Two-Photon Excitation Sets Limit to Entangled Photon Pair Generation from Quantum Emitters

Entangled photon pairs are key to many novel applications in quantum technologies. Semiconductor quantum dots can be used as sources of on-demand, highly entangled photons. The fidelity to a fixed maximally entangled state is limited by the excitonic fine-structure splitting. This work demonstrates that, even if this splitting is absent, the degree of entanglement cannot reach unity when the excitation pulse in a two-photon resonance scheme has a finite duration. The degradation of the entanglement has its origin in a dynamically induced splitting of the exciton states caused by the laser pulse itself. Hence, in the setting explored here, the excitation process limits the achievable concurrence for entangled photons generated in an optically excited four-level quantum emitter.

Figure 1

A Gaussian pulse (left) with finite full width at half maximum (FWHM) excites a quantum dot. In the two-photon resonant excitation scheme, the central frequency of the laser pulse matches half of the ground state-to-biexciton transition energy. The excited quantum dot decays radiatively, following the diamond-shaped cascade. Ideally, either a pair of horizontally ($H$) or vertically ($V$) polarized photons is emitted (center), resulting in a maximally entangled state $|{\mathrm{\Phi }}_{+}⟩$ described by the ideal two-photon density matrix ${\rho }^{2p}$ (right).

Figure 2

Concurrence in dependence of the pulse duration, characterized by its FWHM, for two laser polarizations [horizontal ($H$) filled symbols and diagonal ($D$) open symbols] and three fine-structure splittings $\delta =0$ (blue circles), $1.5\mu \mathrm{eV}$ (green diamonds), and $3\mu \mathrm{eV}$ (red squares). The symbols represent numerical results and lines are the analytic expression Eq. (8) ($H$ polarization: solid line; $D$ polarization: dashed line). For $\delta =0$, the results for $H$ and $D$ polarization are exactly the same. Data points at ${\mathrm{\Delta }}_{\text{FWHM}}=0$ (pentagons) represent calculations with an initially prepared biexciton without optical excitation.

Figure 3

Top: sketches of the four-level system with the laser-induced effects for (a) horizontal polarization and (b) diagonal polarization. Black lines are the unperturbed quantum dot states. Optical transitions are indicated by dashed arrows. The laser effect for horizontally polarized excitation can be interpreted as an ac Stark shift yielding an energetic splitting $E_S$. For a diagonal polarization, the laser-induced interaction can be interpreted as a coupling between the excitons with coupling strength ${\mathcal{V}}_S$. Bottom: examples of resulting density matrices for the two polarizations.