the current in the resistors.

To characterize these unusual lineshapes, we fit each spectrum by a function consisting in the sum of three lines. The first one is a broad Lorentzian line centered on the zero frequency, which accounts for the extra low-frequency noise. The two other lines correspond to the spin-resonance peaks of both $^{85}$Rb and $^{87}$Rb isotopes and are given by

$$C_{zz}(\omega) = A \frac{\gamma(\gamma^2 + \omega^2 + \omega_L^2)}{(\gamma^2 + \omega_L^2 - \omega^2)^2 + 4\gamma^2\omega^2} \quad (1)$$

These specific lineshapes arise from the Bloch equations describing the damped precessions of spins around a magnetic field aligned with the x axis (neglecting the hyperfine coupling), under the action of a white-noise forcing term. The Bloch equations can be written under the matrix form

$$\dot{S} = RS + G\eta, \quad (2)$$

with

$$S = \begin{pmatrix} S_y \\ S_z \end{pmatrix}, \quad R = \begin{pmatrix} -\gamma & \omega_L \\ -\omega_L & -\gamma \end{pmatrix}$$

the deterministic drift matrix, and

$$\eta = \begin{pmatrix} \eta_y \\ \eta_z \end{pmatrix}$$

a vector of independent stochastic variables both with zero mean and unit variance, accounting for the spin fluctuations. The matrix G describes the strength of the spin noise, and we take $G \equiv \sqrt{A\gamma} \times \mathbb{I}$, A being a free parameter in the fit. The variable S obeying such an equation is an Ornstein-Uhlenbeck process, which admits a power spectrum matrix [3,19,24] given by

$$C(\omega) = (R + i\omega\mathbb{I})^{-1}GG^T(R^T - i\omega\mathbb{I})^{-1}. \quad (3)$$

The power spectral density (PSD) of $S_z$ is given by the element $C_{zz}(\omega)$, which yields the result of Eq. (1). These three lines are used to fit our experimental data, as shown in Figs. 1(b) and 1(c).

### A. Density-dependent broadening of the spin noise lines

Using this fitting procedure, more insight into this anomalous lineshape can now be obtained by measuring the half-width at half maximum (HWHM) of both spin-resonance peaks as the temperature of the cell is varied between 80°C and 180°C. This corresponds to an atomic density spanning more than two orders of magnitude, from $n \simeq 10^{12}$ at. cm$^{-3}$ to $4 \times 10^{14}$ at. cm$^{-3}$. Figure 2(a) shows the resulting trend for both isotopes using the same $\Delta/2\pi = 1.5$ GHz optical detuning. The dashed red line is the value of $\gamma_t/2\pi$, which corresponds to a width limited by the transit of the atoms into the laser beam. The purple dash-dotted line is the sum of the transit and the SE collisional rates ($\gamma_t + \gamma_{se}/2\pi$, using the theoretical expression $\gamma_{se} = Kn_{at}\sigma_{se}\bar{v}$, where

<figure><img src="image_4.png" /><figcaption>FIG. 2. (a) Evolution of the half-width at half maximum (HWHM) of the SN peaks of both isotopes as a function of the Rb vapor density for a probe laser of 2 mW and a detuning $\Delta/2\pi = 1.5$ GHz. The shaded areas mark the part of the linewidth $\delta_{DDI}$ which we attribute to the dipole-dipole coupling. (b) Evolution of the fraction of the total noise power contained in the low-frequency component as a function of the vapor density for three different probe powers before the cell. Dashed lines are guides to the eye.</figcaption></figure>

$\sigma_{se} = 2 \times 10^{-14}$ cm$^2$ is the spin-exchange cross section and $K \simeq 0.4$ a factor arising from the nuclear angular momentum of rubidium [3]. We recall that they are the two dominant relaxation mechanisms for the one-body dynamics in typical SNS experiments. In our case, however, the observed HWHM increases from around $\gamma_t/2\pi = 250$ kHz at low density up to almost 2 MHz at the highest temperature for the $^{85}$Rb noise peak. Strikingly, when the density is large enough, the experimental data cannot be explained by the aforementioned relaxation processes, as seen in Fig. 2(a). First, the transit-time broadening depends only on the beam size and the atomic velocity, which increases only very weakly with temperature. Second, the vapor density for which the SE relaxation rate would lead to a broadening of $\approx 2$ MHz is around $5 \times 10^{16}$ at. cm$^{-3}$. This is two orders of magnitude higher than the actual density used in our experiment, eventually ruling out the possibility that transit and SE broadening only are involved. Finally, it cannot be explained by the ground-state population excitation rate induced by the probe laser, which is of the order of 30 kHz and independent of the density. Hence, we introduce $\delta_{DDI}$, a new contribution to the linewidth defined as

$$\frac{\gamma_{tot}}{2\pi} = \frac{\gamma_t + \gamma_{SE}}{2\pi} + \delta_{DDI} \quad (4)$$

and highlighted in the shaded areas in Fig. 2(a). Let us comment Eq. (4). First, if we cast the critical density of $n = 10^{14}$ at. cm$^{-3}$ above which $\delta_{DDI}$ becomes significant into an average interatomic distance, we find
$\langle r \rangle \sim 0.55n^{-1/3} \sim 120$ nm $\sim \lambda/2\pi$. The fact that this distance is smaller than the wavelength of the light (780 nm) suggests that resonant dipole-dipole interaction (DDI) induced by the probe light between the atoms in the vapor is responsible for the emergence of $\delta_{DDI}$. We strengthen this claim in the next sections by developing a model that reproduces the variation of $\delta_{DDI}(n, P, \Delta)$ observed experimentally. Second, while it is always possible to phenomenologically model this broadening by adding a decay rate, it is unclear why such an optical interaction would decrease the ground-state spin coherence lifetime $T_2 = 1/\gamma_{tot}$. The physical meaning of $\delta_{DDI}$ will thus also be clarified by our theoretical model described in Sec. III.

At such high densities, radiation trapping induced by multiple photon scattering is also known to broaden spin resonances [39]. This regime is relevant whenever the optical depth is greater than one [40,41]. However, given the small size of the cell and the detuning used, we reach the regime of unit optical depth only at the highest density of our experiment. We thus chose to neglect the multiple-scattering effects.

## B. Additional low-frequency noise

Let us now describe the low-frequency tail appearing for atomic densities higher than $n = 10^{14}$ at. cm$^{-3}$. This extra component is highlighted in color in Fig. 1(c), which shows a spectrum obtained for a temperature of $T = 175^{\circ}$C ($n = 3 \times 10^{14}$ at. cm$^{-3}$) and a probe power of 2 mW. This noise cannot be attributed to shot noise because we measured it independently and subtracted it from the data, following the procedure described in the SM (Sec. I) [38]. Additionally, the 45 MHz detection bandwidth ensures that the drop in this broadband noise component is not due to the finite bandwidth of the detector. As explained in the previous section, we fit this broad line using a Lorentzian function centered on the zero frequency, with amplitude and width being free parameters.

The precise measurement of its amplitude or its linewidth in a large density range is experimentally difficult because the signal overlaps strongly with the broad SN lines. Hence, we study a more accessible quantity: the ratio of the noise power contained in the low-frequency component over the total spin noise power as a function of the density of the vapor. Figure 2(b) shows that this fraction increases with the vapor density. It is not measurable with precision below $n = 10^{14}$ at. cm$^{-3}$ but increases afterwards up to negligible values. For a 3 mW light beam and a density of $n = 4 \times 10^{14}$ at. cm$^{-3}$ (180°C), more than half of the total power is contained in the low-frequency component. Furthermore, the laser power impinging on the cell has a large impact on the variance of this noise: the fraction of the total spin noise contained in the low-frequency component is three times larger at the highest density for a 3 mW probe power than for 1 mW. As compared with the SN lines described previously, the specificity of this low-frequency noise is that it does not exist at low probe power or low density. This strongly suggests that (i) the low-frequency noise is direct evidence of the dipole-dipole interaction in the vapor, and (ii) it is a spectral feature created by the dynamics of coupled atoms, meaning that it is unlikely to appear in the usual single-particle Bloch equation describing spin noise in noninteracting systems.

<figure><img src="image_5.png" /></figure>

FIG. 3. (a) Single-particle energy diagram considered in our model, quantized in the $x$ direction, along the magnetic field. $\pi$ ($\sigma$) denotes light polarized along the $x$ ($y$) axis. The light is detuned by $\Delta$ from the center of the resonance. (b) Schematics and parameters used to describe the spatial arrangement of the two-body problem: one atom is at the origin of the frame and the spherical coordinates give the position of the second atom.

## III. MICROSCOPIC MODEL OF TWO-BODY SPIN FLUCTUATIONS

In the previous section, we described the anomalous density dependence of the spin noise spectrum. Such new experimental observations suggest the emergence of a strong-coupling regime. In this section, we introduce a simple microscopic two-body model that we further use to validate the claims made in Sec. II.

We propose describing the spin noise in the ensemble as originating from pair of atoms coupled by resonant dipole-dipole interactions. Since we can neglect the SE collisions rate compared with the transit rate, we make the assumption that interhyperfine coherences play no role. We can thus overlook the hyperfine structures of the atoms and consider only one ground-state and one excited state. Moreover, we do not aim at describing higher-order tensorial arrangements of spins [5,7] but only spin orientation. Consequently, it is reasonable to consider only a $J = 1/2$ ground state to simulate the dynamics of spin coherences. The magnetic field is aligned along the $x$ axis and thus splits the ground-state Zeeman sublevels by an energy $\hbar\omega_L$. Figure 3(a) shows a schematic diagram of this configuration, with a quantization axis in the direction of the magnetic field. Since we want to investigate the action of the probe beam on the optical coupling, we also include a $J' = 1/2$ excited state. The light then couples the ground-state Zeeman manifold to the excited one. As the light propagates along the $z$ axis, a linear polarization along the $x$ axis ($\pi$ polarized) couples ground- and excited-state Zeeman sublevels with the same spin projection $m_S$ along the $x$ axis. Orthogonally polarized light ($\sigma$ polarization) couples Zeeman sublevels of opposite values of $m_S$. We recall that, in this picture, the spin noise $\langle S_z(t) \rangle$ arises from fluctuating coherences between the ground-state Zeeman sublevels $|g, -1/2\rangle_x$ and $|g, +1/2\rangle_x$, where $g$ denotes the ground state.

Let us now consider two atoms with the single-atom energy structure described above. The noninteracting part of the two-body Hamiltonian is thus $H_0 = \sum_{i=1,2} H_{SA}^{(i)}$, with $H_{SA}^{(i)} = \hbar\omega_L S_z^{(i)} - D_z^{(i)} \cdot E(r_i, t)$, where $r_i$ denotes the position of the atom $i$, is the single-atom Hamiltonian including Zeeman splitting and light-matter coupling. $D_z^{(i)}$ is the dipole operator of atom $i$. Here, we choose to work in the long-wavelength approximation, i.e., we assume that the atoms are
close enough to neglect the dependence on $\mathbf{r}_i$ of the electric field $\mathbf{E}(\mathbf{r}_i, t) = \mathbf{E}(t)$. We thus neglect the phase difference in the oscillations of the atomic dipoles. This approximation is justified by our experimental observations: the impact of the DDI on the SN spectra is shown to be relevant only when the interatomic distance is smaller than $\lambda/2\pi$, i.e., when the approximation becomes valid. For large interatomic separation, this approximation breaks down: however, as discussed later in the paper, the radiative coupling has no impact in this case. Additionally, for simplicity reasons we work in the thin-medium approximation and do not take into account the absorption of light through the cell. The Rabi frequency used in the simulation thus matches the light power at the entry of the cell.

Let us now include the dipole-dipole interaction between the two atoms. This can be done by considering the interaction of both atoms with the vacuum modes of the electromagnetic (EM) field bath [42–45]. Following the standard derivation of the master equation, we trace out the bath degree of freedom to get the following equation of motion, with $\rho$ a two-particle density matrix:

$$ \dot{\rho} = -\frac{i}{\hbar}[H_0 + V_{dd}, \rho] + D[\rho] + f, \quad (5) $$

where the resonant dipole-dipole interaction Hamiltonian is

$$ V_{dd} = \hbar \sum_{\substack{i \neq j \\ \lambda, \lambda'}} \zeta_{\lambda, \lambda'}(\mathbf{r}_{ij}) \frac{D_{+}^{\lambda, i} D_{-}^{\lambda', j}}{d_0^2}. \quad (6) $$

Here, $D_{+}^{\lambda, i} (D_{-}^{\lambda, i})$ is the excitation (deexcitation) part of the dipole operator of atom $i$ and of polarization component $\lambda = \sigma_+, \sigma_-, z$. We denote by $d_0$ the dipole element for the $J \to J'$ transition and by $\Gamma_0$ the excited-state population relaxation rate. The quantities $\zeta_{\lambda, \lambda'}(\mathbf{r}_{ij})$ appearing on the right-hand side of Eq. (6) are the elements of the tensor $\overline{\zeta}(\mathbf{r}_{ij}) = -\frac{3\Gamma_0}{4} \text{Re}(\overline{\mathcal{G}}(\mathbf{r}_i, \mathbf{r}_j, \omega_0))$, where $\overline{\mathcal{G}}(\mathbf{r}_i, \mathbf{r}_j, \omega_0) = (\overline{\mathbf{I}} + \frac{\nabla \nabla}{k_0^2}) \frac{e^{ik_0 r_{ij}}}{k_0 r_{ij}}$ is the electromagnetic Green’s dyadic. They give the strength of the radiative coupling between a dipole in $\mathbf{r}_i$ polarized along $\boldsymbol{\varepsilon}_{\lambda'}$ and a second one placed in $\mathbf{r}_j$ oscillating in the direction of $\boldsymbol{\varepsilon}_{\lambda}$. Here we denoted by $k_0 = \omega_0/c$ the light wave vector, $\mathbf{r}_{ij} = \mathbf{r}_i - \mathbf{r}_j$ the interatomic distance vector, and $r_{ij}$ its norm. The tensor $\overline{\zeta}(\mathbf{r}_{ij})$ strongly depends on the interatomic geometry, and particularly on the distance between two atoms:

$$ \overline{\zeta}(\mathbf{r}_{ij}) = \frac{3\Gamma_0}{4} \left( \frac{\cos \xi}{\xi^3} + \frac{\sin \xi}{\xi^2} - \frac{\cos \xi}{\xi} \right) \mathbf{I} + \frac{3\Gamma_0}{4} \left( \frac{\cos \xi}{\xi} - \frac{3\cos \xi}{\xi^3} - \frac{3\sin \xi}{\xi^2} \right) \mathbf{n} \otimes \mathbf{n}, \quad (7) $$

where $\xi = k_0 r_{ij}$ and $\mathbf{n} = \mathbf{r}_{ij}/r_{ij}$ is the unit vector in the interatomic direction. The decoherence matrix $D[\rho]$ takes the form

$$ D[\rho] = -\gamma_t(\rho - \rho_0) - \sum_{\substack{i, j \\ \lambda, \lambda'}} \frac{\gamma_{\lambda, \lambda'}(\mathbf{r}_{ij})}{d_0^2} [D_{+}^{\lambda, i} D_{-}^{\lambda', j} \rho + \rho D_{+}^{\lambda, i} D_{-}^{\lambda', j} - 2D_{-}^{\lambda', j} \rho D_{+}^{\lambda, i}], \quad (8) $$

where

$$ \gamma_{\lambda, \lambda'}(\mathbf{r}_{ij}) = \frac{3\Gamma_0}{4} (\boldsymbol{\varepsilon}_{\lambda})^* \cdot \text{Im}(\overline{\mathcal{G}}(\mathbf{r}_i, \mathbf{r}_j, \omega_0)) \cdot \boldsymbol{\varepsilon}_{\lambda'} \quad (9) $$

is the decoherence rate due to the coupling with the electromagnetic field modes. The first term in the definition of $D[\rho]$ accounts for the relaxations induced by the transit of the atoms towards the state $\rho_0 = \rho_{th}^{(1)} \otimes \rho_{th}^{(2)}$, which is that of two uncorrelated atoms at thermal equilibrium. We also include the spin fluctuations by using a noise operator $f$ similar to the one already introduced in Refs. [6, 7, 21]. This Hermitian, traceless operator introduces white-noise fluctuations in the Zeeman population and coherences. This way, we mimic the Poissonian transit of the atoms through the beam, which is the main mechanism of noise in our case. We note that such a “stochastic master equation” has been shown in Ref. [23] to be an equivalent alternative to the quantum-trajectories description of spin noise.

The equation of motion given by Eq. (5) is used to simulate the time evolution of the two-atom density matrix. To correctly take into account the action of the light on the atomic populations and coherences, we start from the uncorrelated thermal state $\rho_0$ and let it evolve towards a steady-state density matrix under the action of the magnetic field and the light beam. After it is reached, we add the fluctuations operator in Eq. (5) and start simulating a set of noisy time traces with a duration $T_0$ where $T_0$ is chosen much longer than any other typical timescale of the system. For each of them, the ensemble average $\langle S_z \rangle = \text{Tr}(\rho S_z)$ of the spin operator $S_z = \sum_{i=1,2} s_i^{(i)}$ of the two-body system is computed at each time, along with its Fourier transform $\hat{F}[S_z](\omega)$. One then accesses the two-body spin noise PSD $\langle |\hat{F}(\omega)^2| \rangle$, $\langle \dots \rangle$ denoting the average over all simulated samples. It should be mentioned here that we do not average over the Doppler effect experienced by the atoms, and comparison with experimental results should thus be done with parameters that keep the ratio $\Delta/\Gamma$ constant, where $\Gamma$ is the optical coherence decay rate of one atom: this allows us to have the same saturation parameter for the experiments and simulations. Nevertheless, since the dipole-dipole interaction depends strongly on the relative positions of the two atoms, we also have to take into account the atomic motion in order to simulate the experimental results. For that purpose, we include random changes in the relative atomic spherical coordinates $(r, \theta, \phi)$ after a time of evolution denoted $\tau_c$. This time can be considered as the time during which the two atoms are close enough for the interaction to exist with a given orientation, before they separate. We assume that the atomic movement is an ergodic process and thus all positions are accessible with equal probabilities in the cell. Following this assumption, the interatomic distance $r$ is picked randomly using the nearest-neighbor probability density function and the average atomic density $n$ [46–48]:

$$ P(r) = 4\pi n r^2 e^{-\frac{4}{3}\pi n r^3}. \quad (10) $$

The two angles $\theta, \phi$ follow uniform probability distributions in $[0, \pi]$ and $[0, 2\pi]$, respectively. The spin noise PSD is then computed using the full simulated time sequence of $\langle S_z \rangle (t)$, during which $T_0/\tau_c$ changes of conformation occurred. We note here that the correlation time $\tau_c$ is an arbitrary free parameter of the model, set to $\tau_c = 100$ ns. At this stage,
<figure><img src="image_6.png" /><figcaption>FIG. 4. (a), (b) Examples of spin noise spectra simulated for densities of $10^{12}$ at. cm$^{-3}$ ($\simeq$75°C) and $4 \times 10^{14}$ at. cm$^{-3}$ ($\simeq$180°C). (c) Simulated evolution of the HWHM of the SN peak as a function of atomic density. Parameters: $\Omega/2\pi = 150$ MHz, $\Delta/2\pi = 300$ MHz, $\gamma_t = 300$ kHz.</figcaption></figure>

we simply note that $\tau_c$ is much larger than the time step $\Delta t$ and much smaller than the simulation duration $T_0$, thus exploring a large number of conformations. A physical interpretation of $\tau_c$ is discussed in more details in the next section.

## IV. RESULTS OF TWO-BODY SIMULATIONS AND DISCUSSION

We now numerically simulate the spin noise spectra associated with two atoms in interaction and discuss the origin of the broadening $\delta_{DDI}$ as well as its scaling laws with the parameters of the system. We then propose a mechanism for the appearance of the low-frequency noise and briefly comment on its potential to probe the dynamics of atomic binaries.

### A. Simulation results and interpretation of the anomalous broadening

Figures 4(a) and 4(b) show examples of SN spectra simulated for $n = 10^{12}$ at. cm$^{-3}$ and $10^{14}$ at. cm$^{-3}$, respectively, with a Rabi frequency $\Omega/2\pi = 150$ MHz (matching the 2 mW probe power used experimentally), a detuning $\Delta/2\pi = 300$ MHz, and a Larmor frequency $\omega_L/2\pi = 9$ MHz. The shapes of both spectra match very well the experimental results of Figs. 1(b) and 1(c). At low densities, the width of the peak equals the transit decoherence rate and corresponds to the usual single-body dynamics. At high densities the simulations precisely reproduce the broadening of the peak up to the MHz range, as well as the appearance of a broad background noise centered near zero frequency. Figure 4(c) shows the evolution of the HWHM of the simulated SN peak with the density. As the latter increases, the average interatomic distance decreases and the resonance broadens. These numerical results are close to the experimental ones highlighted in Fig. 2(a). In particular, the additional linewidth is found to evolve linearly with the density: $\delta_{DDI} \sim n$, as shown in Fig. 4(c), with a proportionality

<figure><img src="image_7.png" /><figcaption>FIG. 5. (a) Circles: experimental HWHM of the $^{85}$Rb spin noise resonance as a function of laser power for three different atomic densities. Triangles: corresponding numerical simulations. (b) Comparison of the experimental broadening of the $^{85}$Rb peak induced by the DDI at two different detunings of $\Delta/2\pi = 1.5$ GHz and $\Delta/2\pi = -8.5$ GHz.</figcaption></figure>

coefficient $\beta = 2 \times 10^{-15}$ MHz cm$^3$. The good agreement with the experiment constitutes a good evidence that the changes in the SN spectra observed at high densities are caused by DDI between atoms.

Let us now discuss how such an atomic interaction can actually induce changes in the dynamics of the ground-state coherences and populations. First, the action of the near-resonant laser is to induce small optical coherences [16,49] and therefore a residual excited-state population [17], which can then give rise to dipole-dipole interaction between an atom in the ground state and another in the excited state (so-called “resonant” DDI). Consequently, the power of the probe beam should have a huge impact on the interaction strength. Figure 5(a) shows the evolution of the HWHM of the $^{85}$Rb spin noise as the laser power increases, for temperatures of $T = 90^{\circ}$C ($n = 1.5 \times 10^{12}$ at. cm$^{-3}$), $160^{\circ}$C ($n = 1.5 \times 10^{14}$ at. cm$^{-3}$), and $180^{\circ}$C ($n = 4 \times 10^{14}$ at. cm$^{-3}$). At low temperatures, the linewidth seems almost independent of the laser power, since the distance between the atoms is too large for the interaction to be relevant. Now, as soon as the average distance becomes shorter than the wavelength: $\langle r \rangle < \lambda/2\pi$ ($T > 150^{\circ}$C, i.e., $n > 1 \times 10^{14}$ at. cm$^{-3}$), the interaction strength becomes non-negligible and one can clearly see a net increase in the peak width as the laser drives the dipoles more efficiently. Such observations are well reproduced by our model [see the triangles in Fig. 5(a)]. The discrepancy between experiments and simulations being larger at the highest density may suggest the limit of our two-body model and the need to include three or more particles.

To explain the effects of the DDI on the spin dynamics, we thoroughly study in the Supplemental Material [38] the spin noise spectra obtained in the static case where the two
atoms are fixed in space and show that the usual SN resonance typical of a one-body evolution now splits into two lines. The strength of this splitting depends on the parameter of the two-body geometry, i.e., the triplet (r, θ, φ) represented in Fig. 3(b). We perform an analytical treatment of the perturbation due to the DDI and show that this splitting is actually due to a lift of degeneracy in the two-body ground-state energy levels because of the DDI. Our calculations demonstrate (see the Supplemental Material [38]) that δ<sub>DDI</sub> scales like

(1) 1/r<sup>3</sup> ∝ n in the limit ⟨r⟩ ≪ λ/2π;

(2) Ω<sup>2</sup>/Δ<sup>2</sup> ∝ P/Δ<sup>2</sup> in the weak driving regime.

However, the two-body geometry varies with time in the vapor due to the thermal motion. Therefore, over a long measurement time, the changes of the two-body conformation included in the simulations perform an average over different geometries, yielding a single broadened peak centered on the single-body SN line. Hence our model reveals that δ<sub>DDI</sub> emerges from the averaging of numerous split spectra and thus should be considered as an inhomogeneous broadening of the spin resonance. This solves the apparent contradiction, noted in Sec. II, that the decoherence channels added by the coupling of the atoms to the EM field modes should not be responsible for the relaxation of spin coherences.

The scaling of δ<sub>DDI</sub> with the parameters of the experiment thus directly comes from the scaling of the static splitting, being averaged over the distributions of (r, θ, φ). Putting all our observations in a single formula, we infer the following scaling for δ<sub>DDI</sub>, valid at low probe power P:

$$ \delta_{\text{DDI}} = \kappa \frac{P}{\Delta^2} f(n) \Gamma_0, \quad (11) $$

with κ a constant and f(n) a function of n that emerges from the average over different two-body geometries that scales with n in the large-density limit. The fact that δ<sub>DDI</sub> decreases when using a large optical detuning is also experimentally verified: Fig. 5(b) shows that the broadening of the stronger <sup>85</sup>Rb peak measured at Δ/2π = 1.5 GHz is around 15 times larger than for a detuning Δ/2π = −8.5 GHz. However, a more detailed study of this point is made difficult by the presence of several hyperfine transitions that contribute to the SN signal. Similarly, Eq. (11) suggest that the broadening should be symmetric for either red or blue detuned light; however, the complex nature of the ground state of both rubidium isotopes prevents us from precisely verifying this prediction.

## B. The low-frequency noise as a probe of a two-atom dynamics

We now propose a mechanism to explain the second unexpected spectral feature, namely, the low-frequency noise. Figure 6 shows two time traces of spin noise simulated for different atomic densities. The plot obtained for n = 10<sup>12</sup> at. cm<sup>-3</sup> [Fig. 6(a)] shows the usual random noise modulated at the Larmor frequency, as expected from the single-particle dynamics. However, the result at n = 4 × 10<sup>14</sup> at. cm<sup>-3</sup>, shown in Fig. 6(b), is drastically different. In our simulations, we randomly vary the two atoms conformation every τ<sub>c</sub> in order to mimic the motion of the atom. This sudden change of conformation forces the system to evolve towards a new and nonzero steady-state represented by the red dotted line in Fig. 6(b) due to dipole-dipole

<figure><img src="image_8.png" /><figcaption>FIG. 6. (a) Typical numerical simulations of a time sequence of the two-body spin component ⟨S<sub>z</sub>⟩(t) at a low density of n = 10<sup>12</sup> at. cm<sup>-3</sup>, a 2 mW probe power and a detuning Δ/2π = 300 MHz. (b) Simulations for a density n = 4 × 10<sup>14</sup> at. cm<sup>-3</sup>. The step-like red dotted line represents the fluctuations of the steady-state ⟨S<sub>z</sub>⟩(st) around which the spins precess.</figcaption></figure>

coupling. Let us denote by ⟨S<sub>z</sub>⟩(st) (t) this steady-state spin value towards which the system relaxes at time t. The fluctuations of the value ⟨S<sub>z</sub>⟩(st) are responsible for the low-frequency noise visible in the experimental spectra in Fig. 1(c) and the simulated spectra of Fig. 4(b). We emphasize here that the value of τ<sub>c</sub>, which physically represents the lifetime of a binary, has been enlarged to 500 ns in Fig. 6 to better illustrate the mechanism. In the rest of the paper, we use the value τ<sub>c</sub> = 100 ns: since the low-frequency noise emerges from the finite lifetime of the two-atom configurations, its linewidth (around 15 MHz) is of the order of ≈1/τ<sub>c</sub>. The value τ<sub>c</sub> = 100 ns has been chosen arbitrarily to match the experimental results.

It is noteworthy that this description makes the low-frequency noise a true signature of a two-body dynamics, in the sense that this extra noise can only be ascribed to the dynamics of a pair of atoms and cannot be accounted for by ad hoc modification of a single-body equation of motion. Indeed, contrary to the line broadening which could look like an effective reduction of the coherence time T<sub>2</sub><sup>*</sup>, this noise can simply not be reproduced by the stochastic Bloch equation (2). This again proves that the spin dynamics in the high-density regime is purely collective and cannot be attributed to a set of identical, independent spins.

However, our model does not allow for a deeper analysis of the binary dynamics for two main reasons. First, the implementation of the atomic motion in the master equation is a highly complex undertaking. Hence, the finite lifetime of
the binary is introduced in a simplistic manner as a free parameter that we tune to reproduce the experimental results. Second, our model is not quantitative (it cannot capture properly the amplitude of the noise spectra) and we have to restrict ourselves to the description of the shape of the spectra. Nevertheless, we emphasize here a number of interesting perspectives for the future. Our results suggest that this spectral component reflects the temporal dynamics of a two-atom system, which depends only on the nature and the range of the interaction. Hence we firmly believe that an in-depth study of this noise can reveal more information beyond the mean-field approximation. Investigating the scaling of the low-frequency noise amplitude and width with the density and laser power is further required to help clarifying nonambiguously its origin. More importantly, as this noise seems to emerge from the correlated dynamics of two particles, we expect possible signatures of atomic correlations in the scaling of the spin noise variance with the atomic density. Nevertheless, the search for a possible departure of the variance from the scaling $\langle S_2^2 \rangle \propto n$, valid for an uncorrelated medium, requires (i) a high-bandwidth detection because of the exceptionally large linewidths of the resonances and (ii) a more quantitative model for the two-atom dynamics, which is left for future work.

## V. CONCLUSION

We have experimentally and theoretically investigated anomalous spin noise spectra arising in a dense alkali vapor. Using a mm-thick vapor cell, we studied the Faraday rotation noise imprinted on a laser tuned near the $D_2$ line of natural rubidium by the spin fluctuations in the vapor. We took advantage of the low optical depths allowed by such a thin cell to probe high atomic densities and induce strong optical interactions within the vapor. In this regime, we showed drastic changes in the spin noise spectra as compared with the lower densities. We exhibited two spectral features that are the hallmark of the resonant dipole-dipole coupling, namely, a density-dependent broadening of the spin noise lines and a broad noise component centered on low frequencies.

Furthermore, we introduced a microscopic model to perform numerical simulations and discuss the experimental results. By considering a system of two atoms coupled by resonant dipole-dipole interaction, we reproduced the unusual experimental lineshapes. Using this model as well as analytical calculations, we interpreted the unusually large linewidths as an inhomogeneous broadening of the spin precession

frequencies due to the dipole-dipole interaction and the atomic motion in the cell. Finally, we have proposed a simple physical picture to explain the emergence of the unexpected low-frequency noise, based on the existence of nonzero-spin steady states for the binaries formed by the dipole-dipole interaction. Eventually, our model suggests that both features are the signature of a genuine two-body dynamics, and cannot be described by the usual mean-field description based on Bloch equations.

A possible consequence of the observed effects is the loss of sensitivity of optically pumped atomic magnetometers (OPMs) operating at high density because of the drastic broadening of the magnetic resonances. Fortunately, no such observations have been reported in these systems so far. We suggest that this might be due to the use of $N_2$ buffer gas, which is known to quench the excited population and thus mitigate the resonant DDI in the vapor.

This study also opens the way to the characterization of many-body spin noise in strongly correlated ensembles using SNS. Studying a rather simple system such as a dense vapor allowed us to access to collective effects beyond the single-particle model. We believe that a proper description of the low-frequency noise could give further insight into the possibility to detect genuine atomic correlation in our experiment. Further research includes a refinement of our model to propose a quantitative analysis of the noise spectra. The inclusion of three or more particles in our simulation is also an exciting prospect and may already be relevant in the regime presented in this paper. Another perspective is to study the possible effect of radiation trapping on the SN lineshapes. This could be done, e.g., by varying the cell dimensions, in order to deconvolve the multiple-scattering effects from the dipole-dipole coupling, for which we expect only a dependence on the mean distance between atoms. Finally, further work includes the search for possible higher-order spin correlators, which could help revealing the quantum or classical nature of the anomalous spin noise described in this work.

## ACKNOWLEDGMENTS

The authors are happy to thank Antoine Browaeys, Thomas F. Cutler, Ifan G. Hughes, and Charles S. Adams, for their help with thin cells. We also benefited from the help of Yassir Amazouz and Garvit Bansal for their early interest in the project, from Sébastien Rousselot for technical assistance, and Thierry Jolicœur for some help with simulations. The authors acknowledge funding by the Labex PALM.

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