Atomic matter-wave revivals with definite atom number in an optical lattice

We study the collapse and revival of interference patterns in the momentum distribution of atoms in optical lattices using a projection technique to properly account for the fixed total number of atoms in the system. We consider the common experimental situation in which weakly interacting bosons are loaded into a shallow lattice, which is suddenly made deep. The collapse and revival of peaks in the momentum distribution is then driven by interactions in a lattice with essentially no tunneling. The projection technique allows to us to treat inhomogeneous (trapped) systems exactly in the case that noninteracting bosons are loaded into the system initially, and we use time-dependent density-matrix renormalization group techniques to study the system in the case of finite tunneling in the lattice and finite initial interactions. For systems of more than a few sites and particles, we find good agreement with results calculated via a naive approach, in which the state at each lattice site is described by a coherent state in the particle occupation number. However, for systems on the order of ten lattice sites, we find experimentally measurable discrepancies to the results predicted by this standard approach.

Figure 1

Plot of the time evolution of the off-diagonal SPDM elements for an initial ideal superfluid state ($U_0=0$) with homogeneous density and without external trapping (${\varepsilon }_i=0$) in a deep optical lattice ($J=0$). The density is chosen as one particle per site ($\left|\beta \right|{}^2=1$), and we show the first collapse and revival for increasing number of particles. We compare results from the exact treatment of Eq. (22) to the naive result from Eq. (3) (solid black line). Periodic boundary conditions are assumed.

Figure 2

The time-evolution of the relative error ${\epsilon }_t$ of the naive approach compared with exact results in the height of the quasimomentum distribution peak (i.e., $n_{q=0}$). The results are for a 1D system with box boundary conditions, resulting in a site-dependent particle density. Results are shown for the time evolution of the error until the first revival at $t=2\pi /U$, and for increasing system sizes ($N=M$, $U_0=0$, $J=0$, ${\varepsilon }_i=0$).

Figure 3

The scaling of the relative error ${\epsilon }_{t=\pi /4U}$ of the naive approach compared with the exact results at the middle of the first collapse $t=\pi /4U$. Results are shown both for periodic and open (box) boundary conditions. To compare the results to Eq. (25), we also plot $1/N$ ($U_0=0$, $J=0$, ${\varepsilon }_i=0$).

Figure 4

Time evolution of the quasimomentum distribution $n_q$ for several tunneling amplitudes $J$ in a system with $M=51$ sites and box boundary conditions. Initially the system is in the exact ground state for $U_0/J_0=0$. Panels (a–c) show exact results obtained numerically by using the TEBD algorithm. Panel (d) shows the evolution of the $n_{q=0}$ peak. The TEBD results for $J=0$ are compared to the exact results obtained from Eq. (21), and to the naive treatment obtained from Eq. (2). The particles are initially noninteracting ($U_0=0$).

Figure 5

Panel (a) compares the initial quasimomentum distribution for noninteracting particles ($U_0=0$) with the one for weakly interacting particles ($U_0/J_0=0.1$). Shown are results for a system with $51$ sites and box boundary conditions. The inset shows the scaling of the $n_{q=0}$ peak as a function of $U_0$. Panel (b) shows a comparison of the time evolution of the $n_{q=0}$ peak for initially noninteracting particles (solid lines) with weakly interacting ones ($U_0/J_0=0.1$).