We handle all the self-adjoint extensions of the minimal Schrödinger operator for the nonrelativistic electron living in the one-dimensional configuration space with a junction. We are interested in every boundary condition corresponding to the individual self-adjoint extension. Thus, we clarify all the types of those boundary conditions of the wave functions of the nonrelativistic electron. We find concrete tunnel-junction formulae for the nonrelativistic electron passing through the junction, which reveals the phase factor caused by the tunneling through the junction. Using this tunnel-junction formula, we propose a mathematical possibility of a tunnel-junction device for qubit.
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.
We handle all the self-adjoint extensions of the minimal Schroedinger operator for the non-relativistic electron living in the one-dimensional configuration space with a junction. We are interested in every boundary condition corresponding to the individual self-adjoint extension. Thus, we clarify all the types of those boundary conditions of the wave functions of the non-relativistic electron. We find a tunnel-junction formula for the non-relativistic electron passing through the junction. Using this tunnel-junction formula, we propose a mathematical possibility of a tunnel-junction device for qubit.
A semiclassical solution of general two-state non-adiabatic transition and tunneling is found analytically within the Wentzel–Kramers–Brillouin (WKB) semiclassical framework associated with the Stokes phenomenon in mathematics. The non-adiabatic scattering matrix is determined by a complex quantity called the Stokes constant, which can be directly connected to the complex transition points of the WKB solution. An accurate and compact analytical solution is found for this Stokes constant which is a function of three parameters, one of which corresponds to the diabatic-to-adiabatic transformation angle that is interpreted as a type of non-adiabatic transition. Numerical examples demonstrate that the present unified analytical semiclassical theory works very well for both non-adiabatic transition and non-adiabatic tunneling. The present analytical semiclassical method can be a very powerful tool for application to multidimensional non-adiabatic dynamic processes.
In recent experiments, carried out at the University of Heidelberg, tunneling rates of ultracold distinguishable fermions out of an optomagnetical trap have been measured. The fermions interact by a tunable short-ranged interaction,\nand the trap is asymmetric making the trapped quantum system\neffectively one-dimensional. In this thesis, a method for calculating the energy levels and tunneling rates of one and two interacting particles out of a very general onedimensional potential well is devised. The method is based on expanding the Schrodinger equation of the system in a complex-momentum basis. This is done utilizing the so-called Berggren completeness relation. Ultimately,\nthe basis expansion leads to a complex symmetric non-Hermitian eigenvalue problem for a large, dense matrix.\nThe general method is applied to a system of trapped, ultracold fermionic atoms in a setup that closely resembles the Heidelberg experiments. The short-ranged interaction is modeled as a point-interaction, and the trap potential\nis regularized at large distances from the interesting region. The obtained energies and decay rates are contrasted to results obtained using the Wentzel-Kramers-Brillouin (WKB) approximation. Notable differences\ncan be observed, and these may be due to insufficiency of the WKB approximation to accurately describe the system in question.