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arXiv:1407.5595 · Physical Review B

Dissipation in Josephson tunneling junctions at low temperatures

Lev N. Bulaevskii, Shi‐Zeng Lin

Published Aug 14, 2014• 0 citations
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Abstract

It is important to know the decoherence mechanism of a qubit based on Josephson junctions. At low temperatures, as quasiparticle concentration becomes exponentially small, one needs to consider energy transfer from tunneling electrons to other degrees of freedom to find dissipation in Josephson junctions and decoherence in qubits. Here we discuss the energy transfer to two-level systems, i.e., the transitions between two different configurations of ions inside an insulating layer separated by a potential barrier. We derive a general equation of motion for the phase difference between two superconducting electrodes and we find a retarded dissipation term due to electromagnetic mechanism and also contribution due to electron tunneling mechanism. Using the equation of motion we calculate the decay of Rabi oscillations and frequency shift in qubits due to the presence of the two-level systems. In the long-time limit our results coincide with those obtained by Martinis et al. [Martinis et al., Phys. Rev. Lett. 95, 210503 (2005)] within the Fermi's Golden Rule approach up to a numerical factor.

DOI
10.1103/physrevb.90.054509

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It is important to know the decoherence mechanism of a qubit based on Josephson junctions. At low temperatures, as quasiparticle concentration becomes exponentially small, one needs to consider energy transfer from tunneling electrons to other degrees of freedom to find dissipation in Josephson junctions and decoherence in qubits. Here we discuss the energy transfer to two-level systems, i.e., the transitions between two different configurations of ions inside an insulating layer separated by a potential barrier. We derive a general equation of motion for the phase difference between two superconducting electrodes and we find a retarded dissipation term due to electromagnetic mechanism and also contribution due to electron tunneling mechanism. Using the equation of motion we calculate the decay of Rabi oscillations and frequency shift in qubits due to the presence of the two-level systems. In the long-time limit our results coincide with those obtained by Martinis et al. [Martinis et al., Phys. Rev. Lett. 95, 210503 (2005)] within the Fermi's Golden Rule approach up to a numerical factor.
Noise and decoherence are major obstacles to the implementation of Josephson junction qubits in quantum computing. Recent experiments suggest that two-level systems (TLS) in the oxide tunnel barrier are a source of decoherence. We explore two decoherence mechanisms in which these two-level systems lead to the decay of Rabi oscillations that result when Josephson junction qubits are subjected to strong microwave driving. (A) We consider a Josephson qubit coupled resonantly to a two-level system, i.e., the qubit and TLS have equal energy splittings. As a result of this resonant interaction, the occupation probability of the excited state of the qubit exhibits beating. Decoherence of the qubit results when the two-level system decays from its excited state by emitting a phonon. (B) Fluctuations of the two-level systems in the oxide barrier produce fluctuations and 1∕f1∕f1∕f noise in the Josephson junction critical current I0{I}_{0}I0​. This in turn leads to fluctuations in the qubit energy splitting that degrade the qubit coherence. We compare our results with experiments on Josephson junction phase qubits.
I studied the Josephson junction system in the context of its usability for quan-tum computation. The zero-voltage state of a Josephson junction biased with constant current has a set of metastable quantum energy levels. In principle, it should be possible to use the two lowest states as a qubit for a quantum computer, provided both the dissipation time and coherence time of this qubit can be made long enough. I describe two methods for isolating a Josephson junction to increase its dissipation time. The rst is a resistive isolation scheme, for which I fabricated and measured a prototype system. I discuss the results of these measurements and show that the constraints of heating in conjunction with high-frequency design require-ments make it unlikely that this isolation method will lead to the coherence times required for quantum computation. I then discuss a resonant isolation method that I used to successfully increase the dissipation time of a Josephson junction qubit. I performed spectroscopic mea-surements at low temperatures to nd the coherence time of resonantly isolated 100 m2 Al/AlOx/Al and Nb/AlOx/Nb junctions with critical currents of roughly 10 A. The results of these measurements reveal two mechanisms beyond spontaneous emission for decoherence: low-frequency current noise and tunneling to the voltage state. To explain these results on decoherence, I present a model based on the Bloch equations. Going beyond a single qubit, I designed a capacitively coupled two-qubit system. Using spectroscopy, I showed that the energy level spacings between the ground state and the excited entangled states of the system agree with those calculated from the Schrodinger equation.
It is frequently observed that even at very low temperatures, the number of quasiparticles in superconducting materials is higher than predicted by standard BCS theory. These quasiparticles can interact with two-level systems, such as superconducting qubits or two-level systems (TLS) in the amorphous oxide layer of a Josephson junction. This interaction leads to decay and decoherence of the TLS, with specific results, such as time dependence, depending on the distribution of quasiparticles and the form of interaction. We study the resulting decay laws for different experimentally relevant protocols.
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 consider a two-level system (TLS) with energy level separation plankvOmega0 inside a Josephson junction. The junction is shunted by a resistor R and is voltage V biased. If the TLS modulates the Josephson energy and/or is optically active, it is Rabi driven by the Josephson oscillations in the running phase regime near the resonance 2eV=plankvOmega0. The Rabi oscillations, in turn, translate into oscillations of current and voltage that can be detected in noise measurements. This effect provides an option to fully characterize the TLS inside Josephson junction and to find the TLS's contribution to the decoherence when the junction is used as a qubit.
We study the Josephson junction between two d -wave superconductors, which is discussed as an implementation of a qubit. We propose an approach to calculate the decoherence time due to an intrinsic dissipative process: quantum tunneling between the two minima of the double-well potential excites nodal quasiparticles, which lead to incoherent damping of quantum oscillations. The decoherence is weakest in the mirror junction, where the contribution of nodal quasiparticles corresponds to the superohmic dissipation and becomes small at small tunnel splitting of the energy level in the double-well potential. For available experimental data, we estimate the quality factor.