Hamiltonian for coupled flux qubits

TL;DR

This paper derives an effective Hamiltonian for two inductively coupled three-Josephson-junction qubits, accounting for inductance effects and quantum fluctuations, leading to a practical model for qubit interactions.

Contribution

It introduces a novel approach to incorporate inductance effects and quantum fluctuations into the Hamiltonian of coupled flux qubits, improving the modeling of their interactions.

Findings

Zero-point fluctuations renormalize Josephson couplings.

Effective four-phase theory captures mutual inductance interactions.

Entanglement effects are manageable in the reduced model.

Abstract

An effective Hamiltonian is derived for two coupled three-Josephson-junction (3JJ) qubits. This is not quite trivial, for the customary "free" 3JJ Hamiltonian is written in the limit of zero inductance L. Neglecting the self-flux is already dubious for one qubit when it comes to readout, and becomes untenable when discussing inductive coupling. First, inductance effects are analyzed for a single qubit. For small L, the self-flux is a "fast variable" which can be eliminated adiabatically. However, the commonly used junction phases are_not_ appropriate "slow variables", and instead one introduces degrees of freedom which are decoupled from the loop current to leading order. In the quantum case, the zero-point fluctuations (LC oscillations) in the loop current diverge as L->0. Fortunately, they merely renormalize the Josephson couplings of the effective (two-phase) theory. In the coupled case, the strong zero-point fluctuations render the full (six-phase) wave function significantly entangled in leading order. However, in going to the four-phase theory, this uncontrollable entanglement is integrated out completely, leaving a computationally usable mutual-inductance term of the expected form as the effective interaction.