Formation dynamics of an entangled photon pair: A temperature-dependent analysis

We theoretically study the polarization entanglement of photons generated by the biexciton cascade in a GaAs/InAs semiconductor quantum dot (QD) located in a nanocavity. A detailed analysis of the complex interplay between photon and carrier coherences and phonons which occurs during the cascade allows us to clearly identify the conditions under which entanglement is generated and destroyed. A quantum state tomography is evaluated for varying exciton fine-structure splittings. Also, by constructing an effective multiphonon Hamiltonian which couples the continuum of the QD-embedding wetting layer states to the quantum confined states, we investigate the relaxation of the biexciton and exciton states. This consistently introduces a temperature dependence to the cascade. Considering typical Stranski-Krastanov grown QDs for temperatures around 80 K the degree of entanglement starts to be affected by the dephasing of the exciton states and is ultimately lost above 100 K.

Figure 1

(Color online) (a) QD band structure with spin states $s=|J;m_j⟩$. Since the light hole (LH) and split-off (SO) states lie energetically well below the heavy hole (HH) states, only the latter and the electron states are relevant for the biexcitonic framework. (b) Cascade of a symmetric QD with degenerate exciton states $|X_{\pm }⟩$. (c) Cascade with a FSS $\delta \ne 0$ (without biexcitonic shift). There are two possible paths: either two photons with vertical $V$ or horizontal $H$ polarization are emitted into a cavity mode ${\omega }_{V/H}^{\text{cav}}$ with a spectral FWHM $\kappa$. For vanishing FSS, $\delta$ the which-path information is lost and the photons are polarization entangled. With a FSS $\delta \ne 0$, the photons are not fully polarization entangled.

Figure 2

(Color online) General system scheme. Two electron-hole pairs in a QD are coupled to the continuum of WL states via an effective LO phonon interaction $p_{cw}^{q_1,q_2}$. The phonons are treated as a thermal bath. The QD has two levels with a conduction and a valence band level and is placed inside a cavity. Here, the carriers interact with photons via the electron-light coupling element $M$, where a photon loss via $\kappa$ is assumed.

Figure 3

(Color online) Wetting layer system scheme. Shown in blue is the parabolic 2D dispersion of the WL holes ${\hslash }^2{\mathbf{k}}^2/\left(2m_h^{\ast }\right)$. Marked on the energy scale are (i) the WL band edge ${\epsilon }_{v0}^{\text{WL}}$ which is energetically separated from the QD $v$ shell ${\epsilon }_v^{\text{QD}}$ by more than a single-phonon energy: $\Delta {\epsilon }_v^{\text{QD}}={\epsilon }_{v0}^{\text{WL}}-{\epsilon }_v^{\text{QD}}>\hslash {\omega }_{\text{LO}}$, (ii) the WL states at ${\mathbf{k}}_{\text{res}}$ resonant with a two-phonon transition ${\epsilon }_{v{\mathbf{k}}_{\text{res}}}^{\text{WL}}={\epsilon }_v^{\text{QD}}+2\hslash {\omega }_{\text{LO}}$ to the QD. The transition from the resonant states to the QD pass through intermediate states at ${\epsilon }_{v\mathbf{k}}^{\text{WL}}$ with probability amplitude $g_{{\mathbf{k}}_2{\mathbf{k}}_1}^{\mathbf{q}}$.

Figure 4

(Color online) The phonon-induced relaxation rate ${\Gamma }_w$ as a function of temperature $T$. At about 80 K, ${\Gamma }_w$ has approached $1{\text{ns}}^{-1}$ and starts to contribute strongly to the decay of the QD states.

Figure 5

(Color online) Equations of motion truncated to the pure cascade scheme. The dark blue quantities represent densities which do not contribute to entanglement, whereas the dark orange and red quantities directly generate a crossing of the different paths in the light orange boxes and are crucial to entanglement.

Figure 6

(Color online) Temporal evolution of the nonintegrated off-diagonal two-photon density-matrix elements at $T=0\text{K}$ [real (solid) and imaginary (dashed) parts] for $V_{\text{ex}}=1\mu \text{eV}$ (orange) and $V_{\text{ex}}=10\mu \text{eV}$ (red). With increasing $V_{\text{ex}}$, the oscillations become more rapid.

Figure 7

(Color online) Averaged two-photon matrix elements at $T=0\text{K}$. Their steady-state values give the quantum-state tomography.

Figure 8

(Color online) Quantum-state tomography for (a) $V_{\text{ex}}=1\mu \text{eV}$ and (b) $V_{\text{ex}}=10\mu \text{eV}$.

Figure 9

(Color online) Plot of the concurrence. (a) displays $C$ at 0 K as a function of $V_{\text{ex}}$. (b) illustrates the influence of the temperature due to the phonon-induced WL influence at $V_{\text{ex}}=0\mu \text{eV}$. Both temperature and $V_{\text{ex}}$ dependences are shown in (c).

Figure 10

(Color online) Dynamics of the cascade $\text{biexciton}\to \text{exciton}$ and one $\text{photon}\to \text{ground}$ state and two photons. Results for $V_{\text{ex}}=0\mu \text{eV}$ at $T=0\text{K}$. $⟨{\widehat{X}}_H^{†}{\widehat{X}}_H{\widehat{a}}_H^{†}{\widehat{a}}_H⟩$ and ${\rho }_{HH}$ are enlarged by a factor of ${10}^4$ and ${10}^5$, respectively. The inset shows the dynamics on a logarithmic scale.