Universal Quantum Localizing Transition of a Partial Barrier in a Chaotic Sea

Generic 2D Hamiltonian systems possess partial barriers in their chaotic phase space that restrict classical transport. Quantum mechanically, the transport is suppressed if Planck’s constant $h$ is large compared to the classical flux, $h\gg \Phi$, such that wave packets and states are localized. In contrast, classical transport is mimicked for $h\ll \Phi$. Designing a quantum map with an isolated partial barrier of controllable flux $\Phi$ is the key to investigating the transition from this form of quantum localization to mimicking classical transport. It is observed that quantum transport follows a universal transition curve as a function of the expected scaling parameter $\Phi /h$. We find this curve to be symmetric to $\Phi /h=1$, having a width of 2 orders of magnitude in $\Phi /h$, and exhibiting no quantized steps. We establish the relevance of local coupling, improving on previous random matrix models relying on global coupling. It turns out that a phenomenological $2\times 2$ model gives an accurate analytical description of the transition curve.

Figure 1

Asymptotic transmitted weight $w_{\infty }$ vs $\Phi /h$ for $\Phi \approx \frac{1}{3000}$ (pluses), $\frac{1}{800}$ (crosses), $\frac{1}{200}$ (circles), $h=\frac{1}{100},\dots ,\frac{1}{6400}$ compared to Eq. (4) (solid line). Insets: Husimi representation of time-evolved wave packet for $\Phi \approx \frac{1}{200}$, $h=\frac{1}{40}$ and $h=\frac{1}{1000}$.

Figure 2

(a) $T^{\prime }(p)$ of the designed map $F$ with $\Phi \approx \frac{1}{200}$. (b) Corresponding phase space with a large chaotic sea (dots) between the regular tori (lines) and zoom with hyperbolic fixed point (black circle), partial barrier (solid green line) separating regions 1 and 2, its preimage (dotted green line) and the turnstile areas of size $\Phi$.

Figure 3

Average eigenstate equipartition measure $⟨w_{12}⟩$ for the designed map (gray crosses, same parameters as in Fig. 1) compared to Eq. (4) (solid line), the BTU model (gray dashed line), and the channel coupling model with discrete (orange circles) and continuous transmissions (light orange line) using 1000 realizations with $N_1=N_2=500$. Inset: Husimi representation of typical eigenstates for $\Phi \approx \frac{1}{200}$, $h=\frac{1}{40}$ (left) and $h=\frac{1}{1000}$ (top right). Illustration of the matrix structure of $U_0$ and $U_c$, where black lines represent unit entries (bottom right).

Figure 4

Asymptotic transmitted weight $w_{\infty }$ using the Husimi measure for the standard map with $K=2.7$ ($\Phi \approx \frac{1}{200}$) and $K=2.9$ ($\Phi \approx \frac{1}{100}$) for $h=\frac{1}{100},\dots ,\frac{1}{51200}$ versus $\Phi /h$ compared to Eq. (4) (solid line). The data are averaged over 15 initial conditions placed in region 1, 10 values of the Bloch phase, and 100 time steps after time ${10}^6$. Data for four different variants [15, 16] of the partial barrier are shown. Inset: One partial barrier (thick solid line) separating regions 1 and 2 for the standard map with $K=2.7$ as well as its preimage (dotted lines).