Structured-Light Magnetometry in a Coherently Controlled Atomic Medium

The gist: A structured-light-based approach for detecting magneto-optical rotation is presented, in which polarization rotation is mapped onto a directly observable spatial degree of freedom.

Introduction and Motivation

Magnetometry plays a central role in diverse areas of science and technology, ranging from geophysics [2] and space research to biomedical imaging [3], materials characterization [4], and quantum information processing [5, 6]. Conventional magnetometers often require cryogenic operation or calibration against external standards, whereas optical and atomic magnetometers exploit magnetic-field-dependent evolution of atomic populations for ultra-high sensitivity at room or near-room temperatures [4, 14]. Magneto-optical rotation (MOR), or the Faraday effect, has attracted considerable attention in atomic magnetometry because it allows the field strength to be inferred from the rotation angle [16, 17]. Recent advances have directed attention toward extending MOR-based magnetometry beyond conventional Gaussian probes by employing structured light fields [22–31]. Traditional atomic magnetometers based on linear Faraday rotation detect magnetic-field-induced polarization rotation using balanced photodetection and electronic demodulation techniques [32].

Theoretical Model

The atomic system considered is an ensemble of cold 87Rb atoms in a tripod-type four-level configuration realized using Zeeman sublevels of the D2 line in the presence of a homogeneous static magnetic field applied along the z-direction [II. THEORETICAL MODEL]. The Hamiltonian for this system is given by Equation (1), where B represents the Zeeman shift due to the applied magnetic field [II. THEORETICAL MODEL]. The atomic dynamics are governed by the Liouville equation for the density operator ρ, with relevant equations of motion provided in Equation (2) [II. THEORETICAL MODEL]. In the steady-state regime under weak-probe approximation, populations are approximated as ρ11 = ρ33 = 1/2 and ρ22 = ρ44 = 0 [II. THEORETICAL MODEL]. The probe-induced coherences are then given by Equation (3), where Re(ρ41) and Re(ρ43) acquire real parts of equal magnitude and opposite sign, while absorption remains suppressed by EIT [III. RESULTS AND DISCUSSION].

Detection Scheme

To detect the MOR angle, a radially polarized Laguerre–Gaussian (LG) probe field interacts with an ensemble of Rubidium atoms subjected to a homogeneous static magnetic field along the z axis [A. Detection of MOR Angle]. The electric field component of the probe beam at the input plane z = 0 is given by Equation (6), where E0 denotes the field amplitude, Ll1m is the generalized Laguerre polynomial, m represents the radial index corresponding to the number of concentric rings in the intensity distribution, and l1 denotes the topological charge [A. Detection of MOR Angle]. The radially polarized LG mode can be expressed as a superposition of right- (RHC) and left-hand circularly (LHC) polarized components: E⃗p = E⃗− + E⃗+ [A. Detection of MOR Angle]. The field propagation through the atomic medium is governed by the coupled Maxwell equations under the slowly varying envelope approximation, described by Equation (9) [A. Detection of MOR Angle].

Measurement and Results

The accumulated phase does not appear directly in the LG intensity, so it is measured by interfering the output probe beam with a reference radially polarized LG beam E⃗ref [A. Detection of MOR Angle]. The interference intensity is given by Equation (13), which simplifies to Eq. (14) when absorption is negligible, showing that the intensity becomes: I(ρ, z) = 2E0p2h / Ll1m(ρ)2 + Ll2m(ρ)2 + Ll1m(ρ)Ll2m(ρ) cos (l1 − l2)ϕ + cos θ(z)/i [A. Detection of MOR Angle]. Here, θ(z) is the MOR angle, as given by Eq. (5). The azimuthal dependence of the intensity arises from the term cos(l1 − l2)ϕ, which leads to a petal-like intensity distribution with l1 − l2 equally spaced bright lobes [A. Detection of MOR Angle]. When a static magnetic field is applied, a differential phase shift is induced between the RHC and LHC polarized components, resulting in a rotation of the petal structure by an angle directly related to θ(z) [III. RESULTS AND DISCUSSION]. The rotation angle can be extracted by tracking the angular displacement of the intensity maxes or by monitoring the field intensity at a fixed point in the transverse plane [A. Detection of MOR Angle].

Sensitivity Analysis

The ultimate performance of an optical magnetometer is fundamentally limited by quantum noise, with photon shot noise being a dominant source [IV. SENSITIVITY]. Under the shot-noise limit, the magnetic-field sensitivity δB is given by Equation (18), where δB = 1/p Nph ∂θ/∂B-1 [IV. SENSITIVITY]. Improving sensitivity primarily requires maximizing the differential angular response ∂θ/∂B [IV. SENSITIVITY]. The analysis shows that optimal operation must remain within the dispersion-dominated EIT region, as this regime enhances dispersion while suppressing absorption [IV. SENSITIVITY]. For an optimal length z = 50 mm and probe strength 0.1γ, Fig. 8d yields a maximum slope ∂θ/∂B ≈ 15 rad/γ, corresponding to a magnetic-field sensitivity of 0.44 nT/√Hz [IV. SENSITIVITY]. The key novelty lies in the spatial transduction of polarization rotation into a directly observable interference pattern, enabling visually accessible and spatially resolved magnetometry at finite magnetic fields [V. CONCLUSION].

Conclusion

This study demonstrates the potential of structured-light-based detection as a robust and alignment-free approach for precision magnetometry [V. CONCLUSION]. The proposed scheme translates polarization rotation into a measurable spatial displacement of interference fringes, eliminating the need for conventional polarimetric detection [V. CONCLUSION]. The shot-noise-limited sensitivity is estimated to range from nT/√Hz to pT/√Hz [V. CONCLUSION]. This findings demonstrate the potential of structured-light-based detection as a robust and alignment-free approach for precision magnetometry, and suggest new opportunities for spatially encoded quantum sensing and optical measurement techniques [V. CONCLUSION].

ACKNOWLEDGMENTS

One of us (P.B.) acknowledges the financial support provided by the Council for Scientific and Industrial Research (CSIR), India, during this work [ACKNOWLEDGMENTS].

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1. Bold header: Spatial Readout of Magnetometry

This system can perform spatially resolved magnetic-field sensing by converting magneto-optical rotation into a directly observable spatial degree of freedom, eliminating the need for traditional polarizers or Stokes-parameter analysis.

2. Bold header: Alignment-Free Magnetic Field Sensing

The approach enables a direct, alignment-free, and spatially resolved measurement of the magnetic field by mapping polarization rotation onto topological rotation, which is achieved by mapping it onto a measurable rotation of the interference pattern.

3. Bold header: Enhanced Sensitivity via Structured Light Control

Sensitivity can be optimized by selecting an optimal control-field strength and propagation length, with results showing that Increasing the propagation distance enhances both the MOR angle and sensitivity as compared to Fig.8a with 8b and Fig.8c with 8d through accumulated birefringence.

4. Bold header: Tunable System Complexity

The system allows for flexible manipulation of structured light because the radial index serves as a tunable parameter that controls the complexity of the interference pattern, enabling users to tailor the field distribution.

5. Bold header: High-Precision Magnetic Field Mapping

The method can quantify magnetic field strength by monitoring the intensity variation at a fixed point in the transverse plane while varying the magnetic field B and the propagation distance z, as shown by Eq. (16).