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.
There exists a large number of experimental and theoretical results supporting the picture of macroscopic qubits implemented by nanoscopic Josephson junctions of three different types -- charge qubit, flux qubit and phase qubit. The standard unified description of such systems is based on the formal quantization of the phenomenological Kirchhoff equations for the corresponding circuits. In this paper a simplified version of the BCS theory for superconductors is used to derive microscopic models for all types of small Josephson junctions. For these models the state-dependent individual tunneling of Cooper pairs couples ground pair states with excited pair states what leads to a more complicated structure of the lowest lying energy levels. In particular, the highly degenerate levels emerge, which act as probability sinks for the qubit. These models allow also for the coupling to phonons as an efficient mechanism of relaxation for all types of junctions. The alternative formulas concerning basic spectral parameters of superconducting qubits are presented and compared with the experimental data. Finally, the question whether small Josephson junctions can be treated as macroscopic quantum systems is briefly discussed.
We describe in this paper how the nonlinear Josephson inductance is the crucial circuit element for all Josephson qubits. We discuss the three types of qubit circuits, and show how these circuits use this nonlinearity in unique manners. We give a brief derivation of the BCS theory, highlighting the appearance of the macroscopic phase parameter. The Josephson equations are derived using standard first and second order perturbation theory that describe quasiparticle and Cooper-pair tunneling. An exact calculation of the Josephson effect then follows using the quasiparticle bound-state theory, and then expand upon this theory to describe quasiparticle excitations as transitions from the ground to excited bound states from nonadiabatic changes in the bias. Although quasiparticle current is typically calculated only for a constant DC voltage, the advantage to this approach is seen where we qualitatively describe quasiparticle tunneling with AC voltage excitations, as appropriate for the qubit state. This section describes how the Josephson qubit is typically insensitive to quasiparticle damping, even to the extent that a phase qubit can be constructed from microbridge junctions.
Superconducting quantum interference devices (SQUIDs), single flux-quantum (SFQ) logic circuits, and quantum Josephson junction circuits have been developed into a family of superconductor integrated circuit, and are widely applied for subtle magnetic-field measurements, energy-efficient computing, and quantum computing, respectively. They are Josephson junction networks composed of Josephson junctions and normal resistor-inductor-capacitor (RLC) components, working with the fluxoid-quantization principle and Josephson effects to achieve unique flux-modulated dynamics and characteristics; they react to the vector potential of magnetic fields rather than the electric potential. However, the conventional circuit diagrams and nodal analysis methods focus on the electric charges flowing though branches and nodes, ignoring dynamics of the magnetic fluxes flowing from loop to loop. This article introduces a general flux-based circuit theory to unify the analyses of Josephson junction circuits and normal RLC circuits. This theory presents a magnetic-flux-generator (MFG) concept to unify Josephson junctions and normal circuit elements, and abstract both Josephson junction circuits and normal RLC circuits as MFG network; it derives a general network equation to describe dynamics of Josephson junction circuits, and invents a kind of magnetic-flux flow (MFF) diagram to depict the working principles of magnetic-flux flows inside Josephson junction circuits. The flux-based theory is complementary to the conventional circuit theories in the design and analysis of superconductor integrated circuits.
Abstract Superconductivity is reformulated as a phenomenon in which a stable velocity field is created by a U (1) phase neglected by Dirac in the Schrödinger 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 not 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 exist in the new one in accordance with a recent experiment (Murani et al 2020 Phys. Rev. X 10 021003).