Orthogonal flexible Rydberg aggregates

We study the link between atomic motion and exciton transport in flexible Rydberg aggregates, assemblies of highly excited light alkali-metal atoms, for which motion due to dipole-dipole interaction becomes relevant. In two one-dimensional atom chains crossing at a right angle adiabatic exciton transport is affected by a conical intersection of excitonic energy surfaces, which induces controllable nonadiabatic effects. A joint exciton-motion pulse that is initially governed by a single energy surface is coherently split into two modes after crossing the intersection. The modes induce strongly different atomic motion, leading to clear signatures of nonadiabatic effects in atomic density profiles. We have shown how this scenario can be exploited as an exciton switch, controlling direction and coherence properties of the joint pulse on the second of the chains [K. Leonhardt et al., Phys. Rev. Lett. 113, 223001 (2014)]. In this article we discuss the underlying complex dynamics in detail, characterize the switch, and derive our isotropic interaction model from a realistic anisotropic one with the addition of a magnetic bias field.

Figure 1

(a) Orthogonal atom chains with one Rydberg dimer each. Atoms 0 and 1 initially share an excitation. Due to the ensuing repulsion (blue arrows) atom 1 reaches the conical intersection at $x_{\text{CI}}$. The colored bars visualize the excitation amplitude on each atom $c_n=\langle {\pi }_n|{\phi }_{\text{rep}}\rangle$, with orange $c_n>0$ and blue $c_n<0$. The origin of the coordinate system is set to the mean initial position of atom 0. (b) The repulsive energy surface $U_{\mathrm{rep}}$ (red) and adjacent surface $U_{\mathrm{mid}}$ (green) of the trimer subunit (atoms 1, 2, and 3) near the CI. (c) and (d) Forces on atom 2 (solid lines) and atom 3 (dashed lines), for the repulsive surface [red, (c)] and adjacent surface [green, (d)]. The insets show atomic positions and the excitation distribution $c_n$ of exciton states and forces for the indicated values of $\mathrm{\Delta }{x}_{12}$, which denotes the distance between atom 1 and the vertical chain. The parameter $p$ controls the degree of symmetry of the trimer, where $p=1$ corresponds to an isosceles trimer configuration. Figure reproduced from [31].

Figure 2

Selected single trajectories $\mathbf{R}\left(t\right)$ of atoms 2 and 3 in (a), (c), and (e) with corresponding time-resolved energy spectra (black lines) and potential energy $U_{s\left(t\right)}\left(\mathbf{R}\left(t\right)\right)$ (colored line) in (b), (d), and (f). For both, trajectories and potential energy, evolution on the repulsive surface is marked with red lines and evolution on the adjacent surface with green lines. (a) and (c) Two characteristic trajectories that stay on the adjacent surface. The two trajectories are almost mirror symmetric around the $x$ axis. (e) A characteristic trajectory jumping back to the repulsive surface after passing the vicinity of the CI. The gray area marks the vicinity of the CI between the adjacent and the repulsive surfaces, ruling the dynamics. The earlier crossings are less relevant for our dynamics here; see [39].

Figure 3

Atomic densities of atoms 2 and 3 on the adjacent surface (a), and on the repulsive surface (b). Atomic positions were binned into one or the other according to the trajectory surface index $s$. The maximum value of the data over both (a) and (b) is set to unity with a common normalization.

Figure 4

Dynamics of atomic motion and electronic populations; total atomic density of atoms 0 and 1 in (a) and atoms 2 and 3 in (b). (c) Adiabatic populations, $\overline{|{\tilde{c}}_n{|}^2}$, (solid lines) and trajectory fractions $f_n={\sum }_i^{N_{\text{traj}}}{\delta }_{n,s_i\left(t\right)}/N_{\text{traj}}$ (dashed dotted lines) on the repulsive surface (red) and adjacent surface (green). Here $\overline{\cdots }$ denotes a trajectory average. (d) Purity $P=\mathrm{Tr}[{\widehat{\sigma }}^2]$ of the reduced electronic density matrix [10, 43]. All results are averaged over ${10}^5$ realizations. The maximum value of the data in (a) and (b) is individually set to one.

Figure 5

Sketch of a seven-atom system combining features of adiabatic entanglement transport and CI dynamics. Three atoms are placed on the horizontal line and four on the vertical one. Atoms 0 and 1 are prepared in a repulsive exciton at the initial time $t_0$, resulting in the excitation distribution shown by orange and blue bars, as described in the caption of Fig. 1. Atom 2 comes close to a CI configuration, when it is at position $x_{\text{CI}}$. Around that moment, the excitation almost exclusively resides on atoms 2, 4, and 5, since all other atoms are far away from this trimer. Atoms 4 and 5 are accelerated at this time on the two different surfaces, already described in the four-atom section. We analyze whether or not the atom pair $(3,4) \left[\right(5,6\left)\right]$ finally contains an exciton-motion pulse propagating in the negative (positive) $y$ direction. This is quantified by the binary entanglement [10] $E_{34} \left[E_{56}\right]$ at the moment that atom $3 \left[5\right]$ reaches the location indicated by a black box, termed “entanglement readout.” We finally use the indicated displacement $\mathrm{\Delta }y$ of the horizontal chain to steer entanglement transport upwards or downwards.

Figure 6

Dynamics of atomic motion and populations. (a) Total atomic density of atom 0 to 2. (b) Total atomic density of atom 3 to 6. (c) Adiabatic populations (solid lines) and fractions (dashed dotted lines) of repulsive surface (red), second most energetic (adjacent) surface (green), and third most energetic surface (yellow). All data are averaged over 1 million realizations. The maximum value of the data in (a) and (b) is individually set to one. To highlight details at lower densities, all values between 0.7 and 1 are represented with white.

Figure 7

Atomic positions from single trajectories for atoms 3–6 (a), (c), and (e) with corresponding eigenenergies in (b), (d), and (f) for $\mathrm{\Delta }y=0\mu \mathrm{m},a_2=9.5\mu \mathrm{m}$. (a) and (c) Two different trajectories ending up on the adjacent surface. (e) A trajectory returning to the repulsive surface after being in the vicinity of the CI. (b), (d), and (f) Time-resolved energy spectra (black lines) and potential energy (colored line), for position trajectories on the left. For both, positions and potential energy, evolution on the repulsive surface is marked with red and evolution on the adjacent surface with green. The trajectories shown in (e) evolve in the end on a third surface, marked with black lines. The gray area marks the CI and its vicinity. For a discussion of the earlier curve crossings see [39].

Figure 8

Qualitatively different dynamics on the repulsive and adjacent surface. Mean atomic positions (green lines), with $p$-excitation probability of the combined repulsive and adjacent surfaces (black-red-yellow-white shades) on each atom of the vertical chain, $\mathrm{\Delta }y=1.5\mu \mathrm{m}$. (a) Case that mostly remains on the adjacent surface, with $a_2=9.5\mu \mathrm{m}$. (b) Dominantly repulsive case with $a_2=20\mu \mathrm{m}$. The excitation probability for atom $n$ is represented by Gaussians of a selected fixed width, normalized to ${\sum }_n{\left|\langle {\pi }_n|{\phi }_{\text{n}}\left(\mathbf{R}\right)\rangle \right|}^2$ (where ${\sum }_n$ runs over the repulsive and adjacent surfaces) and centered on the mean position of atom $n$.

Figure 9

Response of the exciton switch to control parameters. We show the bipartite entanglement transported downwards in $y,{\overline{E}}_{34}$ (left column) and upwards, ${\overline{E}}_{56}$ (right column). Parameters are $\nu =80$ and $d=22\mu \mathrm{m}$. Both entanglement readouts (see Fig. 5) are placed in a distance of $b=0.3d$ from atoms 3 and 6, respectively, on the vertical chain. (a)–(d) Entanglement as a function of $a_1$ and $\mathrm{\Delta }y$, for $a_2=9.5\mu \mathrm{m}$ in (a) and (b) and $a_2=20\mu \mathrm{m}$ in (c) and (d). (e) and (f) Entanglement as a function of $a_2$ and $\mathrm{\Delta }y$, for fixed $a_1=6\mu \mathrm{m}$. The markers $(*,\times ,+)$ highlight extreme cases already presented in [31]: $(*)$ high entanglement transport to atoms (3,4), $(\times )$ high entanglement transport to atoms (5,6), $(+)$ equal entanglement transport towards atoms (3,4) and atoms (5,6).

Figure 10

For the case of Fig. 4, we compare our model of Sec. 2 with one additionally allowing different azimuthal sublevels $m_l$, anisotropic interactions and a magnetic field; see Appendix pp1. In both models we used the parameters $\nu =80,a_2=19\mu \mathrm{m}$ and $d=40\mu \mathrm{m}$. To obtain qualitatively the same dynamics in both, $a_1$ is adjusted individually, compensating quantitative differences in potentials. The anisotropic model uses $a_1=8\mu \mathrm{m}$ and the isotropic one $a_1=11.8\mu \mathrm{m}$. (a) and (c) Atomic density of atom 0 and atom 1. (b) and (d) Atomic density of atoms 2 and 3, where (a) and (c) are for the anisotropic model and (b) and (d) for the isotropic one. To highlight details at lower densities, all values between 0.4 and 1 are represented with white. (e) Adiabatic populations on the (red line) repulsive and (green line) adjacent energy surface and purity (black lines). We compare the anisotropic (solid line) with the isotropic model (dash-dotted line). (f) Cut through the atomic densities at the time indicated by white lines in (b) and (d) for both models, with line styles as in (e).

Figure 11

Sketch of energy splittings for the aggregate states. (a) Without magnetic field, the fine structure separates aggregate states where the $p$ states have total angular momentum $j=1/2$ from the ones with $j=3/2$. (b) An external magnetic field within the strong field regime, where ${\mu }_B·B_z\gg \mathrm{\Delta }{E}_{FS}$, splits the aggregate states into different manifolds with energy gaps ${\mu }_B·B_z$ for $(\mathrm{\Delta }{M}_S,\mathrm{\Delta }{m}_l)=(0,\pm 1)$ and energy gaps of the order of $\mathrm{\Delta }{E}_{FS}$, for $(\mathrm{\Delta }{M}_S,\mathrm{\Delta }{m}_l)=(\pm 1,\mp 2)$ (energy gap between neighboring blue and red lines). The notation ${\{\uparrow \}}^{N_{\uparrow }}{\{\downarrow \}}^{N_{\downarrow }}$ is a short form for a spin configuration with $N_{\uparrow }$ spins oriented upwards $\left(m_s=1/2\right)$ and $N_{\downarrow }$ spins oriented downwards $\left(m_s=-1/2\right)$.

Figure 12

Comparison of energy spectra from different approximations of the complete Hamiltionian, for a double dimer as sketched in Fig. 1 (a). We use $\nu =80$, which yields a fine-structure splitting of $\mathrm{\Delta }{E}_{\mathrm{FS}}=0.15$ MHz [53]. Further parameters are $p=0,d=40\mu \mathrm{m},a_1=8\mu \mathrm{m}$, and $a_2=19\mu \mathrm{m}$. Atoms 2 and 3 are fixed, whereas the positions of atoms 0 and 1 are parametrized, such that $x_0=-x$ for atom 0 and $x_1=a_1+x$ for atom 1. We compare the complete model (solid lines) with different approximate models (dashed dotted lines) in different columns. The approximate models are as follows: (a)–(c) purely isotropic model used for the main results in this paper, with Hamiltonian given in (1); (d)–(f) corrected model with effective Hamiltonian (A28) up to first order in $\alpha$; (g)–(i) same as (d)–(f), but using (A28) up to second order in $\alpha$. We consider three different magnetic field strengths: $B_z=50$ G in the lower row, $B_z=100$ G in the adjacent row, and $B_z=160$ G in the upper row. The energy of infinitely separated atoms is set to zero.