The photoluminescence (PL) spectra of a two-dimensional electron system induced in a Be-δ-doped GaAs/AlGaAs quantum well (QW) with a back gate are measured. The electron density is controlled from 1×10 9 cm −2 to 2.5×10 11 cm −2 by changing the back gate voltage. There is a linear increase in the acceptor PL spectrum around 1.49 eV with an increase in the back gate voltage and the PL disappears from the exciton bound to neutral donors (D 0 X) around 1.51 eV at 1.2×10 10 cm −2 .High controllability of the electron density and absence of scattering by ionized donors make back-gated quantum well (QW) structures advantageous for exploring novel physics in a dilute two-dimensional electron system (2DES). The 2DES induced by the back gate in an undoped GaAs/AlGaAs heterostructure, which shows high mobility at low electron density [1], is promising for investigating not only the physical properties in the lowdensity regime such as a metal insulator transition and Wigner crystallization, but also new device applications such as artificial magnets [2,3]. Transport measurements of such system have already been reported in the low electron density regime [1,4,5]. However, there have been difficulties in measuring the transport properties when the electron density is extremely dilute, since the 2DES tends to localize and become insulating. Optical method is suitable for measuring the electrical properties of the dilute 2DES, because it can reach not only the conductive regime but also the insulating regime. This paper reports the photoluminescence (PL) measurement of the 2DES induced by the back gate. We focus on the PL from the recombination of the electrons with a hole bound to the negatively charged acceptor. The advantage of this method is that the optical selection rule in the recombination is relaxed due to the strong localization of a hole at the acceptor site and the PL spectrum provides a direct method for investigating the energy spectra of electrons in 2DES [6,7,8].The layer structure of the sample is shown in the inset of Fig.1. The layers consist of a Si-doped GaAs substrate, overgrown by molecular beam epitaxy with a 1000 nm heavily-Si-doped GaAs layer, which acts as a back gate, followed by a barrier layer comrising 150 periods of GaAs(2-nm thick)/AlAs(2-nm thick) superlattice and 20 nm of Al 0.3 Ga 0.7 As. Subsequently, 80 nm of GaAs is deposited as a channel, followed by 70 nm of Al 0.3 Ga 0.7 As, and this is capped by 30 nm of undoped GaAs. from the bottom of the channel as acceptors. The sample is processed into a 1 mm×1 mm mesa and an AuGeNi ohmic contact is attached at the mesa edge. A back gate voltage (V b ) is applied between the ohmic contact and the back gate. The PL is measured by exciting the samples at 532 nm with a continuous wave at an induced power density of 25 µWcm −2 at 1.7 K. The PL from the sample is dispersed through a 55-cm monochrometer and detected by a charge-coupled device cooled by liquid nitrogen.Before optical measurements are undertaken, the electron density is estimated from t...