In this paper, we study THz simultaneous wireless information and power transfer (SWIPT) systems. Since coherent information detection is challenging at THz frequencies and Schottky diodes may not be efficient for THz energy harvesting (EH) and information detection, we employ unipolar amplitude shift keying (ASK) modulation at the transmitter (TX) and a resonant tunnelling diode (RTD)-based EH circuit at the receiver (RX) to extract both information and power from the RX signal. We model the dependence of the instantaneous output power at the RX on the instantaneous received power by a non-linear piecewise function, whose parameters are adjusted to fit circuit simulation results. To determine the rate-power tradeoff in THz SWIPT systems, we derive the distribution of the TX signal that maximizes the mutual information between the TX and RX signals subject to constraints on the required average harvested power at the RX and the peak signal amplitude at the TX. Since the computational complexity of maximizing the mutual information may be too high for real-time THz SWIPT systems, for high and low required average harvested powers, we also obtain the suboptimal input signal distribution that maximizes the achievable information rate numerically and in closed form, respectively. Furthermore, based on the obtained results, we propose a suboptimal closed-form TX distribution which also achieves a desired harvested power at the RX. Our simulation results show that a lower reverse current flow and a higher breakdown voltage of the employed RTD are preferable when the input signal power at the RX is low and high, respectively. Finally, we demonstrate that for low and high received signal powers, the rate-power tradeoff of THz SWIPT systems is determined by the peak amplitude of the TX signal and the maximum instantaneous harvested power, respectively.