Density Oscillations Induced by Individual Ultracold Two-Body Collisions

Access to single-particle momenta provides new means of studying the dynamics of a few interacting particles. In a joint theoretical and experimental effort, we observe and analyze the effects of a finite number of ultracold two-body collisions on the relative and single-particle densities by quenching two ultracold atoms with an initial narrow wave packet into a wide trap with an inverted aspect ratio. The experimentally observed spatial oscillations of the relative density are reproduced by a parameter-free zero-range theory and interpreted in terms of cross-dimensional flux. We theoretically study the long-time dynamics and find that the system does not approach its thermodynamic limit. The setup can be viewed as an advanced particle collider that allows one to watch the collision process itself.

Figure 1

Schematic of system and characterization of initial state. (a)(i) We prepare the wave packet of two interacting ${}^6\mathrm{Li}$ atoms (blue cloud) in the ground state of a tight optical tweezer trap (dark red). We additionally ramp on a weak crossed beam optical dipole trap (light red, not to scale). (a)(ii) At $t=0$ we switch off the tweezer trap to quench the trap geometry so that the cloud quickly oscillates in the $x$ and $y$ directions while slowly expanding in the $z$ direction. (a)(iii) After a variable expansion time we record the $z$ positions of both atoms via fluorescence imaging. (b) Schematic energy spectrum of the Hamiltonian $H_{\mathrm{rel},f}$ after the quench, showing a single “molecular branch” and a nearly harmonic spectrum of “scattering states.” Different magnetic fields lead to very different projections of the initial state onto the eigenstates of $H_{\mathrm{rel},f}$. In the weakly interacting regime at large magnetic fields, many scattering states are populated with roughly equal probability. At lower magnetic fields, the projection onto the single molecular state increases, eventually dominating the two-body dynamics. For the experimental parameters, hundreds of levels contribute to the dynamics.

Figure 2

Expectation values of ${\rho }^2$ (solid line) and $z^2$ (dashed line) for $a_s=-4.64a_{\mathrm{HO},z}$ as a function of time. The circle, square, and arrows mark the times that are considered in Figs. 3, 4, and 5, respectively.

Figure 3

Experiment-theory comparison after four two-body collisions. The relative densities $n_{\mathrm{rel}}(z,t)$ are shown for $t=3.5\mathrm{ms}$ and six different scattering lengths. The values of $a_s$ (in units of $a_{\mathrm{HO},z}$) are reported in the upper left corner of each panel. Circles show experimental data and solid lines show convolved theory results. Typical error bars are shown for a subset of the experimental data. Note the different $y$ scales in the panels.

Figure 4

Development of fringe pattern in the relative density during the first collision event ($t\approx T_x/2$) for $a_s=-0.651a_{\mathrm{HO},z}$. (a)–(d) The lines show the theoretically determined unconvolved relative density $n_{\mathrm{rel}}(z,t)$ for the times $t$ reported in the upper left corner of each panel. (e)–(g) The red solid lines schematically show the wave packet at (e) $t=0^{+}$, (f) $t\approx T_x/4$, and (g) $t\approx T_x/2$. The black dashed lines schematically show the equipotential lines of the final trap. The arrows schematically indicate the flux.

Figure 5

Single-particle densities for $a_s=-4.64a_{\mathrm{HO},z}$. (a) The black solid line and purple diamonds with error bars show the convolved theoretical data and experimental results for $t=1.234T_x=2\mathrm{ms}$; the red dashed and green dot-dashed lines show the convolved theoretical data for $t=3T_x$ and $t=4.5T_x$, respectively. (b) The blue solid and red dotted lines show, respectively, the unconvolved and convolved theoretical data for $t=5T_x=0.5T_z$.

Figure 6

Unconvolved theory results for the long-time regime for $a_s=-4.64a_{\mathrm{HO},z}$. The left- and right-hand columns show the relative and single-particle densities, respectively. Panels (a) and (b) show snapshots for $t=4T_z$, while panels (c) and (d) show snapshots for $t=4.25T_z$. Lines in panels (e) and (f) show cycle-averaged observables for $n=1$, $10^2$, and $10^4$. On the scale shown, the lines are essentially indistinguishable. For comparison, the green solid line in (e) shows the relative thermal density.