Quantum Accelerator Modes from the Farey Tree

We show that mode locking finds a purely quantum nondissipative counterpart in atom-optical quantum accelerator modes. These modes are formed by exposing cold atoms to periodic kicks in the direction of the gravitational field. They are anchored to generalized Arnol’d tongues, parameter regions where driven nonlinear classical systems exhibit mode locking. A hierarchy for the rational numbers known as the Farey tree provides an ordering of the Arnol’d tongues and hence of experimentally observed accelerator modes.

Figure 1

Density plot of experimental atomic momentum distributions (measured in a frame falling freely with $g$) after $n=30$ pulses as $T$ is varied in the vicinity of $T_2=133.3\mu \mathrm{s}$, from $124.5\mu \mathrm{s}$ to $142.5\mu \mathrm{s}$ in steps of $0.128\mu \mathrm{s}$ [13]. The labels (a), (b), and (c) indicate $(2,1)$, $(3,1)$, and $(5,2)$ quantum accelerator modes, as predicted by the corresponding Arnol’d tongues depicted in Fig. 3. Labels (d) and (e) indicate $(7,3)$ and $(8,3)$ modes. White lines are the momenta predicted by Eq. (3). Inset shows the evolution of a typical [in this case a $(1,0)$] QAM as a function of pulse number. The color bar indicates the population scale.

Figure 2

Arnol’d tongues’ structure. Different colors identify different tongues associated with different quantum accelerator modes. The dotted line marks the locus of experimentally explored points when $T\sim T_2$; dashed lines bound regions for which $(21,8)$ and $(5,2)$ stable periodic orbits exist, as specified by $|K|>2\pi \sqrt{\mathfrak{p}}|\Omega -\mathfrak{m}/\mathfrak{p}|$.

Figure 3

Operation of the Farey recursion, moving from frames A to D, for determining the experimentally observed $(2,1)$, $(3,1)$, and $(5,2)$ quantum accelerator modes. The dotted line indicates the locus of the experimentally explored points, while the labels (a), (b), and (c) mark the regions in which accelerator modes are observed. The labels correspond to those used in Fig. 1.