Exciton exchange splitting in wide GaAs quantum wells

We determine the exciton exchange splitting in a wide GaAs quantum well. Our method is based on applying a magnetic field parallel to the layers and measuring the oscillator strength ratio of the Zeeman split lines in two linear polarizations. We develop a theoretical model to describe the effect of the magnetic field on the exciton spectrum, and use it to determine the exchange splitting in a 22-nm quantum well to be $22\ifmmode\pm\else\textpm\fi{}3 \ensuremath{\mu}\mathrm{eV}.$ These measurements also allow us to make an accurate determination of the value of $q=0.03,$ the Luttinger parameter which appears in the cubic term of the valence band Zeeman Hamiltonian.