Circuit quantization with time-dependent flux:the parallel-plate SQUID

TL;DR

This paper investigates the quantization of superconducting circuits with time-dependent magnetic flux, demonstrating through simulations that negative and singular capacitances are necessary for accurate modeling of geometries like the parallel-plate SQUID.

Contribution

It extends circuit quantization methods to continuous geometries with time-dependent magnetic fields, highlighting the need for negative and singular capacitances in such scenarios.

Findings

Numerical simulations confirm the necessity of negative capacitances.

Time-dependent magnetic fields induce the need for singular capacitances.

Continuous geometries require generalized capacitance assignments for proper quantization.

Abstract

Quantum circuit theory has emerged as an essential tool for the study of the dynamics of superconducting circuits. Recently, the problem of accounting for time-dependent driving via external magnetic fields was addressed by Riwar-DiVincenzo in their paper - 'Circuit quantization with time-dependent magnetic fields for realistic geometries' in which they proposed a technique to construct a low-energy Hamiltonian for a given circuit geometry, taking as input the external magnetic field interacting with the geometry. This result generalises previous efforts that dealt only with discrete circuits. Moreover, it shows through the example of a parallel-plate SQUID circuit that assigning individual, discrete capacitances to each individual Josephson junction, as proposed by treatments of discrete circuits, is only possible if we allow for negative, time-dependent and even singular capacitances. In this report, we provide numerical evidence to substantiate this result by performing finite-difference simulations on a parallel-plate SQUID. We furnish continuous geometries with a uniform magnetic field whose distribution we vary such that the capacitances that are to be assigned to each Josephson junction must be negative and even singular. Thus, the necessity for time-dependent capacitances for appropriate quantization emerges naturally when we allow the distribution of the magnetic field to change with time.