Multidimensional tunnelling between excited states

A Matrix Element for Chaotic Tunnelling Rates and Scarring Intensities

“…Eq. (24) is the form given in [3] (a factor of 2 appears because we use a different normalisation of the Green's function). There are actually four terms involved, depending on the combinations of directions in which the derivatives in ↔ D n and ↔ D n ′ act.…”

“…For this reason it is of value to pursue an alternative approach, as we do in this section, which steers a course between purely semiclassical and fully quantum calculation to specify individual tunnelling rates. Our discussion follows the methodology of AK [2] and WH [3], whereby each splitting is expressed as an overlap between approximate states localised in each of the wells. As discussed in WH, for chaotic systems, these local approximations are not found using semiclassical approximations.…”

“…Following [3], we simply assume the wavefunction to be known in the allowed region and then use Green's theorem to continue it into the forbidden region, using a semiclassical approximation for the Green's function constructed around complex classical trajectories. Let us define sections Σ L and Σ R = σΣ L cutting through the left and right wells respectively, as illustrated in Fig.…”

“…The important point is that it relates the splitting to an overlap of the localised states that is restricted to the forbidden region of phase space, where the Weyl symbol ofP is nonconstant. The form used in [3],…”

“…We concentrate on energy level splittings between states in a symmetric double well, though the calculation for resonance widths is also summarised. Our calculation is built upon an approach developed by Auerbach and Kivelson [2] and by Wilkinson and Hannay [3], henceforth referred to as AK and WH respectively. They use semiclassical Green's functions, constructed from complex trajectories, to extend wavefunctions from allowed to forbidden regions and thence to calculate spectral tunnelling effects.…”