Free-electron qubits and maximum-contrast attosecond pulses via temporal Talbot revivals

We use laser light and a transmission electron microscope to modulate a free-electron beam into high-contrast electron pulses and free-electron qubits by using temporal Talbot revivals. At large enough propagation distances, the discrete energy sidebands from a laser modulation acquire special phases and group delays that optimize or cancel their time-domain interference, producing a revival or alternatively a pulse train at close to 100% modulation depth. A sequence of two laser interactions at an optimized propagation distance allows us to coherently control adjacent energy sidebands in amplitude and phase in the way of a qubit. The use of continuous-wave laser light provides these modulations at almost the full brightness of the beam source. Free electrons under large-distance laser control are therefore a promising tool for ultrafast material characterizations or investigations of free-electron quantum mechanics.

Figure 1

Quantum revivals and sub-light-cycle electron microscopy at maximum contrast. (a) Concept for exploiting quantum revivals for generating attosecond electron pulses and qubits. A laser field (red) modulates an electron beam (blue) at a dielectric membrane (orange). Upon further propagation, the wave packet transforms into pulses and afterward restores back to the initial form, followed by further pulse compressions and revivals. (b) Evolution of the energy spectrum and sideband phases. Upper panel: Initial phases. Middle panel: Phases at the half-revival. Bottom panel: Phases at the full revival. (c) Wave packets and pulses in the time domain. A Gaussian envelope is added for depicting a limited temporal coherence. (d) Simulated quantum carpet ${\left|\mathrm{\Psi }\right|}^2$ of an electron wave packet at a kinetic energy of 75 keV. $L$ is the propagation distance, and $t$ is the local time. (e) Electron pulse duration $\mathrm{\Delta }\tau$ (solid) and temporal contrast (dashed) as a function of $L$. Pulse duration is defined as the full width at half maximum of the peak above the adjacent local minimum [8, 9].

Figure 2

Concept and experiment for attosecond pulse formation and free-electron quantum optics. (a) Schematic of the instrument. A continuous-wave (CW) laser (red) interacts with electron wave packets $\mathrm{\Psi }$ (blue) in a transmission electron microscope at two interaction regions (orange), one for temporal compression and one for diagnostics or qubit formation. The optical phase difference $\mathrm{\Delta }\phi$ is controlled with a piezoelectric actuator. Electron energy spectra are dispersed with a magnetic spectrometer on a screen (green). (b)–(d) Measurement results for an electron energy of 120 keV (left panels) and 75 keV (right panels). (b) The laser-unaffected reference spectrum is independent of laser delay $\mathrm{\Delta }\phi$. (c) An electron spectrum after two interactions with laser light as a function of $\mathrm{\Delta }\phi$. (d) Difference between the signal spectrum from (c) and the reference spectrum from (b). The intensity is normalized to the decrease in the zero-loss peak. Dashed lines are indications of symmetry. The laser powers at the two membranes are 4 W for 120 keV electrons and 3 W for 75 keV electrons, corresponding to electric field strengths of 2 and 1.7 MV/m, respectively. (e) Measurement and analysis of the ±1 photon-order sideband intensities $I_{\pm 1}$ for 120 keV (left panel) and 75 keV (right panel). Intensities are normalized to the laser-unaffected zero-loss peak. (f) Electron currents for the conditions of short-pulse formation (left) and maximum contrast (right). The blue arrows show the pulse contrast $C_{\max }$ (peak above background). (g) Analytical predictions obtained by fitting probability density of Eq. (A16) to the experimental data of (d).

Figure 3

Experimental evidence for realization of a free-electron qubit. (a) Bloch sphere. State $|0\rangle$ denotes an electron wave function in the −1 energy sideband, and state $|1\rangle$ denotes an electron wave function in the +1 sideband. In the experiment, the first laser-electron interaction (red) creates from the zero-loss peak a symmetric sideband pattern. Free propagation for half the temporal Talbot time (solid blue) rotates the phases and simultaneously prepares a time-controlled wave function. The second laser interaction (green) creates the qubit and simultaneously rotates it around the Bloch sphere in dependence of the optical laser phase delay $\mathrm{\Delta }\phi$. Additional free-space propagation or adjusting the timing of electron emission can rotate the qubit state around the $z$ axis (dashed blue). Note that the transient Bloch sphere with smaller radius (red arrow) simply indicates less electron probability in the sidebands and not a statistical incoherence. (b) Measured electron spectra for the four particular qubit states depicted in the Bloch sphere. Phases are discussed in the text. The zero-loss peak [constant with delay time; see Fig. 2] is subtracted for clarity.