Abstract Recent advances in transmission electron microscopy (TEM) have opened the path toward spin resonance spectroscopy with single-spin sensitivity. To assess this potential, we investigate the quantum precision limits for sensing magnetic moments with free-electron probes. Using a scattering model where an electron wavepacket interacts with a localized spin, we study two metrological tasks: estimating the magnitude of the magnetic moment and discriminating the presence of a spin. The sensitivity for a given measurement setting is generally determined by the classical Fisher information, which we benchmark against the quantum bound optimized over all measurements. We find that conventional TEM imaging can saturate the quantum bound when backaction of the probe electron onto the spin state is negligible. We also find that, when backaction is relevant, one could do better by realizing a measurement of the electron's orbital angular momentum state. These results establish the quantum limits of spin sensing in TEM and guide the development of future experiments probing individual electron spins or nanoscale ensembles of nuclear spins.
Abstract Recent advances in transmission electron microscopy (TEM) have opened the path toward spin resonance spectroscopy with single-spin sensitivity. To assess this potential, we investigate the quantum precision limits for sensing magnetic moments with free-electron probes. Using a scattering model where an electron wavepacket interacts with a localized spin, we study two metrological tasks: estimating the magnitude of the magnetic moment and discriminating the presence of a spin. The sensitivity for a given measurement setting is generally determined by the classical Fisher information, which we benchmark against the quantum bound optimized over all measurements. We find that conventional TEM imaging can saturate the quantum bound when backaction of the probe electron onto the spin state is negligible. We also find that, when backaction is relevant, one could do better by realizing a measurement of the electron's orbital angular momentum state. These results establish the quantum limits of spin sensing in TEM and guide the development of future experiments probing individual electron spins or nanoscale ensembles of nuclear spins.
Coherent spin resonance techniques, such as nuclear and electron spin resonance spectroscopy, have revolutionized non-invasive imaging by providing spectrally resolved information about spin dynamics. Motivated by the recent emergence of electron microscopy methods capable of sensing microwave-excitations, we establish a theoretical framework for Spin Resonance Spectroscopy (SRS) in transmission electron microscopy (TEM). This technique combines microwave pump fields with focused electron probe beams to enable state-selective spin imaging at the atomic scale. Using scattering theory, we model the interaction between free-space electrons and electron spin systems, capturing both elastic and inelastic processes. The strongest effect of the spin system on the free electron is a magnetic phase shift. Our simulations demonstrate that phase shifts from individual electron spins are detectable in both image mode and diffraction mode. In principle, differential measurements under microwave control allow the extraction of local resonance frequencies that are influenced by the surrounding spin environment. By evaluating the Classical Fisher Information (CFI), we identify imaging conditions that maximize the signal-to-noise ratio (SNR), showing how defocus and beam width affect the measurement sensitivity. These findings establish a foundation for integrating SRS with high-resolution TEM, bridging spin spectroscopy and atomic-scale imaging, and enabling new capabilities in quantum spin research and nanoscale materials characterization.
Scattering experiments with energetic particles, such as free electrons, have been historically used to reveal the quantum structure of matter. However, realizing coherent interactions between free-electron beams and solid-state quantum systems has remained out of reach, owing to their intrinsically weak coupling. Realizing such coherent control would open up opportunities for hybrid quantum platforms combining free electrons and solid-state qubits for coincident quantum information processing and nanoscale sensing. Here, we present a framework that employs negatively charged nitrogen-vacancy centers (NV − ) in diamond as quantum sensors of a bunched electron beam. We develop a Lindblad master equation description of the magnetic free-electron–qubit interactions and identify spin relaxometry as a sensitive probe of the interaction. Experimentally, we integrate a confocal fluorescence microscopy setup into a microwave-bunched electron beam line. We monitor charge-state dynamics and assess their impact on key sensing performance metrics (such as spin readout contrast), defining safe operating parameters for quantum sensing experiments. By performing T 1 relaxometry under controlled electron beam exposure, we do not resolve a measurable reduction in T 1 within experimental uncertainty, and instead establish an upper bound on the free-electron–spin coupling strength. Our results establish NV − centers as quantitative probes of free electrons, providing a metrological benchmark for free-electron–qubit coupling under realistic conditions, and chart a route toward solid-state quantum control with electron beams.
Scattering experiments with energetic particles, such as free electrons, have been historically used to reveal the quantum structure of matter. However, realizing coherent interactions between free-electron beams and solid-state quantum systems has remained out of reach, owing to their intrinsically weak coupling. Realizing such coherent control would open up opportunities for hybrid quantum platforms combining free electrons and solid-state qubits for coincident quantum information processing and nanoscale sensing. Here, we present a framework that employs negatively charged nitrogen-vacancy centers (NV-) in diamond as quantum sensors of a bunched electron beam. We develop a Lindblad master equation description of the magnetic free-electron--qubit interactions and identify spin relaxometry as a sensitive probe of the interaction. Experimentally, we integrate a confocal fluorescence microscopy setup into a microwave-bunched electron beam line. We monitor charge-state dynamics and assess their impact on key sensing performance metrics (such as spin readout contrast), defining safe operating parameters for quantum sensing experiments. By performing T1 relaxometry under controlled electron beam exposure, we establish an upper bound on the free-electron--spin coupling strength. Our results establish NV- centers as quantitative probes of free electrons, providing a metrological benchmark for free-electron--qubit coupling under realistic conditions, and chart a route toward solid-state quantum control with electron beams.
The characterization of electric and magnetic dipole moments (EDM and MDM) in quantum systems is central to fundamental physics and quantum sensing. While EDM searches provide powerful probes of CP violation within and beyond the Standard Model, precise MDM estimation is crucial for high-precision magnetometry and the development of quantum sensors. In this work, we address the ultimate precision limits for separate and simultaneous estimation of both dipole moments in a generic two-level system coupled to electromagnetic fields. We analyze three classes of quantum probes/strategies: unitary and depolarizing dynamics, and thermal equilibrium states. For each, we derive the quantum Fisher information (matrix), identify optimal probes, and determine the ideal operating conditions, such as evolution times and temperatures, that maximize estimation precision. We further assess the compatibility and sloppiness of the statistical models, showing that orthogonal dipole moments configurations enable joint estimation of EDM and MDM, whereas parallel configurations are intrinsically sloppy, permitting only the estimation of a single parameter combination. Our results provide a unified metrological framework for estimation schemes ranging from neutron EDM searches to molecular magnetometry, and highlight the distinct roles of coherence, noise, and thermalization in multiparameter quantum sensing of dipole moments.
Free-electron quantum optics has recently emerged as a promising platform for coherent electron-light interactions, yet its potential for quantum-enhanced metrology remains largely unexplored. Here, we investigate the fundamental precision limit of coupling-strength estimation in multi-electron photon-induced near-field electron microscopy (PINEM), demonstrating that the quantum Fisher information scales quadratically with electron number. We then propose an experimentally accessible scheme combining squeezed-light probing and optical homodyne detection that attains Heisenberg scaling with respect to electron number, provided the electrons exhibit nonzero inter-electron coherence, while surpassing conventional PINEM sensing in photon-number enhancement as well. We further analyze the effect of imperfect electron preparation, showing that electron phase noise is converted into a fluctuating optical displacement and eventually limits the precision. To overcome this limitation, an auxiliary optical mode coupled to the same electron ensemble measures the displacement noise in the signal mode, allowing its conditional suppression and recovery of the Heisenberg scaling. Our work establishes multi-electron coherence as a resource for free-electron quantum metrology and opens the route to precision measurements of nanoscale electron-light interactions.
What are the ultimate limits of accuracy for techniques in transmission electron microscopy (TEM)? For example, for a given electron dose, what is the ultimate accuracy that can be achieved for the atomic structure of a material? And what determines this accuracy? In this work, we show how quantum estimation theory can be used to provide some quantitative answers to such “big” questions. Quantum estimation theory (also known as quantum metrology theory) is concerned with measurements made at the quantum level. In recent years, the theory has been applied to an increasingly broad range of scientific techniques [1,2]. Here, we apply it to the problem of multiple electron scattering from an atomic structure in a TEM [3]. We focus on cases where the parameters of interest are those which characterize the sample, an extremely relevant and very general scenario. We report the following: We have obtained a quantitative prediction of the ultimate accuracy allowed by quantum mechanics for the precision of atom coordinates within a sample. As expected, this ultimate precision is a function of the electron dose (among other factors), and so this result has important consequences for the precision achievable for the atomic structure of radiation-sensitive materials. We demonstrate that the ultimate (quantum) limit for a simultaneous estimation of multiple sample parameters can be achieved under the condition of weak electron scattering, such as when the phase-object or weak-phase-object approximation is applicable. We demonstrate that this same quantum limit cannot be achieved under the condition of strong multiple scattering, due to the incompatibility of the projection measurements required. Hence strong multiple scattering precludes the quantum limit (with few exceptions) - a fascinating result. In the theory, a TEM experiment (which could be imaging, diffraction, holography, etc.) is divided into three stages: the incident electron beam |ψ0⟩, the beam scattered by the sample S|ψ0⟩, and the beam incident on the electron detector AS|ψ0⟩. Here, S and A are mathematical operators that account, respectively, for the scattering by the sample and the post-sample electron optics (e.g., lens aberrations). This description applies to many TEM modalities, which allows the theory to make general predictions. Fig. 1 shows the specific case of TEM imaging. As will be shown, the quantum limit for an arbitrary set of sample parameters can be achieved by phase-contrast imaging under the Zernike phase condition. Fig. 2 shows a quantitative prediction of the ultimate precision achievable for the lateral coordinate of an atom in a material, as measured in the TEM. Higher doses allow better precision, as expected. For a given dose, better precision is possible for heavier atoms, and for lower beam energies. The results show a strong and unexpected "threshold" effect which occurs at the onset of the asymptotic regime (where the dose is high enough for the statistics to become Gaussian). Finally, while our main focus here is TEM, the generality of the theory means that several of our important conclusions extend to other techniques which utilize the scattering of coherent radiation, such as coherent x-ray scattering in a synchrotron and neutron scattering. Optics for TEM imaging, showing the mathematical notation relevant to this work. Ultimate limits for the precision of a (lateral) atom position measured using TEM. Results are shown for two representative atomic species and beam energies 30-300 keV. Squares and diamonds show results from maximum likelihood estimation (MLE) applied to Monte Carlo image simulations.