Quantum-mechanical cumulant dynamics near stable periodic orbits in phase space: Application to the classical-like dynamics of quantum accelerator modes

We formulate a general method for the study of semiclassical-like dynamics in stable regions of a mixed phase space, in order to theoretically study the dynamics of quantum accelerator modes. In the simplest case, this involves determining solutions, which are stable when constrained to remain pure-state Gaussian wave packets, and then propagating them using a cumulant-based formalism. Using this methodology, we study the relative longevity, under different parameter regimes, of quantum accelerator modes. Within this attractively simple formalism, we are able to obtain good qualitative agreement with exact wave-function dynamics.

Figure 1

(Color online) Poincaré sections determined by Eq. (36) (white dots), superimposed on Wigner functions [34] corresponding to single wave packets of the form given in Eq. (1) (color density plots). Means and variances determined by Eq. (60) for $\tilde{k}=2$ and (a) $ϵ=-0.2$, (b) $ϵ=0.2$. Units are dimensionless.

Figure 2

Comparison of $e^{-{\tilde{\sigma }}^2}$ (solid line) with (a) $f_{-}\left({\tilde{\sigma }}^2\right)={(2\pi \Omega ∕\tilde{k})}^2-{\overline{\sigma }}^2-\sqrt{{\left({\tilde{\sigma }}^2\right)}^2+{[1-{(2\pi \Omega ∕\tilde{k})}^2]}^2}$ and (b) $f_{+}\left({\tilde{\sigma }}^2\right)={(2\pi \Omega ∕\tilde{k})}^2-{\tilde{\sigma }}^2+\sqrt{{\left({\tilde{\sigma }}^2\right)}^2+{[1-{(2\pi \Omega ∕\tilde{k})}^2]}^2}$, for values of ${(2\pi \Omega ∕\tilde{k})}^2$ equal to 0 (points), 0.2 (circles), 0.4 (crosses), 0.6 (plusses), 0.8 (asterisks), and 1 (squares). Note that $f_{-}=e^{-{\tilde{\sigma }}^2}$ only when ${\tilde{\sigma }}^2=0$, and that $f_{+}=e^{-{\tilde{\sigma }}^2}$ only when ${\tilde{\sigma }}^2=0$ and ${(2\pi \Omega ∕\tilde{k})}^2=1$.

Figure 3

(Color online) Number of iterations of Eq. (51) evolved by a Gaussian stable fixed point such that $\mid \mathcal{J}\mid <\pi$. Black indicates absence of $ϵ$-classical stable solutions [18], white absence of Gaussian stable solutions. Numbers label the contours where $\mid \mathcal{J}\mid <\pi$ for that number of iterations (the number of iterations is capped at 100). The solid line marks the average experimental laser intensity ${\varphi }_d=0.8\pi$, dashes demarcate its experimental range $(0.3\pi -1.2\pi )$ [4]. Units are dimensionless.

Figure 4

(Color online) As in Fig. 3, but for exact wave-packet evolutions, propagating initial conditions from Eq. (67) with the time-evolution operator of Eq. (8) $(\beta =0)$. Units are dimensionless.