Quantum and classical dynamics of a Bose-Einstein condensate in a large-period optical lattice

We experimentally investigate diffraction of a ${}^{87}\text{R}\text{b}$ Bose-Einstein condensate from a one-dimensional optical lattice. We use a range of lattice periods and timescales, including those beyond the Raman-Nath limit. We compare the results to numerical solutions of the Gross-Pitaevskii equation and classical calculations, with quantitative and qualitative agreement, respectively. The classical calculations predict that the envelope of the time-evolving diffraction pattern is shaped by caustics: singularities in the phase-space density of classical trajectories. This behavior becomes increasingly clear as the lattice period grows.

Figure 1

(Color) Concatenated absorption images of diffraction patterns, showing the evolving momentum distribution at four lattice periods $d$. The momentum is scaled by $k_{\text{max}}=z_{\text{max}}M/t_{\text{TOF}}\hslash$ where $z_{\text{max}}$ is the measured maximum amplitude of the diffraction pattern after time of flight. The calculated initial Thomas-Fermi radius in the lattice direction, $\widehat{z}$, ranges from $26\mu \text{m}$ to $29\mu \text{m}$, and in the transverse directions, from $6\mu \text{m}$ to $10\mu \text{m}$. The numerical solutions of the GP equation were convolved with initial system sizes to model the effects of mean-field-driven expansion during TOF. $t_{\text{RN}}$ is given by Eq. (2) and is the time at which the Raman-Nath approximation is expected to substantially fail.

Figure 2

(Color) Splitting of diffraction orders ($1.8\mu \text{m}$ lattice period) for pulse durations beyond the Raman-Nath regime. The effect is more pronounced the longer the pulse duration and two examples are indicated with white boxes.

Figure 3

Calculated projections of a uniform wave function onto even-parity, zero-quasimomentum Bloch states of a $30E_{\text{R}}$ lattice with periods of $1.8\mu \text{m}$, 3.5, 6.5, and $9.3\mu \text{m}$. Each point corresponds to one such Bloch state. The horizontal axis gives the energy separation to that state’s even-parity, higher-energy neighbor, normalized to the separation of the lowest energy pair, and the vertical axis gives the fractional occupation of the state. The difference between consecutive energy levels first decreases with increasing $n$, then increases as states become unbound. This explains the sharp kinks at the last data point (the first unbound state).

Figure 4

(Color) Calculated evolution of the probability density ${\left|\Psi \left(z,t\right)\right|}^2$ within a single site of a lattice (potential shown in red) with period $d=1.8\mu \text{m}$ and depth $30E_{\text{R}}$. Positions $z/d=-0.5$ and 0.5 correspond to the edges of the well. When $T_{\text{pulse}}/T_{\text{ho}}\approx 0.25$ there is a large increase in density at the center of the well, at which time a large number of high-momentum states are occupied.

Figure 5

Classical motion of particles in a sinusoidal potential. The curves correspond to the trajectories of particles starting at different points in the potential, each with zero initial momentum, $p_z=0$. The smallest amplitude oscillations correspond to the smallest period, $T_{\text{ho}}$. The oscillation period diverges for particles starting increasingly near the top of the sinusoidal potential. The dashed curve indicates the first caustic, a classical analog to the first apparent high-density feature in our quantum system.

Figure 6

(Color) Classical phase-space evolution of atoms in a single well of a sinusoidal potential. Each black curve depicts the phase-space distribution at a specific time. The lips of a single well are located at positions $z=\pm d/2$. The phase-space evolution is characterized by fixed points at the lips and rotation approaching the harmonic frequency near $z=0$.