Lyapunov spreading of semiclassical wave packets for the Lorentz gas: Theory and applications

We consider the quantum-mechanical propagator for a particle moving in a $d$-dimensional Lorentz gas, with fixed, hard-sphere scatterers. To evaluate this propagator in the semiclassical region, and for times less than the Ehrenfest time, we express its effect on an initial Gaussian wave packet in terms of quantities analogous to those used to describe the exponential separation of trajectories in the classical version of this system. This result relates the spread of the wave packet to the rate of separation of classical trajectories, characterized by positive Lyapunov exponents. We consider applications of these results, first to illustrate the behavior of the wave-packet autocorrelation functions for wave packets on periodic orbits. The autocorrelation function can be related to the fidelity, or Loschmidt echo, for the special case that the perturbation is a small change in the mass of the particle. An exact expression for the fidelity, appropriate for this perturbation, leads to an analytical result valid over very long time intervals, inversely proportional to the size of the mass perturbation. For such perturbations, we then calculate the long-time echo for semiclassical wave packets on periodic orbits.

Figure 1

Particle-fixed frame of reference at time $t=0$.

Figure 2

Particle-fixed frames of reference: $({\zeta }^{\prime },{\eta }^{\prime })$ at time 0 and $(\zeta ,\eta )$ at $t$.

Figure 3

Free flight time evolution of $\rho$ and $\sigma$: (a) classical case, $\epsilon =0$; (b) quantum case, $\epsilon >0$.

Figure 4

Wave-packet size, $\sigma$, real radius of curvature, $\rho$, and $\epsilon$ are shown as functions of time, $t$, for a two-disk periodic orbit. Disk radii $a=1$, center-to-center separation $R=3$, de Broglie wavelength $\mathrm{ƛ}={10}^{-7}$. The corresponding Lyapunov exponent $\lambda ∕v\approx 1.32$. Initial wave-packet size ${\sigma }_0=2\times {10}^{-4}$ and ${\rho }_0=10$. The particle is located in the middle between the two disks at $t=0$. Exponential trends are shown for plots of $\sigma$ and $\epsilon$. All distances are measured in units of $a$.

Figure 5

A Gaussian wave packet is shown at time $t=0$ (centered about point $O$), and at a later time $t=nT+\mathrm{\Delta }∕v$ (centered about point $P$). Points $O$ and $P$ lie on the same periodic orbit, and are separated in time by $n(=0,1,2,\mathrm{\dots },N)$ periods, $T$, of the periodic orbit, plus a short time interval $\mathrm{\Delta }∕v$. The separation distance $\mathrm{\Delta }$ is assumed to be sufficiently small in order for the initial and final wave packets to overlap significantly.

Figure 6

Revival peaks of the wave-packet autocorrelation function, $C\left(t\right)$, for the same two-disk periodic orbit as in Fig. 4: $a=1$ and $R=3$. Particles de Broglie wavelength $\mathrm{ƛ}={10}^{-7}$. The initial wave packet is characterized by ${\sigma }_0={\sigma }_{\parallel 0}=2\times {10}^{-4}$ and ${\rho }_0={\rho }_{\parallel 0}=10$. The exponential trend is indicated by a straight line.