Hyperfine interaction limits polarization entanglement of photons from semiconductor quantum dots

Excitons in quantum dots are excellent sources of polarization-entangled photon pairs, but a quantitative understanding of their interaction with the nuclear spin bath is still missing. Here we investigate the role of hyperfine energy shifts using experimentally accessible parameters and derive an upper limit to the achievable entanglement fidelity. Our results are consistent with all available literature, indicate that spin noise is often the dominant process limiting the entanglement in InGaAs quantum dots, and suggest routes to alleviate its effect.

Figure 1

Spectrum of an InGaAs QD (QD1) under two-photon excitation (TPE). The sketch depicts the biexciton-exciton (XX-X) cascade. TPE is performed by tuning the energy $E_{\mathrm{L}}$ of a pulsed laser to half of the XX energy. Here, the XX level lies $E_{\mathrm{B}}=2.5\mathrm{meV}$ below the energy of two uncorrelated excitons energy (2X). The X states are split by the constant fine-structure splitting $S$ and also by the inhomogeneous Overhauser shift (OS) with a normally distributed amplitude with a standard deviation $\sigma$.

Figure 2

(a) Setup used to determine the entanglement fidelity of the QD photon source. A 50:50 nonpolarizing beam splitter (BS) is placed directly after the source. Waveplates before and polarizers (P) after the BS set the measurement basis for XX and X. The emitted photons are spectrally filtered (F) before impinging on avalanche photodiodes (APD). (b) Measured fidelity values for different QDs (squares) plotted against the respective FSS values. The red dots show the expected fidelity when only accounting for the FSS and the lifetime, and the blue area depicts the range of expected values when including fluctuating Overhauser shifts according to the values of the electron $T_2^{*}$ reported in the literature.

Figure 3

(a) Measured fidelity values of QD1p as a function of strain-dependent $S$. The red line shows the expected fidelity for $\sigma =0$ (no spin noise), and the blue area depicts the expected fidelity range when including OS. The sketch shows the piezoelectric actuator used for tuning $S$. (b) Real and imaginary parts of the two-qubit density matrix, measured by state tomography on QD1p [$T_1=430\left(4\right)\mathrm{ps}, S=0.4\left(2\right)\mu \mathrm{eV}$, and $k=0.990\left(5\right)$] with 16 measurement bases, calculated using the maximum-likelihood approach with a resulting fidelity of 0.85(1), a purity of 0.74(2), and a concurrence of 0.69(2). (c) Modeled density matrix for QD1p, assuming $\sigma =0.41\mu \mathrm{eV}$ ($T_2^{*}=1.6\mathrm{ns}$), with a resulting fidelity of 0.89, a purity of 0.81, and a concurrence of 0.79.

Figure 4

(a) Measured maximum $f$ from InGaAs QDs with $S\approx 0$ from the indicated references, compared against the values expected from Eq. (7), taking into account the reported $T_1$. The leftmost data point refers to QD1p. (b) Measured concurrence from the references indicated on the $x$ axis. The measurements are compared against the values calculated by numerically propagating Eq. (4) from 0 up to the used coincidence window (indicated in the graph in picoseconds), taking into account the reported values for $T_1$ and $S$. The leftmost data point refers to QD1p at its lowest value of $S$.