Entangled photon sources based on semiconductor quantum dots: The role of pure dephasing

We theoretically investigate the production of polarization-entangled photons through the biexciton cascade decay in a single semiconductor quantum dot. To accomplish a high degree of entanglement, despite the exciton fine-structure splitting, one must either energetically align the two exciton states by means of external fields or erase the which-path information by postselection or time reordering of photons. Here we show that in the latter schemes the photon state becomes deteriorated through dephasing processes in the solid, and the degree of entanglement remains low.

Figure 1

(Color online) (a) Level scheme consisting of biexciton state $u$, exciton states polarized along $x$ and $y$, and ground state $g$. The biexciton energy is reduced by the biexciton binding $\Delta$ and the exciton states are energetically separated by the fine-structure splitting $\delta$. $H$ and $V$ denote the polarizations of the emitted photons. (b) Schematic of creation of entangled photons. The biexciton first decays by emitting a photon with polarization $H$ or $V$, leaving behind a single exciton $x$ or $y$ in the dot, which decays in a second step. The ambiguity of the biexciton decay, through either of the two ideally spin-degenerate exciton states, translates to an entanglement of the emitted photons (Ref. 3). Because of the fine-structure splitting, the degree of entanglement is largely diminished. Several experimental implementations and proposals exist for overcoming this deficiency: (c) the states can be brought back to degeneracy by means of external fields (Ref. 5), (d) spectral filtering can be used to postselect only those photons whose energy contains no which-path information (Ref. 4), with $h\left(\omega \right)$ being the filter function, or (e) photons of different generations of the cascade can be entangled by bringing the biexciton transition in resonance with the exciton one (Refs. 16, 17).

Figure 2

(Color online) Schematic of relation between quantum dot cascade decay and two-photon density matrix. (a) Diagonal element of the two-photon density matrix ${\rho }_{HH,HH}^{\left(2\right)}$ resulting from the decay of the biexciton $u$ into the $x$ polarized exciton and subsequently into the ground state $g$. The dipole operators $d$ mediate the electric-field amplitudes $E$, as described by Eq. (4). We have indicated the states involved in the dipole transition rather than the polarization and color associated with the emitted photon. Wiggled lines indicate emitted photons, with polarization $H$ or $V$ depending on the orientation. Two electric-field amplitudes build up a photon intensity $I_{HH}$ at the detector (half circles). The two-photon density matrix is the correlation between measurements at the two detectors, as described in Eq. (1). The diagonal element ${\rho }_{HH,HH}^{\left(2\right)}$ is the result of averaging over the arrival times of the intensity $I_{HH}$ of the first (red) and second (blue) photons. (b) Off-diagonal element of the two-photon density matrix ${\rho }_{HH,VV}^{\left(2\right)}$ resulting from the decay of the biexciton $u$ into a superposition of the $x$ and $y$ excitons. (c) ${\rho }_{HH,VV}^{\left(2\right)}$ with a filter modifying the electric-field amplitudes arriving at the detectors, according to Eq. (6). (d) Time reordering by delaying the first generation of photons by a constant time, indicated by the thick black line between mirrors.

Figure 3

(Color online) (a) Concurrence as a function of fine-structure splitting and for different dephasing rates, as computed from the expressions given in Table . The solid line corresponds to the unfiltered case. The dashed (dotted) lines report results for $\delta \omega =1\mu \text{eV}$ $\left(10\mu \text{eV}\right)$ which are plotted for positive (negative) fine-structure splittings only. We use a radiative decay rate of ${\gamma }_r=1.6\mu \text{eV}$ which corresponds to a lifetime of about 1 ns. (b) Concurrence as a function of filter width $\delta \omega$ and for $\delta =10\mu \text{eV}$. In presence of dephasing the concurrence remains low despite filtering. The insets report density maps of the concurrence as a function of filter width and dephasing rate.