Experimental observation of the avoided crossing of two $S$-matrix resonance poles in an ultracold atom collider

In quantum mechanics, collisions between two particles are captured by a scattering matrix which describes the transfer from an initial entrance state to an outgoing final state. Analyticity of the elements of this $S$ matrix enables their continuation onto the complex energy plane and opens up a powerful and widely used framework in scattering theory, where bound states and scattering resonances for a physical system are ascribed to $S$-matrix poles. In the Gedankenexperiment of gradually changing the potential parameters of the system, the complex energy poles will begin to move, and in their ensuing flow, two poles approaching will interact. An actual observation of this intriguing interaction between scattering poles in a collision experiment has, however, been elusive. Here, we expose the interplay between two scattering poles relating to a shape resonance and a magnetically tunable Feshbach resonance by studying ultracold atoms with a laser-based collider. We exploit the tunability of the Feshbach resonance to observe a compelling avoided crossing of the poles in their energies which is the hallmark of a strongly coupled system.

Figure 1

Scenario for atomic scattering with two $S$-matrix poles simultaneously in play. (a) Relevant atomic-basis potentials for cold $d$-wave collisions between two ${}^{87}\mathrm{Rb}$ atoms in the $|F=1,m_F=1\rangle$ hyperfine state [25]. The difference between the $|2,0\rangle +|2,2\rangle$ and $|2,1\rangle +|2,1\rangle$ potentials is not discernible for the window shown. The energy scale also renders the $\ell =2$ centrifugal barriers invisible, but the red diamond shows the classical turning point inside the barrier for the $|1,1\rangle +|1,1\rangle$ channel at threshold (dashed line). (b) Scattering state wave functions in the $|1,1\rangle +|1,1\rangle$ entrance channel [blue, with dark blue highlighting a $d$-wave shape resonance (SR)]. For an external magnetic field $B=927.5$ G a bound state (BS) resides below threshold of the open channel: the orange wave function shows its $|2,0\rangle +|2,2\rangle$ component; the other significant component (not shown), $|2,1\rangle +|2,1\rangle$, is very similar. Upon increasing the external magnetic field, the position of the bound state moves up in energy as indicated by the arrow. It eventually crosses the $|1,1\rangle +|1,1\rangle$ threshold to become a quasibound state corresponding to a Feshbach resonance in this channel. (c) The bound state and the shape resonance correspond to $S$-matrix poles on the complex energy plane and the magnetic field will tune their relative positions.

Figure 2

$d$-wave scattering in our optical collider. (a) $d$-wave angular emission pattern which dominates the distribution of elastically scattered atoms in the collider experiment. (b) Optical collider procedure. Atomic clouds confined by laser beams are steered to collide in free space while subject to a magnetic field. The resulting $d$-wave dominated scattering halo is detected using absorption imaging.

Figure 3

Fraction of scattered atoms as a function of both magnetic field and collision energy. Experimentally acquired data are shown as blue dots while the orange lines are curve fits of Beutler-Fano lines shapes given by Eq. (2). The surface is an estimate of the scattered fraction based on coupled-channels predictions of the elastic cross section. Insets: Fano profiles at selected collision energies spanning the shape resonance, with error bars showing the standard deviation.

Figure 4

(a) Rb $S$-matrix pole trajectories as functions of magnetic field with dots spaced every 2 G. The two poles can be seen to move quickly when outside of the vicinity of 200–$400\mu \mathrm{K}$. Inset above shows the observed scattering at three fixed energies. The narrow Fano profile at $156\mu \mathrm{K}$ is a result of the orange pole moving past quickly. At $303\mu \mathrm{K}$, the broad Fano dip is the combined effect of the blue pole leaving, and the orange pole arriving and their intermediate interference. The resonance at $697\mu \mathrm{K}$ is due to the pass-by of the blue pole. (b) The avoided crossing nature of the trajectories of (a) highlighted by viewing the real energy position of the poles in the magnetic field. The region below threshold is shaded grey, and the data in this region are derived from direct calculations of bound state positions in the coupled channels.

Figure 5

Measured and calculated cross section at three magnetic fields, demonstrating the movement and restoration of the shape resonance during the avoided crossing. The color of the curves corresponds to the arrows at the right of Fig. 4.