Electron-Enabled Nanoparticle Diffraction

Abstract

We propose a scheme for generating high-mass quantum superposition states of an optically pre-cooled, levitated nanoparticle through electron diffraction at its sub-nanometer crystal lattice. When a single electron undergoes Bragg diffraction at a free-falling nanoparticle, momentum conservation implies that the superposition of Bragg momenta is imprinted onto the relative coordinate between electron and nanoparticle, which entangles their wavefunctions. By imaging the electron interferogram, one maps the nanoparticle state onto a superposition of Bragg momenta, as if it was diffracted by its own lattice. This results in a coherent momentum splitting approximately 1000 times greater than what is achievable with two-photon recoils in conventional standing-wave gratings. Self-interference of the nanoparticle can thus be observed within drastically shorter free-fall times in a time-domain Talbot interferometer configuration, significantly relaxing source requirements and alleviating decoherence from environmental factors such as residual gas and thermal radiation. Shorter interference times also allow for a recapture of the nanoparticle within its initial trapping volume, facilitating its reuse in many rapid experimental duty cycles. This opens new possibilities for experimental tests of macroscopic quantum effects within a transmission electron microscope.

Figures (3)

• Figure 1: Proposed scheme of electron-enabled nanoparticle diffraction consisting of three steps: (a) a nanoparticle of mass $M$ is cooled close to its motional ground state in, e.g., an optical trap, acting as a highly localised matter wave source released into free fall when the trap is switched off. After a sufficient buildup of coherent delocalisation over the time $t_0$, (b) a triggered single-electron wave packet impinges and Bragg-diffracts off the particle's lattice structure. As each imparted Bragg momentum comes with an equivalent recoil, the wavefunctions of particle and electron are now entangled. The electron passes a mask on the back focal plane (BFP) of an objective lens (OL), which selects Bragg orders corresponding to a lattice period $d$ (Inset: experimental diffraction image of a silicon crystal, illustrating the intensity distribution when aligned along the relevant zone axis, and the selection of Bragg orders of periodicity $d=192\,$pm). The selected orders are recombined on an image plane by the projector lens (PL) system and an energy filter is used to exclude inelastic events. The detector at the image plane records the electron's position $x$. This effectively maps the particle wavefunction into an $x$-dependent superposition of Bragg momenta. They are allowed to interfere over another free-fall time $t$, before (c) the position $X$ is measured and the particle re-trapped. Given free-fall times of at least one Talbot time, $t,t_0 \gtrsim T_M$, stable interference fringes of magnified period $D=d(1+t/t_0)$ in the relative coordinate $X-xD/d$ will form over many valid repetitions (i.e., whenever the electron is detected).
• Figure 2: Exemplary fringe patterns for an aligned silicon nanocrystal of mass $M=2\times 10^9\,$amu, based on electron Bragg diffraction at $\pm(1\bar{1}0), \pm(2\bar{2}0)$ and plotted against the relative coordinate $X-xD/d$ with respect to the detected electron position $x$. The particle is released and expands freely for the time $t_0 = T_M = 192\,\mu$s before the electron diffraction. In (a), we plot the position distribution of the particle as a function of time $t$ after diffraction, normalised to its maximum value at each $t$. In (b), we show the hypothetical shadow pattern for a classical particle transmitted by a classical aperture. (c) Position distributions at $t=T_M$ [dotted lines in (a) and (b)] corresponding to quantum interference (red solid) and classical shadow fringes (black solid), in units relative to the maximum of the classical pattern. The dashed line indicates the Gaussian distribution of a freely evolved particle without diffraction.
• Figure 3: Fringe patterns for an imperfectly aligned silicon nanocrystal of mass $M=2\times 10^9\,$amu, based on the settings of Fig. 2 in the main text. Rather than perfect alignment, we here assume that the crystal orientation is uncertain, but filtered through narrow Gaussian pinhole apertures ($\xi=10^{-3}$) of the Bragg transmission mask. The fringe patterns shown here are conditioned on the detection of the electron at $y=R_M/2$. Detection at smaller (greater) $|y|$ results in higher (lower) fringe contrast. It is most likely to find $|y| \lesssim R_M$.