Figure 1

Spatially periodic phase-space portraits (black dots) of the system given by Eq. (1) [6] showing a large island of regular motion and a chaotic sea in each spatial unit cell with $p\in [-1/2,1/2]$. Eigenstates of the corresponding quantum system are projected onto phase space (colored density). (a) The regular island is transporting as indicated by the arrows. All eigenstates, such as the one shown, spread over the regular islands as well as the chaotic sea. These amphibious eigenstates ignore the classical structures even though Planck’s constant $h$ is smaller than the size of the island. For comparison we show in (b) and (c) a system with regular islands of similar size that are not transporting [6]. Two different types of eigenstates—(b) chaotic and (c) regular—exist, in agreement with the semiclassical eigenfunction hypothesis.Reuse & Permissions

Figure 2

The weight of the amphibious eigenstate from Fig. 1a in each unit cell, $\rho (x)={\int }_{x-1/2}^{x+1/2}d{x}^{\prime }|\psi (x^{\prime }){|}^2$, shows an overall exponential decay with strong fluctuations. The localization length is several orders of magnitude bigger than in the case of the system of Fig. 1b with nontransporting islands (inset). It is quite well estimated by $\exp (-|x|/\lambda )$ (dashed line), with $\lambda =v/\gamma$ determined from the tunneling rate $\gamma$ out of the regular island with velocity $v=1$.Reuse & Permissions

Figure 3

Quantum evolution of a wave packet (gray shade) from $t=0$ (left) to $t=1$ (right) for the system of Fig. 1a where classically a regular island moves one unit cell to the right in one time period. Although the underlying amphibious eigenstates ignore the classical phase-space structure they conspire such that the initial dynamics of wave packets follows the semiclassical expectation.Reuse & Permissions

Figure 4

A typical eigenstate for a system with a symmetric phase space and islands transporting in opposite directions (arrows in lower inset) [25]. Several regions are projected onto phase space (insets) demonstrating the mixed type of this eigenstate, with some regions being mainly regular, some mainly chaotic, and some amphibious. Each type appears at many locations of the eigenstate (short arrows).Reuse & Permissions