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arXiv:2608.15130

Stochastic Liouville-transport theory of light-atom interaction noise in thermal atomic vapors

Shaoxin Yuan, Bin Wu, Mingyong Jing, Chaoyang Hu, Yan Peng, Tingting Li, Xingya Li, Wenguang Yang, Junyao Xie, Zongkai Liu, Hao Zhang, Linjie Zhang, Liantuan Xiao, Suotang Jia

quant-phphysics.atom-phphysics.comp-phphysics.opticsSubmitted Aug 15, 2026• Announced Aug 18, 2026• 0 citations
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Abstract

Atom-light interaction noise can limit thermal-vapor sensing. Existing theories often treat internal-state dynamics, finite-mode atomic motion, and stochastic renewal separately, obscuring their coupled contributions to measured noise. We develop a general stochastic Liouville-transport theory, tested against polarization-resolved resonant Cs D$_2$ spectra. Joint experiment-theory analysis identifies atom-light noise below approximately 100 kHz as transit-dominated. Ballistic motion through the finite Gaussian mode modulates both the coupling-weighted effective atom number and trajectory-dependent Rabi coupling, producing predominantly common-mode noise. Boundary renewal introduces atoms with independently sampled ground-state sublevels, generating differential population fluctuations with opposite effects on the circular channels. Under an applied longitudinal magnetic field, experiment and theory show the same qualitative nonmonotonic change in common-mode suppression, supporting Zeeman redistribution of the channel responses. The framework can analyze noise in other thermal-atom sensors, including Rydberg-atom electric-field measurements.

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Spin Noise Spectroscopy Beyond Thermal Equilibrium and Linear Response

Ph. Glasenapp, Nikolai A. Sinitsyn, Luyi Yang +5 more

Per the fluctuation-dissipation theorem, the information obtained from spin fluctuation studies in thermal equilibrium is necessarily constrained by the system's linear response functions. However, by including weak radio frequency magnetic fields, we demonstrate that intrinsic and random spin fluctuations even in strictly unpolarized ensembles can reveal underlying patterns of correlation and coupling beyond linear response, and can be used to study nonequilibrium and even multiphoton coherent spin phenomena. We demonstrate this capability in a classical vapor of (41)K alkali atoms, where spin fluctuations alone directly reveal Rabi splittings, the formation of Mollow triplets and Autler-Townes doublets, ac Zeeman shifts, and even nonlinear multiphoton coherences.

arXiv:1407.2895•Physical Review Letters•Oct 6, 2014•50 citations
Atom-light interaction noise can limit thermal-vapor sensing. Existing theories often treat internal-state dynamics, finite-mode atomic motion, and stochastic renewal separately, obscuring their coupled contributions to measured noise. We develop a general stochastic Liouville-transport theory, tested against polarization-resolved resonant Cs D2_22​ spectra. Joint experiment-theory analysis identifies atom-light noise below approximately 100 kHz as transit-dominated. Ballistic motion through the finite Gaussian mode modulates both the coupling-weighted effective atom number and trajectory-dependent Rabi coupling, producing predominantly common-mode noise. Boundary renewal introduces atoms with independently sampled ground-state sublevels, generating differential population fluctuations with opposite effects on the circular channels. Under an applied longitudinal magnetic field, experiment and theory show the same qualitative nonmonotonic change in common-mode suppression, supporting Zeeman redistribution of the channel responses. The framework can analyze noise in other thermal-atom sensors, including Rydberg-atom electric-field measurements.
Coherent-population-trapping resonance in vapor cell is a quantum interference effect that appears in the two-photon transitions between the ground-state hyperfine levels of alkali-metal atoms and is often utilized in miniature and centimeter-scale clock devices. To quantitatively understand and predict the performance of this phenomenon, it is necessary to consider the transitions and relaxations between all hyperfine Zeeman sublevels involved in the different excitation processes of the atom. In this paper, we constructed a computational multilevel atomic model of the Liouville density-matrix equation for 32 Zeeman sublevels involved in the D1{D}_{1}D1​ line of 133Cs^{133}\mathrm{Cs}133Cs irradiated by two frequencies with circularly polarized components and then simulated the amplitude and shape of the resonance spectrum of the transmitted light through centimeter-scale Cs-vapor cells. We show that the numerical solutions of the equation and analytical investigations adequately explain a variety of the characteristics observed in the experiment.
We present a continuation of a general theoretical approach to noisy laser-atom interactions based on a special Markov-chain description of external fluctuations. This chain, composed of n independent two-state jump processes, forms a non-Gaussian stochastic process from which a detailed discussion of the atomic response can be given. We show that the n-component Markov chain also has the property of pre-Gaussian noise, because in the limit of n → ∞ it converges to a Gaussian stochastic process. We exhibit solutions for the field spectrum and the atomic-fluorescence spectrum for phase, frequency, and amplitude pre-Gaussian fluctuations of the driving-laser light.
There has been much interest in the influence of the laser noise on the atomic system. So far theoretical works have been performed on calculation of the mean values and the variance of the fluctuated atomic populations, and verified through observation of the fluctuating resonance fluorescence from the atomic vapors. Yabuzaki et al. have observed the excess intensity noise of the resonant light from a diode laser propagated through alkali atomic vapor. They have measured the intensity power spectrum of the light when the laser was tuned near the D2 line of Cs and observed all corresponding hyperfine-splitting resonances in the excited state and Zeeman splitting resonances within a hyperfine level of the ground state. Walser et al. have theoretically studied the intensity fluctuation of laser light propagated through a weakly absorbing medium. The calculated power spectrum of a phase-diffusing field has been found qualitative agreement with the experimental results by Yabuzaki et al. on hyperfine-splitting resonances. Recently, Mitsui has reported a new type of noise spectroscopy with a highly stabilized diode laser. He has investigated the spontaneous noise appeared in optical detection of magnetic resonance of Rb atoms. In this note we report the first observation of hyperfine spectra in the ground state of K and K appearing as the intensity fluctuation at hyperfine splittings of 462MHz and 254MHz of circularly polarized light from a diode laser transmitted through a sample cell. We have also observed the Zeeman hyperfine spectra of K in a static magnetic field. We show that the spectra are strongly dependent of the direction of the static magnetic field. Their splittings as functions of the strength of static magnetic field are compared with the Breit–Rabi formula. The experimental apparatus is quite simple as shown in Fig. 1. The diode laser used was an ordinary Fabry–Perot type with output power of 5mW, operating near the D1 line (769.9 nm) of K atoms. The output of the laser was circularly polarized and applied to a cell containing K vapor at a temperature 90 C, corresponding atomic density to be about 1:15 10 cm 3. The transmitted light was detected by an avalanche photo-diode having frequency response up to 1GHz. The output of the photo-diode was applied to a rf frequency analyzer to obtain the power spectrum of the fluctuation of the transmitted light intensity. A static magnetic field is provided by two pairs of Helmholtz coils mutually perpendicular and about 90 cm in diameter. The strength of the magnetic field is calibrated by using magnetic resonance signals of optically pumped Rb atoms. Figure 2 shows the frequency spectrum of the intensity fluctuation of transmitted light observed when the laser was tuned to the transition from the state F 1⁄4 1 in the ground state to the state F0 1⁄4 1 in the excited state by adjusting the driving current of the laser diode. Sharp resonance signals can be seen at the hyperfine splittings of K and K, 462 and 254MHz, respectively, on the background of broad noise spectrum. When a weak static magnetic field H0 is applied to the K vapor, the hyperfine spectra of K splits into several lines (the effective gyro-magnetic ratio of K is known to be 1⁄4 2 700 kHz/G). Figures 3(a) and 3(b) show the observed Zeeman hyperfine spectra of K for
The ground-state Hanle effect (GSHE) in alkali-metal atomic vapor using a single circularly polarized light wave underlies one of the most reliable and simple techniques of modern atomic magnetometry. This effect causes a narrow (subnatural-width) resonance in the light wave intensity passing through the vapor cell. The GSHE-based sensors typically operate in the so-called spin-exchange relaxation-free (SERF) regime to reduce the resonance linewidth. However, this regime requires a relatively high temperature of vapors (approximately equal to $150{\phantom{\rule{0.16em}{0ex}}}^{\ensuremath{\circ}}\mathrm{C}$), leading to high heat release and power consumption of the sensor head. In addition, without applying special measures, the SERF regime significantly limits the dynamic range of measurements. Here we study a pump-probe scheme involving a single elliptically polarized light wave and a polarimetric detection technique. The wave is in resonance with two adjacent optical transitions in the cesium D1{D}_{1}D1​ line ($\ensuremath{\lambda}\phantom{\rule{0.16em}{0ex}}\ensuremath{\approx}\phantom{\rule{0.16em}{0ex}}894.5$ nm) due to their overlap in the presence of a buffer gas (130 Torr neon). Using a small ($V\ensuremath{\approx}0.1\phantom{\rule{4pt}{0ex}}{\mathrm{cm}}^{3}$) glass vapor cell, we demonstrate the possibility to observe subnatural-width resonances with a high contrast-to-width ratio (up to 45%/mG) in low-temperature ($60{\phantom{\rule{0.16em}{0ex}}}^{\ensuremath{\circ}}\mathrm{C}$) operation due to strong light-induced circular dichroism in the medium. Based on the $\mathrm{\ensuremath{\Lambda}}$ scheme of atomic energy levels, we obtain explicit analytical expressions for the shape of the resonance line. The model reveals a linewidth narrowing effect due to openness of the level scheme. The observed bright features are unusual for magneto-optical atomic spectroscopy as openness is commonly considered to be an undesirable effect that degrades resonance characteristics. Measuring the noise voltage, we estimate the sensitivity of the magnetic-field measurements to be 1.8 pT/Hz\mathrm{pT}/\sqrt{\mathrm{Hz}}pT/Hz​ with a sensitivity of 60 fT/Hz\mathrm{fT}/\sqrt{\mathrm{Hz}}fT/Hz​ in the photon-shot-noise limit. In general, the results contribute to the theory of GSHE resonances and also can be applied to development of a low-temperature high-sensitivity miniaturized magnetic-field sensor with an extended dynamic range.
Conventional practice of spatially resolved detection in diffusion-coupled thermal atomic vapors implicitly treat localized responses as mutually independent. However, in this study, it is shown that observable correlations are governed by the intrinsic spatiotemporal covariance of a global spin-fluctuation field, such that spatial separation specifies only overlapping statistical projections rather than independent physical components. A unified field-theoretic description is established in which sub-ensembles are defined as measurement-induced statistical projections of a single stochastic field. Within this formulation, sub-ensemble correlations are determined by the covariance operator, inducing a natural geometry in which statistical independence corresponds to orthogonality of the measurement functionals. For collective spin fluctuations described by a diffusion-relaxation Ornstein-Uhlenbeck stochastic field, the covariance spectrum admits only a finite set of fluctuation modes in a bounded domain, imposing an intrinsic, field-level limit on the number of statistically distinguishable sub-ensembles. The loss of sub-ensemble independence is formalized through the notion of spatial sampling overlap, which quantifies the unavoidable statistical coupling arising from shared access to common low-order fluctuation modes. While multi-channel atomic magnetometry provides a concrete physical setting in which these constraints become explicit, the framework applies generically to diffusion-coupled stochastic fields.
Per the fluctuation-dissipation theorem, the information obtained from spin fluctuation studies in thermal equilibrium is necessarily constrained by the system's linear response functions. However, by including weak radio frequency magnetic fields, we demonstrate that intrinsic and random spin fluctuations even in strictly unpolarized ensembles can reveal underlying patterns of correlation and coupling beyond linear response, and can be used to study nonequilibrium and even multiphoton coherent spin phenomena. We demonstrate this capability in a classical vapor of (41)K alkali atoms, where spin fluctuations alone directly reveal Rabi splittings, the formation of Mollow triplets and Autler-Townes doublets, ac Zeeman shifts, and even nonlinear multiphoton coherences.