# Superconducting qubits beyond the dispersive regime

## TL;DR

This paper introduces a new formalism for analyzing superconducting circuits beyond the dispersive regime, enabling accurate modeling of qubit interactions as circuits scale up and detuning decreases.

## Contribution

The authors develop a formalism that diagonalizes superconducting circuit Hamiltonians beyond the dispersive regime, allowing for accurate analysis of qubit interactions at small or negligible detuning.

## Key findings

- Formalism reproduces perturbative results in the dispersive regime.
- Formalism extrapolates to regimes with small or zero detuning.
- Closed-form formulas for circuit characteristics like dressed frequencies and Kerr couplings.

## Abstract

Superconducting circuits consisting of a few low-anharmonic transmons coupled to readout and bus resonators can perform basic quantum computations. Since the number of qubits in such circuits is limited to not more than a few tens, the qubits can be designed to operate within the dispersive regime, where frequency detuning are much stronger than coupling strengths. However, scaling up the number of qubits will bring the circuit out of this regime and invalidates current theories. We develop a formalism that allows to consistently diagonalize superconducting circuit hamiltonian beyond dispersive regime. This will allow to study qubit-qubit interaction unperturbatively, therefore our formalism remains valid and accurate at small or even negligible frequency detuning; thus our formalism serves as a theoretical ground for designing qubit characteristics for scaling up the number of qubits in superconducting circuits. We study the most important circuits with single- and two-qubit gates, i.e. a single transmon coupled to a resonator and two transmons sharing a bus resonator. Surprisingly our formalism allows to determine the circuit characteristics, such as dressed frequencies and Kerr couplings, in closed-form formulas that not only reproduce perturbative results but also extrapolate beyond the dispersive regime and can ultimately reproduce (and even modify) the Jaynes-Cumming results at resonant frequencies.