Coherence and degree of time-bin entanglement from quantum dots

We report a study on coherence of excitation of single quantum dots. We address the coherent excitation of biexcitons, the process that is indispensable for deterministic photon pair generation in quantum dots. Based on theoretical modeling we optimized the duration of the excitation pulse in our experiment to minimize the laser-induced dephasing and increase the biexciton-to-background single-exciton occupation probability. An additional effect of this approach is a high degree of time-bin entanglement with a concurrence of up to 0.78(6) and a 0.88(3) overlap with a maximally entangled state.

Figure 1

Schematic of time-bin entanglement. The quantum-dot system (QD) is excited by two consecutive pulses obtained from an unbalanced Michelson interferometer shown on the left. The relative phase between these pulses is ${\varphi }_P$. The state analysis is performed using additional two interferometers, one for the exciton photon and the other for the biexciton photon. These two interferometers have their respective phases, ${\varphi }_X$ and ${\varphi }_{XX}$. The photons are detected upon leaving the analysis interferometers using detectors $D_X$ and $D_{XX}$. The level scheme depicts the quantum dot excited resonantly from the ground state $|g\rangle$ to the biexciton state $|b\rangle$ using a two-photon excitation. A pulsed laser populates the biexciton via a virtual level (dashed gray line). The system decays emitting a biexciton-exciton photon cascade ($X{X}_V$ and $X_V$ or $X{X}_H$ and $X_H$).

Figure 2

Rabi oscillations (theory) for constant dephasing as a function of ${\mathrm{\Omega }}^2\sigma$. The emission probabilities for the biexciton, $P_b$ (solid), and the exciton, $P_x$ (dashed), levels are shown. The Rabi oscillation damping is strongly dependant on the length of the excitation pulse.

Figure 3

(a) Biexciton emission probability, $P_b$ (theory), for different dephasing models. We observe a strong damping of the Rabi oscillations even at moderate ${\gamma }_{I_0}$. From this it is clear that the amplitude of the intensity-dependant dephasing rate plays a much greater role than the exponent $n_p$. (b) Emission probability for a biexciton, $P_b$, and exciton photon, $P_x$, as a function of the laser pulse area for linear (solid line) and quadratic (dashed line) intensity-dependent dephasing compared to the experimental data, respectively. The error bars are smaller than symbols. The theoretical parameters, ${\gamma }_{I_0}$, are obtained by fitting the ratio of the first maximum and minimum of the Rabi cycle.

Figure 4

Relative probability for biexciton vs single exciton excitation in the quantum dot as a function of ${\mathrm{\Omega }}^2\sigma$ for two different pulse lengths (upper plot). The choice of ${\mathrm{\Omega }}^2\sigma$, which is proportional to the energy per pulse, as $x$ axis allows for easier comparison of the maxima. We see an optimum ratio of about 8 at a still moderate excitation rate. The lower plot gives the biexciton photon emission probability as a function of ${\mathrm{\Omega }}^2\sigma$.

Figure 5

Examples of (a) real and (b) imaginary parts of the reconstructed density matrix. Measurements used to obtain this density matrix were performed using 12-ps excitation pulses while the emission probability was kept at $6\%$. Here, $E$ (early) and $L$ (late) denote the measurement basis.