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arXiv:cond-mat/0603640 · Physical Review B

Kinetics of the superconducting charge qubit in the presence of a quasiparticle

Roman M. Lutchyn, L. I. Glazman, A. I. Larkin

Published Aug 30, 2006• 58 citations
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Abstract

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 ${g}_{T}$ 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 $T$ to the Josephson energy ${E}_{J}$.

DOI
10.1103/physrevb.74.064515

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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 \ensuremathΓin\ensuremath−1{\ensuremath{\Gamma}}_{\text{in}}^{\ensuremath{-}1}\ensuremathΓin\ensuremath−1​ and quasiparticle relaxation time \ensuremathτ\ensuremath{\tau}\ensuremathτ. The former is determined by the dimensionless normal-state conductance gT{g}_{T}gT​ of the Josephson junction and one-electron level spacing \ensuremathδr{\ensuremath{\delta}}_{r}\ensuremathδr​ in the reservoir (\ensuremathΓin\ensuremath∼gT\ensuremathδr)({\ensuremath{\Gamma}}_{\text{in}}\ensuremath{\sim}{g}_{T}{\ensuremath{\delta}}_{r})(\ensuremathΓin​\ensuremath∼gT​\ensuremathδr​), and the latter is due to the electron-phonon interaction. We find that phase coherence is damped on the time scale of \ensuremathΓin\ensuremath−1{\ensuremath{\Gamma}}_{\text{in}}^{\ensuremath{-}1}\ensuremathΓin\ensuremath−1​. The qubit energy relaxation depends on the ratio of the two characteristic times \ensuremathτ\ensuremath{\tau}\ensuremathτ and \ensuremathΓin\ensuremath−1{\ensuremath{\Gamma}}_{\text{in}}^{\ensuremath{-}1}\ensuremathΓin\ensuremath−1​ and also on the ratio of temperature TTT to the Josephson energy EJ{E}_{J}EJ​.
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.
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 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 have analyzed and measured the quantum coherent dynamics of a circuit containing two coupled superconducting charge qubits. Each qubit is based on a Cooper pair box connected to a reservoir electrode through a Josephson junction. Two qubits are coupled electrostatically by a small island overlapping both Cooper pair boxes. Quantum state manipulation ofthe qubit circuit is done by applying non-adiabatic voltage pulses to the common gate. We read out each qubit by means of probe electrodes connected to Cooper pair boxes through high-Ohmic tunnel junctions. With such a setup the measured pulse-induced probe currents are proportional to the probability for each qubit to have an extra Cooper pai1r after the manipulation. As expected from theory and observed experimentally the measured pulse-induced current in each probe has two frequency components whose position on the frequency axis can be externally controlled. This is a result ofthe inter-qubit coupling which is also responsible for the avoided level crossing that we observed in the qubits' spectra. Our simulations show that in the absence of decoherence and with a rectangular pulse shape the system remains entangled most ofthe time reaching maximally entangled states at certain instances.
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.