Exploring vibronic dynamics near a sloped conical intersection with trapped Rydberg ions

Abstract

We study spin-phonon coupled dynamics in the vicinity of a sloped conical intersection created by laser coupling the electronic (spin) and vibrational degrees of freedom of a pair of trapped Rydberg ions. We show that the shape of the potential energy surfaces can be engineered and controlled by exploiting the sideband transitions of the crystal vibration and dipole-dipole interactions between Rydberg ions in the Lamb-Dicke regime. Using the sideband transition, we realize a sloped conical intersection whose cone axis is only tilted along one spatial axis. When the phonon wavepacket is located in the potential minimum of the lower potential surface, the spin and phonon dynamics are largely frozen owing to the geometric phase effect. When starting from the upper potential surface, the electronic and phonon states tunnel to the lower potential surface, leading to a partial revival of the initial state. In contrast, the dynamics drastically change when the initial wavepackets are away from the conical intersection. The initial state is revived, and is almost entirely irrelevant to whether it is from the lower or upper potential surface. Complete Rabi oscillations of the adiabatic states are found when the wavepacket is initialized on the upper potential surface. The dynamics occur on the microsecond and nanometer scales, implying that Rydberg ions provide a platform for simulating nonadiabatic processes in the vicinity of a sloped conical intersection.

Figures (8)

• Figure 1: (a) CI in a system of two trapped Rydberg ions using the sideband resolved laser excitation. (b) Each Rydberg ion is modeled as a two-level system, laser excited from the ground state $\ket{g}$ to the Rydberg state $\ket{r}$. The two ions are coupled to a standing wave laser $E(\Omega_{0},\Delta)$ with Rabi frequency $\Omega_{0}$ and detuning $\Delta$. The dipole-dipole interaction $V_{DD}$ couples the vibration to the Rydberg state. (c) Sloped CIs. (d) Peaked CI, where the upper PES is entirely above the lower PES in the vicinity of the CI. In (c) and (d), the profiles of the CI along $Q_y=0$ are shown. Here we consider Sr$^+$ ions with mass $m=87.9\times1.66\times 10^{-27}\,\text{Kg}$, $\omega_x=2\pi\,\text{MHz}$, $\omega_y=2\pi\times 1.6 \,\text{MHz}$, $F_0/\hbar=- 1441.99\,\text{MHz/{$\mu$m}}$, $G_0/\hbar=88.8572\,\text{MHz/{$\mu$m}}$ and $\Delta/\hbar=-12.29\,\text{MHz}$. The position of the CI is $(Q_y^*,q_x^*)=(0,0)$.
• Figure 2: (a) Time evolution of the diabatic and adiabatic state populations. The blue and orange dotted curves show the populations of the diabatic states $\ket{rr}$ and $\ket{+}$, respectively. The green and red curves show the populations of the adiabatic states $\ket{\varphi_{+}(Q,q)}$ and $\ket{\varphi_{-}(Q,q)}$, respectively. (b) Time evolution of the average position $\langle q_x(t)\rangle$ (red curve), together with its projections on the lower PES $\langle q_x(t)\rangle_{-}$ (dotted blue curve) and upper PES $\langle q_x(t)\rangle_{+}$ (solid blue curve). Here the initial state is $\ket{\psi_0(q_x,Q_y)}=\phi(q_x,Q_y) \ket{\varphi_{-}(q_x,Q_y)}$. $q_x^0=-17.5\,\text{nm}$, $Q_y^0=0\,\text{nm}$ and all other parameters are same as Fig. \ref{['fig_diagram']}.
• Figure 3: Dynamics of nuclear densities. From top to bottom: the nuclear density on the lower PES, the nuclear density on the upper PES, and the full nuclear density. Red contours indicate the shape of $E_{-}$ in the top and bottom panels, The middle panel indicates the shape of $E_{+}$. The red star marks the position of the CI. The left and right column correspond to snapshots at $t = 0.6~\mu\text{s}$ and $t = 1~\mu\text{s}$, respectively.
• Figure 4: (a) Time evolution of the diabatic and adiabatic state populations. The blue and orange dotted curves show the populations of the diabatic states $\ket{rr}$ and $\ket{+}$, respectively. The green and red curves show the populations of the adiabatic states $\ket{\varphi_{+}(Q,q)}$ and $\ket{\varphi_{-}(Q,q)}$, respectively.(b) Time evolution of the average position $\langle q_x(t)\rangle$ (red curve), together with its projections on the lower PES $\langle q_x(t)\rangle_{-}$ (dotted blue curve) and the upper PES $\langle q_x(t)\rangle_{+}$ (solid blue curve). Here we have taken $\ket{\psi_0(q_x,Q_y)}=\phi(q_x,Q_y) \ket{\varphi_{-}(q_x,Q_y)}$ where $q_x^0=-42 \,\text{nm}$, $Q_y^0=0\,\text{nm}$ and all other parameters are as in Fig. \ref{['fig_diagram']}.
• Figure 5: Dynamics of the nuclear densities on the lower PES, upper PES, and the full nuclear density (top to bottom). Initially the system is prepared on the lower PES. The red contours indicate the shape of $E_{-}$ in the top and bottom panels, while in the middle panel, they indicate the shape of $E_{+}$. The red star marks the position of the CI. The left and right columns correspond to snapshots at $t = 0.2~\mu\text{s}$ and $t = 0.4~\mu\text{s}$, respectively. Initially $q_x^0=-42 \,\text{nm}$, $Q_y^0=0\,\text{nm}$ and all other parameters are as in Fig. \ref{['fig_diagram']}.