Superconductivity is reformulated as a phenomenon in which a stable velocity field is created by a
U(1) phase neglected by Dirac in the Schroedinger representation of quantum mechanics. The neglected phase gives rise to a
U(1) gauge field expressed as the Berry connection from many-body wave functions. The inclusion of this gauge field transforms the standard particle-number non-conserving formalism of superconductivity to a particle-number conserving one with many results of the former unaltered. In other words, the new formalism indicates that the current standard one is an approximation that effectively takes into account this neglected
U(1) gauge field by employing the particle-number non-conserving formalism. Since the standard and new formalisms are physically different, conflicting results are predicted in some cases. We reexamine the Josephson relation and show that a capacitance contribution of the Josephson junction to the
U(1) phase is missing in the standard formalism, and inclusion of it indicates that the standard theory actually does one agree with the experiment while the new one does. It is also shown that the dissipative quantum phase transition in Josephson junctions predicted in the standard theory does not exit in the new one in accordance with the recent experimental result.
Recently, a new theory of superconductivity has been put forward that attributes the origin of superconductivity to the appearance of a non-trivial Berry connection from many-electron wave functions. This theory reproduces the major results of the BCS theory with conserving the particle number, and predicts the single-electron supercurrent tunneling across the Josephson junction with keeping the correct Josephson relation. We re-examine the quantization of superconducting qubit circuits by taking into account the above development, and show that the dynamical variables used in the standard theory, the flux nodes relating to the voltage, should be replaced by those relating to the electromagnetic vector potential. The fact that the Josephson junction tunneling allows the single-electron supercurrent tunneling is the reason for the existence of excited single electrons in superconducting qubits with Josephson junctions. We predict that it will be avoided by weakening the coupling between two superconductors in the Josephson junction.
The new superconductivity theory that attributes the
U(1) superconductivity phase to a Berry phase arising from many-body wave functions is applied to the circuit quantization for superconducting qubits. The phase-charge duality required for the occurrence of superconductivity in the standard theory becomes irrelevant in the new theory, and the absence of a dissipative quantum phase transition in Josephson junctions is explained. It is shown that a charge-decaying term leads to the compact phase description, and the appearance of Shapiro steps is explained without introducing normal current.
Josephson junctions are the central nonlinear element of superconducting qubits, yet their standard description is reached through a chain of approximations. We propose a purely quantum-mechanical model of the junction, formulated directly in Hilbert space: a charged particle moves on a one-dimensional loop, and the junction is a point at which the wave function obeys self-adjoint boundary conditions, labelled by a coupling strength
λ, which controls the transparency of the junction, and a phase jump
δ, which plays the role of the superconducting phase difference. For a loop with a single junction we solve the spectral problem and show that the ground-state momentum expectation, identified with the supercurrent, factorises exactly as
⟨p^0⟩=Ic(δ,λ)sinδ. For opaque junctions
Ic becomes independent of
δ and the first Josephson relation
I=Icsinδ is recovered; for strongly attractive coupling, where the ground state is bound to the junction, the
sinδ law is also recovered, but with an exponentially suppressed amplitude. For more transparent junctions, however, the model \emph{predicts} deviations from the pure
sinδ law:
Ic acquires a dependence on
δ already at order
λ−2, compatible with the higher Josephson harmonics recently found necessary to describe superconducting qubit spectra, and a fully transparent contact yields a current linear in
δ. When the phase varies in time, a gauge transformation shows that the voltage drop across the junction is proportional to
δ˙, so that the second Josephson relation emerges without being imposed. The boundary-condition approach thus reproduces both Josephson relations from first principles and predicts corrections to them, offering a route towards higher-fidelity models of superconducting qubits.
Superconductivity is a phenomenon where electrical current flows without friction. The current standard theory for it is the BCS (Bardeen-Cooper-Schrieffer) theory, which explains it as due to the energy gap formation by the electron-pairing, and the key ingredient for the supercurrent generation is the gauge symmetry breaking brought about by it. It was thought that superconductivity was fully understood by this standard theory; however, the discovery of superconductivity in cuprates in 1986 changed this situation, showing a number of experimental results that contradict the standard theory. It is also notable that the standard theory contradicts in the supecurrent carrier mass in the London moment; the predicted mass is an effective mass, while the experimental value is the free electron mass. The above contradictions suggest the necessity for a fundamental revision for the theory of superconductivity. Here we show that the required revision may be achieved by using the Berry phase formalism. It was developed after the establishment of the BCS theory, and provides a way to detect emergent gauge fields. A self-referencing phase factor on the wave function detected by the Berry phase formalism explains the supercurrent generation in the conventional and cuprate superconductors. It gives rise to a gauge field that enables the gauge symmetry breaking in the standard theory interpretation.
In the design and investigation of superconducting qubits and related devices, a lumped-element circuit model is the standard theoretical approach. However, many important physical questions lie beyond its scope, such as the behavior of circuits with strong Josephson junctions carrying substantial currents and the properties of very small superconducting devices. By performing gauge transformations on self-consistent solutions of the Bogoliubov--de Gennes equations, we develop here a formalism that treats Josephson couplings nonperturbatively. We apply the formalism to (a) show that Fermi sea effects can contribute to the effective capacitance of small charge qubits; (b) demonstrate an asymmetry in clockwise and counterclockwise current states in small rf superconducting quantum interference device qubits; and (c) provide a microscopic wave function of superconducting Schr\"odinger cats suitable for computing the number of entangled electrons.