Poincaré Husimi representation of eigenstates in quantum billiards

For the representation of eigenstates on a Poincaré section at the boundary of a billiard different variants have been proposed. We compare these Poincaré Husimi functions, discuss their properties, and based on this select one particularly suited definition. For the mean behavior of these Poincaré Husimi functions an asymptotic expression is derived, including a uniform approximation. We establish the relation between the Poincaré Husimi functions and the Husimi function in phase space from which a direct physical interpretation follows. Using this, a quantum ergodicity theorem for the Poincaré Husimi functions in the case of ergodic systems is shown.

Figure 1

Examples of eigenstates ${\psi }_n\left(\mathit{q}\right)$, shown to the left, and to the right their Poincaré Husimi functions $h_n(q,p)$. In (a) an eigenstate $\left(n=1952\right)$ localizing around a regular orbit for the limaçon billiard at $\epsilon =0.3$ is shown. In (b) and (c) two eigenstates for the cardioid billiard are shown ($n=1817$ and $n=1277$).

Figure 2

Plot of ${\mathcal{H}}_k(q,p)$ for $k=125$ using the first 2000 eigenstates in the limaçon billiard of odd symmetry at $\epsilon =0.3$. In (a) the result for ${\mathcal{H}}_k(q,p)$ using definition (12) for $h_n(q,p)$ is shown and in (b) a corresponding ${\tilde{\mathcal{H}}}_k(q,p)$ using definition (14) is displayed. In addition to the symmetry related dips at $(q,p)=(0,0)$ and $(L∕2,0)$ one clearly sees the variation in $p$ direction in both cases and in (b) we, moreover, observe a variation in $q$.

Figure 3

Comparison of the uniformized asymptotic behavior $F_k\left(p\right)$, see Eq. (26), with $\mid \tilde{b}{\mid }^2∕\text{Im}b=1$ and for $k=10,30,500$. The asymptotic semicircle behavior is reached slowly.

Figure 4

The full curve shows a section of ${\mathcal{H}}_k(q,p)$ at $q=3.0$ with $k=125$ for the desymmetrized limaçon billiard, see Fig. 2a, and the second line is the uniformized mean behavior. The remaining deviations are caused by higher order corrections.

Figure 5

Illustration of a Gaussian beam as given by Eq. (38) inside the limaçon billiard at $\epsilon =0.3$.