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arXiv:cond-mat/0310425 · Physical Review B

Hamiltonian for coupled flux qubits

Alec Maassen van den Brink

Published Feb 8, 2005• 15 citations
arXivHTMLPDF

Abstract

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 $L$. 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 $L$, 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 ($LC$ 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.

DOI
10.1103/physrevb.71.064503

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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 $L$. 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 $L$, 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→0L\ensuremath{\rightarrow}0L\ensuremath→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.
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 0.50.50.5, the tunnel splitting between the ground and first excited states decreases rapidly, and the device becomes progressively less useful as a qubit.
An optimal effective current operator for flux qubit has been investigated by taking account of the inductive effects of the circuit loop. The whole system is treated as two interacting subsystems: one is the inductance-free flux qubit consisting of three Josephson junctions and the other a high frequency LCLCLC oscillator. As the composite system hardly affords one excessively high-energy LCLCLC photon, an effective theory for the inductive flux qubit providing its physical variable operators has been achieved, which can take account of the inductive effects but does not include the additional degree of freedom for the LCLCLC oscillator. Considering the tradeoff between simplicity and accuracy, it has been revealed that the optimal effective current operator resulting in an error only on the order of L3/2{L}^{3/2}L3/2 provides an approximation of high accuracy, which is also verified numerically.
We propose a scheme for tunable coupling of phase-coupled flux qubits. The phase-coupling scheme can provide a strong coupling strength of the order of Josephson coupling energy of Josephson junctions in the connecting loop, while the previously studied inductive coupling scheme cannot provide due to small mutual inductance and induced currents. We show that, in order to control the coupling, we need two dc SQUID's in the connecting loop and the control fluxes threading the dc SQUID's must be in opposite directions. The coupling strength is analytically calculated as a function of the control flux at the coresonance point.
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.
The basic two cell qubit contains four josephson junctions, two residing on the conductor common to the two cell loops, and one on each loop's perimeter. In the limit of zero inductance the qubit potential is described by two phase variables and two orthogonal external magnetic flux parameters (the symmetric and anti symmetric combinations of external flux). The barrier height of the effective double well potential is controlled by the symmetric external flux and the potential tilt by the anti symmetric flux. We will discuss the effect of non-zero inductance and defects on the qubit energy levels, circulating currents, matrix elements, and on the relaxation and phase decoherence times.