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arXiv:1902.04734 · Physical review. B./Physical review. B

Circuit quantization in the presence of time-dependent external flux

Xinyuan You, James A. Sauls, Jens Koch

Published May 21, 2019• 87 citations
arXivHTMLPDF

Abstract

Circuit quantization links a physical circuit to its corresponding quantum Hamiltonian. The standard quantization procedure generally assumes any external magnetic flux to be static. Time dependence naturally arises, however, when flux is modulated or when flux noise is considered. In this case, application of the existing quantization procedure can lead to inconsistencies. To resolve these, we generalize circuit quantization to incorporate time-dependent external flux.

DOI
10.1103/physrevb.99.174512

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Circuit quantization links a physical circuit to its corresponding quantum Hamiltonian. The standard quantization procedure generally assumes any external magnetic flux to be static. Time dependence naturally arises, however, when flux is modulated or when flux noise is considered. In this case, application of the existing quantization procedure can lead to inconsistencies. To resolve these, we generalize circuit quantization to incorporate time-dependent external flux.
Circuit quantization is an extraordinarily successful theory that describes the behavior of quantum circuits with high precision. The most widely used approach of circuit quantization relies on introducing a classical Lagrangian whose degrees of freedom are either magnetic fluxes or electric charges in the circuit. By combining nonlinear circuit elements (such as Josephson junctions or quantum phase slips), it is possible to build circuits where a standard Lagrangian description (and thus the standard quantization method) does not exist. Inspired by the mathematics of symplectic geometry and graph theory, we address this challenge, and present a Hamiltonian formulation of nondissipative electrodynamic circuits. The resulting procedure for circuit quantization is independent of whether circuit elements are linear or nonlinear, or if the circuit is driven by external biases. We explain how to rederive known results from our formalism, and provide an efficient algorithm for quantizing circuits, including those that cannot be quantized using existing methods. Published by the American Physical Society 2024
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.
Time-dependent electromagnetic drives are fundamental for controlling complex quantum systems, including superconducting Josephson circuits. In these devices, accurate time-dependent Hamiltonian models are imperative for predicting their dynamics and designing high-fidelity quantum operations. Existing numerical methods, such as black-box quantization (BBQ) and energy-participation ratio (EPR), excel at modeling the static Hamiltonians of Josephson circuits. However, these techniques do not fully capture the behavior of driven circuits stimulated by external microwave drives, nor do they include a generalized approach to account for the inevitable noise and dissipation that enter through microwave ports. Here, we introduce numerical techniques that leverage classical microwave simulations, efficiently executable in finite-element solvers, to obtain the time-dependent Hamiltonian of microwave-driven superconducting circuits with arbitrary geometries under charge, flux, or mixed electromagnetic modulation. Importantly, our techniques do not rely on a lumped-element description of the superconducting circuit, in contrast to previous approaches to tackling this problem. We demonstrate the versatility of our approach by characterizing the driven properties of realistic circuit devices in complex electromagnetic environments, including coherent dynamics due to charge and flux modulation, as well as drive-induced relaxation and dephasing. Our techniques offer a powerful toolbox for optimizing circuit designs and advancing practical applications in superconducting quantum computing.
Time-dependent external flux is ubiquitous across circuit quantum electrodynamics and calls for accurate modeling. Contrary to long-held belief, it is recently predicted---but not yet verified---that the allocation of time-dependent flux cannot be arbitrary in the Hamiltonian. This study provides an experimental verification of this prediction by applying fast flux pulses to a fluxonium qubit and taking measurements of the quantum state following the pulses. Given the widespread use of time-dependent flux in experiments with superconducting qubits, verification of this theory is critical to our understanding of circuit quantum electrodynamics.
Abstract Quantum circuit theory has become a powerful and indispensable tool to predict the dynamics of superconducting circuits. Surprisingly however, the question of how to properly account for a time-dependent driving via external magnetic fields has hardly been addressed so far. Here, we derive a general recipe to construct a low-energy Hamiltonian, taking as input only the circuit geometry and the solution of the external magnetic fields. We find that the interplay of geometry and field distribution leads to a much richer circuit dynamics than commonly anticipated, already in devices as simple as the superconducting quantum interference device (SQUID). These dynamics can be captured by assigning negative, time-dependent or even momentarily singular capacitances to the Josephson junctions. Negative capacitances give rise to a strong enhancement of the qubit relaxation rates, while time-dependent capacitances lead to a finite Berry phase.