The quantization of a charge qubit. The role of inductance and gate capacitance

TL;DR

This paper derives a quantum mechanical Hamiltonian for a charge qubit including inductance and gate capacitance, analyzing how these factors influence the qubit's current operator and energy levels, especially in coupled qubit circuits.

Contribution

It provides a self-consistent derivation of the charge qubit Hamiltonian with inductance and gate capacitance, highlighting their effects on current operators and qubit energies.

Findings

The current operator has nonzero nondiagonal matrix elements in charge and eigenstate bases.

Interaction with the LC resonator affects qubit energy levels.

Junction asymmetry and gate capacitance significantly influence the current matrix elements and energies.

Abstract

The Hamiltonian of a charge qubit, which consists of two Josephson junctions is found within well known quantum mechanical procedure. The inductance of the qubit is included from the very beginning. It allows a selfconsistent derivation of the current operator in a two state basis. It is shown that the current operator has nonzero nondiagonal matrix elements both in the charge and the eigenstate basis. It is also shown that the interaction of the qubit with its own LC resonator has a noticeable influence on the qubit energies. The influence of the junctions asymmetry and the gate capacitance on the matrix elements of the current operator and on the qubit energies are calculated. The results obtained in the paper are important for the circuits where two or more charge qubits are coupled with the aid of inductive coil.