A DC SQUID with Josephson junctions characterized by nonsinusoidal current- phase relation is being considered as a basis for a phase qubit. It has been shown that the second and third harmonic components in the current-phase relation are able to provide a double-well potential and the energy level splitting. The threshold condition for the double- well formation has been determined taking into account the impact of both harmonics. The splitting of the ground state energy level has been calculated as a function of the harmonic amplitudes for different ratio s of characteristic Josephson energy EC to the Coulomb energy EQ0. It has been shown that the gap value comes to about 7EQ0 with increase of the ratio s. No external field needed, no bias current required and no circular currents are the major advantages of such a qubit.
Purpose The purpose of this paper is to employ an alternative numerical approach to analyze the characteristics of superconducting charge qubit based on a single Cooper pair box (SCB), also to study the influence of the bias current. Design/methodology/approach The paper starts with the circuit model of a charge qubit system based on Josephson junction using Hamiltonian formalism. Corresponding Schrodinger eigenvalue problem with periodic boundary condition is converted to the Mathieu type eigenvalue problem. By applying finite difference technique, energy spectrum of charge qubit is obtained and the solutions in the lowest band are obtained in the form of Bloch waves whose superposition provides a wave packet to investigate the effect of bias current to the Coulomb blockade. Findings The paper identifies a periodic tridiagonal Hermitian matrix form of the eigenvalue problem that is believed to be a special eigenvalue problem. The paper emphasizes that Schrodinger formalism is very useful to model superconducting qubits systems. The investigations indicate that the bias current strongly influences the Coulomb blockade and expectation value of supercurrent (as well as number of Cooper pairs) are affected by gate voltage and energy scale. Research limitations/implications In the present study, Schrodinger eigenvalue problem is time independent, therefore, current‐voltage characteristics of the charge qubit system could not be considered. The solution technique applied here can also be used to apply other type of Josephson junction based qubits and circuits. Practical implications The paper includes theoretical findings for the development of superconducting qubit that can be valuable for experimentalist. The result obtained in this study is useful for the comparison of experimental study with the expectation value of number of Cooper pairs as function of gate voltage. Working parameters of a SCB can be determined from the findings. Originality/value This paper fulfils the contribution of the numerical study of Schrodinger equation for the investigation of superconducting qubits under the influence of bias current.
Abstract Approaches to developing large-scale superconducting quantum processors must cope with the numerous microscopic degrees of freedom that are ubiquitous in solid-state devices. State-of-the-art superconducting qubits employ aluminium oxide (AlO x ) tunnel Josephson junctions as the sources of nonlinearity necessary to perform quantum operations. Analyses of these junctions typically assume an idealized, purely sinusoidal current–phase relation. However, this relation is expected to hold only in the limit of vanishingly low-transparency channels in the AlO x barrier. Here we show that the standard current–phase relation fails to accurately describe the energy spectra of transmon artificial atoms across various samples and laboratories. Instead, a mesoscopic model of tunnelling through an inhomogeneous AlO x barrier predicts percent-level contributions from higher Josephson harmonics. By including these in the transmon Hamiltonian, we obtain orders of magnitude better agreement between the computed and measured energy spectra. The presence and impact of Josephson harmonics has important implications for developing AlO x -based quantum technologies including quantum computers and parametric amplifiers. As an example, we show that engineered Josephson harmonics can reduce the charge dispersion and associated errors in transmon qubits by an order of magnitude while preserving their anharmonicity.
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.