Exact time evolution of the pair distribution function for an entangled two-electron initial state

Based on the correlated ground-state wave function of an exactly solvable interacting one-dimensional two-electron model Hamiltonian we address the switch-off of confining and interparticle interactions to calculate the exact time-evolving wave function from a prescribed correlated initial state. Using this evolving wave function, the time-dependent pair probability function $\mathcal{R}(x_1,x_2,t)\equiv n_2(x_1,x_2,t)/\left[n(x_1,t)n(x_2,t)\right]$ is determined via the pair density $n_2(x_1,x_2,t)$ and single-particle density $n(x,t)$. It is found that $\mathcal{R}(0,0,t=\infty )=\mathcal{R}(0,0,t=0)>1$, and $\mathcal{R}(x_1,x_2,t^{*})=1$ at a finite $t^{*}$ for $\Lambda \ne 0$ interparticle interaction strength in the initial two-electron model. By expanding $n(x,t)$ in an infinite sum of closed-shell products of time-dependent normalized single-particle states and time-dependent occupation numbers $P_k(\Lambda ,t)$, the von Neumann entropy $S(\Lambda ,t)=-{\sum }_{k=0}^{\infty }{P}_k\left(t\right)\ln P_k\left(t\right)$ is calculated as well. The such-defined information entropy is zero at $t^{*}\left(\Lambda \right)$ and its maximum in time is $S(\Lambda ,t=\infty )=S(\Lambda ,t=0)$.

Figure 1

Calculated dimensionless ratio, defined via ${\Omega }_t/{\Omega }_e\left(t\right)$, as a function of time measured in atomic units. The solid curve refers to the repulsive interparticle interaction, while the dashed one to the attractive case. See the text for further details on numerical values used in data generation.

Figure 2

The dimensionless probability function $\mathcal{R}(0,0,t)$ (i.e., the pair distribution function at the origin $x_1=x_2=0$, as a function of time $t$). The same parameters are used as in Fig. 1. The solid and dashed curves refer to repulsive and attractive interparticle interactions, respectively.

Figure 3

The information-theoretic entropy $S\left(t\right)$, as a function of time $t$ measured in atomic units. The same parameters are used as in Figs. 1 and 2. The solid curve refers to the repulsive interparticle interaction, while the dashed one to the attractive case.