Controlling both the amplitude and the phase of the superconducting quantum order parameter ψ in nanostructures is important for next-generation information and communication technologies. The lack of electric resistance in superconductors, which may be advantageous for some technologies, hinders convenient voltage-bias tuning and hence limits the tunability of ψ at the microscopic scale. Here, we demonstrate the local tunability of the phase and amplitude of ψ, obtained by patterning with a single lithography step a Nb nano-superconducting quantum interference device (nano-SQUID) that is biased at its nanobridges. We accompany our experimental results by a semi-classical linearized model that is valid for generic nano-SQUIDs with multiple ports and helps simplify the modelling of non-linear couplings among the Josephson junctions. Our design helped us reveal unusual electric characteristics with effective zero inductance, which is promising for nanoscale magnetic sensing and quantum technologies.

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