Exploring the Many-Body Dynamics Near a Conical Intersection with Trapped Rydberg Ions

Conical intersections between electronic potential energy surfaces are paradigmatic for the study of nonadiabatic processes in the excited states of large molecules. However, since the corresponding dynamics occurs on a femtosecond timescale, their investigation remains challenging and requires ultrafast spectroscopy techniques. We demonstrate that trapped Rydberg ions are a platform to engineer conical intersections and to simulate their ensuing dynamics on larger length scales and timescales of the order of nanometers and microseconds, respectively; all this in a highly controllable system. Here, the shape of the potential energy surfaces and the position of the conical intersection can be tuned thanks to the interplay between the high polarizability and the strong dipolar exchange interactions of Rydberg ions. We study how the presence of a conical intersection affects both the nuclear and electronic dynamics demonstrating, in particular, how it results in the inhibition of the nuclear motion. These effects can be monitored in real time via a direct spectroscopic measurement of the electronic populations in a state-of-the-art experimental setup.

Figure 1

Conical intersection in a system of two Rydberg ions. (a) Two ions are confined by a harmonic trapping potential in the $X–Z$ plane. From the ground state $|g⟩$, each ion is excited to the two Rydberg states $|\downarrow ⟩=|nS⟩$ and $|\uparrow ⟩=|nP⟩$. Because of different polarizabilities, ions in Rydberg states experience a state-dependent trapping potential along $X$, with frequencies ${\omega }_X^{\downarrow ,\uparrow }$, and different equilibrium positions. When excited to the pair states $|{\pi }_1⟩=|\uparrow \downarrow ⟩$ and $|{\pi }_2⟩=|\downarrow \uparrow ⟩$, ions interact through the exchange interaction $V_{\mathrm{ex}}$, whose strength depends on their separation $r$. A static offset electric field $\mathcal{E}$ allows us to control the transverse equilibrium position of the ions. (b) The interplay between the exchange interaction and the state-dependent confinement shifts the energies of the states $|{\pi }_1⟩$ (dashed curve) and $|{\pi }_2⟩$ (solid curve). This results in a conical intersection in the potential energy surfaces $U_{-}$ (red) and $U_{+}$ (blue) which, in the reference frame of the center of mass of the system, occurs at the ions’ relative equilibrium separation $r_{x,z}^0$. See text for details.

Figure 2

Potential energy surfaces in the branching plane. Plot of the eigenvalues of Eq. (4) (i.e., the PESs) $U_{\pm }(\mathit{q})$ as a function of the displacements $q_x$ and $q_z$. (a) For $V_{\mathrm{ex}}^0/h=0$, a CI occurs at ${\mathit{q}}^{*}=(0,0)$ while for (b) $V_{\mathrm{ex}}^0/h=2\pi \times 0.3\mathrm{MHz}$ the CI is shifted to ${\mathit{q}}^{*}=(0,-V_{\mathrm{ex}}^0/F_z^0)$ (see text for details). Here, we considered ${}^{88}{\mathrm{Sr}}^{+}$ Rydberg ions, with $m=87.9\times 1.66\times {10}^{-27}\mathrm{kg}$, and set ${\omega }_X=2\pi \times 1.6\mathrm{MHz}$, ${\omega }_Z=2\pi \mathrm{MHz}$, $\mathcal{E}=2.529\mathrm{V}/\mathrm{m}$ (resulting in $X^0=-0.024\mu \mathrm{m}$), ${\rho }_{\downarrow }=8.9\times {10}^{-31}{\mathrm{C}}^2{\mathrm{m}}^2/\mathrm{J}$, ${\rho }_{\uparrow }=-3.8\times {10}^{-30}{\mathrm{C}}^2{\mathrm{m}}^2/\mathrm{J}$ (corresponding to Rydberg $|nS⟩$ and $|nP⟩$ states with $n=50$), and $F_z^0/h=2\pi \times 20\mathrm{MHz}/\mu \mathrm{m}$.

Figure 3

Dynamics of the diabatic and adiabatic populations. Time evolution of the populations $n_k(t)$ of the diabatic states $|{\pi }_1⟩$ (blue curve) and $|{\pi }_2⟩$ (yellow curve) in the (a) BO approximation, (b) exact dynamics with $V_{\mathrm{ex}}^0/h=0$, and (c) exact dynamics with $V_{\mathrm{ex}}^0/h=2\pi \times 0.3\mathrm{MHz}$. In (b),(c), the populations of the adiabatic states, ${\tilde{n}}_{-}(t)$ (red curves) and ${\tilde{n}}_{+}(t)$ (green curves) are also shown. (b) The complete destructive interference between the two paths surrounding the CI results in the freezing of the nuclear motion in the c.m. reference frame. (c) A finite $V_{\mathrm{ex}}^0$ shifts the CI and breaks the perfect symmetry between the two paths observed in (b). As a consequence, the nuclear wave packet can move to the other local minimum of $U_{-}(\mathit{q})$ and the diabatic populations $n_k(t)$ swap with each other. The bottom row illustrates some snapshots of the dynamics of the nuclear density $\mathcal{N}(\mathit{q},t)$ in the c.m. reference frame associated with the various time evolutions shown in the upper row (see the corresponding marks). Here, red contours display the behavior of $U_{-}(\mathit{q})$, with lighter tones corresponding to larger values, while the cross highlights the position of the CI. The arrows in the first and last plots are a sketch of the weight of the nuclear wave packets encircling the opposite sides of the CI. In all panels, the system is initialized in the state $|{\psi }_0(\mathit{q})⟩$, with $q_{x,\mathrm{ss}}^0=-0.011\mu \mathrm{m}$. Other parameters are as in Fig. 2.