Resonance-assisted tunneling in four-dimensional normal-form Hamiltonians

Nonlinear resonances in the classical phase space lead to a significant enhancement of tunneling. We demonstrate that the double resonance gives rise to a complicated tunneling peak structure. Such double resonances occur in Hamiltonian systems with an at least four-dimensional phase space. To explain the tunneling peak structure, we use the universal description of single and double resonances by the four-dimensional normal-form Hamiltonians. By applying perturbative methods, we reveal the underlying mechanism of enhancement and suppression of tunneling and obtain excellent quantitative agreement. Using a minimal matrix model, we obtain an intuitive understanding.

Figure 1

Phase space of the single coupled resonance shown in a 3D hyperplane for ${\theta }_2=0$ fixed. Regular structures are shown as 1D black curves for four planes with constant polyad number. The eigenstate $|{\psi }_{\mathit{0}}\rangle$ maximally localizing on the classical quantizing torus ${\mathit{I}}_{\mathit{0}}=\left(0,0\right)$ (white line) is shown in a 3D Husimi phase-space representation (normal scale, see color bar) (a) for $1/h=0.3450$ and (b) for $1/h=0.3979$. The region $\mathrm{\Lambda }$, Eq. (9), trivially extended in the ${\theta }_1$ direction, is shown in light blue. For comparison, the squared amplitude of the eigenstate $|{\psi }_{\mathit{0}}(I_1,I_2){|}^2$ is shown in action-space representation (normal and logarithmic scale). The resonance center line, Eq. (3), is shown as dashed line.

Figure 2

Weight $w_{\mathit{0}}$ of state $|{\psi }_{\mathit{0}}\rangle$, see Eq. (12), as function of the inverse effective Planck constant $1/h$ for the single coupled resonance shown as orange bullets. The perturbative prediction (18) is shown as blue crosses.

Figure 3

Scheme of perturbation theory for the single coupled resonance. The action grid ${\mathit{I}}_{\mathit{m}}$ is depicted as gray points, the quantizing torus ${\mathit{I}}_{\mathit{0}}=\left(0,0\right)$ as green point. The dashed line shows the position of the resonance center line; see Eq. (3). The circle indicates the level set of constant energy $H_0\left({\mathit{I}}_{\mathit{0}}\right)$. The blue shaded area shows the region $\mathrm{\Lambda }$; see Eq. (9). The red line indicates the shortest path $\mathrm{\Gamma }$. (a) The nonresonant case for $1/h=0.1314$ and $\left|\mathrm{\Gamma }\right|=2$. (b) The resonant case for $1/h=0.1638$ and $\left|\mathrm{\Gamma }\right|=3$.

Figure 4

Phase space of the double resonance shown in a 3D hyperplane for ${\theta }_2=0$ fixed. Regular structures are shown as 1D black curves, whereas chaotic structures are shown as points. The eigenstate $|{\psi }_{\mathit{0}}\rangle$ maximally localizing on the classical quantizing torus ${\mathit{I}}_{\mathit{0}}=\left(0,0\right)$ (white line) is shown in a 3D Husimi phase-space representation (normal scale, see color bar) (a) for $1/h=0.375$, (b) for $1/h=0.1991$, and (c) for $1/h=0.1591$. The region $\mathrm{\Lambda }$, Eq. (21), trivially extended in the ${\theta }_1$ direction, is shown in light blue. For comparison, the squared amplitude of the eigenstate $|{\psi }_{\mathit{0}}(I_1,I_2){|}^2$ is shown in action-space representation (normal and logarithmic scale). The resonance center lines, Eq. (3), are shown as dashed lines.

Figure 5

Weight $w_{\mathit{0}}$ of state $|{\psi }_{\mathit{0}}\rangle$, see Eq. (12), as function of the inverse effective Planck constant $1/h$ for the double resonance is shown as orange bullets. The perturbative prediction (27), only using the shortest paths, is shown as blue crosses.

Figure 6

Perturbative prediction using Eq. (27) incorporating an increasing number of paths sorted by the absolute value of their contribution $|{\lambda }_{\mathit{m}}^{\mathrm{\Gamma }}|$. Prediction for one path is shown as crosses, for two paths as circles, and for all paths as squares.

Figure 7

Scheme of perturbation theory for the double resonance. The action grid ${\mathit{I}}_{\mathit{m}}$ is depicted as gray points and the quantizing torus ${\mathit{I}}_{\mathit{0}}=\left(0,0\right)$ as green point. The dashed lines show the position of the two resonance center lines; see Eq. (3). The circle indicates the level set of constant energy $H_0\left({\mathit{I}}_{\mathit{0}}\right)$. The blue shaded area shows the region $\mathrm{\Lambda }$; see Eq. (21). In panel (a) for $1/h=0.1991$, five paths with an resonant intermediate state are shown, and in panel (b) for $1/h=0.1591$, two resonant paths with different intermediate state are shown. In Figs. 4 and 4, both states are shown in a Husimi representation. In panel (c) for $1/h=0.1465$, two canceling paths are shown, and in panel (d) for $1/h=0.2244$, six nonresonant paths are shown.

Figure 8

Matrix model: (a) Weight $w_0$ as a function of $\hslash$ obtained via Eq. (29) shown as orange line and $w_{0,\text{pert}}$ calculated using Eq. (32) as blue line. (b) Energies ${\tilde{E}}_i$ (solid lines) of the states $|{\psi }_i\rangle$ as a function of $\hslash$ in black, red, green, and gray for $i=0,1,2,3$. The vertical dashed lines indicate the values of $\hslash$ for the cases of enhancement and suppression.