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arXiv:0706.0522 · Physical Review B

Energy relaxation of a superconducting charge qubit via Andreev processes

Roman M. Lutchyn, L. I. Glazman

Published Sep 11, 2007• 1 citation
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Abstract

We study fundamental limitations on the energy relaxation rate of a superconducting charge qubit with a large-gap Cooper-pair box, ${\ensuremath{\Delta}}_{b}>{\ensuremath{\Delta}}_{r}$. At a sufficiently large mismatch between the gap energies in the box ${\ensuremath{\Delta}}_{b}$ and in the reservoir ${\ensuremath{\Delta}}_{r}$, ``quasiparticle poisoning" becomes ineffective even in the presence of nonequilibrium quasiparticles in the reservoir. The qubit relaxation still may occur due to higher-order (Andreev) processes. In this paper, we evaluate the qubit energy relaxation rate ${T\phantom{\rule{0.2em}{0ex}}}_{1}^{\ensuremath{-}1}$ due to Andreev processes.

DOI
10.1103/physrevb.76.104507

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We study fundamental limitations on the energy relaxation rate of a superconducting charge qubit with a large-gap Cooper-pair box, \ensuremathΔb>\ensuremathΔr{\ensuremath{\Delta}}_{b}>{\ensuremath{\Delta}}_{r}\ensuremathΔb​>\ensuremathΔr​. At a sufficiently large mismatch between the gap energies in the box \ensuremathΔb{\ensuremath{\Delta}}_{b}\ensuremathΔb​ and in the reservoir \ensuremathΔr{\ensuremath{\Delta}}_{r}\ensuremathΔr​, ``quasiparticle poisoning" becomes ineffective even in the presence of nonequilibrium quasiparticles in the reservoir. The qubit relaxation still may occur due to higher-order (Andreev) processes. In this paper, we evaluate the qubit energy relaxation rate T1\ensuremath−1{T\phantom{\rule{0.2em}{0ex}}}_{1}^{\ensuremath{-}1}T1\ensuremath−1​ due to Andreev processes.
We investigate the energy and phase relaxation of a superconducting qubit caused by a single quasiparticle. In our model, the qubit is an isolated system consisting of a small island (Cooper-pair box) and a larger superconductor (reservoir) connected with each other by a tunable Josephson junction. If such a system contains an odd number of electrons, then even at lowest temperatures a single quasiparticle is present in the qubit. Tunneling of a quasiparticle between the reservoir and the Cooper-pair box results in the relaxation of the qubit. We derive master equations governing the evolution of the qubit coherences and populations. We find that the kinetics of the qubit can be characterized by two time scales---quasiparticle escape time from the reservoir to the box ${\ensuremath{\Gamma}}_{\text{in}}^{\ensuremath{-}1}$ and quasiparticle relaxation time $\ensuremath{\tau}$. The former is determined by the dimensionless normal-state conductance gT{g}_{T}gT​ of the Josephson junction and one-electron level spacing ${\ensuremath{\delta}}_{r}$ in the reservoir $({\ensuremath{\Gamma}}_{\text{in}}\ensuremath{\sim}{g}_{T}{\ensuremath{\delta}}_{r})$, and the latter is due to the electron-phonon interaction. We find that phase coherence is damped on the time scale of ${\ensuremath{\Gamma}}_{\text{in}}^{\ensuremath{-}1}$. The qubit energy relaxation depends on the ratio of the two characteristic times $\ensuremath{\tau}$ and ${\ensuremath{\Gamma}}_{\text{in}}^{\ensuremath{-}1}$ and also on the ratio of temperature TTT to the Josephson energy EJ{E}_{J}EJ​.
We study the kinetics of the quasiparticle capture and emission process in a small superconducting island (Cooper-pair box) connected by a tunnel junction to a massive superconducting lead. At low temperatures, the charge on the box fluctuates between two states, even and odd in the number of electrons. Assuming that the odd-electron state has the lowest energy, we evaluate the distribution of lifetimes of the even- and odd-electron states of the Cooper-pair box. The lifetime in the even-electron state is an exponentially distributed random variable corresponding to a homogenous Poisson process of ``poisoning'' the island with a quasiparticle. The distribution of lifetimes of the odd-electron state may deviate from the exponential one. The deviations come from two sources---the peculiarity of the quasiparticle density of states in a superconductor and the possibility of quasiparticle energy relaxation via phonon emission. In addition to the lifetime distribution, we also find spectral density of charge fluctuations generated by capture and emission processes. The complex statistics of the quasiparticle dwell times in the Cooper-pair box may result in strong deviations of the noise spectrum from the Lorentzian form.
Superconducting qubits based on Josephson junctions have emerged as a leading hardware platform in the quest to realize large-scale quantum computers. However, qubit decoherence remains a major hurdle on the path toward fault-tolerant quantum computing. In particular, excess quasiparticle (QP) excitations constitute a significant source of dissipation in superconducting quantum devices. The empirically observed QP density in these systems is orders of magnitude larger than that predicted by Bardeen-Cooper-Schrieffer (BCS) theory. Moreover, QP poisoning due to high-energy particle impacts can cause correlated errors that simultaneously affect multiple qubits, thereby impeding quantum error correction. Although QP-induced decoherence has been extensively studied in widely implemented transmon qubits, its impact on fluxonium qubits, a leading alternative architecture, remains far less explored. In this thesis, we investigate energy relaxation in fluxonium qubits induced by controlled QP poisoning using an on-chip Josephson injector junction that generates Cooper-pair-breaking phonons and photons. We separately measure the relaxation and excitation rates, instead of the conventional energy relaxation time. For specific injection parameters, we observe an apparent reversal of the rates, with the excitation rate exceeding the relaxation rate. We further measure the rates as a function of external flux to disentangle the effects of QP tunneling across the small junction and the junction array superinductor. Across all measured samples, differing in qubit spectra and injection conditions, we infer roughly an order of magnitude higher QP density at the small junction than in the array junctions. Our findings advance the understanding of QP-induced decoherence in fluxonium qubits and provide insight for reconciling discrepancies in reported bounds on QP densities, potentially enabling the development of QP mitigation strategies tailored to fluxonium 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.
Superconducting circuits exhibit quantum properties on a macroscopic scale, and are natural candidates for solid state quantum computing. Their low-energy physics can be described in terms of the phase of the order parameter, a single collective degree of freedom associated with billions of coherently paired electrons. In practice, however, on top of the superfluid condensate there are single-particle excitations (quasiparticles) with a continuous energy spectrum; the quasiparticles are coupled to the phase degree of freedom. The presence of quasiparticles in the system sets serious constraints on the performance of superconducting charge qubits. In this thesis, we study the kinetics of superconducting quantum circuits, and discuss the fundamental limitations on the energy and phase relaxation times in the presence of 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.