Figure 1

Poincaré section for a classical particle in an amplitude-modulated optical standing wave. Momentum and position (one well of the standing wave) of the particle along the standing wave are plotted stroboscopically with the stroboscopic period being equal to the modulation period. The central region consists of small amplitude motion. Chaos (dotted region) separates this region from two period-1 regions of regular motion (represented in the Poincaré section as sets of closed curves) located left and right of the center along momentum $p=0$. Further out in momentum are two stable regions of motion known as librations. At the edges are bands of regular motion corresponding to above barrier motion. It is plotted for modulation parameter $\epsilon =0.20$ and scaled well depth $\kappa =1.20$.Reuse & Permissions

Figure 2

Stroboscopic mean momentum as a function of the interaction time with the modulated standing wave measured in modulation periods, $n$, for modulation parameter $\epsilon =0.29$, scaled well depth $\kappa =1.66$, modulation frequency $\omega ∕2\pi =250\mathrm{kHz}$, and a phase shift $\phi =0.21\times 2\pi$. (a) and (b) correspond to the two different interaction times $n+0.25$ and $n+0.75$ modulation periods, respectively. Results from the dynamic evolution of the Gross-Pitaevskii equation are plotted as a solid line (circles) and the experimental data are plotted as a dashed line (diamonds).Reuse & Permissions

Figure 3

Position representation of the atomic wave function as a function of the number of modulation periods calculated using the Gross-Pitaevskii equation. The wave packet is plotted stroboscopically at a phase so it is located classically approximately at its highest point in the potential well. The position of the standing wave wells is also shown (dotted line). Dynamical tunneling can be observed. At $t=5T$ most atoms have tunneled into the other period-1 region of regular motion. The position axis is given in Fermi units (scaled by the mean Thomas-Fermi diameter).Reuse & Permissions

Figure 4

Momentum distributions as a function of the interaction time with the modulated standing wave calculated using numerical solutions of the Gross-Pitaevskii equation for the same parameters as Fig. 3. Initially atoms have mostly negative momentum $(t=0.25T)$. After approximately five modulation periods most atoms populate a state with positive momentum, therefore having undergone dynamical tunneling. Corresponding experimental data from Ref. 1 is also included as insets.Reuse & Permissions

Figure 5

Mean momentum as a function of the interaction time with the modulated standing wave measured in modulation periods, $n$, for $\epsilon =0.28$, $\kappa =1.49$, $\omega ∕2\pi =250\mathrm{kHz}$, and $\phi =0.22\times 2\pi$. The points are plotted stroboscopically with an interaction time of $n+0.25$ modulation periods which corresponds to turning off the standing wave at maximum. Results from the dynamic evolution of the Gross-Pitaevskii equation are plotted as a solid line (circles) and the experimental data is plotted as a dashed line (diamonds).Reuse & Permissions

Figure 6

Mean momentum as a function of the interaction time with the modulated standing wave measured in modulation periods, $n$, for modulation parameter $\epsilon =0.30$, scaled well depth $\kappa =1.82$, modulation frequency $\omega ∕2\pi =222\mathrm{kHz}$, and phase shift $\phi =0.21\times 2\pi$. The points are plotted stroboscopically with an interaction time of $n+0.25$ modulation periods which corresponds to turning off the standing wave at maximum. Results from the dynamic evolution of the Gross-Pitaevskii equation are plotted as a solid line (circles) and the experimental data is plotted as a dashed line (diamonds).Reuse & Permissions

Figure 7

Mean momentum as a function of the interaction time with the modulated standing wave measured in modulation periods, $n$, using the same parameters as Fig. 6 but the modulation parameter $\epsilon$ is reduced from 0.30 to 0.28 for the theoretical simulation. A much better fit is obtained.Reuse & Permissions

Figure 8

Stroboscopic evolution of mean-field energy, the kinetic energy, and the potential energy. The energies are given in $\mathrm{Hz}$ (energy is scaled with Planck’s constant). For comparison the mean momentum is also shown as a function of the interaction time with the standing wave (in units of modulation periods). The solid (circles) and dashed (diamonds) curves correspond to the two different interaction times $n+0.25$ and $n+0.75$ modulation periods, respectively. The evolution is plotted for modulation parameter $\epsilon =0.29$, scaled well depth $\kappa =1.66$, modulation frequency $\omega ∕2\pi =250\mathrm{kHz}$, and a phase shift $\phi =0.21\times 2\pi$.Reuse & Permissions

Figure 9

Phase space representations of two Floquet states for $\vartheta =0$ that are involved in the dynamical tunneling are shown in (a) and (b). The Floquet states correspond to the experimental parameters: modulation parameter $\epsilon =0.29$, scaled well depth $\kappa =1.66$, and modulation frequency $\omega ∕2\pi =250\mathrm{kHz}$, which were utilized to obtain the experimental results shown in Fig. 2. A third state, shown in (c) also has significant overlap with the initial experimental state.Reuse & Permissions

Figure 10

Quasi-eigenenergy spectrum for parameters: modulation parameter $\epsilon =0.29$, scaled well depth $\kappa =1.66$, and modulation frequency $\omega ∕2\pi =250\mathrm{kHz}$. The quasi-eigenenergy is measured in reduced energy units and it is a function of the Bloch angle. The Bloch angle is equal to the quasi-momentum multiplied with the spatial period $\lambda ∕2$ of the lattice. Each line corresponds to one Floquet state labeled by $n$ [see Eq. (13)]. The quasi-eigenenergy of most Floquet states is strongly dependent on the quasi-momentum but it is not the case for the states of the tunneling doublet. Recall that the quasi-momentum $\vartheta$ is proportional to the phase taken by a state when spatially translated by $\lambda ∕2$. In other words, fixing $\vartheta$ imposes on the states some conditions on the boundary of one elementary cell $[q,q+\lambda ∕2]$. If a state is localized deep inside a cell, changing the boundary conditions (i.e., $\vartheta$) will not affect the state so much and the corresponding quasienergies appear as curves that are approximately parallel to the $\vartheta$ axis. However, one should expect a strong $\vartheta$-dependence for a state that spreads over at least $\Delta q\sim \lambda ∕2$.Reuse & Permissions

Figure 11

Phase space representations of two Floquet states whose presence can lead to the occurrence of dynamical tunneling. The Floquet states correspond to the experimental parameters: modulation parameter $\epsilon =0.30$, scaled well depth $\kappa =1.82$, and modulation frequency $\omega ∕2\pi =222\mathrm{kHz}$, which were utilized to obtain the experimental results shown in Fig. 6.Reuse & Permissions

Figure 12

Mean atomic momentum as a function of the interaction time with the standing wave plotted for a range of the initial phase shift $\phi$ of the standing wave. The solid line (circles) corresponds to an interaction time with the standing wave of $n+0.25$ modulation periods and the dashed line (diamonds) corresponds to an interaction time of $n+0.75$ modulation periods.Reuse & Permissions

Figure 13

Mean momentum as a function of the interaction time with the modulated standing wave for different values of the scaled well depth $\kappa$ (modulation parameter $\epsilon =0.29$, modulation frequency $\omega ∕2\pi =250\mathrm{kHz}$, and phase shift $\phi =0.21\times 2\pi$). The solid line (circles) corresponds to an interaction time with the standing wave of $n+0.25$ modulation periods and the dashed line (diamonds) corresponds to an interaction time of $n+0.75$ modulation periods.Reuse & Permissions

Figure 14

Tunneling frequency as a function of the scaled well depth $\kappa$ (modulation parameter $\epsilon =0.29$, modulation frequency $\omega ∕2\pi =250\mathrm{kHz}$, and phase shift $\phi =0.21\times 2\pi$) taken from Fig. 13. Diamonds and squares denote low and high frequency components, respectively.Reuse & Permissions

Figure 15

Floquet spectrum as a function of the scaled well depth $\kappa$ (modulation parameter $\epsilon =0.29$, modulation frequency $\omega ∕2\pi =250\mathrm{kHz}$, and phase shift $\phi =0.21\times 2\pi$. The Floquet states with maximum overlap are marked by black bullets, white bullets are used for second most overlap, and white squared bullets show third most overlap. Phase space representations (Husimi functions) of some of the Floquet states for different values of $\kappa$ are also shown.Reuse & Permissions

Figure 16

Tunneling frequency as a function of the modulation parameter $\epsilon$ (scaled well depth $\kappa =1.66$, modulation frequency $\omega ∕2\pi =250\mathrm{kHz}$, and phase shift $\phi =0.21\times 2\pi$). The error bars result from the readout uncertainty of the tunneling oscillations from the simulations. The high frequency component that is present in the data for $0.00<\epsilon <0.18$ is not shown.Reuse & Permissions

Figure 17

Dominant tunneling frequency as a function of the modulation frequency $\omega ∕2\pi$ for modulation parameter $\epsilon =0.29$, scaled well depth $\kappa =1.66$, and phase shift $\phi =0.21\times 2\pi$ which corresponds to experimental results shown in Fig. 2. As expected, the overall tendency consists of a decreasing of the tunneling frequency as the modulation frequency increases and therefore as the scaled Planck’s constant decreases, however, the transition is not as smooth as one might expect.Reuse & Permissions