Transient concurrence for copropagating entangled bosons and fermions

Abstract

Transient dynamics of copropagating entangled bosons and fermions remains an unexplored aspect of quantum mechanics. We investigate how entanglement affects the spatiotemporal evolution of the particles using a modified version of the quantum shutter model. We derive a transient concurrence to characterize momentum-space entanglement, and show that it modulates the interference correlation of the joint probability density, allowing us to visualize the active regions where probabilistic bunching and antibunching phenomena emerge. Furthermore, we derive analytical expressions that reveal a direct connection between entanglement and the characteristic oscillations of the Hanbury-Brown and Twiss (HBT) effect, highlighting the modulation of the phenomenon by quantum concurrence. This work introduces a temporal indicator of entanglement for a system of two coherent copropagating modes establishing a direct relationship with HBT-type interference patterns, providing a theoretical framework to explore the manifestation of entanglement in transient regimes.

Figures (2)

• Figure 1: (a) Density plot of the transient concurrence $\mathscr{C}(\Psi)$ calculated from Eq. (\ref{['eq_18']}) as a function of both the coefficient $\xi$ and time $t$ using the shutter model in the calculation of $|\Phi_{A}\Phi_{B}|$. As we can see in the upper panel (b), a section of the map at a fixed time, say $t_0=0.324$ ps (horizontal dashed red line), exhibits a curve with the characteristic shape of the concurrence of bipartite systems for a stationary entangled pure state walborn2006. On the other hand, as we see in the right panel (c), a vertical cut in the map (vertical dashed red line) at a fixed value of $\xi$, say $\xi_0=1 / \sqrt{2}$, exhibits the emblematic profile of a diffraction in time pattern PhysRev.88.625. Parameters: $a = 10.0$ nm, $b = 11.0$ nm, $\alpha = 1.0 k_0$, $\beta = 1.01 k_0$, with $k_0 = 1.449~\mathrm{nm}^{-1}$. The reference value of $k_0$ comes from $k_0=\sqrt{2 m E_0}/\hbar$, where we chose the arbitrary reference energy $E_0 = 0.08~\mathrm{eV}$ and the mass $m = m_e$ ($m_e$ is used here only as a reference mass).
• Figure 2: On the left panel, we plot in (a) to (e) the symmetric (blue line) and antisymmetric (red line) joint probability densities as functions of time for the indicated fixed values of $\Delta x=b-a$ calculated from Eq. (\ref{['Sol_Ms']}). Parameters: $\chi=\xi_0=1 / \sqrt{2}$, $a = 100$ nm (varying $b$), $\alpha = 2.00 k_0$, $\beta = 2.05 k_0$ (the parameters $k_0$ and $m$ are the same as in Fig. \ref{['Fig_1']}). Also included here is the transient concurrence as a function of time (shaded gray curves). (f) Joint probability density as a function of $\Delta x$ at a fixed time $t_0=0.4$ ps, comparing the symmetric case (blue line) with the anti-symmetric case (red line). (g) Correlation interference calculated with Eq. (\ref{['interference_c']}) (orange line) plotted as a function of $\Delta x$ compared to the exact calculation form Eq. (\ref{['interference']}) (green line).