Hanbury Brown and Twiss interference of electrons in free space from independent needle tip sources

TL;DR

This work addresses the challenge of disentangling fermionic antibunching due to Pauli exclusion from Coulomb repulsion in two-electron Hanbury Brown and Twiss interferometry in free space. It develops a two-tip electron-source geometry, derives a quantum-path-based $G^{(2)}(\delta)$ neglecting charge, and shows how spin and source statistics shape the interference pattern. A classical Coulomb-dip estimate is then used to demonstrate that, with two independent tips, numerous fermionic fringes can be observed within the Coulomb dip, enabling unambiguous separation of quantum and classical effects. The results point to practical routes for electron-based imaging and correlation spectroscopy, and invite further quantum treatments of Coulomb interactions in fermionic HBT interferometers.

Abstract

We investigate two-electron interference in free space using two laser-triggered needle tips as independent electron sources, a fermionic realisation of the landmark Hanbury Brown and Twiss interferometer. We calculate the two-electron interference pattern in a quantum path formalism taking into account the fermionic nature and the spin configuration of the electrons. We also estimate the Coulomb repulsion in the setup in a semiclassical approach. We find that antibunching resulting from Pauli's exclusion principle and repulsion stemming from the Coulomb interaction can be clearly distinguished.

Figures (4)

• Figure 1: Setup of the system. Two independent needle tip sources at positions $\mathbf{R}_1$ and $\mathbf{R}_2$ with distance $d$ are illuminated by short laser pulses to emit electrons via the photo effect. In the far field, the electrons are coincidentally detected by detectors at positions $\mathbf{r}_1$ and $\mathbf{r}_2$. The resulting correlation pattern $G^{(2)}(\mathbf{r}_1,\mathbf{r}_2)$ exhibits two-particle interference due to the interference of different two-electron quantum paths. Inset: The path difference for two electrons emitted by different sources but detected at the same detector $j$ is given by $d\sin(\theta_j)$.
• Figure 2: Quantum paths and resulting second-order spatial correlation functions $G^{(2)}(\delta)$. (a) Different possible quantum paths of the electrons - distinguished according to equal spin ($s=s'$) vs. different spin ($s\neq s'$) as well as single-fermion emitters (SFE) emitting a single photon at maximum vs. multi-fermion emitters (MFE), where two electrons can be emitted by a single source. Note that the latter is forbidden for equal spins by the Pauli principle. (b) For single-particle emitters and neglecting different spins, i.e., concentrating on the two upper left quantum paths of (a), fermionic and bosonic correlations exhibit two-particle interference with visibility one, but with a $\pi$ phase shift showing opposite interfering behaviour. This is due to the different commutativtiy of fermionic and bosonic particles. (c) Including different spin settings leads to an offset as shown in blue (dashed) for single-fermion emitters (SFE) due to the additional two quantum paths top right in (a). For multi-fermion emitters (MFE), the two green (dot-dashed) quantum paths displayed bottom right in (a) additionally add to the offset. For the plots, we assume a Poissonian particle distribution. For a better comparison with the bosonic case, we plot in (b) and (c) $G^{(2)}(\delta)$ with $4p_1^2|C(\boldsymbol{\mathdutchcal{k}}_1)|^2|C(\boldsymbol{\mathdutchcal{k}}_2)|^2 = 2$.
• Figure 3: Scheme for the classical estimation of the Coulomb repulsion. (a) Two electrons are emitted with parallel momentum $\mathbf{k}$ towards the detection screen. Due to the Coulomb force, the two electrons repel each other (blue dotted trajectories) leading to an increased separation of the electrons and an estimated dip width $z_{\mathrm{dip}}$. (b) In the centre-of-mass frame, the problem can be reduced to the one-dimensional movement of an electron in a Coulomb potential.
• Figure 4: Two-electron spatial correlation pattern including Coulomb repulsion, plotted for perfect spin polarization ($s=s'$), fixed magnitude of the wave vector $k=1e11\per m$ ($\lambda_{dB}=6.28e-11m$), a fixed distance between the needle tip electron sources and the detection screen $D=1 m$, but varying distance between the tips $d$. (a) $d=0.01nm$ mimicking a single tip setup; here the oscillation due to Coulomb repulsion and fermionic anti-correlation is of the same order of magnitude. (b) $d=10nm$ for a realistic two tip setup; here the fermionic oscillations can be clearly differentiated from the Coulomb dip. As in Fig. \ref{['fig:Paths_and_Patterns']}, we plot $G^{(2)}(\delta)$ with $4p_1^2|C(\boldsymbol{\mathdutchcal{k}}_1)|^2|C(\boldsymbol{\mathdutchcal{k}}_2)|^2 = 2$.