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arXiv:0803.3102 · Physical Review B

Kinetics of nonequilibrium quasiparticle tunneling in superconducting charge qubits

Matthew D. Shaw, Roman M. Lutchyn, Per Delsing, P. M. Echternach

Published Jul 1, 2008• 104 citations
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Abstract

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.

DOI
10.1103/physrevb.78.024503

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Physical Review B•Aug 1, 2007•25 citations
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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 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​.
In superconducting qubits the lifetime of quantum states cannot be prolonged arbitrarily by decreasing temperature. At low temperature quasiparticles tunneling between electromagnetic environment and superconducting islands takes the condensate state out of equilibrium due to charge imbalance. We obtain the tunneling rate from a phenomenological model of non-equilibrium, where nonequilibrium quasiparticle tunnelling stimulates a temperature-dependent chemical potential shift in superconductor. As a result we obtain a non-monotonic behavior for relaxation rate as function of temperature. Depending on the fabrication parameters for some qubits the lowest tunneling rate of nonequilibrium quasiparticles can take place only near the onset temperature below which nonequilibrium quasiparticles dominate over equilibrium one. Our theory also indicates that such tunnelings can influence the probability of transitions in qubits through a coupling to the zero-point energy of phase fluctuations.
We have directly measured non-equilibrium quasiparticle tunneling in the time domain as a function of temperature and RF carrier power for a pair of charge qubits based on the single Cooper-pair box, where the readout is performed with a multiplexed quantum capacitance technique. We have extracted an effective electron temperature for each applied RF power, using the data taken at the lowest power as a reference curve. This data has been fit to a standard T 5 electron heating model, with a qualitative correspondence with established material parameters.
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.