Perturbation-free prediction of resonance-assisted tunneling in mixed regular-chaotic systems

For generic Hamiltonian systems we derive predictions for dynamical tunneling from regular to chaotic phase-space regions. In contrast to previous approaches, we account for the resonance-assisted enhancement of regular-to-chaotic tunneling in a nonperturbative way. This provides the foundation for future semiclassical complex-path evaluations of resonance-assisted regular-to-chaotic tunneling. Our approach is based on a new class of integrable approximations which mimic the regular phase-space region and its dominant nonlinear resonance chain in a mixed regular-chaotic system. We illustrate the method for the standard map.

Figure 1

(a) Regular-to-chaotic decay rate ${\gamma }_0$ versus $1/h$ for the standard map at $\kappa =3.4$. The numerically determined rates (gray dots) are compared to (the sum of incoherent terms of) the predictions of Eq. (2) [(red) triangles] and Eq. (3) [(blue) squares]. (b) Phase space with regular orbits (lines) and a chaotic orbit (dots) including a 6:2 nonlinear resonance chain. (c) Like (b) with an integrable approximation [(red) lines] on top.

Figure 2

[(a) and (b)] Husimi representation of $\left|I_n\right.〉$ for the standard map at $\kappa =3.4$ for (a) $n=0$ and (b) $n=6$ at $1/h=53$. Regular tori (gray lines) and chaotic orbits (dots) illustrate the phase space. The quantizing tori of $H_{r:s}$ for $V_{r:s}=0$ are shown by a thick (white) line. (c) Approximate mode $\left|m_{\text{int}}\right.〉$ for $m=0$.

Figure 3

Decay rates for the standard map at (a) $\kappa =2.9$, (b) $\kappa =3.4$, and (c) $\kappa =3.5$ versus the inverse effective Planck constant $1/h$. Numerically determined rates (dots) are compared to predicted rates, using Eq. (2) [(red) triangles] and Eq. (3) [(blue) squares]. The insets show the corresponding phase space with regular tori [(gray) lines] and chaotic orbits (dots) with tori of the integrable approximation [(red) lines].

Figure 4

Decay rates for the standard map at (a) $\kappa =2.9$, (b) $\kappa =3.4$, and $\kappa =3.5$ versus the inverse effective Planck constant $1/h$. Numerically determined rates (dots) are compared to rates, predicted from incoherent terms according to Eq. (29), with ${\mathrm{\Gamma }}_m^{\text{inc}}\left(1\right)$ [(red) pluses] and ${\mathrm{\Gamma }}_m^{\text{inc}}\left(0\right)$ [(blue) crosses]. We further show predictions according to Eq. (2) [(gray) triangles] and Eq. (3) [(gray) squares].

Figure 5

Decay rates for the standard map at (a) $\kappa =2.9$, (b) $\kappa =3.4$, and (c) $\kappa =3.5$ versus the inverse effective Planck constant $1/h$. Numerically determined rates (dots) are compared to rates, predicted perturbatively according to Eq. (33) with ${\mathrm{\Gamma }}_m^{\text{per}}\left(1\right)$ [(red) pluses] and ${\mathrm{\Gamma }}_m^{\text{per}}\left(0\right)$ [(blue) crosses]. We further show the prediction based on incoherent terms, according to Eq. (29) with ${\mathrm{\Gamma }}_m^{\text{inc}}\left(1\right)$ [(gray) pluses] and ${\mathrm{\Gamma }}_m^{\text{inc}}\left(0\right)$ [(gray) crosses].

Figure 6

Numerically determined decay rates ${\gamma }_0$ of the standard map at $\kappa =3.4$ versus the inverse effective Planck constant $1/h$ for $q_l=0.26$ [(gray) dots] and $q_l=0.1$ [(magenta) squares]. [(b) and (c)] Phase space with shaded areas showing the leaky regions corresponding to $q_l$.

Figure 7

Same as Fig. 6 for $\kappa =2.9$ with $q_l=0.27$ [(gray) dots] and $q_l=0.1$ [(magenta) squares].

Figure 8

For the standard map at $\kappa =3.4$ with $1/h=55$ we compare the position representation of (a) ${\left|\langle q|m\rangle \right|}^2$ to $|\langle q|m_{\text{int}}\rangle {|}^2$, (b) $|\langle q|m^{\prime }\rangle {|}^2$ to $|\langle q|m_{\text{int}}\rangle {|}^2$, and (c) $|\langle q|\widehat{U}{|m\rangle |}^2$ to $|\langle q|\widehat{U}|m_{\text{int}}{\rangle |}^2$, depicting them with (gray) dots and (magenta) squares, respectively, for $m=m^{\prime }=m_{\text{int}}=0$. The dashed lines mark the positions $q_l$ and $1-q_l$ of the leaky region, as given in the text.

Figure 9

Error analysis for the standard map at $\kappa =3.4$. [(a)–(c)] The numerically determined rates [(gray) dots] and (d) the numerically determined phases $\text{Arg}\left(〈I_n|0〉\right)$ for $n=0,1,2$ (lines) are shown versus the inverse effective Planck constant $1/h$. (a) The contributions ${\gamma }_{0,n}^{\text{diag}}$, Eq. (35), are shown by lines. (b) The reduced prediction, Eq. (34) with $n,n^{\prime }\in \{0,6,12\}$ is shown by (red) triangles. (c) The contributions ${\gamma }_{0,n}^{\text{diag}}$ of Eq. (35) (lines) are compared to ${\mathrm{\Gamma }}_{0,n}^{\text{diag}}\left(1\right)$ of Eq. (30) (markers) for $n=0,6,12$. (d) The phases $\text{Arg}\left(〈I_n|0〉\right)$ (lines) and $\text{Arg}\left(〈I_n|0_{\text{int}}〉\right)$ (markers) are compared for $n=0,6,12$. (Phases close to zero are slightly shifted for $n=6,12$ for visibility.)

Figure 10

For the standard map at $\kappa =3.4$ we show (a) the fit of ${\mathcal{H}}_0^{\prime }\left(I\right)$ (line) to the actions and frequencies of the regular region $(\overline{J},\overline{\omega })$ (crosses). (b) The function ${\mathcal{H}}_0\left(I\right)$ is shown as a (red) line. The two (black) dots show $(I_m,{\mathcal{H}}_0\left(I_m\right))$ and $(I_{m+kr},{\mathcal{H}}_0\left(I_{m+kr}\right))$ at $1/h=53$. [(a) and (b)] The dashed line shows the position of ${\overline{J}}_{\text{max}}$.

Figure 11

Decay rates ${\gamma }_0$ for the standard map at $\kappa =3.4$ versus the inverse effective Planck constant. Numerically determined rates [(gray) circles] are compared to predicted rates according to Eq. (2) [(colored) symbols] based on three slightly different integrable approximations.