Two-electron coherence and its measurement in electron quantum optics

Engineering and studying few-electron states in ballistic conductors is a key step towards understanding entanglement in quantum electronic systems. In this Rapid Communication, we introduce the intrinsic two-electron coherence of an electronic source in quantum Hall edge channels and relate it to two-electron wave functions and to current noise in a Hanbury Brown–Twiss interferometer. Inspired by the analogy with photon quantum optics, we propose to measure the intrinsic two-electron coherence of a source using low-frequency current correlation measurements at the output of a Franson interferometer. To illustrate this protocol, we discuss how it can distinguish between a time-bin-entangled pure state and a statistical mixture of time-shifted electron pairs.

Figure 1

Contributions to the two-electron detection probability by detectors $D_1$ and $D_2$. Full lines represent direct probability amplitudes, whereas dotted lines represent complex conjugated ones. The electronic system is decomposed into the Fermi sea $\left(F\right)$ and a source $\left(S\right)$ emitting electron and/or hole excitations. The two-electron detection probability contains four types of contributions.

Figure 2

(a) Normalized density plot of $\mathrm{\Delta }{\mathcal{G}}_S^{\text{(2e)}}(t_1,t_2|t_1,t_2)$ as a function of $\overline{t}=(t_1+t_2)/2$ and $\tau =t_1-t_2$ for three Landau excitations of duration ${\tau }_{\text{e}}$ and energy ${\omega }_{\text{e}}=3/{\tau }_{\text{e}}$ separated by $\mathrm{\Delta }t=3{\tau }_{\text{e}}$. (b) Integral $\int \mathrm{\Delta }{\mathcal{G}}_S^{\text{(2e)}}d\overline{t}$ i.e., average correlations as a function of $\tau$ only. Vanishing at $\tau =0$ is a manifestation of Pauli principle.

Figure 3

Franson interferometer setup used to access two-particle coherence in the time domain. The left $\left(L\right)$ and right $\left(R\right)$ MZI are pierced by magnetic fluxes ${\mathrm{\Phi }}_{L,R}$ and have time delays $\delta t_{L,R}$. An electron source is connected to the channel $1s$, whereas channels $2s,L^{\prime }\text{in}$ and $R^{\prime }\text{in}$ are grounded. The quantity of interest is the current correlation $\langle i_{L\mathrm{out}}\left(t\right)i_{R\mathrm{out}}\left(t^{\prime }\right)\rangle$.

Figure 4

Density plots for real [panels (a) and (c)] and imaginary [panels (b) and (d)] parts of the averaged excess two-electron coherence as a function of $(\delta t_L,\delta t_R)$, extracted from the ${\mathrm{\Phi }}_L+{\mathrm{\Phi }}_R$ dependence of the low-frequency correlation (13). Panels (a) and (b): signal for a pair of Landau excitations of duration ${\tau }_{\text{e}}$ and energy ${\omega }_{\text{e}}=3/{\tau }_{\text{e}}$ separated by $\mathrm{\Delta }t=3{\tau }_{\text{e}}$. The plot is identical for a statistical mixture of two time-shifted identical electron pairs. Panels (c) and (d): signal expected for a quantum superposition of two such pairs in two time bins, separated by $\mathrm{\Delta }{T}_{\text{tb}}=7{\tau }_{\text{e}}$.