Entangled photons from quantum-dot-cavity systems under non-Markovian decoherence by pulsed excitation

Abstract

Cascaded emission from the biexciton state of a quantum dot results in polarization entangled photon pairs. However, modelling this system becomes challenging when photon emission is cavity-mediated due to the large Hilbert space and non-Markovian nature of its phonon-induced decoherence. Here, we introduce an algorithm that reduces the computational cost of the numerically exact process tensor method for non-Markovian dynamics simulations when the environmental coupling operator has degenerate eigenvalues, making calculations of the non-Markovian dynamics of large systems feasible. We compute the degree of entanglement of photon pairs generated by pulsed two-photon resonant excitation and find surprisingly good agreement between the numerically exact results and those calculated using the approximate polaron master equation method, permitting an efficient exploration of trends across system parameters.

Figures (5)

• Figure 1: Schematic of the pulsed QD-cavity-phonon system under consideration. The quantum dot is excited from the ground $G$ to biexciton $XX$ state by two-photon excitation with a Gaussian laser pulse at frequency equal to half the biexciton frequency. Cascaded emission from $XX$ to $G$ via the exciton states $X_{H/V}$ results in polarization-entangled photon pairs. A bimodal cavity tuned to half the biexciton frequency enhances the direct two-photon emission process, meanwhile lattice vibrations in the substrate result in a non-Markovian decoherence.
• Figure 2: Illustration of the process tensor method for time-evolving the state of an open system under non-Markovian decoherence. In the standard scenario, shown in the first row, the system $S_1$ is coupled to an environment, and the influence of the environment is captured by a process tensor, which is constructed from an infinite network of elementary tensors (blue rectangles), and subsequently compressed and contracted (blue circles). This process tensor, alongside the free system propagator (pink rectangles), is then used to time-evolve the density matrix $\rho$, with finite-time boundary conditions imposed (blue triangles). In the second row we consider a system-environment coupling operator with degenerate eigenvalues. This corresponds to a system that can be divided into two subsystems; a subsystem $S_1$ that interacts with the environment and a subsystem $S_2$ that does not. The process tensor can then be decomposed into a smaller process tensor acting on the first subsystem (black tensor legs) and a part resulting in trivial evolution of the other subsystem (gray tensor legs).
• Figure 3: The degree of entanglement, measured by the concurrence, of photons emitted by the QD-cavity-phonon system following initialization in the biexciton ($XX$) state, as calculated using three different methods; the numerically exact uniTEMPO algorithm, the approximate polaron master equation (PME) and the case of no phonon environment. Calculations performed in the full Hilbert space are compared to those in the two-excitation subspace (denoted 5LS).
• Figure 4: Dynamics of the QD-cavity-phonon system under pulsed two-photon excitation of the biexciton state from ground, calculated as before using three different methods; the numerically exact uniTEMPO, approximate PME, and the no-phonon case. Top: populations of the biexciton ($XX$), exciton ($X$), and ground ($G$) states, with the Gaussian laser pulse overlaid in gray. Bottom: concurrence is used as a measure of the entanglement of emitted photon pairs.
• Figure 5: The concurrence of entangled photon pairs emitted by the QD-cavity-phonon system, following pulsed two-photon excitation of the biexciton state, is shown as a function of pulse width for three different phonon bath temperatures. The numerically exact uniTEMPO method is used to verify the approximate PME result for the 10 K temperature series. Also shown are the maximal concurrences for each temperature, which are computed for the system initialised in the biexciton state and represented by stars.