Toward Atomic-Resolution Quantum Measurements with Coherently Shaped Free Electrons

Free electrons provide a powerful tool for probing material properties at atomic resolution. Recent advances in ultrafast electron microscopy enable the manipulation of free-electron wave functions using laser pulses. It would be of great importance if one could combine the spatial resolution of electron microscopes with the ability of laser pulses to probe coherent phenomena in quantum systems. To this end, we propose a novel concept that leverages free electrons that are coherently shaped by laser pulses to measure quantum coherence in materials. We develop the quantum theory of interactions between shaped electrons and arbitrary qubit states in materials, and show how the postinteraction electron energy spectrum enables measuring the qubit state (on the Bloch sphere) and the decoherence or relaxation times $({\mathit{T}}_2/{\mathit{T}}_1)$. Finally, we describe how such electrons can detect and quantify superradiance from multiple qubits. Our scheme can be implemented in ultrafast transmission electron microscopes (UTEM), opening the way toward the full characterization of the state of quantum systems at atomic resolution.

Figure 1

Coherently shaped free electrons as high-resolution probes of coherence in quantum systems. (a) An ultrafast transmission electron microscope where (b) electrons are coherently shaped by a PINEM interaction, creating a superposition of energies spaced by integer multiples of the driving laser frequency $\omega$, chosen so the gap between a certain pair of electron energies matches the qubit energy $\hslash {\omega }_0$. (c) The qubit is excited to $1/\sqrt{2}|g⟩+i/\sqrt{2}|e⟩$ and interacts with the electron after time delay $\tau$. (d) The postinteraction electron is measured as a function of $\tau$, using electron energy loss spectroscopy (EELS). This scheme enables measuring different qubit properties, e.g., extracting $T_2$ from the asymmetry of the spectrum.

Figure 2

Extraction of the relaxation (${\mathit{T}}_1$) and decoherence (${\mathit{T}}_2$) times from a sequence of delayed electron energy spectra. (a) A laser pulse excites the qubit to $|e⟩$, which then relaxes back. $T_1$ is extracted from the sequence of electron energy loss spectroscopy (EELS) measurements of unshaped electrons probing the qubit at different delays $\tau$. $⟨E_{\mathrm{gain}}⟩$ is the average energy gain. (b) A laser pulse excites the qubit to $1/\sqrt{2}(|g⟩+i|e⟩)$, which then precesses while decohering. $T_2$ is extracted from the sequence EELS measurements of shaped electrons probing the qubit at different delays $\tau$.

Figure 3

A scheme for measuring the qubit state on the Bloch sphere: (a) The electron is coherently shaped by a laser pulse with a tunable optical path length that changes the relative phase ${\varphi }_e$ between the electron’s energy states. The same laser (or another one that is phase-locked) excites the qubit to a state defined by angles ${\theta }_a$ and ${\varphi }_a$ on the Bloch sphere. (b) The initial electron wave function with relative phase ${\varphi }_e$. (c) The resulting EELS spectra for different ${\varphi }_e$. We extract the qubit state from the difference between the maximal and minimal EELS peaks. ${\varphi }_g$ is the phase of the interaction constant $g$ and $\mathrm{\Phi }={\varphi }_a-{\varphi }_e-{\varphi }_g$.

Figure 4

Free-electron-qubit interactions for studying superradiance and mapping the local density of photonic states. (a) To exemplify the concept, we place qubits on a membrane surrounding a 10 nm Au nanoparticle, which varies the LDOS and $T_1$ as a function of position. (b) Comparison of the time decay of normalized $⟨E_{\mathrm{gain}}⟩$ for different qubit positions (the same curves are also shown schematically at the four bottom plots). (c) The emission from a single qubit (orange) vs superradiance from multiple qubits (blue).