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arXiv:cond-mat/0301470 · Physical review. B, Condensed matter

Double-well potentials in current qubits

Mun Dae Kim, Dongkwan Shin, Jongbae Hong

Published Oct 7, 2003• 17 citations
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Abstract

The effective potentials of the rf-SQUID and three-Josephson junction loop with a penetrating external magnetic flux are studied. Using the periodic boundary condition for the phase evolution of the wave function of Cooper pairs, we obtain new periodic potentials with cusp barriers in contrast with the usual smooth double-well potential. The tunneling through the cusp barrier becomes dominant for a parameter regime, where the self-inductance of the superconducting loop and the Josephson coupling energy are large. Calculating the tunneling frequency we see that the cusp potential may offer a more efficient qubit.

DOI
10.1103/physrevb.68.134513

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The effective potentials of the rf-SQUID and three-Josephson junction loop with a penetrating external magnetic flux are studied. Using the periodic boundary condition for the phase evolution of the wave function of Cooper pairs, we obtain new periodic potentials with cusp barriers in contrast with the usual smooth double-well potential. The tunneling through the cusp barrier becomes dominant for a parameter regime, where the self-inductance of the superconducting loop and the Josephson coupling energy are large. Calculating the tunneling frequency we see that the cusp potential may offer a more efficient qubit.
The Hamiltonian of a superconducting thin loop interrupted by a double-barrier Josephson junction is derived. The parameter ε, which takes account of the difference in the Josephson coupling of the two single junctions in the SISIS structure, is seen to allow variation of the energy gap ΔE01 between the ground state and the first excited flux number state of the system. An additional parameter γ, describing the coupling between the outer electrodes of the double-barrier Josephson junction, is introduced. Persistent currents in the system are studied in terms of the usual parameter α, representing the ratio between the Josephson energy and the magnetic energy, and of the externally applied flux.
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.
Fluxons in a superconducting loop can be coherently coupled by quantum phase slips occurring at a weak link such as a Josephson junction. If Cooper pair tunneling at the junction occurs through a resonant level, then $2\ensuremath{\pi}$ quantum phase slips are suppressed, and fluxons are predominantly coupled by $4\ensuremath{\pi}$ quantum phase slips. We analyze this scenario by computing the coupling between fluxons as the level is brought into resonance with the superconducting condensate. The results indicate that the $4\ensuremath{\pi}$-dominated regime can be observed directly in the transition spectrum for circuit parameters typical of a fluxonium qubit. We also show that if the inductive energy of the loop is much smaller than the plasma frequency of the junction, then the low-energy Hamiltonian of the circuit is dual to that of a topological superconducting island. These findings can inform experiments on bifluxon qubits as well as the design of novel types of protected qubits.
At low temperature and low damping, the magnetic flux of a rf Superconducting QUantum Interference Device (SQUID) acts as a macroscopic quantum variable whose dynamic is described by a double well potential. A rf SQUID consists of a superconducting loop interrupted by a thin insulating barrier, known as a Josephson junction. The modified rf SQUID design discussed in this thesis has independent, in situ, controls that allow varying the barrier height between the potential wells and their relative tilt. This high level of control of the potential together with the capability to implement an experimental setup optimized to reduce coupling to external sources of noise, made this system ideal not only for tests of quantum transition at a macroscopic level, but also as a scalable solid state device for quantum computation. A variety of measurements of quantum phenomena are reported in this thesis, including microwave level spectroscopy, coherent oscillations, and macroscopic resonant tunneling (MRT). These measurements are compared with theoretical models to study the possible sources of decoherence in a rf SQUID qubit. The analysis confirmed that the main source of decoherence is not due to coupling to the external environment, but it can be linked to intrinsic low frequency flux noise. Work from many groups over the last several years has shown that the performance of a superconductive qubit is limited by coupling to spurious defects in the materials used for the device fabrication. In this work we analyze the possibility of using the MRT as a tool for future studies of effects of materials and fabrication process in the realization of improved qubits, and we also explore the feasibility for measurements of quantum phenomena that involve fast pulsing of the potential barrier height.
Superconducting quantum circuits derive their nonlinearity from the Josephson energy-phase relation. Besides the fundamental cos⁡φ\cosφcosφ term, this relation can also contain higher Fourier harmonics cos⁡(kφ)\cos(kφ)cos(kφ) corresponding to correlated tunneling of kkk Cooper pairs. The parity of the dominant tunneling process, i.e.~whether an odd or even number of Cooper pairs tunnel, results in qualitatively different properties, and controlling this opens up a wide range of applications in superconducting technology. However, access to even-dominated regimes has remained challenging and has so far relied on complex multi-junction or all-hybrid architectures. Here, we demonstrate a simple "harmonic parity qubit" (HPQ); an element that combines two aluminum-oxide tunnel junctions in parallel to a gate-tunable InAs/Al nanowire junction forming a SQUID, and use spectroscopy versus flux to reconstruct its energy-phase relation at 85 gate voltage points. At half flux quantum, the odd harmonics of the Josephson potential can be suppressed by up to two orders of magnitude relative to the even harmonics, producing a double-well potential dominated by even harmonics with minima near ±π/2\pmπ/2±π/2. The ability to control harmonic parity enables supercurrent carried by pairs of Cooper pairs and provides a new building block for Fourier engineering in superconducting circuits.
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.