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arXiv:cond-mat/0611692 · Physical Review B

Kinetics of quasiparticle trapping in a Cooper-pair box

Roman M. Lutchyn, L. I. Glazman

Published May 22, 2007• 29 citations
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Abstract

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.

DOI
10.1103/physrevb.75.184520

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Bulletin of the American Physical Society•Mar 16, 2006•0 citations
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.
We directly observe low-temperature nonequilibrium quasiparticle tunneling in charge qubits based on the single-Cooper-pair box. We measure even- and odd-state dwell-time distributions as a function of temperature, and interpret these results using a kinetic theory. While the even-state lifetime is exponentially distributed, the odd-state distribution is more heavily weighted to short times, implying that odd-to-even tunnel events are not described by a homogenous Poisson process. The mean odd-state dwell time increases sharply at low temperature, which is consistent with quasiparticles tunneling out of the island before reaching thermal equilibrium.
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 effect of quantum charge fluctuations on the discrete spectrum of charge states of a small superconducting island (Cooper-pair box) connected to a large finite-size superconductor by a tunnel junction. In particular, we calculate the reduction of the even-odd energy difference $\ensuremath{\delta}E$ due to virtual tunneling of electrons across the junction. We show that the renormalization effects are important for understanding the quasiparticle ``poisoning'' effect because $\ensuremath{\delta}E$ determines the activation energy of a trapped quasiparticle in the Cooper-pair box. We find that renormalization of the activation energy depends on the dimensionless normal-state conductance of the junction gT{g}_{T}gT​ and becomes strong at gT⪢1{g}_{T}⪢1gT​⪢1.
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.
A superconducting Coulomb-blockade electrometer was used to measure the Coulomb staircase of an Al∕AlOx∕Al\mathrm{Al}∕\mathrm{Al}{\mathrm{O}}_{x}∕\mathrm{Al}Al∕AlOx​∕Al Cooper-pair box from a temperature of 30to300mK30\phantom{\rule{0.3em}{0ex}}\text{to}\phantom{\rule{0.3em}{0ex}}300\phantom{\rule{0.3em}{0ex}}\mathrm{mK}30to300mK. At the lowest temperature, the Coulomb staircase displays effects from nonequilibrium quasiparticles. As the temperature is increased, an initial decrease is found in the width of the odd steps in the staircase, which corresponds to a reduction in the probability of having a quasiparticle on the island of the box. Above 180mK180\phantom{\rule{0.3em}{0ex}}\mathrm{mK}180mK, the width of the odd steps increases, eventually producing a staircase with 1e1e1e features. We develop a steady-state model of the system and find that the presence of quasiparticles at low temperature is consistent with the assumption of Aumentado et al. [Phys. Rev. Lett. 92, 066802 (2004)] that nonequilibrium quasiparticles are generated in the leads. Above 180mK180\phantom{\rule{0.3em}{0ex}}\mathrm{mK}180mK, our results are consistent with the quasiparticle states of the island being thermally populated.
box in the presence of quasiparticle poisoning. The model uses a master equation approach to flnd the probabilities for the box to be in the even or odd state. The transition rates between the two states are calculated assuming a flxed number of non-equilibrium quasiparticles in the leads and island of the box. We fabricated Al/AlOx/Al devices with a Cooper pair box capacitively coupled to an SET and measured the charge on the box for SET bias currents ranging from about 1 pA to 1 nA. We flnd good agreement between the theory and measurements in the temperature range from 60 mK to 300 mK. For large SET bias the poisoning in the Cooper pair box increases and the charge staircase develops additional features. Our model is capable of qualitatively reproducing the features induced by the measuring SET.