Expansion of a Quantum Gas Released from an Optical Lattice

We analyze the interference pattern produced by ultracold atoms released from an optical lattice, commonly interpreted as the momentum distributions of the trapped quantum gas. We show that for finite times of flight the resulting density distribution can, however, be significantly altered, similar to a near-field diffraction regime in optics. We illustrate our findings with a simple model and realistic quantum Monte Carlo simulations for bosonic atoms and compare the latter to experiments.

Figure 1

(a) Momentum distributions for a one-dimensional lattice with parabolic distribution of the occupation numbers calculated using Eq. (5) (solid line: expansion time $t=20\mathrm{ms}$; dashed line: expansion time $t=100\mathrm{ms}$; dotted-dashed line: expansion time $t\to \infty$). (b) Evolution of the peak amplitude $A$ with expansion time $t/t_{\mathrm{FF}}$. The dashed line shows the expected near-field scaling in one dimension $A\propto t/t_{\mathrm{FF}}$. The number of sites is $2N_{\mathrm{TF}}+1=61$ for (a) and (b). (c) Evolution of the width of the diffraction peaks with expansion time. The width has been normalized to the separation between two adjacent diffraction peaks for convenience. The circles show the experimental measurements and the solid line a fit by a hyperbola $\propto 1/t$, as expected in the near field.

Figure 2

Results from quantum Monte Carlo simulations. On the left column, we show a horizontal cut through the ToF distributions calculated using Eq. (3) for a finite expansion time $t=14\mathrm{ms}$ (solid line) compared to a cut through the profile calculated for $t\to \infty$ (dashed line). Units for $n_{\perp }$ are arbitrary. The insets show directly the two-dimensional ToF distributions for $t=14\mathrm{ms}$. On the right column, we show the in-trap density profiles for reference. The lattice depths are $V_0=12E_R$ (a),(d), $15E_R$ (b),(e), and $17E_R$ (c),(f), respectively.

Figure 3

Visibility of the interference pattern as defined in Eq. (6). The dashed and dotted-dashed lines show the quantum Monte Carlo result for infinite and finite ($t=14\mathrm{ms}$) expansion times, assuming perfect experimental resolution. The solid line is computed for $t=14\mathrm{ms}$ accounting for finite experimental resolution. The simulations were performed at constant temperature $T=J/k_B$.