We develop a method of non-perturbative optical control over adiabatic remagnetisation of the nuclear spin system and apply it to verify the spin temperature concept in GaAs microcavities. The nuclear spin system is shown to exactly follow the predictions of the spin-temperature theory, despite the quadrupole interaction that was earlier reported to disrupt nuclear spin thermalisation. These findings open a way to deep cooling of nuclear spins in semiconductor structures, with a prospect of realisation of nuclear spin-ordered states for high fidelity spin-photon interfaces.The concept of nuclear spin temperature is one of the cornerstones of the nuclear magnetism in solids 1,2 . It has made possible realisation of the cryogenic cooling into the microKelvin range 3 and observation of nuclear spin ordering in metals and insulators 4,5 . Such degree of control of the nuclear spin system (NSS) in semiconductor heterostructures would allow enhancing the efficiency of spin-based information storage and processing 6-9 . However, proving the validity of the spin temperature concept for semiconductor nano-and microstructures is challenging due to the lack of techniques capable of precise sensing of weak nuclear magnetisation in a small volume. In addition, recent experiments showed that in quantum dots, where strong quadrupole-induced local fields have been reported, nuclear spin temperature failed to establish 10 . In this context, NSS thermalisation sensing in semiconductor heterostructures is one the central issues for both fundamental questions related to the realisation of nuclear spin-ordered states, and for potential applications, such as high fidelity spin-photon interfaces 6-9 .The basic postulates of the spin temperature theory are illustrated in Fig. 1(a). It is assumed that during the characteristic time T 2 determined by spin-spin interactions the NSS reaches the internal equilibrium. This means that properties of the NSS are governed by a single parameter, the spin temperature Θ N . When this temperature is made different from the lattice temperature Θ L (e.g. by the optical pumping), the thermalisation of the NSS with the crystal lattice usually requires a much longer characteristic time T 1 . Fig.1(b) illustrates one of the main predictions of the spin temperature theory: if the NSS is subjected to a slowly varying magnetic field, such that dB/dt < B L /T 2 , then Θ N and the nuclear spin polarisation P N change obeying universal expressions:(1) Here γ N is the gyromagnetic ratio of the nuclear spin I, angular brackets denote the averaging over all nuclear species, k B is the Boltzman constant, and Θ N i is the spin temperature at strong magnetic field B i >> B L , where B L is the local field induced by the fluctuating nuclear spins. These generic relations are based on the principle of entropy conservation in a thermodynamic system during adiabatic process. They constitute the basis for the nuclear spin cooling by adiabatic demagnetisation, a widely used cryogenic technique 4,11-13 . The nuclear spin tem...