In 1909, Millikan showed that the charge of electrically isolated systems is quantized in units of the elementary electron charge e. Today, the persistence of charge quantization in small, weakly connected conductors allows for circuits where single electrons are manipulated, with applications in e.g. metrology, detectors and thermometry. However, quantum fluctuations progressively reduce the discreteness of charge as the connection strength is increased. Here we report on the full quantum control and characterization of charge quantization. By using semiconductor-based tunable elemental conduction channels to connect a micrometer-scale metallic island, the complete evolution is explored while scanning the entire range of connection strengths, from tunnel barrier to ballistic contact. We observe a robust scaling of charge quantization as the square root of the residual electron reflection probability across a quantum channel when approaching the ballistic critical point, which also applies beyond the regimes yet accessible to theory. At increased temperatures, the thermal fluctuations result in an exponential suppression of charge quantization as well as in a universal square root scaling, for arbitrary connection strengths, in agreement with expectations. Besides direct applications to improve single-electron functionalities and for the metal-semiconductor hybrids emerging in the quest toward topological quantum computing, the knowledge of the quantum laws of electricity will be essential for the quantum engineering of future nanoelectronic devices.
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