Simulating Microwave-Controlled Spin Imaging with Free-Space Electrons

Abstract

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.

Figures (10)

• Figure 1: (a) Schematic representation of the transmission electron microscope setup for image acquisition, highlighting a zoomed-in view of the sample region positioned between the objective lens pole pieces. A spin sample is placed in a bias magnetic field $\boldsymbol{B}_0$, generated by the pole pieces, with the sample subjected to microwave (MW) pulses and probed by an electron beam within the microscope. This configuration enables precise read-out of spin dynamics through the combination of MW excitation and high-resolution electron beam probing. (b) The Zeeman effect is illustrated, showing the splitting of spin states, where states with $s = \pm1/2$ are energetically separated. Transitions between the spin states are achieved through resonant MW pulses. (c) The state preparation process begins with the spin's magnetic moment $\boldsymbol{\mu} = g_{\rm e} \mu_{\rm B} \langle\hat{\boldsymbol{\sigma}}\rangle/2 \approx -\mu_{\rm B} \langle\hat{\boldsymbol{\sigma}}\rangle$ initially aligned along the bias magnetic field. A resonant $\pi/2$ pulse tilts the spin into the equatorial plane of the Bloch sphere, initiating precession. The precession is visualized by purple arrows encircling the equator. (d) Graphical representation of the interaction between a precessing spin and an electron plane wave beam characterized by momentum $\mathrlap{\raisebox{-0.35pt}{$\space{\mathchar'26\mkern-9mu}$}}h\boldsymbol{k}$: The electron beam propagates under the influence of the bias field, and its interaction with the spin is captured by the detector. The image displayed here corresponds to image mode, though data acquisition can also be performed in diffraction mode. This dual-observation approach provides a consistent basis for analyzing spin dynamics using complementary imaging techniques.
• Figure 2: Inelastic scattering amplitudes in position representation: (a) $\tilde{\beta}_{z,+}(\boldsymbol{r})$ and (b) $\tilde{\beta}_{z,-}(\boldsymbol{r})$ plotted over the plane transverse to the beam axis for a fixed value of the distance to the sample plane $z = 1.7~\Delta r_\perp=800~\text{\AA}$. The amplitudes correspond to vortex beam components with orbital angular momenta of $- \mathrlap{\raisebox{-0.35pt}{$\space{\mathchar'26\mkern-9mu}$}}h$ and $+ \mathrlap{\raisebox{-0.35pt}{$\space{\mathchar'26\mkern-9mu}$}}h$ arising from the spin-flip transitions ${\hat{\sigma}}_{z,+}$ and ${\hat{\sigma}}_{z,-}$. The complex-valued fields are visualized using a standard color-hue convention: the hue represents the local phase, while the brightness scales with the absolute magnitude. These magnitudes are presented in arbitrary units and jointly normalized to a global maximum value of 1. When the spin state aligned along $\boldsymbol{{\rm e}}_y$ is conserved, the post-interaction electron state remains pure. Panels (c) and (d) show the amplitude (normalized to unity at the origin in arbitrary units) and phase in radians of the corresponding electron wavefunction represented in the plane transverse to the beam axis at $z$ (see Appendix \ref{['app:coherent_wavefunction']}). The simulation assumes an incident Gaussian beam with a FWHM of $0.11~\text{\textmu}$m ($\Delta k_\perp=4.22\times 10^{-6} k_{z,0}$ for $E_{\rm kin} = 200$ keV). The images depict the beam center, where the strongest spin-electron interaction signatures appear. The amplitude shows a smooth parabolic radial profile with an asymmetric modulation due to the magnetic dipole field of the spin, while the phase reflects the Aharonov-Bohm--type shift accumulated along distinct semiclassical paths Haslinger_2024. Interference features in both amplitude and phase arise from the representation of the wavefunction at non-zero $z$.
• Figure 3: (I) Expectation values of the Pauli operators for a spin initially anti-aligned with the bias magnetic field and driven by a $\pi/2$ MW pulse, shown as a function of detuning $\delta/\omega_0$ for a fixed Rabi frequency $\omega_1 = 0.01\omega_0$. Markers indicate detuning values corresponding to the spin states used for imaging. (II) Differential probability density in angular space with $\boldsymbol{\vartheta}=(\vartheta_x, \vartheta_y)$, and (III) in position space at a defocus of $z_d = 1.7~\Delta r_\perp=800~\text{\AA}$, illustrating the spin-dependent interaction with a free electron beam [FWHM $=0.11~\text{\textmu}\mathrm{m}$ ($\Delta k_\perp=4.22\times 10^{-6} k_{z,0}$)] following the same $\pi/2$ MW pulse. Results are shown for detunings (a) zero detuning, (b) $\delta = 0.025~\omega_0$, (c) $\delta = 0.050~\omega_0$, and (d) $\delta = 0.075~\omega_0$. The color scale represents the normalized intensity variation relative to a far off-resonant excitation ($\delta_0 = 10\omega_0$), corresponding to the spin in its ground state (anti-aligned with the bias field), and is normalized to unity for maximum variation in the on-resonance case.
• Figure F.1: Zernike image mode: differential probability density from the interaction between a free electron and a single spin driven by a $\pi/2$ pulse, for varying detunings. Electron beam FWHM: $0.11~\text{\textmu m}~( \Delta k_\perp=4.22\times 10^{-6} k_{z,0})$; defocus: $z_d = 0~\text{\AA}$. (a) $\delta = 0.000~\omega_0$, (b) $\delta = 0.025~\omega_0$, (c) $\delta = 0.050~\omega_0$, (d) $\delta = 0.075~\omega_0$.
• Figure F.2: CFI as a function of the defocus distance $z_d$ and the half-side length $x_{\rm max}$ of the square detection region $X = [-x_{\rm max}, x_{\rm max}]\times [-x_{\rm max}, x_{\rm max}]$. The defocus is normalized to the transverse spread $\Delta r_\perp = 470$ Å ($\Delta k_\perp=4.22\times 10^{-6} k_{z,0}$). The trends are shown for (a) standard defocused imaging and (b) Zernike phase imaging. In (a), the CFI increases with both defocus and detection size, though the values remain generally lower than the Zernike maximum CFI. The red curve marks the saturation threshold; to the right of this curve, expanding the detection region ($x_{\rm max}$) for a fixed defocus yields no additional information. In contrast, the Zernike configuration (b) exhibits a maximum at zero defocus ($z_d=0$) and increases monotonically with the detection region size.