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arXiv:cond-mat/0307568 · Physical Review B

Resonant microwave properties of a voltage-biased single-Cooper-pair transistor

L. Y. Gorelik, S. I. Kulinich, R. I. Shekhter, M. Jonson

Published Mar 24, 2004• 4 citations
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Abstract

We consider the microwave dynamics and transport properties of a voltage-biased single-Cooper-pair transistor. The dynamics is shown to be strongly affected by interference between multiple microwave-induced interlevel transitions. As a result the magnitude and direction of the dc Josephson current are extremely sensitive to small variations of the bias voltage and to changes in the frequency of the microwave field.

DOI
10.1103/physrevb.69.094516

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We consider the microwave dynamics and transport properties of a voltage-biased single-Cooper-pair transistor. The dynamics is shown to be strongly affected by interference between multiple microwave-induced interlevel transitions. As a result the magnitude and direction of the dc Josephson current are extremely sensitive to small variations of the bias voltage and to changes in the frequency of the microwave field.
We analyze how decoherence appears in the $I\text{\ensuremath{-}}V$ characteristics of a voltage-biased single-Cooper-pair transistor. Especially the effect on resonant single or several Cooper-pair tunneling is studied. We consider both a symmetric and an asymmetric transistor. As a decoherence source we use a small resistive impedance $(\text{Re}[Z(\ensuremath{\omega})]⪡{R}_{Q}=h/4{e}^{2})$ in series with the transistor, which provides both thermal and quantum fluctuations of the voltage. Additional decoherence sources are quasiparticle tunneling across the Josephson junctions and quantum fff noise caused by spurious charge fluctuators nearby the island. The analysis is based on a real-time diagrammatic technique which includes Zeno-type effects in the charge transport, where the tunneling is slowed down due to strong decoherence. As compared to the Pauli-master-equation treatment of the problem, the present results are more consistent with experiments where many of the predicted sharp resonant structures are missing or weakened due to decoherence.
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​.
We have studied the microwave response of a single Cooper-pair transistor (CPT) coupled to a lumped-element microwave resonator. The resonance frequency of this circuit, fr{f}_{r}fr​, was measured as a function of the charge ng{n}_{g}ng​ induced on the CPT island by the gate electrode and the phase difference across the CPT, ${\ensuremath{\varphi}}_{B}$, which was controlled by the magnetic flux in the superconducting loop containing the CPT. The observed ${f}_{r}({n}_{g},{\ensuremath{\varphi}}_{B})$ dependencies reflect the variations of the CPT Josephson inductance with ng{n}_{g}ng​ and ${\ensuremath{\varphi}}_{B}$ as well as the CPT excitation when the microwaves induce transitions between different quantum states of the CPT. The results are in excellent agreement with our simulations based on the numerical diagonalization of the circuit Hamiltonian. This agreement over the whole range of ng{n}_{g}ng​ and ${\ensuremath{\varphi}}_{B}$ is unexpected, because the relevant energies vary widely, from 0.1 to 3 K. The sensitivity of the CPT as an electrometer is peaked when the CPT excitation level approaches that of the microwave resonator.
A study of the Andreev level current through a weakly voltage-biased single-mode superconducting quantum point contact under microwave irradiation is presented. The applied microwave field induces resonant transitions between the current carrying Andreev states localized to the contact region, which results in an increase of the dc subgap current by several orders of magnitude. The applied voltage-bias couples to the superconductor phase difference across the junction, which will evolve in time. In turn the Andreev levels will oscillate adiabatically within the superconductor energy gap. Depending on the frequency of the applied field there can be either one or two resonances per oscillation period. When considering the case of two resonances the calculated dc-current exhibits an interference pattern as a function of inverse voltage. A measurement of the oscillation period would make it possible to reconstruct the Andreev level spectrum. Further, a detailed study is conducted concerning the microwave-induced current as a function of voltage bias, and the amplitude and frequency of the applied field. In realistic experimental situations there are equilibrium fluctuations of the applied voltage bias and sometimes impurity induced noise in the contact transparency. The amount of dephasing induced on the interference pattern by these fluctuations is estimated. In most cases the interference pattern will be subject to some amount of decay. However, there are parameter values for which this decay will be small. The decay originating from equilibrium fluctuations is predicted to have a non-monotonic dependence on the position of the resonances, while the decay induced by impurities should increase linearly when the separation between the resonances is increased. This difference could be used to distinguish between the two sources. The results also indicate that it should be possible to experimentally investigate the coupling between Andreev states and their environment, explicitly the impedance which the contact "sees" and the coupling strength between impurities and Andreev states.
This chapter investigates the quantum versus classical dynamics of a microwave cavity-coupled-Cooper-pair transistor (CPT) system, where a variable applied dc bias causes the system to tunably self-oscillate via the ac Josephson effect. An essential feature of the system is that the dc bias does not significantly affect the high quality factor of the cavity mode to which the CPT predominantly couples. The classical, nonlinear dynamics of the system exhibits chaotic, as well as aperiodic motions depending on the initial conditions and the nature and strengths of the damping/noise forces. The corresponding quantum dynamics exhibits such phenomena as dynamical tunnelling and the generation of nonclassical states from initial classical states. Obviating the need for an external ac-drive line, which typically is harder to noise filter than a dc bias line, the self-oscillating system described here has considerable promise for demonstrating macroscopic quantum dynamical behaviour.