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Introduction For artificial synapses whose strengths are assumed to be bounded and can only be updated with finite precision, achieving optimal memory consolidation using primitives from classical physics leads to synaptic models that are too complex to be scaled in-silico . Here we report that a relatively simple differential device that operates using the physics of Fowler-Nordheim (FN) quantum-mechanical tunneling can achieve tunable memory consolidation characteristics with different plasticity-stability trade-offs. Methods A prototype FN-synapse array was fabricated in a standard silicon process and was used to verify the optimal memory consolidation characteristics and used for estimating the parameters of an FN-synapse analytical model. The analytical model was then used for large-scale memory consolidation and continual learning experiments. Results We show that compared to other physical implementations of synapses for memory consolidation, the operation of the FN-synapse is near-optimal in terms of the synaptic lifetime and the consolidation properties. We also demonstrate that a network comprising FN-synapses outperforms a comparable elastic weight consolidation (EWC) network for some benchmark continual learning tasks. Discussions With an energy footprint of femtojoules per synaptic update, we believe that the proposed FN-synapse provides an ultra-energy-efficient approach for implementing both synaptic memory consolidation and continual learning on a physical device.