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Physical Review B

Quantum theory of three-junction flux qubit with non-negligible loop inductance: Towards scalability

Timothy L. Robertson, B. L. T. Plourde, P. A. Reichardt, T. Hime, C.-E. Wu, John L. Clarke

Published May 30, 2006• 23 citations
Publisher

Abstract

The three-junction flux qubit (quantum bit) consists of three Josephson junctions connected in series on a superconducting loop. We present a numerical treatment of this device for the general case in which the ratio ${\ensuremath{\beta}}_{Q}$ of the geometrical inductance of the loop to the kinetic inductance of the Josephson junctions is not necessarily negligible. Relatively large geometric inductances allow the flux through each qubit to be controlled independently with on-chip bias lines, an essential consideration for scalability. We derive the three-dimensional potential in terms of the macroscopic degrees of freedom, and include the possible effects of asymmetry among the junctions and of stray capacitance associated with them. To find solutions of the Hamiltonian, we use basis functions consisting of the product of two plane wave states and a harmonic oscillator eigenfunction to compute the energy levels and eigenfunctions of the qubit numerically. We present calculated energy levels for the relevant range of ${\ensuremath{\beta}}_{Q}$. As ${\ensuremath{\beta}}_{Q}$ is increased beyond $0.5$, the tunnel splitting between the ground and first excited states decreases rapidly, and the device becomes progressively less useful as a qubit.

DOI
10.1103/physrevb.73.174526

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The three-junction flux qubit (quantum bit) consists of three Josephson junctions connected in series on a superconducting loop. We present a numerical treatment of this device for the general case in which the ratio \ensuremathβQ{\ensuremath{\beta}}_{Q}\ensuremathβQ​ of the geometrical inductance of the loop to the kinetic inductance of the Josephson junctions is not necessarily negligible. Relatively large geometric inductances allow the flux through each qubit to be controlled independently with on-chip bias lines, an essential consideration for scalability. We derive the three-dimensional potential in terms of the macroscopic degrees of freedom, and include the possible effects of asymmetry among the junctions and of stray capacitance associated with them. To find solutions of the Hamiltonian, we use basis functions consisting of the product of two plane wave states and a harmonic oscillator eigenfunction to compute the energy levels and eigenfunctions of the qubit numerically. We present calculated energy levels for the relevant range of \ensuremathβQ{\ensuremath{\beta}}_{Q}\ensuremathβQ​. As \ensuremathβQ{\ensuremath{\beta}}_{Q}\ensuremathβQ​ is increased beyond 0.50.50.5, the tunnel splitting between the ground and first excited states decreases rapidly, and the device becomes progressively less useful as a qubit.
We present the design of a superconducting flux qubit with a large loop inductance. The large loop inductance is desirable for coupling between qubits. The loop is configured into a gradiometer form that could reduce the interference from environmental magnetic noise. A combined Josephson junction, i.e., a DC-SQUID is used to replace the small Josephson junction in the usual 3-JJ (Josephaon junction) flux qubit, leading to a tunable energy gap by using an independent external flux line. We perform numerical calculations to investigate the dependence of the energy gap on qubit parameters such as junction capacitance, critical current, loop inductance, and the ratio of junction energy between small and large junctions in the flux qubit. We suggest a range of values for the parameters.
We present the design of a superconducting flux qubit with a large loop inductance. The large loop inductance is desirable for coupling between qubits. The loop is configured into a gradiometer form that could reduce the interference from environmental magnetic noise. A combined Josephson junction, i.e., a DC-SQUID is used to replace the small Josephson junction in the usual 3-JJ (Josephaon junction) flux qubit, leading to a tunable energy gap by using an independent external flux line. We perform numerical calculations to investigate the dependence of the energy gap on qubit parameters such as junction capacitance, critical current, loop inductance, and the ratio of junction energy between small and large junctions in the flux qubit. We suggest a range of values for the parameters.
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.
An effective Hamiltonian is derived for two coupled three-Josephson-junction (3JJ) qubits. This is not quite trivial, for the customary ``free'' 3JJ Hamiltonian is written in the limit of zero inductance LLL. Neglecting the self-flux is already dubious for one qubit when it comes to readout, and becomes untenable when discussing inductive coupling. First, inductance effects are analyzed for a single qubit. For small LLL, the self-flux is a ``fast variable,'' which can be eliminated adiabatically. However, the commonly used junction phases are not appropriate ``slow variables,'' and instead one introduces degrees of freedom that are decoupled from the loop current to leading order. In the quantum case, the zero-point fluctuations (LCLCLC oscillations) in the loop current diverge as $L\ensuremath{\rightarrow}0$. While their effect thus formally dominates over the classical self-flux, it merely renormalizes the Josephson couplings of the effective (two-phase) theory. In the coupled case, the strong zero-point fluctuations render the full (six-phase) wave function significantly entangled in leading order. However, in going to the four-phase theory, this uncontrollable entanglement is integrated out completely, leaving a computationally usable mutual-inductance term of the expected form as the effective interaction.
For gradiometric three-Josephson-junction flux qubits, we perform a systematic study on the tuning of the minimal transition frequency, the so-called qubit gap. By replacing one of the qubit's Josephson junctions by a dc superconducting quantum interference device (SQUID), the critical current of this SQUID and, in turn, the qubit gap can be tuned in situ by a control flux threading the SQUID loop. We present spectroscopic measurements demonstrating a well-defined controllability of the qubit gap between zero and more than 10 GHz. This is important for tuning the qubit into and out of resonance with other superconducting quantum circuits in scalable architectures, while still operating it at its symmetry point with optimal dephasing properties. The experimental data agree very well with model calculations based on the full qubit Hamiltonian. From a numerical fit, we determine the Josephson coupling and the charging energies of the qubit junctions. The derived values agree well with those measured for other junctions fabricated on the same chip. We also demonstrate the biasing of gradiometric flux qubits near the symmetry point by trapping an odd number of flux quanta in the gradiometer loop. In this way, we study the effect of the significant kinetic inductance, thereby obtaining valuable information for the qubit design.
Abstract. For gradiometric three-Josephson-junction flux qubits, we perform a systematic study on the tuning of the minimal transition frequency, the so-called qubit gap. By replacing one of the qubit’s Josephson junctions by a dc SQUID, the critical current of this SQUID and, in turn, the qubit gap can be tuned in situ by a control flux threading the SQUID loop. We present spectroscopic measurements demonstrating a well-defined controllability of the qubit gap between zero and more than 10 GHz. This is important for tuning the qubit into and out of resonance with other superconducting quantum circuits in scalable architectures, while still operating it at its symmetry point with optimal dephasing properties. The experimental data agree very well with model calculations based on the full qubit Hamiltonian. From a numerical fit, we determine the Josephson coupling and the charging energies of the qubit junctions. The derived values agree well with those measured for other junctions fabricated on the same chip. We also demonstrate the biasing of gradiometric flux qubits near the symmetry point by trapping an odd number of flux quanta in the gradiometer loop. In this way, we study the effect of the significant kinetic inductance, thereby obtaining valuable information for the qubit design.