We report the first studies of luminescence spectra from asymmetric double quantum wells of GaAs with widths around 100 Å differing by only one (2.8 Å) or two monolayers. Studies of the position and intensity of the direct and indirect exciton lines suggest the existence of acoustic phonon-assisted tunnelling between exciton states separated by a few meV. At temperatures 10-20 K the electron tunnelling process is fast enough to maintain thermal equilibrium between these direct and indirect excitons which are connected with holes in the same quantum well.
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