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arXiv:0810.4722 · Physical Review B

Classical analysis of capacitively coupled superconducting qubits

James A. Blackburn, Jeffrey E. Marchese, Matteo Cirillo, Niels Grønbech‐Jensen

Published Feb 17, 2009• 12 citations
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Abstract

An electrical circuit consisting of two capacitively coupled inductive loops, each interrupted by a Josephson junction, is analyzed through the classical resistively and capacitively shunted junction (RCSJ) model. The same circuit has recently been studied experimentally and the results were used to demonstrate quantum-mechanical entanglement in the system by observing the correlated states of the two inductive loops after initial microwave perturbations. Our classical analysis shows that the observed phenomenon exists entirely within the classical RCSJ model, and we provide a detailed intuitive description of the transient dynamics responsible for the observations.

DOI
10.1103/physrevb.79.054516

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An electrical circuit consisting of two capacitively coupled inductive loops, each interrupted by a Josephson junction, is analyzed through the classical resistively and capacitively shunted junction (RCSJ) model. The same circuit has recently been studied experimentally and the results were used to demonstrate quantum-mechanical entanglement in the system by observing the correlated states of the two inductive loops after initial microwave perturbations. Our classical analysis shows that the observed phenomenon exists entirely within the classical RCSJ model, and we provide a detailed intuitive description of the transient dynamics responsible for the observations.
We provide an alternative interpretation of experimental results that were represented as demonstrating entanglement between two macroscopic quantum Josephson oscillators. We model the experimental system using the well-established classical equivalent circuit of a resistively and capacitively shunted Josephson junction. Simulation results are used to generate the corresponding density matrix which shows features quite similar to the previously published matrix generated from experimental data. Since our data are generated by a classical model, we therefore submit that state tomography cannot be used to determine absolutely whether or not quantum entanglement has taken place. Analytical arguments are given for why the classical analysis provides an adequate explanation of the experimental results.
A complete analysis of the decoherence properties of a Josephson junction qubit is presented. The qubit is of the flux type and consists of two large loops forming a gradiometer, one small loop, and three Josephson junctions. We develop a complete, quantitative description of the inductances and capacitances of the circuit. Including two stray capacitances makes the quantum mechanical modeling of the system five dimensional. To make the required calculation tractable, we devise a general Born-Oppenheimer approximation which reduces the effective dimensionality in the calculation to one. Contributions to relaxation (T1)({T}_{1})(T1​) and dephasing $({T}_{\ensuremath{\phi}})$ arising from two different control circuits, one coupled to the small loop and one coupled to a large loop, is computed. We explore T1{T}_{1}T1​ and ${T}_{\ensuremath{\phi}}$ along an optimal line in the space of applied fluxes; along this ``S line'' we see significant and rapidly varying contributions to the decoherence parameters, primarily from the circuit coupling to the large loop.
We present an analytical and computational study of resonances and transient responses in a classical Josephson junction system. A theoretical basis for resonances in a superconducting loop with three junctions is presented, outlining both the direct relationship between the dynamics of single- and multijunction systems and the direct relationships between observations of the classical counterparts to Rabi oscillations, Ramsey fringes, and spin-echo oscillations in this class of systems. We show simulation data along with analytical analyses of the classical model, and the results are related to previously reported experiments conducted on three junction loops. We further investigate the effect of off-resonant microwave perturbations to, e.g., the Rabi-type response of the Josephson system, and we relate this response back to the nonlinear and multivalued resonance behavior previously reported for a single Josephson junction. The close relationships between single and multijunction behaviors demonstrate the underlying dynamical mechanism for a whole class of classical counterparts to expected quantum-mechanical observations in a variety of systems, namely, the resonant and transient behavior of a particle in an anharmonic potential well with subsequent escape.
We study the dynamics of a quantum superconducting circuit which consists of a Josephson charge qubit, coupled capacitively to a current biased Josephson junction. Under certain conditions, the eigenstates of the qubit and the junction become entangled. We obtain the time evolution of these states in the limit of weak coupling. Rabi oscillations occur, as a result of the spontaneous emission and re-absorption of a single oscillation quantum in the junction. We discuss a possible way to experimentally determine the quantum state of the junction and hence observe the Rabi oscillations.
We have studied both theoretically and experimentally how an LC series circuit connected in parallel to a Josephson junction influences the Josephson dynamics. The presence of the shell circuit introduces two energy scales, which in specific cases, can strongly differ from the plasma frequency of the isolated junction. Josephson junctions were manufactured using the Nb/Al-AlOx/Nb fabrication technology with various on-chip LC shunt circuits. Spectroscopic measurements in the quantum limit show excellent agreement with theory taking into account the shunt inductance and capacitance in the resistively and capacitively shunted junction model. The results clearly show that the dynamics of the system are two-dimensional, resulting in two resonant modes of the system. These findings have important implications for the design and operation of Josephson junction based quantum bits.
Abstract Josephson junctions based on superconducting materials are fundamental components for quantum detection, quantum communication and quantum computers. An accurate behavioral model of Josephson junctions is the prerequisite for predicting various superconducting circuits. In this study, We present a resistively and capacitively shunted junction (RCSJ) model-based behavioral-level model for the current-voltage characteristics of Josephson junctions. This model accurately predicts the current-voltage characteristics and their temperature dependencies of Josephson junctions made of different materials under three typical working modes: underdamped voltage-driven, overdamped current-driven, and underdamped current-driven. Additionally, it forecasts the critical current and superconducting energy gap characteristics with respect to temperature, as well as the constraint relationship between the shunt resistance, superconducting energy gap, and critical current. Comparing the measured data with the simulation predictions, the model has an average accuracy of 85.9%, which illustrates the reliability of the model.