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Physica Scripta

A dual-pulse heterodyne detection scheme for tunable temporal weak-value amplification towards unshielded Faraday magnetometry

Md Tousif Alam, Surya Narayan Sahoo, Nandini Mondal, Hudha P M, A K Mohapatra

Published Oct 16, 2026• 0 citations
Publisher

Abstract

Abstract When a polarized optical pulse propagates through a Faraday medium, the two circular polarization eigenmodes accumulate differential phase shifts and differential group delay. These two eigenmodes thus become distinguishable by the arrival time of photons, which can serve as a pointer state in the weak-measurement (WM) framework. Here, we describe the Faraday effect as a von Neumann-type interaction, coupling the polarization degree of freedom of light to the temporal envelope of the pulse. Under suitable pre- and post-selection of nearly orthogonal linear polarizations, the two delayed circular components interfere spectrally. This produces an amplified spectral centroid shift, corresponding to the imaginary weak value of the circular polarization component, which is highly sensitive to the small phase shifts induced by the Faraday effect. We propose detecting this spectral shift using heterodyne detection by measuring the centroid of the radio-frequency beat spectrum, a method suitable for atomic magnetometry. While this single-pulse approach may be viable where steep dispersion near resonance yields considerable group delay, it would be ineffective in most far-off-resonant magnetometry where group delay is insignificant. We propose to overcome this challenge by employing two oppositely circularly polarized pulses with a controllable phase difference and a synthetic temporal delay. These engineered input pulses provide tunable sensitivity by allowing weak-value amplification for detecting small magnetic field variations, even in the presence of an ambient magnetic field. Crucially, because the post-selection process prevents detector saturation, it allows for the use of significantly higher optical power, up to the atomic saturation level. For far-off-resonant Faraday magnetometry, we estimate that this detection scheme could potentially allow 10–100 femtotesla-level ( fT Hz − 1 ) photon-shot-noise-limited sensitivity for thermal alkali-vapors depending on detuning, with microtesla-level tunable dynamic range. Hence, this WM-based heterodyne spectral centroid readout scheme can mitigate technical noise, paving the way for high-sensitivity alkali-vapor magnetometry in unshielded environments.

DOI
10.1088/1402-4896/aeb586

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Abstract When a polarized optical pulse propagates through a Faraday medium, the two circular polarization eigenmodes accumulate differential phase shifts and differential group delay. These two eigenmodes thus become distinguishable by the arrival time of photons, which can serve as a pointer state in the weak-measurement (WM) framework. Here, we describe the Faraday effect as a von Neumann-type interaction, coupling the polarization degree of freedom of light to the temporal envelope of the pulse. Under suitable pre- and post-selection of nearly orthogonal linear polarizations, the two delayed circular components interfere spectrally. This produces an amplified spectral centroid shift, corresponding to the imaginary weak value of the circular polarization component, which is highly sensitive to the small phase shifts induced by the Faraday effect. We propose detecting this spectral shift using heterodyne detection by measuring the centroid of the radio-frequency beat spectrum, a method suitable for atomic magnetometry. While this single-pulse approach may be viable where steep dispersion near resonance yields considerable group delay, it would be ineffective in most far-off-resonant magnetometry where group delay is insignificant. We propose to overcome this challenge by employing two oppositely circularly polarized pulses with a controllable phase difference and a synthetic temporal delay. These engineered input pulses provide tunable sensitivity by allowing weak-value amplification for detecting small magnetic field variations, even in the presence of an ambient magnetic field. Crucially, because the post-selection process prevents detector saturation, it allows for the use of significantly higher optical power, up to the atomic saturation level. For far-off-resonant Faraday magnetometry, we estimate that this detection scheme could potentially allow 10–100 femtotesla-level ( fT Hz − 1 ) photon-shot-noise-limited sensitivity for thermal alkali-vapors depending on detuning, with microtesla-level tunable dynamic range. Hence, this WM-based heterodyne spectral centroid readout scheme can mitigate technical noise, paving the way for high-sensitivity alkali-vapor magnetometry in unshielded environments.
Generally a measurement yields (eigen)values that can be distinguished and recorded by a suitable measuring device. Aharanov, Albert, and Vaidman (AAV) considered the case of weak measurements in quantum mechanics, where the eigenvaluespectrum is not properly resolved. AAV showed that in this case it is helpful to choose a suitable postselection state, which should be almost orthogonal to the initial state, and which is able to give rise to arbitrarily large expectation values. Here, we discuss how weak value amplification can be adapted to measure the optical activity of chiral liquids. The hallmark of optical activity is that the liquid has different refractive indices for right and left circularly polarized light, which causes the rotation of the plane of polarization of a linearly polarized light beam traversing the solution. This small difference (<< 10-7) in refractive indices may, not only be detected in transmission (polarimetry), but can also be detected in refraction. A linearly polarized light beam incident at an interface between a chiral and an achiral medium will split into its two circular polarization components, as the two components refract with different angles of refraction. A position sensitive detector may be used to register the difference in beam positions. For chemical and pharmaceutical applications it is of interest to measure small optical activities (with correspondingly small refractive index differences) in minute liquid samples. In order to increase the sensitivity of the measurement it becomes necessary to increase the separation between the two split circular beam components. We show that conventional weak value amplification, as reported in the literature before, does not provide sufficient information to determine the handedness of a chiral liquid. Instead we describe how weak value amplification combined with a heterodyne detection scheme can measure both the concentration and the handedness of the solution. The Faraday Effect in a glass prism is used as a model system for optical activity. The linear polarized beam is split into its circular polarization components at the interface.. To amplify these small angular displacements (typically a few nanoradians) of the two circular beam components we have implemented a weak measurement with a suitable post-selection scheme. It is shown, that the signal is proportional to the circular birefringence and furthermore sensitive to the sign of the optical activity.
Abstract Weak measurement (WM) with state pre- and post-selection can amplify otherwise undetectable small signals and thus has potential in precision measurement applications. Although frequency measurements offer the hitherto highest precision due to the stable narrow atomic transitions, it remains a long-standing interest to develop new schemes to further escalate their performance. Here, we demonstrate a WM-enhanced correlation spectroscopy technique capable of narrowing the resonance linewidth down to 0.1 Hz in a room-temperature atomic vapour cell. The potential of this technique for precision measurement is demonstrated through weak magnetic-field sensing. By judiciously pre- and post-selecting frequency-modulated input and output optical states in a nearly orthogonal manner, a sensitivity of 7 fT Hz−1/2 at a low frequency near DC is achieved using only one laser beam with 15 µW of power. Additionally, our results extend the WM framework to a non-Hermitian Hamiltonian and shed new light on metrology and bio-magnetic field sensing.
Weak measurement (WM) with state pre- and post-selection can amplify otherwise undetectable small signals and thus promise great potentials in precision measurements. Although frequency measurements offer the hitherto highest precision owing to stable narrow atomic transitions, it remains a long-standing interest to develop new schemes to further escalate their performance. Here, we propose and demonstrate a WM-enhanced spectroscopy technique which is capable of narrowing the resonance to 0.1 Hz in a room-temperature atomic vapor cell. Potential of this technique for precision measurement is demonstrated through weak magnetic field sensing. By judiciously pre- and post-selecting frequency-modulated input and output optical states in a nearly-orthogonal manner, a sensitivity of 7 fT/Hz\text{7 fT/}\sqrt{\text{Hz}}7 fT/Hz​ near DC is achieved, using only one laser beam of $\text{7 }\text{\mu W}$ power. Additionally, our results extend the WM framework to a non-Hermitian Hamiltonian, and shed new light in metrology and bio-magnetic field sensing applications.
We show how two circular polarization components of a linearly polarized pulse, propagating through a coherently driven dilute atomic vapor, can be well resolved in a time domain by weak measurement. The slower group velocity of one of the components due to electromagnetically induced transparency leads to a differential group delay between the two components. For a low number density, this delay may not be large enough to temporally resolve the two components. We show how this can be enhanced in terms of the mean time of arrival of the output pulse through a postselected polarizer. We demonstrate the idea with all the analytical and numerical results, with a specific example of alkali atoms.
We demonstrate experimentally a scheme to measure small temporal delays, much smaller than the pulse width, between optical pulses. Specifically, we observe an interference effect, based on the concepts of quantum weak measurements and weak value amplification, through which a sub-pulse-width temporal delay between two femtosecond pulses induces a measurable shift of the central frequency of the pulse. The amount of frequency shift, and the accompanying losses of the measurement, can be tailored by postselecting different states of polarization. Our scheme requires only spectrum measurements and linear optics elements, hence greatly facilitating its implementation. Thus it appears to be a promising technique for measuring small and rapidly varying temporal delays.