Time-dependent approach to many-particle tunneling in one dimension

By employing the time-dependent approach, we investigate a quantum tunneling decay of many-particle systems. We apply it to a schematic one-dimensional three-body model with a heavy-core nucleus and two valence protons. We calculate the decay width for two-proton emission from the survival probability, which well obeys the exponential decay law after a sufficient time. The effect of the correlation between the two emitted protons is also studied by observing the time evolution of the two-particle density distribution. It is shown that the pairing correlation significantly enhances the probability for the simultaneous diproton decay.

Figure 1

The core-proton potential used in our calculations. The initial state at $t=0$ is constructed with a modified potential $V_{\mathrm{mod}}\left(x\right)$ given by Eq. (7), which is shown by the dashed line. For $t>0$, the potential is changed to the original potential $V\left(x\right)$, given by Eq. (3) as shown by the solid line. We also show the wave function for the bound single particle (s.p.) state of the modified potential at $\epsilon =2.71\mathrm{MeV}$, which is shown by the dotted line.

Figure 2

Overlaps between the initial state ${\Psi }_0$ and the eigenfunctions ${\tilde{\Psi }}_k$ of the original three-body Hamiltonian as a function of the corresponding eigenenergies ${\tilde{E}}_k$. The solid and the dashed lines correspond to the cases of $g=20$ and $0\mathrm{MeV}\mathrm{fm}$, respectively. Notice that the initial state is the lowest eigenstate of the modified Hamiltonian $H_{\mathrm{mod}}$ with the eigenenergy of 4.56 MeV (5.42 MeV) for $g=20\mathrm{MeV}\mathrm{fm}$ ($g=0\mathrm{MeV}\mathrm{fm}$).

Figure 3

(a) The survival probability as a function of time $t$ defined by Eq. (21). (b) The decay width defined by Eq. (22). These are plotted for several values of $g$ indicated in the figure.

Figure 4

(a) The decay energy $E_0$ of the three-body system as a function of the strength of the pairing interaction $g$. The dashed line indicates the asymptotic kinetic energy of a bound diproton. (b) The decay width estimated at $t=1200\mathrm{fm}/\mathrm{c}$.

Figure 5

The time evolution of the density distribution $\rho (t,x_1,x_2)$ in the two-dimensional $(x_1,x_2)$ plane calculated with the pairing strength of $g=20\mathrm{MeV}\mathrm{fm}$. Panels (a)–(c) correspond to the density at $t=0,300$, and $600\mathrm{fm}/c$, respectively.

Figure 6

The flux distributions $\mathit{j}(t,x_1,x_2)=j_1(t,x_1,x_2){\mathit{e}}_1+j_2(t,x_1,x_2){\mathit{e}}_2$, where ${\mathit{e}}_i$ is the unit vector in the two-dimensional $(x_1,x_2)$ plane. These are obtained with $g=20\mathrm{MeV}\mathrm{fm}$ at (a) $t=300\mathrm{fm}/c$ and (b) $600\mathrm{fm}/c$ and are plotted in arbitrary units.

Figure 7

Same as Fig. 5 but for $g=0\mathrm{MeV}\mathrm{fm}$.

Figure 8

Same as Fig. 6 but for $g=0\mathrm{MeV}\mathrm{fm}$.

Figure 9

The four regions in the $(x_1,x_2)$ plane used to calculate the partial probabilities shown in Fig. 10. The boundaries of each region are at $x_1=\pm 16$ and $x_2=\pm 16\mathrm{fm}$.

Figure 10

The time evolution of the partial probabilities in the regions defined in Fig. 9 for $g=0$, 20, and $32\mathrm{MeV}\mathrm{fm}$. The total decay probability $P_{\mathrm{tot}}\equiv 1-P_4=P_1+P_2+P_3$ is also shown by the dot-dashed lines.