Two-Body Kapitza-Dirac Scattering of One-Dimensional Ultracold Atoms

TL;DR

This work develops a numerically exact two-body description of Kapitza–Dirac scattering for two contact-interacting atoms in a 1D harmonic trap under a pulsed optical lattice. By leveraging the Busch solution for the interacting relative motion and coupling it to center-of-mass dynamics, the authors construct and diagonalize a truncated Hamiltonian to obtain full time evolution, densities, and momentum distributions. They systematically map how interaction strength, lattice depth, lattice momentum, and pulse duration shape diffraction patterns and real-/momentum-space correlations, and they contrast the exact dynamics with a sudden-approximation impulse. The results provide a rigorous few-body benchmark for interacting Kapitza–Dirac scattering and deliver quantitative guidance for using Kapitza–Dirac protocols to probe strongly correlated ultracold atoms, with potential extensions to higher dimensions and more complex many-body systems.

Abstract

Kapitza-Dirac scattering, the diffraction of matter waves from a standing light field, is widely utilized in ultracold gases, but its behavior in the strongly interacting regime is an open question. Here we develop a numerically-exact two-body description of Kapitza-Dirac scattering for two contact-interacting atoms in a one-dimensional harmonic trap subjected to a pulsed optical lattice, enabling us to obtain the numerically exact dynamics. We map how interaction strength, lattice depth, lattice wavenumber, and pulse duration reshape the diffraction pattern, leading to an interaction-dependent population redistribution in real and momentum-space. By comparing the exact dynamics to an impulsive sudden-approximation description, we delineate the parameter regimes where it remains accurate and those, notably at strong attraction and small lattice wavenumber, where it fails. Our results provide a controlled few-body benchmark for interacting Kapitza-Dirac scattering and quantitative guidance for Kapitza-Dirac-based probes of ultracold atomic systems.

Figures (11)

• Figure 1: (a) Two atoms (blue and orange spheres) in a harmonic trap at $t=0$, interacting via a contact interaction of strength $g$ at the trap center. (b) For $t>0$, an additional optical lattice potential is applied on top of the harmonic confinement. The total potential (solid) and lattice potential (dashed) are shown separately for clarity.
• Figure 2: (a) Time evolution of the one-body density of two interacting particles with interaction strength $g = -5\hbar\omega\ell$ in a lattice with $k_{\mathrm{lat}} = 4/\ell$ and lattice depth $U_{0} = -1000\hbar\omega$. (b) Corresponding momentum distribution; dotted lines indicate $k = \pm 2k_{\mathrm{lat}}$ and $k = \pm 4k_{\rm{lat}}$. (c) Two-body correlation function. In all panels, the dashed lines indicate the time slices at $t = 0$, $t = 2.5\times10^{-3}/\omega$ and $t = 2.0\times10^{-2}/\omega$, for which the corresponding line profiles are displayed in panels (d)--(f).
• Figure 3: (a) Time evolution of the one-body density of two interacting particles with interaction strength $g = 5\hbar\omega\ell$ in a lattice with $k_{\mathrm{lat}} = 4/\ell$ and lattice depth $U_{0} = -1000\hbar\omega$. (b) Corresponding momentum distribution; dotted lines indicate $k = \pm 2k_{\mathrm{lat}}$ and $k = \pm 4k_{\rm{lat}}$. (c) Two-body correlation function. In all panels, the dashed lines indicate the time slices at $t = 0$, $t = 2.5\times10^{-3}/\omega$ and $t = 2.0\times10^{-2}/\omega$, for which the corresponding line profiles are displayed in panels (d)--(f).
• Figure 4: (a) Time evolution of the one-body density of two interacting particles with interaction strength $g =-5\hbar\omega\ell$ in a lattice with $k_{\mathrm{lat}} = 6/\ell$ and lattice depth $U_{0} = -1000\hbar\omega$. (b) Corresponding momentum distribution; dotted lines indicate $k = \pm 2k_{\mathrm{lat}}$. (c) Two-body correlation function. In all panels, the dashed lines indicate the time slices at $t = 0$, $t = 2.5\times10^{-3}/\omega$ and $t = 2.0\times10^{-2}/\omega$, for which the corresponding line profiles are displayed in panels (d)--(f).
• Figure 5: (a) Time evolution of the one-body density of two interacting particles with interaction strength $g =5\hbar\omega\ell$ in a lattice with $k_{\mathrm{lat}} = 6/\ell$ and lattice depth $U_{0} = -1000\hbar\omega$. (b) Corresponding momentum distribution; dotted lines indicate $k = \pm 2k_{\mathrm{lat}}$. (c) Two-body correlation function. In all panels, the dashed lines indicate the time slices at $t = 0$, $t = 2.5\times10^{-3}/\omega$ and $t = 2.0\times10^{-2}/\omega$, for which the corresponding line profiles are displayed in panels (d)--(f).