Coupling free electrons to a trapped-ion quantum computer

Abstract

Freely propagating electrons may serve as quantum probes that can become coherently correlated with other quantum systems, offering access to advanced metrological resources. We propose a setup that coherently couples free electrons in an electron microscope to a trapped-ion quantum processor, enabling non-destructive, quantum-coherent detection and the accumulation of information across multiple electrons. Our analysis shows that single electrons can induce resolvable qubit excitations, establishing a platform for practical applications such as quantum-enhanced, dose-efficient electron microscopy.

Figures (5)

• Figure 1: (a) A transmission electron microscope (TEM). Within a plane conjugate to the specimen plane, a trapped-ion quantum computer is inserted. (b) The electron $e^-$ couples to the ionic center-of-mass degree of freedom via the Coulomb interaction (scattering matrix $\hat{S}$). If the ion is prepared in a superposition of coherent states, the electron induces a relative phase shift between these states. (c) The figure shows a simplified level diagram of ^40Ca+ with the relevant transitions used to encode the qubit. The center-of-mass degree of freedom of the ion is coupled to the 4S$_{1/2}$ and 3D$_{5/2}$ states of the valence electron (qubit degree of freedom) via the application of bichromatic laser pulses (red- and blue-detuned transitions in the figure). The rapidly decaying fluorescent transition from 4P$_{1/2}$ to 4S$_{1/2}$ is used to perform projective readout of the qubit state.
• Figure 2: (a) The phase shift $\Delta\phi(b)$ (Eq. \ref{['eq:action_on_coherent']}), assuming $\delta_{{\mathbf{r_\perp}}} \ll R_0$, of the combined electron-ion wavefunction due to the interaction with impact parameter $b$ compared to the corresponding phase shift $\Delta\phi(0)$ when the electron is focused to the center of the harmonic trap. (b) The probability $P_\text{scat}=1-\abs*{ \mathcal{S}({\mathbf{r}}_\perp,\boldsymbol{\upalpha}) }^2$, assuming $\delta_{{\mathbf{r_\perp}}} \ll R_0$, for the ion to be scattered out of the state $\ket{\boldsymbol{\upalpha}}_\text{cmi}$ upon impact of an electron as a function of the impact parameter $b$.
• Figure 3: The plot shows the probabilities to find the qubit in the state $\ket{1}$ after the interaction with an electron of different energies as a function of the cat state size $\alpha$. It is assumed that the qubit is initially prepared in the state $\ket{0}$. The dashed line indicates $\abs*{\boldsymbol{\upalpha}}=6.5$, which has already been demonstrated experimentally wu2025infraredabsorptionspectroscopysingle.
• Figure 4: The change in the final electronic state for a focused electron beam of 100 eV ($v_\text{el}\sim 3$) measured in terms of $\eta$ (Eq. \ref{['eq:electronic_overlap']}) as a function of the relative probe size $\sqrt{\chi}=\sqrt{2}\delta_{{\mathbf{r_\perp}}}/R_0$ and the impact parameter $b =\abs*{{\mathbf{b}}}$.
• Figure 5: Top: Comparison of the expected $F(\phi)$ given by $n^2(1-\epsilon)^n$ for $\epsilon=0.01$ to the SQL $n$ and the HL $n^2$ as a function of the number of electrons $n$. Bottom: Depending on the probability of electron loss $\epsilon$, we show two functions. First, the ideal number of electrons, which is given by rounding $n^*=\frac{-1}{\log(1-\epsilon)}$. Second, inserting this value back into the expected ratio of obtained $F(\phi)$ and SQL, $g_{\text{FI}}(n^*)$, which quantifies the quantum advantage.