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Journal of Physics Conference Series

Energy-band dynamics in a current-biased Josephson junction probed by incoherent Cooper-pair tunneling

Juha Leppäkangas, E. V. Thuneberg

Published Feb 1, 2009• 1 citation
Publisher

Abstract

We analyze how the dynamics in a current-biased mesoscopic Josephson junction can be probed by an additional smaller Josephson junction (probe). The probe is connected to a small island in between the Josephson junction and a resistor R >> R Q = h /4e 2 . The current through the resistor results in an effective current bias to the larger JJ. We model the steady state properties of the system under thermal and quantum fluctuations of the feed current. We use a density matrix approach based on the Born-Markov equations treating the current fluctuations perturbatively. The probe is also treated perturbatively assuming the regime of incoherent Cooper-pair tunneling. We find that the I - V characteristics across the probe shows traces of energy bands, Bloch oscillations and Zener tunneling in the current-biased Josephson junction.

DOI
10.1088/1742-6596/150/2/022050

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We analyze how the dynamics in a current-biased mesoscopic Josephson junction can be probed by an additional smaller Josephson junction (probe). The probe is connected to a small island in between the Josephson junction and a resistor R >> R Q = h /4e 2 . The current through the resistor results in an effective current bias to the larger JJ. We model the steady state properties of the system under thermal and quantum fluctuations of the feed current. We use a density matrix approach based on the Born-Markov equations treating the current fluctuations perturbatively. The probe is also treated perturbatively assuming the regime of incoherent Cooper-pair tunneling. We find that the I - V characteristics across the probe shows traces of energy bands, Bloch oscillations and Zener tunneling in the current-biased Josephson junction.
We analyze tunneling of Cooper pairs across voltage biased asymmetric single-Cooper-pair transistors. Also tunneling of Cooper pairs across two capacitively coupled Cooper-pair boxes is considered, when the capacitive coupling and Cooper pair tunneling are provided by a small Josephson junction between the islands. The theoretical analysis is done at subgap voltages, where the current-voltage characteristics depend strongly on the macroscopic eigenstates of the island(s) and their coupling to the dissipative environment. As the environment we use an impedance which satisfies $\mathrm{Re}[Z(\ensuremath{\omega})]⪡{R}_{Q}$ and a few LCLCLC oscillators in series with $Z(\ensuremath{\omega})$. The numerically calculated $I\text{\ensuremath{-}}V$ curves are compared with experiments where the quantum states of mesoscopic SQUIDs are probed with inelastic Cooper pair tunneling. The main features of the observed $I\text{\ensuremath{-}}V$ data are reproduced. Especially, we find traces of band structure in the higher excited states of the Cooper-pair boxes as well as traces of multiphoton processes between two Cooper-pair boxes in the regime of large Josephson coupling EJ⪢EC{E}_{J}⪢{E}_{C}EJ​⪢EC​.
The quantum-mechanical effects typical for single atoms or molecules can be reproduced in micrometer-scale electric devices. In these systems the essential component is a small Josephson junction (JJ) consisting of two superconductors separated by a thin insulator. The quantum phenomena can be controlled in real time by external signals and have a great potential for novel applications. However, their fragility on uncontrolled disturbance caused by typical nearby environments is a drawback for quantum information science, but a virtue for detector technology. Motivated by this we have theoretically studied transistor kind of devices based on single-charge tunneling through small JJs. A common factor of the research is the analysis of the interplay between the coherent Cooperpair (charge carriers in the superconducting state) tunneling and incoherent environmental processes. In the first work we calculate the current due to incoherent Cooper-pair tunneling through a voltage-biased small JJ in series with large JJs and compare the results with recent experiments. We are able to reproduce the main experimental features and interpret these as traces of energy levels and energy bands of the mesoscopic device. In the second work we analyze a similar circuit (asymmetric single-Cooper-pair transistor) but under the assumption that the Cooper-pair tunneling is mainly coherent. This predicts new resonant transport voltages in the circuit due to higher-order processes. However, no clear traces of most of them are seen in the experiments, and similar discrepancy is present also in the case of the symmetric circuit. We continue to study this problem by modeling the interplay between the coherent and incoherent processes more accurately using a density-matrix approach. By this we are able to demonstrate that in typical conditions most of these resonances are indeed washed out by strong decoherence caused by the environment. We also analyze the contribution of three typical weakly interacting dissipative environments: electromagnetic environment, spurious charge fluctuators in the nearby insulating materials, and quasiparticles. In the last work we model the dynamics of a currentbiased JJ perturbed by a smaller JJ using a similar density-matrix approach. We demonstrate that the small JJ can be used also as a detector of the energy-band dynamics in a current biased JJ. The method is also used for modeling the charge transport in the Bloch-oscillating transistor.
We have measured the current-voltage characteristics of a Josephson junction with tunable Josephson energy EJ{E}_{J}EJ​ embedded in an inductive environment provided by a chain of SQUIDs. Such an environment induces localization of the charge on the junction, which results in an enhancement of the zero-bias resistance of the circuit. We explain this result quantitatively in terms of the Bloch band dynamics of the localized charge. This dynamics is governed by charge diffusion in the lowest Bloch band of the Josephson junction as well as by Landau-Zener transitions out of the lowest band into the higher bands. In addition, the frequencies corresponding to the self-resonant modes of the SQUID array exceed the Josephson energy EJ{E}_{J}EJ​ of the tunable junction, which results in a renormalization of EJ{E}_{J}EJ​, and, as a consequence, an increase of the effective bandwidth of the lowest Bloch band.
Quantum fluctuations in a current-biased Josephson junction, described in terms of the RCSJ model, are considered. The fluctuations of the voltage and phase across the junction are assumed to be initiated by equilibrium current fluctuations in the shunting resistor. This corresponds to low enough temperatures, when fluctuations of the normal current in the junction itself can be neglected. We used the quantum Langevin equation in terms of random variables related to the limit cycle of the nonlinear Josephson oscillator. This allows to go beyond the perturbation theory and to calculate the widths of the Josephson radiation lines.
We consider quantum fluctuations in a current biased overdamped Josephson junction in the regime when the bias current is larger than the critical current. The fluctuations of the voltage and phase across the junction are assumed to be initiated by equilibrium current fluctuations in the shunting resistor. This corresponds to low enough temperatures, when fluctuations of the normal current in the junction itself can be neglected. Quantum effects are important when the resistor temperature is low compared to the Josephson frequency, even in the case when noise is measured at frequencies lower than the temperature. We used the quantum Langevin equation in terms of random variables related to the limit cycle of the nonlinear Josephson oscillator. This allows to go beyond the perturbation theory and calculate the widths of the Josephson radiation lines.