Electro-optical properties of Rydberg excitons

We show how to compute the electro-optical functions (absorption, reflection, and transmission) when Rydberg exciton-polaritons appear, including the effect of the coherence between the electron-hole pair and the electromagnetic field. With the use of the real density matrix approach, numerical calculations applied for the ${\mathrm{Cu}}_2\mathrm{O}$ crystal are performed. We also examine in detail and explain the dependence of the resonance displacement on the state number and applied electric field strength. We report a fairly good agreement with recently published experimental data.

Figure 1

Oscillator strengths as a function of exciton number $n$. Logarithmic scale is applied. Dashed line marks the linear regression for $n^{-3}$ relation.

Figure 2

The bulk electroabsorption of a ${\mathrm{Cu}}_2\mathrm{O}$ crystal calculated from the imaginary part of the susceptibility in the energetic region of $n=3–10$ excitonic states, for the electric field strengths $F=15\mathrm{V}/\mathrm{cm}$ and $F=0$. The logarithmic scale is applied. Insets show the absorption spectrum near selected states.

Figure 3

The bulk electroabsorption of a ${\mathrm{Cu}}_2\mathrm{O}$ crystal, in the energetic region of $n=3–10$ excitonic states, for the electric field $F=15\mathrm{V}/\mathrm{cm}$. The logarithmic scale is applied. The contributions of $P$ and $F$ excitons are shown separately.

Figure 4

The difference $\mathrm{\Delta }\alpha =\alpha \left(F\right)-\alpha \left(0\right)$, for $F=15\mathrm{V}/\mathrm{cm}$ and in the range of $n=4–10$ excitonic states.

Figure 5

The bulk electroabsorption of ${\mathrm{Cu}}_2\mathrm{O}$ crystal calculated from imaginary part of susceptibility in the energetic region of $n=8–10$ excitonic states, for two electric field strengths $F=15\mathrm{V}/\mathrm{cm}$ and $F=0$. The logarithmic scale is applied. There is some overlap in the identified states.

Figure 6

(a) The same as in Fig. 2, in the energetic region of $n=7$ excitons, without the electric field, (b) for the field strength 50 V/cm.

Figure 7

(a) The changes in the absorption, as a function of the applied field strength, for the range (0, 50) V/cm, (b) the same for three chosen values of the field strength.

Figure 8

Absorption spectrum of ${\mathrm{Cu}}_2\mathrm{O}$ crystal, in the energetic region of $n=7–10$ excitonic states as a function of the applied field strength.

Figure 9

The real part of susceptibility of ${\mathrm{Cu}}_2\mathrm{O}$ crystal in the energetic region of $n=4–10$ excitonic states, for the electric field $F=15\mathrm{V}/\mathrm{cm}$ and $F=0$. The logarithmic scale is applied. Insets show absorption spectrum near selected states.

Figure 10

(a) The reflection coefficient of a ${\mathrm{Cu}}_2\mathrm{O}$ crystal, in the energetic region of $n=8–10$ excitonic states, for two values of the electric field. (b) The difference $\mathrm{\Delta }R=R\left(F\right)-R(F=0)$ for $F=15\mathrm{V}/\mathrm{cm}$. (c) The transmissivity $T$, and (d) $\mathrm{\Delta }T=T\left(F\right)-T\left(0\right)$, for $F=15\mathrm{V}/\mathrm{cm}$.