Wave packet autocorrelation functions for quantum hard-disk and hard-sphere billiards in the high-energy, diffraction regime

We consider the time evolution of a wave packet representing a quantum particle moving in a geometrically open billiard that consists of a number of fixed hard-disk or hard-sphere scatterers. Using the technique of multiple collision expansions we provide a first-principle analytical calculation of the time-dependent autocorrelation function for the wave packet in the high-energy diffraction regime, in which the particle’s de Broglie wavelength, while being small compared to the size of the scatterers, is large enough to prevent the formation of geometric shadow over distances of the order of the particle’s free flight path. The hard-disk or hard-sphere scattering system must be sufficiently dilute in order for this high-energy diffraction regime to be achievable. Apart from the overall exponential decay, the autocorrelation function exhibits a generally complicated sequence of relatively strong peaks corresponding to partial revivals of the wave packet. Both the exponential decay (or escape) rate and the revival peak structure are predominantly determined by the underlying classical dynamics. A relation between the escape rate, and the Lyapunov exponents and Kolmogorov-Sinai entropy of the counterpart classical system, previously known for hard-disk billiards, is strengthened by generalization to three spatial dimensions. The results of the quantum mechanical calculation of the time-dependent autocorrelation function agree with predictions of the semiclassical periodic orbit theory.

Figure 1

The absolute value of the ratio $J_l\left(x\right)∕H_l^{\left(1\right)}\left(x\right)$ as a function of $x∕l$ for two cases $l=10$ and $l=50$. The ratio significantly differs from zero only for $x\gtrsim l$.

Figure 2

Two-disk billiard. The circular wave packet is initially located on the classical periodic orbit distance $r_1$ away from the center of disk “1” and distance $r_2$ away from the center of disk “2” with $r_1+r_2=R$.

Figure 3

Wave packet autocorrelation function $C\left(t\right)$ vs $vt∕R$ for the two-disk billiard. Parameters of the system are as follows: $a=\sigma =1$, $R={10}^4$, $r_1=r_2=R∕2$, and $\bar{\lambda }={10}^{-2}$. The straight line shows exponential decay with the rate given by the classical two-disk Lyapunov exponent ${\lambda }^{\left(2\right)}$. The decay is shown for times $t$ greater than the Ehrenfest time $t_E\approx R∕2v$.

Figure 4

Three-disk isosceles billiard. The circular wave packet is initially located distance $r_1$ away from disk 1 and distance $r_2$ away from disk 2 with $r_1+r_2=R$. Disk 3 is distance $\alpha R$ away from disks 1 and 2. The “equilateral” billiard case corresponds to $\alpha =1$.

Figure 5

The autocorrelation function vs $vt∕R$ for the three-disk equilateral billiard. Parameters of the system are as follows: $a=\sigma =1$, $R={10}^4$, $r_1=r_2=R∕2$, and $ƛ={10}^{-2}$. The straight line shows exponential decay with the rate given by ${\gamma }^{\left(3\right)}$. The decay is shown for times $t$ greater than the Ehrenfest time $t_E\approx R∕2v$.

Figure 6

(a) First two “bands” of poles for the case of $\alpha =5∕2$ and $R∕a={10}^4$. (b) Magnification of the first band, (c) magnification of the second band. Dots correspond to (exact) values of the poles numerically computed directly from Eq. (51), while crosses show the same poles approximated by Eq. (53).

Figure 7

The autocorrelation function as a function of time for the isosceles three-disk billiard with $\alpha =5∕2$. Disks 1 and 2 are separated by distance $R={10}^{4}a$. The wave packet of the de Broglie wavelength $\lambda ={10}^{-2}a$ is initially located as shown in Fig. 4 with $r_1=r_2=R∕2$. The dotted and the solid straight lines represent $e^{-{\lambda }^{\left(2\right)}t}$ and $e^{-{\gamma }_{\alpha }^{\left(3\right)}t}$ decays, respectively.

Figure 8

Peaks of the autocorrelation function for the equilateral three-disk billiard calculated in accordance with Eqs. (63, 65, 67). The dashed line shows $e^{-{\gamma }^{\left(3\right)}t}$ decay, with ${\gamma }^{\left(3\right)}$ given by Eq. (47). The radii of the disks constituting the billiard equal $a=1$, and the disk center-to-center separation is $R={10}^4$. This figure is to be compared with Fig. 5.

Figure 9

Peaks of the autocorrelation function as predicted by Eq. (63) for the isosceles three-disk billiard with $\alpha =5∕2$. The dashed line shows $e^{-{\gamma }_{\alpha }^{\left(3\right)}t}$ decay, with ${\gamma }_{\alpha }^{\left(3\right)}$ given by Eq. (55), while the dotted lines show the trend of the $e^{-{\lambda }^{\left(2\right)}t}$ decay, with ${\lambda }^{\left(2\right)}$ defined by Eq. (37). The billiard is parametrized by $a=1$ and $R={10}^4$. This figure is to be compared with Fig. 7.

Figure 10

Peaks of the autocorrelation function for the three-disk billiard with $\alpha =\sqrt{2}$, $\beta =2$, $a=1$, and $R={10}^4$. The solid line represents $e^{-{\lambda }_{\beta }^{\left(2\right)}t}$ decay, with ${\lambda }_{\beta }^{\left(2\right)}$ calculated according to Eq. (87); the dashed line corresponds to $e^{-{\lambda }^{\left(2\right)}t}$ decay, with ${\lambda }^{\left(2\right)}$ given by Eq. (37).

Figure 11

The tetrahedral four-sphere billiard. The wave packet is initially placed between spheres 1 and 2, with its average momentum directed toward sphere 2.

Figure 12

Peaks of the wave packet autocorrelation function for the three scattering systems: two-sphere (circles), three-sphere equilateral (triangles), and four-sphere tetrahedral (diamonds) billiards. In all cases the radius of the sphere scatterers is $a=1$, and the sphere center-to-center separation is $R={10}^4$. The semiclassical decay rates for these billiards are calculated according to Eqs. (90, 95, 101), and presented in the figure by the dotted, dashed, and solid lines, respectively.

Figure 13

Schematic picture of a classical periodic orbit ${\gamma }^{\prime }$ together with an exponentially spreading tube of infinitesimally close trajectories. The initial cross section of the tube $A_{{\gamma }^{\prime }}\left(0\right)$ gets magnified to the value $A_{{\gamma }^{\prime }}\left(t\right)$, given by Eq. (114), after time $t$ equal to the period of the orbit ${\gamma }^{\prime }$.