Packet narrowing and quantum entanglement in photoionization and photodissociation

The narrowing of electron and ion wave packets in the process of photoionization is investigated, with the electron-ion recoil taken fully into account. Packet localization of this type is directly related to entanglement in the joint quantum state of the electron and ion, and to Einstein-Podolsky-Rosen localization. Experimental observation of such packet-narrowing effects is suggested via coincidence registration by two detectors, with a fixed position of one and varying position of the other. A similar effect, typically with an enhanced degree of entanglement, is shown to occur in the case of photodissociation of molecules.

Figure 1

The relative-motion probability density $\mid {\Psi }_{\mathrm{rel}}{\mid }^2$ (21) in dependence on $\rho =(r_{\mathrm{rel}}-vt)∕\Delta r_{\mathrm{rel}}^{\left(0\right)}$ at (a) $\zeta =0.01$ and (b) $\zeta =20$.

Figure 2

The density distribution of the one-dimensional equivalent of $\mid \Psi {\mid }^2$ given in Eq. (26), where $x_e$ and $x_i$ are the one-dimensional electron and ion coordinates. The left picture corresponds to ${\Psi }_{\text{c.m.}}$ and ${\Psi }_{\text{rel}}$ of Eq. (26) given by Eqs. (8) and (21), with ${\vec{r}}_e$ and ${\vec{r}}_i$ substituted by $x_e$ and $x_i$. In the right picture ${\Psi }_{\text{c.m.}}$ and ${\Psi }_{\text{rel}}$ are modeled by sharp-edged flat functions, with boundaries of their localization regions shown by the dashed lines. The difference between the single-particle and coincidence-scheme widths of the electron wave packets is illustrated by the difference between the $cd$ and $ab$ distances in the right picture. This difference and the value of the entanglement parameter are seen to be large for a large aspect ratio of the shaded region. We have used $m_e∕m_i=0.2,\eta =0.5$, and ${\gamma }_{I}t=4$ for illustration.

Figure 3

Electron (a) and ion (b) wave-packet widths in the schemes of single-particle and coincidence measurements with $m_e∕m_i=0.1$.

Figure 4

Plot of the entanglement parameter $R$ as a function of $\eta \left(t\right)$ with $m_e∕m_i=0.1;{\eta }_{\ast }$ is the stability point (42) at which $\eta \left(t\right)\equiv \text{const}=\sqrt{\mu ∕M}$. The insets give the corresponding plots of the one-dimensional analog $\mid \Psi (x_e,x_i,t){\mid }^2$ from Fig. 2. The axes of the three insets have been rescaled so as to show the details more clearly. The regions where the entanglement parameter is large are clearly seen to correspond to a large aspect ratio of the shaded areas.

Figure 5

(a) shows the time-dependent widths of the center-of-mass and relative-motion wave packets, in units of $\Delta r_{\mathrm{rel}}^{\left(0\right)}$, and (b) shows the control parameter $\eta \left(t\right)$. We have taken ${\eta }_0=0.05$ and $m_e∕M=0.1$.

Figure 6

The same as in Fig. 5 but with ${\eta }_0=0.5$.

Figure 7

Sketch showing relative influences of $\mid {\Psi }_{\mathrm{rel}}{\mid }^2$ and $\mid {\Psi }_{\text{c.m.}}{\mid }^2$ on experimental measurements of localization. The ion is taken as the origin of coordinates, and Eq. (20) shows that a circle of radius $r_e=vt$ limits the range of the electron coordinate. The crescent-shaped shaded areas are the regions where the wave function $\mid {\Psi }_{\mathrm{rel}}{\mid }^2$ is relatively large (one-third of the maximum or more) at a given time $t$, and the sizes of the black spots indicate the widths of $\mid {\Psi }_{\text{c.m.}}{\mid }^2$ for three different electron positions where experiments might be done. In each case a cross marks the corresponding center of mass. The black spots can take different sizes relative to the size of the shaded area depending on the value of the control parameter $\eta \left(t\right)$, as defined in Eq. (41). A large value of the entanglement parameter $R$ is obtained for ${\vec{r}}_e$ located where there is a well-localized $\mid {\Psi }_{\text{c.m.}}{\mid }^2$ (i.e., a small black spot) combined with a large value of $\mid {\Psi }_{\mathrm{rel}}{\mid }^2$. In the case when the black spot is located far outside the shaded area there is no overlap between $\mid {\Psi }_{\mathrm{rel}}{\mid }^2$ and $\mid {\Psi }_{\text{c.m.}}{\mid }^2$ and the total wave function vanishes.

Figure 8

Entanglement parameter $R$ for two dissociating molecular fragments with $M_1=M_2$ (left) and the same dissociation curve plotted vs $\ln \left(\eta \right)$ on the right, where the corresponding photoionization curve is included for comparison, with its very different mass ratio $m_i={10}^4{m}_e$.