Superposition, entanglement and non-locality constitute fundamental features of quantum physics. The fact that quantum physics does not follow the principle of local causality1–3 can be experimentally demonstrated in Bell tests4 performed on pairs of spatially separated, entangled quantum systems. Although Bell tests, which are widely regarded as a litmus test of quantum physics, have been explored using a broad range of quantum systems over the past 50 years, only relatively recently have experiments free of so-called loopholes5 succeeded. Such experiments have been performed with spins in nitrogen–vacancy centres6, optical photons7–9 and neutral atoms10. Here we demonstrate a loophole-free violation of Bell’s inequality with superconducting circuits, which are a prime contender for realizing quantum computing technology11. To evaluate a Clauser–Horne–Shimony–Holt-type Bell inequality4, we deterministically entangle a pair of qubits12 and perform fast and high-fidelity measurements13 along randomly chosen bases on the qubits connected through a cryogenic link14 spanning a distance of 30 metres. Evaluating more than 1 million experimental trials, we find an average S value of 2.0747 ± 0.0033, violating Bell’s inequality with a P value smaller than 10−108. Our work demonstrates that non-locality is a viable new resource in quantum information technology realized with superconducting circuits with potential applications in quantum communication, quantum computing and fundamental physics15.

We present interferometeric measurements of the f = 1 to f = 2 inter-hyperfine scattering lengths in a single-domain spinor Bose-Einstein condensate of 87 Rb. The inter-hyperfine interaction leads to a strong and state-dependent modification of the spin-mixing dynamics with respect to a non-interacting description. We employ hyperfine-specific Faraday-rotation probing to reveal the evolution of the transverse magnetization in each hyperfine manifold for different state preparations, and a comagnetometer strategy to cancel laboratory magnetic noise. The method allows precise determination of inter-hyperfine scattering length differences, calibrated to intra-hyperfine scattering length differences. We report (a (12) 3 −a (12) 2 )/(a (1) 2 −a (1) 0 ) = −1.27(15) and (a (12) 1 −a (12) 2 )/(a (1) 2 −a (1) 0 ) = −1.31(13), limited by atom number uncertainty. With achievable control of atom number, we estimate precisions of ≈0.3 % should be possible with this technique.

We describe a comagnetometer employing the f = 1 and f = 2 ground state hyperfine manifolds of a 87 Rb spinor Bose-Einstein condensate as co-located magnetometers. The hyperfine manifolds feature nearly opposite gyromagnetic ratios and thus the sum of their precession angles is only weakly coupled to external magnetic fields, while being highly sensitive to any effect that rotates both manifolds in the same way. The f = 1 and f = 2 transverse magnetizations and azimuth angles are independently measured by non-destructive Faraday rotation probing, and we demonstrate a 44.0(8) dB common-mode rejection in good agreement with theory. We show how spin-dependent interactions can be used to inhibit 2 → 1 hyperfine relaxing collisions, extending to ∼ 1 s the transverse spin lifetime of the f = 1, 2 mixtures. The technique could be used in high sensitivity searches for new physics on sub-millimeter length scales, precision studies of ultra-cold collision physics, and angle-resolved studies of quantum spin dynamics.