Interactions between Rydberg excitons in ${\mathrm{Cu}}_2\mathrm{O}$

Highly excited states of excitons in cuprous oxide have recently been observed at a record quantum number of up to $n=25$. Here, we evaluate the long-range interactions between pairs of Rydberg excitons in ${\mathrm{Cu}}_2\mathrm{O}$, which are due to direct Coulomb forces rather than short-range collisions typically considered for ground-state excitons. A full numerical analysis is supplemented by the van der Waals asymptotics at large exciton separations, including the angular dependence of the potential surfaces.

Figure 1

(a) Sketch of a pair of excitons $i$ and $j$, consisting of electrons at ${\mathbf{r}}_e^{\left(i\right),\left(j\right)}$ and holes at ${\mathbf{r}}_h^{\left(i\right),\left(j\right)}$. The center-of-mass coordinates are indicated by ${\mathbf{R}}_{i,j}$, the exciton separation by $R_{ij}$. The coordinate system is aligned with $\widehat{\mathbf{z}}$. (b) Potential energy curves centered around the $15p-15p$ asymptote as a function of the exciton-exciton distance $R_{ij}$. The gray solid lines show the result of a diagonalization of the two-exciton Hamiltonian including the dipole-dipole interaction operator given by Eq. (9). At sufficiently large distances, these numerical results agree well with the expected van der Waals interaction potentials shown by the colored dashed lines. They correspond to the van der Waals interaction of exciton pair states with total angular momentum quantum numbers of $\left|M\right|=0$ (black), $\left|M\right|=1$ (red), and $\left|M\right|=2$ (green).

Figure 2

$C_6$ values for the (a) $s-s$, (b) $p-p$, and (c) $d-d$ asymptotes. The fits are for extrapolation for $n\ge 12$. Colored lines denote the families of $M$ with $\left|M\right|=0$ (black), $\left|M\right|=1$ (red), $\left|M\right|=2$ (green), $\left|M\right|=3$ (blue), and $\left|M\right|=4$ (brown).

Figure 3

At $n=11$ a near Förster resonance in the channel $np+np\to (n-1)d+(n+1)d$ leads to outlying points in the otherwise quite homogeneous range of $C_6\left(n\right)$. (a) The Förster defect $\delta$ of the given channel passes zero near $n=11$. Taking the repulsive part of the interaction as an example, the numerical solution (blue dots) is compared with the long-range asymptote with $\left|M\right|=0$ (black), $\left|M\right|=1$ (red), and $\left|M\right|=2$ (green) for $n=11$ (b) and $n=12$ (c). The color code denotes the relative $p$-component of each asymptote. Despite the near resonance at $n=11$, their difference is small as long as $R\gtrsim 1\mu \mathrm{m}$ since the dominant channels fall into the van der Waals regime.

Figure 4

Overlap of the optically active pair state ${\varphi }_a$ with the asymptotic molecular eigenstates $|\mu \rangle ,O=|\langle {\varphi }_a|\mu \rangle {|}^2$. Here, for illustration purposes, we chose $|{\varphi }_a\rangle =|n11,n11\rangle$ for ${\sigma }^{+}$-light and the eigenstates of the $p-p$ asymptote (labeled as listed in Table 1). Note that antisymmetric states 2 and 5 do not couple to the excitation laser.