Tunable and Robust Long-Range Coherent Interactions between Quantum Emitters Mediated by Weyl Bound States

Long-range coherent interactions between quantum emitters are instrumental for quantum information and simulation technologies, but they are generally difficult to isolate from dissipation. Here, we show how such interactions can be obtained in photonic Weyl environments due to the emergence of an exotic bound state whose wave function displays power-law spatial confinement. Using an exact formalism, we show how this bound state can mediate coherent transfer of excitations between emitters, with virtually no dissipation and with a transfer rate that follows the same scaling with distance as the bound state wave function. In addition, we show that the topological nature of Weyl points translates into two important features of the Weyl bound state, and, consequently, of the interactions it mediates: first, its range can be tuned without losing the power-law confinement, and, second, they are robust under energy disorder of the bath. To our knowledge, this is the first proposal of a photonic setup that combines simultaneously coherence, tunability, long range, and robustness to disorder. These findings could ultimately pave the way for the design of more robust long-distance entanglement protocols or quantum simulation implementations for studying long-range interacting systems.

Figure 1

(a) Schematics of the analyzed system. Blue (red) shallow cylinders represent sites belonging to the $A$ ($B$) sublattice. Solid and dashed arrows correspond to positive and negative first-neighbor hoppings, $J$, respectively. $M$ denotes the sublattice frequency offset, whereas $a$ is the lattice constant. (b) Band structure of the system shown in (a), as calculated for $M=0$. The frequency of the Weyl point (${\omega }_W$) is marked with an arrow. (c) Density of states, $D(\omega )$, associated to the bath with $M=0$.

Figure 2

(a) Exact dynamics of the emitter’s excited state population, $|C_e(t){|}^2$, calculated for $M=0$, $M=J$, and $M=2J$ (shown as purple, orange, and green lines, respectively). The dashed line stands for the Markovian prediction. Inset shows $|C_e(t\to \infty ){|}^2$ versus $M$ (the cases displayed in the main figure are marked with the corresponding color code). In all displayed results, $\mathrm{\Delta }={\mathrm{\Delta }}_c$ and $g=0.5J$ is assumed. (b) Left, center, and right upper panels display the three-dimensional distribution of the normalized photonic component of the Weyl bound state for $M=0$, $J$, and $2J$, respectively. Lower panels show cuts of the corresponding spatial distributions along the $x$, $y$ directions and $z$ direction, as a function of the distance $d$ to the quantum emitter. In these panels, blue (red) bars identify the sites belonging to the $A$ ($B$) sublattice, respectively. For each value of $M$, the emitter is detuned from the Weyl frequency by the associated critical value ${\mathrm{\Delta }}_c$.

Figure 3

(a),(b),(c) Frequency of the population’s exchange experienced by two vertically aligned emitters as a function of the separation between them for $M=0$, $J$, and $2J$, respectively. Both emitters feature $\mathrm{\Delta }={\mathrm{\Delta }}_c$ and are coupled to sites belonging to the $A$ sublattice. Gray dashed lines display the power law behavior associated to the Weyl bound state’s confinement in the single emitter case. Insets show the dynamics of the excited state populations corresponding to two emitters ($|C_1(t){|}^2$ and $|C_2(t){|}^2$, solid and dotted lines, respectively), coupled to two different lattice sites with relative position ${\mathbf{r}}_2-{\mathbf{r}}_1=a\widehat{z}$. In this calculation $g=0.5J$ is assumed.

Figure 4

(a),(b),(c) Photonic component of the Weyl bound state (WBS) along the $x$ direction for $M=0$ in the presence of a random onsite energy disorder characterized by the strength value $dM=0.2J,0.4J$, and $dM=0.6J$, respectively. Blue and red crosses depict the average value of the normalized WBS’s photonic component for $A$ and $B$ sites, while shadow areas span their corresponding standard deviation. For comparison, the gray dashed line displays the inverse square power law found in Fig. 3.