Unravelling the Structures in the van der Waals Interactions of Alkali Rydberg Atoms

TL;DR

This work addresses the need to understand and tailor van der Waals interactions between alkali Rydberg atoms. It introduces an angular-channel framework that separates radial (strength) and angular (orientation) contributions, incorporating the energy-defect structure to predict $C_6(\theta)$ efficiently. The key finding is that Förster resonances with $n_1 \neq n_2$ produce rich, highly anisotropic interaction potentials and sign changes that can be exploited to engineer specific Hamiltonians, with rubidium as the primary example and cesium data provided in the Appendix. The approach is computationally faster than brute-force methods and is integrated into the ARC package, enabling rapid searches for pair states that meet experimental constraints and enabling tailored interaction potentials for quantum information and simulation applications.

Abstract

Rydberg atoms are used in a wide range of applications due to their peculiar properties like strong dipolar and van der Waals interactions. The choice of Rydberg state has a huge impact on the strength and angular dependence of the interactions, and so a detailed understanding of the underlying processes and resulting properties of the interactions is therefore key to select the most suitable states for experiments. We study the van der Waals interactions in alkali-metal atoms in detail and highlight the structures which allow an understanding and exploitation of the various interaction properties. A particular theme is the identification of Förster resonances with $n_1 \neq n_2$, which offer interaction potentials with a wide range of properties that make them particularly interesting for experimental applications. A second theme is a focus on the underlying structures that shape the angular dependency and sign of the interactions. This understanding - instead of brute-force calculations - allows for a much simpler and more systematic search for suitable pair states. These insights can be used for the selection of tailored interaction potentials subject to experimental constraints and requirements. We use rubidium as an example species in this work and also provide data for cesium and pair states that are coupled via two- or three-photon transitions, i.e. up to $F$ states, in the Appendix.

Figures (38)

• Figure 1: Dipole-dipole interactions. (a) shows the angular dependency of $\hat{V}_{dd}$ for processes with $\Delta M = 0, \pm 1, \pm 2$. Shaded areas have positive sign. (b) Definition of the relative position $(R, \theta, \phi)$ in a spherical coordinate system with the quantisation axis $\textbf{q}\ ||\ z$. (c) Single-atom (top) and pair-state (bottom) picture of the dipole-coupled system. $\Delta$ indicates the energy difference (defect) between the pair states.
• Figure 2: Structures in $\mathbf{C_6(\theta)}$ values of $\mathbf{\ket{n_1P_{1/2}, n_2P_{1/2}}}$ in rubidium. (a) shows the absolute value of $C_6(\theta)$ and (b) the sign of $C_6(\theta)$ for $\ket{n_1P_{1/2}, n_2P_{1/2}}$ pair states in rubidium (red: $C_6 > 0$, blue: $C_6 < 0$). The 2D maps on the LHS in (a) and (b) show $C_6(\theta)$ at $\theta=0$ while the maps on the RHS show the respective same but for $\theta = \pi/2$. The top row plot shows a cross-section through the corresponding 2D maps at the dashed lines where $n_2 = 60$. Several lines of strong resonances (bright lines in a) can be seen, which are accompanied by a sign (colour) change in (b). The difference in $C_6$ for $\theta = 0, \pi/2$ is strongly pronounced at e.g. $n_1 \approx n_2$, which is indicated by the open triangles/full circles having different values. Order-of-magnitude and sign maps for other $\ket{n_1 L^\prime_{j_1}, n_2 L^{\prime\prime}_{j_2}}$ pair states for Rb and Cs can be found in Appendices \ref{['app:otherStateMapsRb']} and \ref{['app:otherStateMapsCs']} respectively.
• Figure 3: Angular momentum channels for $\mathbf{\ket{P_{1/2}, P_{1/2}}}$ states. For initial and final state being $\ket{P_{1/2}, P_{1/2}}$, one has four angular momentum channels with intermediate states of the type (a) $\ket{S_{1/2}, S_{1/2}}$, (b) $\ket{S_{1/2}, D_{3/2}}$ and its permutation, and (c) $\ket{D_{3/2}, D_{3/2}}$. The resulting angular dependency of each channel is shown along $\theta$ for $m_1 = m_2$ (solid) and $m_1 \neq m_2$ (dashdot) with $\tilde{m}_i = m_i$. (d) Level schematics showing the possible $m_i \to m_i^\prime \to \tilde{m}_i$ pathways fulfilling the condition $m_i = \tilde{m}_i$, i.e. the pathways contributing to the solid lines in the respective polar plots of the angular channels. One can see that different components of $\hat{V}_{dd}$ in equation \ref{['eqn:dipoleInteractionSpherical']} contribute differently to the different channels. The different total angular momentum changes $\Delta M$ of the two-atom system are shown separately, and different possible paths have different linestyles. The different $m_j$ paths available for every channel for a given set $(m_i, \tilde{m}_i)$ carry different angular dependencies and therefore result in different overall channel angular dependencies.
• Figure 4: Relevant energy defect structures for angular momentum channels of $\mathbf{\ket{n_1 P_{1/2}, n_2 P_{1/2}}}$ states in rubidium. For the case of $\ket{n_1 P_{1/2}, n_2 P_{1/2}}$ states, there are four possible angular channels: $\ket{S_{1/2}, S_{1/2}}$, $\ket{S_{1/2}, D_{3/2}}$, $\ket{D_{3/2}, S_{1/2}}$, $\ket{D_{3/2}, D_{3/2}}$, as shown in (a). For each angular channel, the principal quantum numbers $n_1$, $n_2$ can change up or down as the initial state couples to intermediate pair states of the form $\ket{(n_1 + \Delta n_1) L^\prime_{j_1^\prime}, (n_2 + \Delta n_2) L^{\prime\prime}_{j_2^\prime}}$. In (b) we show contributions of each angular channel to $C_6$. The data is color coded as in (a) with only the $\ket{S_{1/2}, D_{3/2}}$ channel in bold. The bottom two panels of (b) show the energy defect for $\Delta n_1 = 1$ and varying $\Delta n_2$, causing multiple lines of resonance. Note that we see that even $\Delta n_2 = -5$ contributes to the $C_6$ value. Panel (c) shows a breakdown of the contribution from different $\Delta n_i$ values for each angular channel. For the $\ket{S_{1/2}, S_{1/2}}$ channel we consider $\Delta n_1 = 0,\ 1$, see the first column in (c) and Appendix \ref{['app:FoersterResonanceLines']} for more details. In the plot we allow $\Delta n_2$ to take values up to $\pm 4$, as indicated by the colour. For the $\ket{S_{1/2}, D_{3/2}}$ channel, it matters whether the first or second atom has a non-zero $\Delta n$. The cases with $\Delta n_2 = -1$ and $\Delta n_1 = 1$ lead to Förster resonances, with the latter also shown in detail in panel (b). The $\ket{D_{3/2}, D_{3/2}}$ channel is analogous to the $\ket{S_{1/2}, S_{1/2}}$ channel. For the symmetric channels $\ket{S_{1/2}, S_{1/2}}$ and $\ket{D_{3/2}, D_{3/2}}$, swapping atoms 1 and 2 means that the figures are simply reflected about the dashed line. Lists containing the $\Delta n_i$ combinations that cause Förster resonances for different channels of initial pair states up to $\ket{n_1 F_J, n_2 F_{J^\prime}}$ are listed in Appendix \ref{['app:FoersterResonanceLines']}.
• Figure 5: Channel structure and resulting angular dependency for $\mathbf{\ket{n_1 P_{1/2}, 60 P_{1/2}}}$ states in rubidium. The upper panel shows $C_6(\theta)$ in (GHz $\upmu$m$^6$) for $m_1 = m_2 = \pm j$ and $\tilde{m}_i = m_i$ at two different angles, $\theta = 0, \pi /2$ (solid and open symbols, respectively). The lower panel shows the corresponding structure of the four angular channels with positive (negative) channel values indicated by closed (open) symbols. The separate Förster resonances of the channels from Figure \ref{['fig:energyDefect']} are clearly visible as local peaks. The four insets in the lower plot show the angular dependencies of each channel for $m_1 = m_2 = \pm j$ and $\tilde{m}_i = m_i$ (blue: $\ket{S_{1/2}, S_{1/2}}$, green: $\ket{S_{1/2}, D_{3/2}}$, yellow: $\ket{D_{3/2}, S_{1/2}}$, red: $\ket{D_{3/2}, D_{3/2}}$). It is interesting to note that the interaction strengths for e.g. $n_1 = 52,\ n_2 = 60$ are larger than for $n_1 = n_2 = 60$ and the angular anisotropy is significantly less pronounced, as can also be seen in the insets above the main plot showing $C_6(\theta)$. Due to the resulting interaction properties, these Förster resonances with $n_1 \neq n_2$ are very interesting for potential experimental applications. States like e.g. $n_1 = 48,\ n_2 = 60$ show Förster zeroes Walker2005, which can be exploited for orders-of-magnitude differences in interaction strength.