Origin of the enhancement of tunneling probability in the nearly integrable system

Abstract

The enhancement of tunneling probability in the nearly integrable system is closely examined, focusing on tunneling splittings plotted as a function of the inverse of the Planck's constant. On the basis of the analysis using the absorber which efficiently suppresses the coupling, creating spikes in the plot, we found that the splitting curve should be viewed as the staircase-shaped skeleton accompanied by spikes. We further introduce renormalized integrable Hamiltonians and explore the origin of such a staircase structure by investigating the nature of eigenfunctions closely. It is found that the origin of the staircase structure could trace back to the anomalous structure in tunneling tail which manifests itself in the representation using renormalized action bases. This also explains the reason why the staircase does not appear in the completely integrable system.

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

“…On the other hand, the authors of of Ref. [38] observe non-vanishing contributions I n |m also for n = m + kr even for a near-integrable situation, where an excellent integrable approximations exist. They argue that nonvanishing I n |m should always occur and claim their treatment is beyond the current framework of resonanceassisted tunneling.…”

“…On the other hand, for our example a prediction of γ 0 where the remainder is smaller than the decay rate based on a reduced set of basis states |I n is only possible when summing over many additional contributions, even including n = m + kr. The precise origin of such contributions γ diag m,n with n = m + kr is currently under debate [38]: From the framework of resonanceassisted tunneling [19,20,36,40], we expect that the overlap I n |m vanishes for n = m + kr. Hence, one might argue that the contributions γ diag m,n with n = m+kr arise in our example only because our basis |I n is insufficiently accurate to decompose |m according to the theoretical expectation of resonance-assisted tunneling.…”

“…Motivated by these applications regular-to-chaotic tunneling is also a field of intense theoretical research [12,[23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39], which is mainly focused on periodically driven model systems with one degree of freedom. Here, a major achievement is the combination of (i) direct [30,35] and (ii) resonance-assisted [17][18][19][20] regular-tochaotic tunneling in a single prediction [36,40].…”

Quantum manifestations of classical nonlinear resonances

“…In the limit N → ∞ and for fixed l, the semiclassical expression works when k → 0. In this limit, we are moving closer and closer to the 2:1 resonance's center and then higher-order resonances, which can generally modify tunneling rates in a drastic manner [10][11][12][13][14]27], practically do not enter into consideration.…”

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

“…This was recently demonstrated in experiments for microwave resonators [29] and optical microcavities [21] by varying the frequency or the shape of the boundary. A lot of progress has been made concerning the theoretical understanding of resonance-assisted tunneling [9,12,20,23,[26][27][28][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49] mostly concentrating on systems effectively described by two-dimensional (2d) maps. One of the key tools for the theoretical description is the mapping of the system with nonlinear resonances to a universal pendulum-like 2d normal-form Hamiltonian, arising from secular perturbation theory [26,28] or normal form theory [50].…”