Quasibound states in a triple Gaussian potential

We derive the transmission probabilities and delay times, and identify quasibound state structures in an open quantum system consisting of three Gaussian potential energy peaks, a system whose classical scattering dynamics we show to be chaotic. Such open quantum systems can serve as models for nanoscale quantum devices and their wave dynamics are similar to electromagnetic wave dynamics in optical microcavities. We use a quantum web to determine energy regimes for which the system exhibits the quantum manifestations of chaos, and we show that the classical scattering dynamics contains a significant amount of chaos. We also derive an exact expression for the non-Hermitian Hamiltonian whose eigenvalues give quasibound state energies and lifetimes of the system.

Figure 1

Scattering potential. (a) Symmetric triple peak: cylindrical coordinates. (b) Symmetric triple peak: Cartesian coordinates. (c) Asymmetric triple peak: cylindrical coordinates. (d) Asymmetric triple peak: Cartesian coordinates.

Figure 2

Symmetric potential. The real part of the $S$ matrix for $U_1=0.0$ and $V_1=20.1$ and $E=10$. Due to the threefold rotation symmetry of the potential and the reflection symmetry, coupling only occurs between angular momenta $\ell \to \ell \pm 3$ together with $\ell \to -\ell$. (Darker squares indicate larger real values, red indicates positive numbers, blue indicates negative numbers.)

Figure 3

Symmetric potential. The magnitude of the scattering matrix elements (a) $|S_{0,0}|$, (b) $|S_{1,1}|$, (c) $|S_{2,2}|$, and (d) $|S_{3,3}|$ for $U_1=0.0$ and $V_1=20.1$.

Figure 4

Symmetric potential. The magnitude of the scattering matrix elements (a) $|S_{0,1}|$, (b) $|S_{0,2}|$, (c) $|S_{0,3}|$ for $U_1=0.0$ and $V_1=20.1$.

Figure 5

A selection of reaction region states for $V=20.1$ and $U=0.0$. (a) $\lambda =0.535$, $L^2=1.279$; (b) $\lambda =1.072$, $L^2=0.599$; (c) $\lambda =2.203$, $L^2=3.049$; (d) $\lambda =4.84$, $L^2=80.512$; (e) $\lambda =7.014$, $L^2=8.259$; (f) $\lambda =19.548$, $L^2=12.432$.

Figure 6

Wigner-Smith delay time plots. (a) $\mathrm{U}=0.0,V=20.1$; (b) $U=0.5$, $V=16.8$.

Figure 7

Asymmetric potential. The real part of the $S$ matrix for $U_1=0.5$ and $V_1=16.8$ and $E=10$. Because there is no rotational symmetry, all values of the angular momentum $\ell$ can be coupled. (Darker squares indicate larger real values, red indicates positive numbers, blue indicates negative numbers.)

Figure 8

Asymmetric potential. The magnitude of the scattering matrix elements (a) $|S_{0,0}|$, (b) $|S_{1,1}|$, (c) $|S_{2,2}|$, and (d) $|S_{3,3}|$ for $U_1=0.5$ and $V_1=16.8$.

Figure 9

Asymmetric potential. The magnitude of the scattering matrix elements (a) $|S_{0,1}|$, (b) $|S_{0,2}|$, (c) $|S_{0,3}|$ for $U_1=0.5$ and $V_1=16.8$.

Figure 10

A selection of reaction region states for $V=16.8$ and $U=0.5$. (a) $\lambda =0.510$, $L^2=1.380$; (b) $\lambda =1.180$, $L^2=2.656$; (c) $\lambda =2.547$, $L^2=14.807$; (d) $\lambda =3.963$, $L^2=63.625$; (e) $\lambda =14.120$, $L^2=37.989$; (f) $\lambda =23.509$, $L^2=107.873$.

Figure 11

Quantum web composed of $L_i^2$ versus ${\lambda }_i$ for (a) $U_1=0.0$ and $V_1=1.0$; (b) $U_1=0.0$ and $V_1=20.1$; and (c) $U_2=0.5$ and $V_2=16.8$. In these plots, we only show values of $0\le L^2\le 110$ and ${\lambda }_i<30$.

Figure 12

A scattering trajectory and the ScSOS for an integrable single Gaussian scattering potential with potential energy height $V=20$. (a) The configuration space picture of a single scattering trajectory with $E=5$, initial impact parameter $b=-0.3$, and angle $\alpha =0$. (b) The ScSOS for a range of initial values of impact parameter $b$. (The change in shading of the orbit, from dark to light, corresponds to entry and successive reentry of the same orbit.)

Figure 13

The ScSOSs for the symmetric triple peak potential for a range of initial impact parameters. (a) $E=5$. (b) $E=10$. (c) $E=20$. (d) $E=30$.

Figure 14

The configuration space tracks of two scattering trajectories for the symmetric triple peak potential. (a) A chaotic trajectory with energy $E=5$. (b) A period-3 trajectory for $E=20$. (The change in shading of the orbits, from dark to light, corresponds to entry and successive reentry of the same orbit.)

Figure 15

All eigenvalues of $H_{\mathrm{eff}}$ with $\text{Im}\left(E\right)\ge -0.12$. (a) Symmetric potential; (b) asymmetric potential.