Effects of classical stochastic webs on the quantum dynamics of cold atomic gases in a moving optical lattice

We introduce and investigate a system that uses temporal resonance-induced phase-space pathways to create strong coupling between an atomic Bose-Einstein condensate and a traveling optical lattice potential. We show that these pathways thread both the classical and quantum phase space of the atom cloud, even when the optical lattice potential is arbitrarily weak. The topology of the pathways, which form weblike patterns, can by controlled by changing the amplitude and period of the optical lattice. In turn, this control can be used to increase and limit the BEC's center-of-mass kinetic energy to prespecified values. Surprisingly, the strength of the atom-lattice interaction and resulting BEC heating of the center-of-mass motion is enhanced by the repulsive interatomic interactions.

Figure 1

Isodensity surface (red) of a BEC that is confined in a 3D harmonic potential and subject to a plane-wave potential [high (low) potential energy shown in green (blue)] traveling along the $z$ direction, corresponding to the lowest frequency of the harmonic trap.

Figure 2

Comparison between the energy per atom, $E\left(t\right)$, in the BEC calculated vs time $t$, shown in units of ${\tau }_z$, for resonant and nonresonant plane-wave driving and for different strengths of interatomic interaction: Dark-blue solid (labeled 3) and dashed (labeled 2) curves show $R=1$ resonant heating for noninteracting $(g=0)$ and interacting atoms, respectively; red (labeled 4) and light-blue (labeled 5) curves show nonresonant heating for $R=0.95$ and $R=\sqrt{2}$, respectively ($g=0$ in both cases); green curve (labeled 1) shows $R=2$ heating for $g=0$ and $V_o=1.3\hslash {\omega }_z$. $V_o=0.3\hslash {\omega }_z$ for all other curves. In all cases, $N={10}^4$ and $k_c=0.5/l_x$.

Figure 3

(a, i–iv) Position-momentum $(z,p_z$) phase-space Wigner functions (normalized moduli, $\left|W\right|/N$, plotted) showing the time evolution calculated for a noninteracting atom cloud with resonant $R=1$ plane-wave driving. (i) $t=0$, cloud at rest; ($\mathrm{ii})t={\tau }_z$, cloud has been excited to a larger phase-space radius, $\rho =\sqrt{z^2+p_z^2}$, along the $\langle p_z\rangle =0$ pathway; (iii) $t=7{\tau }_z$, cloud reaches, and scatters from, a ring-shaped dynamical barrier; (iv) $t=17{\tau }_z$, cloud begins to contract along the $\langle p_z\rangle =0$ pathway, reducing its mean $\rho$ value and energy. (b, i–iv) Phase-space evolution calculated for an interacting atom cloud ($g=g_0$) at times (i) $t=0$, cloud at rest but repulsive interactions produce a larger initial phase-space spread than for $g=0$; (ii) $t=6{\tau }_z$, cloud is excited as in (a,ii) except that the interactions have distorted the phase-space distribution; (iii) $t=17{\tau }_z$, the cloud reaches the ring-shaped dynamical barrier and scatters from it, becoming fragmented; (iv) $t=25{\tau }_z$, cloud continues to evolve in phase space with no reduction in its energy. Color bar shows normalized values of the Wigner function modulus.

Figure 4

(a) Classical Poincaré section (dots) calculated for an ${}^{87}\mathrm{Rb}$ atom driven by a plane wave. Blue and red dashed curves show, respectively, the location of the radial and first ring-shaped filaments of the stochastic web that encloses the islands of stability shown in black. (b)–(d) Wigner functions (normalized moduli plotted: color bar lower right) that are periodically time averaged over $40{\tau }_z$ for $R=1$ with (b) $g=0$ and (c) $g=g_0$, and for $R=2$ with (d) $g=0$. For all panels, $V_o=0.3\hslash {\omega }_z$. (a)–(c) $k_c=0.5/l_z$; (d) $k_c=0.75/l_z$.

Figure 5

Maximum energy of atom clouds calculated vs wave vector, $k_c$ (shown multiplied by $l_z$), of the plane-wave driving potential. Blue curve shows $E_{\mathrm{ring}}^1$ values given by Eq. (5). Black (red) symbols are maximum energies calculated using the GPE with $g=g_0$ ($g=0$). Insets: Wigner functions (moduli plotted) calculated for $R=1$ and $k_c=$ (a) $0.4/l_z$, (b) $0.75/l_z$ peak around the classical stochastic webs in phase space.

Figure 6

Color map (scale right) showing the total energy $E$ of each atom in an ${}^{87}\mathrm{Rb}$ BEC calculated vs time (in units of ${\tau }_z$) and the amplitude $V_o$ (in units of $\hslash {\omega }_z$) of a plane-wave driving potential with $k_c=0.5l_z$. The map is shown both as a surface plot (upper) and a 2D projection (lower). Along the dashed curve in the projection, the BEC's energy is 90% of the energy limit given by Eq. (5).

Figure 7

Atom density profiles calculated for an ${}^{87}\mathrm{Rb}$ BEC with $g=g_o$ shown in the $x=0$ plane for $R=$ (a) 1, (b) 2 at times $t=$ (i) $5.00{\tau }_z$, (ii) $5.25{\tau }_z$, (iii) $5.50{\tau }_z$, (iv) $5.75{\tau }_z$, (v) $6.00{\tau }_z$. (a, i–v) Large-amplitude center-of-mass oscillation during which the BEC maintains a spatially coherent form. (b, i–v) There is no center-of-mass oscillation, but significant deformation and fragmentation of the atom cloud.