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arXiv:1101.1341 · Physical Review Letters

Quasiparticle Relaxation of Superconducting Qubits in the Presence of Flux

Gianluigi Catelani, J. Koch, Luigi Frunzio, Robert J. Schoelkopf, Michel Devoret, L. I. Glazman

Published Feb 16, 2011• 167 citations
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Abstract

Quasiparticle tunneling across a Josephson junction sets a limit for the lifetime of a superconducting qubit state. We develop a general theory of the corresponding decay rate in a qubit controlled by a magnetic flux. The flux affects quasiparticles tunneling amplitudes, thus making the decay rate flux-dependent. The theory is applicable for an arbitrary quasiparticle distribution. It provides estimates for the rates in practically important quantum circuits and also offers a new way of measuring the phase-dependent admittance of a Josephson junction.

DOI
10.1103/physrevlett.106.077002

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Quasiparticle tunneling across a Josephson junction sets a limit for the lifetime of a superconducting qubit state. We develop a general theory of the corresponding decay rate in a qubit controlled by a magnetic flux. The flux affects quasiparticles tunneling amplitudes, thus making the decay rate flux-dependent. The theory is applicable for an arbitrary quasiparticle distribution. It provides estimates for the rates in practically important quantum circuits and also offers a new way of measuring the phase-dependent admittance of a Josephson junction.
As low-loss nonlinear elements, Josephson junctions are the building blocks of superconducting qubits. The interaction of the qubit degree of freedom with the quasiparticles tunneling through the junction represents an intrinsic relaxation mechanism. We develop a general theory for the qubit decay rate induced by quasiparticles, and we study its dependence on the magnetic flux used to tune the qubit properties in devices such as the phase and flux qubits, the split transmon, and the fluxonium. Our estimates for the decay rate apply to both thermal equilibrium and nonequilibrium quasiparticles. We propose measuring the rate in a split transmon to obtain information on the possible nonequilibrium quasiparticle distribution. We also derive expressions for the shift in qubit frequency in the presence of quasiparticles.
Quasiparticles are an important decoherence mechanism in superconducting qubits, and can be described with a complex admittance that is a generalization of the Mattis-Bardeen theory. By injecting nonequilibrium quasiparticles with a tunnel junction, we verify qualitatively the expected change of the decay rate and transition frequency in a phase qubit. With their relative change in agreement to within 4 % of prediction, the theory can be reliably used to infer quasiparticle density. We describe how settling of the decay rate may allow determination of whether qubit energy relaxation is limited by nonequilibrium quasiparticles.
In superconducting qubits, the interaction of the qubit degree of freedom with quasiparticles defines a fundamental limitation for the qubit coherence. We develop a theory of the pure dephasing rate ${\ensuremath{\Gamma}}_{\ensuremath{\phi}}$ caused by quasiparticles tunneling through a Josephson junction and of the inhomogeneous broadening due to changes in the occupations of Andreev states in the junction. To estimate ${\ensuremath{\Gamma}}_{\ensuremath{\phi}}$, we derive a master equation for the qubit dynamics. The tunneling rate of free quasiparticles is enhanced by their large density of states at energies close to the superconducting gap. Nevertheless, we find that ${\ensuremath{\Gamma}}_{\ensuremath{\phi}}$ is small compared to the rates determined by extrinsic factors in most of the current qubit designs (phase and flux qubits, transmon, fluxonium). The split transmon, in which a single junction is replaced by a SQUID loop, represents an exception that could make possible the measurement of ${\ensuremath{\Gamma}}_{\ensuremath{\phi}}$. Fluctuations of the qubit frequency leading to inhomogeneous broadening may be caused by the fluctuations in the occupation numbers of the Andreev states associated with a phase-biased Josephson junction. This mechanism may be revealed in qubits with small-area junctions, since the smallest relative change in frequency it causes is of the order of the inverse number of transmission channels in the junction.
Quasiparticles are a prominent decoherence source in superconducting qubits, but their effects are notoriously difficult to isolate in fluxonium. Unlike the transmon, fluxonium is insensitive to offset charge, precluding charge-parity detection of quasiparticle tunneling. We address this challenge by measuring the excitation and de-excitation rates in a fluxonium qubit under controlled on-chip quasiparticle injection and isolating the quasiparticle-induced contribution by subtracting the background transition rates. We show that to accurately model the external magnetic flux dependence of the quasiparticle-induced transition rates, it is necessary to account for the superconducting gap asymmetry across the Josephson junctions. A comparison between theory and experiment constrains the relative quasiparticle densities in the small junction and the junction array and helps explain previously reported discrepancies between the bounds on the quasiparticle densities inferred for these two circuit elements.
We study decoherence in superconducting qubits due to quasiparticle tunneling which is enhanced by two known deviations from the equilibrium BCS theory. The first process corresponds to tunneling of an already existing quasiparticle across the junction. The quasiparticle density is increased, e.g., because of an effective quasiparticle doping of the system. The second process is quasiparticle tunneling by breaking of a Cooper pair. This can happen at typical energies of superconducting qubits if there is an extended quasiparticle density inside the gap. We calculate the induced energy decay and pure dephasing rates in typical qubit designs. Assuming the lowest reported value of the nonequilibrium quasiparticle density in Aluminum, we find for the persistent-current flux qubit decay times of the order of recent measurements. Using the typical subgap density of states in Niobium we also reproduce observed decay times in the corresponding Niobium flux qubits.
We analyze the decay of quantum oscillations in a charge qubit consisting of a Cooper pair box connected by a Josephson junction to a finite-size superconductor. We concentrate on the contribution of quasiparticles in the superconductors to the decay rate. Passing of a quasiparticle through the Josephson junction leads to the escape of the qubit out of its Hilbert space, and thus determines the decay rate of quantum oscillations. We find the temperature dependence of the quasiparticle contribution to the decay rate for open and isolated systems. The former case is realized if a normal-state trap is included in the circuit, or if just one vortex resides in the qubit; we find exponential suppression of the rate, $\ensuremath{\Gamma}\ensuremath{\propto}\mathrm{exp}(\ensuremath{-}\ensuremath{\Delta}∕T)$, at low temperatures (here, $\ensuremath{\Delta}$ is the superconducting gap). In a superconducting qubit isolated from the environment $\ensuremath{\Gamma}\ensuremath{\propto}\mathrm{exp}(\ensuremath{-}2\ensuremath{\Delta}∕T)$ if the number of electrons is even, while for an odd number of electrons the decay rate remains finite in the limit $T\ensuremath{\rightarrow}0$. We estimate $\ensuremath{\Gamma}$ for realistic parameters of a qubit.