Conical Intersections in an Ultracold Gas

We find that energy surfaces of more than two atoms or molecules interacting via transition dipole-dipole potentials generically possess conical intersections (CIs). Typically only few atoms participate strongly in such an intersection. For the fundamental case, a circular trimer, we show how the CI affects adiabatic excitation transport via electronic decoherence or geometric phase interference. These phenomena may be experimentally accessible if the trimer is realized by light alkali atoms in a ring trap, whose interactions are induced by off-resonant dressing with Rydberg states. Such a setup promises a direct probe of the full many-body density dynamics near a CI.

Figure 1

Conical intersections in ensembles of dipole-dipole interacting atoms or molecules. (a) Randomly placed particles (*) and location of a selected conical intersection (red dot) in their vicinity (large grey spheres, $R_{\max }=0.2d_{\max }$, see text). (b) Potential energy surfaces $U_k(\Delta v\mathbf{V}+\Delta w\mathbf{W})$ near the CI in (a) for $\mu =1$. The modes $\mathbf{V}$ ($\mathbf{W}$) shown as red (black) arrows in (a) span the CI’s branching plane, see Eq. (3). (c) Mean connectivity $N_{\mathrm{conn}}$ (solid line), as explained in the text, and its standard deviation (dotted lines). (d) Histogram of participation numbers for $R_{\max }=0.35d_{\max }$.

Figure 2

Ring trimer. (a) Three atoms confined to a ring with radius $R$ in the $x\mathrm{\text{−}}y$ plane, akin to a triatomic molecule, see, e.g., 20. (b) Born-Oppenheimer surfaces for the electronic Hamiltonian (1). From top to bottom $U_{\mathrm{rep}}$, $U_{\mathrm{mid}}$ $U_{\mathrm{att}}$. For better visibility we shifted the repulsive (attractive) surface up (down) by $\Delta U=0.37\mathrm{MHz}$, and magnified the middle surface by a factor $f_m=5$. At the marked CI location, the original surfaces $U_{\mathrm{rep}}$ and $U_{\mathrm{mid}}$ are degenerate.

Figure 3

(a) Total atomic density $n(\theta )$, where $n(\theta )d\theta$ is the probability to find any of the three atoms in [$\theta ,\theta +d\theta$) on the ring [23]. Note that the $\theta$ axis is periodic; hence, the highest and lowest particles are near neighbors and initially repel. The configuration passes the CI location when all three interparticle distances are equal, around $t=0.17\mathrm{ms}$. (b) Adiabatic populations ${\tilde{p}}_{\mathrm{rep}}$ (solid, black line), ${\tilde{p}}_{\mathrm{mid}}$ (solid, red line), electronic populations $p_1=p_3$ (dashed, blue line), $p_2$ (dashed, red line) and total population (dotted, black line). (c) Purity $P=\mathrm{Tr}[{\widehat{\sigma }}^2]$ of the reduced electronic density matrix, described in the text.

Figure 4

Intersection circumnavigation from initial momentum superposition. The resulting interference pattern manifests the geometrical phase. (a) Shown is the density $|{\tilde{\varphi }}_{\mathrm{rep}}{|}^2$ at times $t_1=0$, $t_2=0.24\mathrm{ms}$, $t_3=0.48\mathrm{ms}$. The white lines are classical trajectories on the repulsive surface. Segregation of different time samples is provided by the white dashed ellipses. (b) Magnification of the area within the violet square in (a). A $\pi$ shift of the pattern due to the geometrical phase is visible between the part of the wave packet that has circumnavigated the CI (⊗) and the part that has not. Black lines in the background are isocontours of $U_{\mathrm{rep}}$.