Coulomb and exchange interaction effects on the exact two-electron dynamics in the Hong-Ou-Mandel interferometer based on Hall edge states

The electronic Hong-Ou-Mandel interferometer in the integer quantum Hall regime is an ideal system to probe the building up of quantum correlations between charge carriers and it has been proposed as a viable platform for quantum computing gates. Using a parallel implementation of the split-step Fourier method, we simulated the antibunching of two interacting fermionic wave packets impinging on a quantum point contact. Numerical results of the exact approach are compared with a simplified theoretical model based on one-dimensional scattering formalism. We show that, for strongly localized wave packets in a full-scale geometry, the Coulomb repulsion dominates over the exchange energy, this effect being strongly dependent on the energy broadening of the particles. We define analytically the spatial entanglement between the two regions of the quantum point contact, and obtain quantitatively its entanglement-generation capabilities.

Figure 1

Top view of the HOM interferometer with the potential landscape reproducing the QPC (blue region) and the integrated single-particle density probability at $t=0$ (red wave packets). $\alpha$ and $\beta$ are the initial counterpropagating states for $\sigma =20$ nm, while $T$ and $B$ label the top and bottom regions we defined in the HOM (separated by the diagonal white line).

Figure 2

Conditional probability (red contour lines) for particle 1 $P^{*}(x_1,y_1,x_2^{*},y_2^{*},t^{\prime })={\left|\mathrm{\Psi }(x_1,y_1,x_2^{*},y_2^{*},t^{\prime })\right|}^2$ at $t^{\prime }=0.3$ ps (left column) and $t=0.9$ ps (right column) for the selected positions (green dots) of particle 2, $(x_2^{*},y_2^{*})=(-11.5,50)$ nm (first row) and $(x_2^{*},y_2^{*})=(50,-11.5)$ nm (second row), in the potential landscape of the HOM. Note that, due to Pauli exclusion principle, the wave function always vanishes at $(x_1,y_1)=(x_2,y_2)$. By selecting different couples of coordinates $(x_2^{*},y_2^{*})$, the conditional probability shows the evolution of one of the two or both single-particle wave packets, as indicated by the arrows.

Figure 3

Numerical simulations of the exchange-symmetry driven HOM interferometer. (a) Bunching probability of the two-fermion state in absence of Coulomb interaction for different spatial dispersions $\sigma$. (Inset) Numerical fit of Eq. (15) (black dashed line) with the stationary bunching probabilities in Fig. 3 at $t=1.2$ ps ($▴$). This provides the geometrical parameter in Eq. (10) for our QPC, $a=60\pm 1$ nm. (b) Unconditional probability of particle 1 $P(x_1,y_1,t^{\prime })=\int \int d{x}_{2}d{y}_2{\left|\mathrm{\Psi }(x_1,y_1,x_2,y_2,t^{\prime })\right|}^2$ at different time steps $t^{\prime }$ in the potential landscape of the HOM. The snapshots show that the stationary regime is achieved between 0.9 and 1.2 ps. Note that due to exchange correlation, the unconditional probability of particle 1 shows the evolution of both wave packets.

Figure 4

(Top) 1D model for the two-particle scattering at the QPC: two wave packets of plane waves with opposite central momentum impinge on a 1D potential barrier. (Bottom) Bunching (empty dots) and antibunching (full dots) probabilities for an antisymmetric (red) and symmetric (black) two-particle wave function with $\sigma =20$ nm. Only exchange symmetry is present.

Figure 5

(a) Bunching probability as a function of the initial displacement $\mathrm{\Delta }y$ between two indistinguishable wave packets ($\sigma =15$ nm) with the zero (blue) and nonzero (green) Coulomb interaction $V_{12}$ with $d=1$ nm. For $V_{12}=0$, the numerical data (blue dots) are compared to Eq. (15) (blue line) with $\alpha =60$ nm, while for $V_{12}\ne 0$ numerical data (green dots) are fit by the equation $g\left(x\right)$ (green dashed line) as explained in the main text. The fit provides an effective broadening ${\sigma }_{\mathrm{eff}}=21.75\pm 0.03$ nm and an effective geometrical parameter $a_{\mathrm{eff}}=84.0\pm 0.1$ nm. (b) Stationary bunching probability ($t=1.2$ ps) in presence of long-range Coulomb interaction without exchange symmetry for different $d$ parameters in Eq. (16) and $\sigma =20$ nm. (c) Bunching probability in presence of long-range Coulomb interaction without exchange symmetry and (d) with exchange symmetry for different $\sigma$ and $d=1$ nm. (e) Comparison between the stationary bunching probability of two distinguishable electrons (yellow line) and indistinguishable electrons (green line) with long-range Coulomb interaction. The stationary bunching probability of two indistinguishable electrons in presence of a screened Coulomb interaction is also reported (blue circle) for the case $\sigma =20$ nm and a damping ${\sigma }_c=5$ nm, see Eq. (17).

Figure 6

Entanglement between $T$ and $B$ regions in the antibunched configuration. (a) Von Neumann entropy for different $\sigma$ and $d=1$ nm in presence of exchange symmetry only; the dotted black lines in the inset show where the full scale wave function is projected with respect to the top view of the potential profile. (Bottom) Comparison between the stationary Von Neumann entropy in the different scenarios.