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arXiv:1410.8715 · Physical Review B

Qubit dephasing due to quasiparticle tunneling

Sebastian Zanker, Michael Marthaler

Published May 7, 2015• 12 citations
arXivHTMLPDF

Abstract

We study dephasing of a superconducting qubit due to quasiparticle tunneling through a Josephson junction. While qubit decay due to tunneling processes is well understood within a golden rule approximation, pure dephasing due to BCS quasiparticles gives rise to a divergent golden rule rate. We calculate qubit dephasing due to quasiparticle tunneling beyond lowest-order approximation in coupling between qubit and quasiparticles. Summing up a certain class of diagrams, we show that qubit dephasing due to purely longitudinal coupling to quasiparticles leads to dephasing $\ensuremath{\sim}exp[\ensuremath{-}x(t)]$ where $x(t)\ensuremath{\propto}{t}^{3/2}$ for short-time scales and $x(t)\ensuremath{\propto}t\phantom{\rule{0.16em}{0ex}}ln(t)$ for long-time scales.

DOI
10.1103/physrevb.91.174504

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We study dephasing of a superconducting qubit due to quasiparticle tunneling through a Josephson junction. While qubit decay due to tunneling processes is well understood within a golden rule approximation, pure dephasing due to BCS quasiparticles gives rise to a divergent golden rule rate. We calculate qubit dephasing due to quasiparticle tunneling beyond lowest-order approximation in coupling between qubit and quasiparticles. Summing up a certain class of diagrams, we show that qubit dephasing due to purely longitudinal coupling to quasiparticles leads to dephasing \ensuremath∼exp[\ensuremath−x(t)]\ensuremath{\sim}exp[\ensuremath{-}x(t)]\ensuremath∼exp[\ensuremath−x(t)] where x(t)\ensuremath∝t3/2x(t)\ensuremath{\propto}{t}^{3/2}x(t)\ensuremath∝t3/2 for short-time scales and x(t)\ensuremath∝tln(t)x(t)\ensuremath{\propto}t\phantom{\rule{0.16em}{0ex}}ln(t)x(t)\ensuremath∝tln(t) for long-time scales.
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.
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.
The fluxonium qubit has arisen as one of the most promising candidate devices for implementing quantum information in superconducting devices, since it is both insensitive to charge noise (like flux qubits) and insensitive to flux noise (like charge qubits). Here, we investigate the stability of the quantum information to quasiparticle tunneling through a Josephson junction. Microscopically, this dephasing is due to the dependence of the quasiparticle transmission probability on the qubit state. We argue that on a phenomenological level the dephasing mechanism can be understood as originating from heat currents, which are flowing in the device due to possible effective temperature gradients, and their sensitivity to the qubit state. The emerging dephasing time is found to be insensitive to the number of junctions with which the superinductance of the fluxonium qubit is realized. Furthermore, we find that the dephasing time increases quadratically with the shunt-inductance of the circuit which highlights the stability of the device to this dephasing mechanism.
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.
We describe in this paper how the nonlinear Josephson inductance is the crucial circuit element for all Josephson qubits. We discuss the three types of qubit circuits, and show how these circuits use this nonlinearity in unique manners. We give a brief derivation of the BCS theory, highlighting the appearance of the macroscopic phase parameter. The Josephson equations are derived using standard first and second order perturbation theory that describe quasiparticle and Cooper-pair tunneling. An exact calculation of the Josephson effect then follows using the quasiparticle bound-state theory, and then expand upon this theory to describe quasiparticle excitations as transitions from the ground to excited bound states from nonadiabatic changes in the bias. Although quasiparticle current is typically calculated only for a constant DC voltage, the advantage to this approach is seen where we qualitatively describe quasiparticle tunneling with AC voltage excitations, as appropriate for the qubit state. This section describes how the Josephson qubit is typically insensitive to quasiparticle damping, even to the extent that a phase qubit can be constructed from microbridge junctions.
We study quasiparticle tunneling in Josephson tunnel junctions embedded in an electromagnetic environment. We identify tunneling processes that transfer electrical charge and couple to the environment in a way similar to that of normal electrons, and processes that mix electrons and holes and are thus creating charge superpositions. The latter are sensitive to the phase difference between the superconductors and are thus limited by phase diffusion even at zero temperature. We show that the environmental coupling is suppressed in many environments, thus leading to lower quasiparticle decay rates and better superconductor qubit coherence than previously expected. Our approach is nonperturbative in the environmental coupling strength.