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.
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.
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.
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.
Recent experiments on superconducting flux qubits, consisting of a superconducting loop interrupted by Josephson junctions, have demonstrated quantum coherence between two different quantum states. The state of the qubit is measured with a superconducting quantum interference device (SQUID). Such measurements require the SQUID to have high resolution while exerting minimal backaction on the qubit. By designing shunts across the SQUID junctions appropriately, one can improve the measurement resolution without increasing the backaction significantly. Using a path-integral approach to analyze the Caldeira-Leggett model, we calculate the narrowing of the distribution of the switching events from the zero-voltage state of the SQUID for arbitrary shunt admittances, focusing on shunts consisting of a capacitance Cs and resistance Rs in series. To test this model, we fabricated a dc SQUID in which each junction is shunted with a thin-film interdigitated capacitor in series with a resistor, and measured the switching distribution as a function of temperature and applied magnetic flux. After accounting for the damping due to the SQUID leads, we found good agreement between the measured escape rates and the predictions of our model. We analyze the backaction of a shunted symmetric SQUID on a flux qubit. For the given parameters of our SQUID and realistic parameters for a flux qubit, at the degeneracy point we find a relaxation time of $113\phantom{\rule{0.3em}{0ex}}\ensuremath{\mu}\mathrm{s}$, which limits the decoherence time to $226\phantom{\rule{0.3em}{0ex}}\ensuremath{\mu}\mathrm{s}$. Based on our analysis of the escape process, we determine that a SQUID with purely capacitive shunts should have narrow switching distributions and no dissipation.
Superconducting quantum interference devices (SQUIDs), single flux-quantum (SFQ) logic circuits, and quantum Josephson junction circuits have been developed into a family of superconductor integrated circuit, and are widely applied for subtle magnetic-field measurements, energy-efficient computing, and quantum computing, respectively. They are Josephson junction networks composed of Josephson junctions and normal resistor-inductor-capacitor (RLC) components, working with the fluxoid-quantization principle and Josephson effects to achieve unique flux-modulated dynamics and characteristics; they react to the vector potential of magnetic fields rather than the electric potential. However, the conventional circuit diagrams and nodal analysis methods focus on the electric charges flowing though branches and nodes, ignoring dynamics of the magnetic fluxes flowing from loop to loop. This article introduces a general flux-based circuit theory to unify the analyses of Josephson junction circuits and normal RLC circuits. This theory presents a magnetic-flux-generator (MFG) concept to unify Josephson junctions and normal circuit elements, and abstract both Josephson junction circuits and normal RLC circuits as MFG network; it derives a general network equation to describe dynamics of Josephson junction circuits, and invents a kind of magnetic-flux flow (MFF) diagram to depict the working principles of magnetic-flux flows inside Josephson junction circuits. The flux-based theory is complementary to the conventional circuit theories in the design and analysis of superconductor integrated circuits.
Abstract Quantum circuit theory is a powerful tool to describe superconducting circuits. In its language, quantum phase slips (QPSs) are considered to be the exact dual to the Josephson effect. This duality renders the integration of QPS junctions into a unified theoretical framework challenging. As we argue, different existing formalisms may be inconsistent, and the correct inclusion of time-dependent flux driving requires introducing a large number of auxiliary, nonphysical degrees of freedom. We resolve these issues by describing QPS junctions as inductive rather than capacitive elements, and reducing the Hilbert space to account for a compact superconducting phase. Our treatment provides an approach to circuit quantization exclusively in terms of node-flux-node variables, and eliminates spurious degrees of freedom. Finally, the inductive treatment reveals the possibility of a voltage-dependent renormalization of the QPS amplitude, by accounting for spatial variations of the electric field built up across the junction.
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