Generation of entangled-photon pairs from biexcitons in CuCl thin films: Nano-to-bulk crossover regime

We have constructed a theoretical framework of biexciton-resonant hyperparametric scattering for pursuit of high-power and high-quality generation of entangled-photon pairs. Our framework is applicable to the nano-to-bulk crossover regime where the center-of-mass motions of excitons and biexcitons are confined and material surroundings and the polarization correlation of generated photons can be considered. We have analyzed the generation of ultraviolet entangled photons from CuCl film with and without dielectric microcavity and revealed that in the nano-to-bulk crossover regime we generally get a high performance from the viewpoint of statistical accuracy and the generation efficiency can be enhanced by the optical cavity while maintaining the high performance. The nano-to-bulk crossover regime has a variety of degrees of freedom to control the entangled-photon generation, and the scattering spectra explicitly reflect quantized exciton-photon coupled modes in finite structures.

Figure 1

The biexciton-resonant hyperparametric scattering (RHPS) is depicted on biexciton, exciton-polariton, and longitudinal exciton dispersion curves. Biexcitons are resonantly created by two-photon absorption, and entangled-photon pairs are emitted when the biexcitons decay into the lower exciton-polariton branch. This emission appears in the scattering spectrum as two peaks called LEP and HEP (lower and higher energy polaritons) as seen in Fig. 3. Due to the conservation of energy and of wave vectors, the positions of the two peaks depend on scattering angle,[10, 11] and the entangled two photons are emitted symmetrically about the pump beam as shown in the inset. Two additional peaks called ${\mathrm{M}}_{\mathrm{T}}$ and ${\mathrm{M}}_{\mathrm{L}}$ in scattering spectra originate from the biexciton decay into transverse and longitudinal exciton levels, respectively.

Figure 2

(a) Center-of-mass wave functions of excitons and biexcitons in a CuCl film. Simple sinusoidal functions vanishing at surfaces are supposed. (b) Cavity structure considered in Figs. 9 and 10. The Bragg mirrors consist of ${\mathrm{PbBr}}_2$ and ${\mathrm{PbF}}_2$. On the transmission side, a high reflectance is achieved by a mirror with 16 periods to suppress the leakage of photons in this direction. On the incident side, only 4 periods are supposed to guarantee rapid radiative decay of entangled photons.

Figure 3

Forward scattering spectra from a CuCl film with thickness $d=7\mu \mathrm{m}$. The film exists in vacuum, and the pump beam is perpendicular to it. The pump frequency ${\omega }_{\mathrm{in}}$ corresponds to the two-photon absorption involving biexcitons. The results for scattering angles $\theta =0^{°}$, ${30}^{°}$, and ${60}^{°}$ are plotted with different lines as functions of scattering frequency $\omega$.

Figure 4

The scattering spectra of horizontal (H) and vertical (V) polarizations are shown. The film thickness is $d=7\mu \mathrm{m}$, and the scattering angle is $\theta ={60}^{°}$. The other parameters are the same as in Fig. 3.

Figure 5

(a) Polarization-resolved scattering spectra for thickness $d=200\hspace{0.5em}\mathrm{nm}$ and scattering angle $\theta ={60}^{°}$. The film exists in vacuum and the pump frequency is tuned to resonantly excite the $m=6$ biexciton state. (b) Dashed lines are the exciton-polariton dispersion relation in bulk crystal. The horizontal bars represent the exciton-photon coupled modes with V polarization in the film with $d=200\hspace{0.5em}\mathrm{nm}$. The bar length is the sum of radiative and nonradiative decay rates and the center is the resonance frequency. (c) The H-polarized modes are plotted. Because of the breaking of translational invariance in the $z$ direction and the nonzero scattering angle, the longitudinal excitons are also optically allowed.

Figure 6

The two-photon coincidence (signal) intensity is plotted as a function of scattering frequency $\omega$ of one of emitted photons. The spectra are resolved by the polarization of the two photons. CuCl films with thicknesses $d=7\hspace{0.5em}\mu \mathrm{m}$ and $200\hspace{0.5em}\mathrm{nm}$ are considered, and the scattering angle is $\theta ={60}^{°}$. The same pump power is supposed for both thicknesses. (a) The polarization directions of two photons are resolved in H and V axes. The collinear pairs ($\mathit{HH}$ and $\mathit{VV}$) are correlated, and the cross-linear pairs ($\mathit{HV}$ and $\mathit{VH}$) are not generated from the lowest biexciton level in a film. (b) The polarization directions are resolved for circular polarization basis. Not only the pairs of left (L) and right (R) circularly polarizations are obtained, but $\mathit{LL}$ and $\mathit{RR}$ pairs are also generated for $\theta >0$. The spectra of $\mathit{LR}$ and $\mathit{RL}$ are the same, and those of $\mathit{RR}$ and $\mathit{LL}$ are also the same. The spectra of 200 nm are magnified by factor 80 in both (a) and (b). Since the spectra are symmetrical about ${\omega }_{\mathrm{in}}$, only the high-frequency side $\omega >{\omega }_{\mathrm{in}}$ is shown.

Figure 7

(a) The generation efficiency (signal intensity) from a CuCl film with thickness $d=200\hspace{0.5em}\mathrm{nm}$ is plotted as a function of scattering frequency $\omega$. The biexciton state $m=6$ is resonantly excited, and the scattering angle is $\theta =0^{°}$. (b) The spectrum of the corresponding performance $P$. The generation efficiency and the performance are normalized in the same manner as in Fig. 8. (c) The exciton-photon coupled modes for $\theta =0^{°}$ in the film are plotted, while $\theta ={60}^{°}$ in Fig. 5b.

Figure 8

Thickness dependencies of (a) generation efficiency $S/{I_{\mathrm{in}}}^2$ and (b) performance $P$. The scattering angle is $\theta =0^{°}$ and the frequencies are ${\omega }_{1/2}={\omega }_{\mathrm{in}}\pm 0^{+}$. To suppress the interference effect, outside medium is a dielectrics with ${\varepsilon }_{\mathrm{bg}}^{\mathrm{ex}}$. The results for ${\gamma }_{\mathrm{ex}}=0$, 0.1, 0.5, and $1.0\hspace{0.5em}\mathrm{meV}$ are plotted with different lines. The performance is normalized to the ideal quantity.

Figure 9

Thickness dependencies of (a) generation efficiency $S/{I_{\mathrm{in}}}^2$ and (b) performance $P$. The black lines represent the results for a bare CuCl film existing in vacuum. The scattering frequencies are ${\omega }_{1/2}={\omega }_{\mathrm{in}}\pm 0^{+}$, and the angle is $\theta =0^{°}$ (forward). The gray lines represent the results for DBR cavity embedding a CuCl layer shown in Fig. 2b. The mode frequency of the optical cavity is tuned to ${\omega }_{\mathrm{T}}$, and the scattering is backward with $\theta ={180}^{°}$. In both cases, ${\gamma }_{\mathrm{ex}}=0.5\hspace{0.5em}\mathrm{meV}$. The performance is normalized in the same manner as in Fig. 8.

Figure 10

(a) The generation efficiency (signal intensity) is plotted as a function of scattering frequency $\omega$. A CuCl film with thickness $d=72\hspace{0.5em}\mathrm{nm}$ is embedded in the DBR cavity considered in Fig. 9, and the other parameters are also the same. (b) The spectrum of the corresponding performance $P$. (c) The exciton-photon coupled modes in the film are plotted in the same manner as in Fig. 5b.