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Transport phenomena and decoherence in short Josephson junction arrays

J. Jussi Toppari

Published Jan 1, 2003• 1 citation
OpenAlex

Abstract

In this thesis transport phenomena in the arrays consisting of two (SSET) or three (CPP) or more Josephson junctions are discussed. A quantitative theory of adiabatic Cooper pair transport in phase biased arrays is developed. The theory predicts that the quantum inaccuracy of the Cooper pair pumping in arrays with a small number of junctions is very large. The effects due to inhomogeneous arrays or nonideal gating sequences are also quantitatively treated. It is also shown explicitly how the pumped charge in the Cooper pair pump can be understood as a partial derivative of Berry’s phase with respect to the phase difference \(\varphi\) across the array. This makes possible to obtain information about Berry’s phase by measuring the pumped current in the CPP. Also the decoherence time \(\tau_{\varphi}\), which is one of the central quantities in quantum computing, is quantitatively estimated in a dissipative electromagnetic environment of the circuit. This method allows comparison of the suitability of a system as a quantum bit. Also a direct measurement of \(\tau_{\varphi}\) as a crossover between coherent and incoherent pumping in the single Cooper pair pump is suggested. In experiments with Cooper pair pump it is demonstrated in practise how the CPP can be used as a turnstile with help of dissipation. Also a regular pumping experiment is performed. To explain the twofold behavior of the CPP, sometimes yielding \(e\)-periodicity while sometimes \(2e\)-periodicity, the energy-minimisation model is developed. It is shown also experimentally that \(\text{Nb}\) based junctions with high critical temperature (\(T_c \approx 8.5\text{ K}\)) and Josephson coupling can be fabricated using the regular self-alignment technique. The measured \(\text{Al/Nb/Al}\) SSETs show a clear signature of resonant tunnelling of Cooper pairs combined with elastic cotunnelling of quasiparticles, \(q\text{-MQT}\), through the barrier of \(\Delta_{\text{Nb}}\).

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In this thesis transport phenomena in the arrays consisting of two (SSET) or three (CPP) or more Josephson junctions are discussed. A quantitative theory of adiabatic Cooper pair transport in phase biased arrays is developed. The theory predicts that the quantum inaccuracy of the Cooper pair pumping in arrays with a small number of junctions is very large. The effects due to inhomogeneous arrays or nonideal gating sequences are also quantitatively treated. It is also shown explicitly how the pumped charge in the Cooper pair pump can be understood as a partial derivative of Berry’s phase with respect to the phase difference (\varphi) across the array. This makes possible to obtain information about Berry’s phase by measuring the pumped current in the CPP. Also the decoherence time (\tau_{\varphi}), which is one of the central quantities in quantum computing, is quantitatively estimated in a dissipative electromagnetic environment of the circuit. This method allows comparison of the suitability of a system as a quantum bit. Also a direct measurement of (\tau_{\varphi}) as a crossover between coherent and incoherent pumping in the single Cooper pair pump is suggested. In experiments with Cooper pair pump it is demonstrated in practise how the CPP can be used as a turnstile with help of dissipation. Also a regular pumping experiment is performed. To explain the twofold behavior of the CPP, sometimes yielding (e)-periodicity while sometimes (2e)-periodicity, the energy-minimisation model is developed. It is shown also experimentally that (\text{Nb}) based junctions with high critical temperature ((T_c \approx 8.5\text{ K})) and Josephson coupling can be fabricated using the regular self-alignment technique. The measured (\text{Al/Nb/Al}) SSETs show a clear signature of resonant tunnelling of Cooper pairs combined with elastic cotunnelling of quasiparticles, (q\text{-MQT}), through the barrier of (\Delta_{\text{Nb}}).
This thesis presents the results of an experimental study on single-charge effects in nanoscale Josephson junctions and Cooper pair transistors (CPTs). In nanoscale Josephson junctions the charging energy EC becomes significant at sub-Kelvin temperatures and single-charge effects, such as the Coulomb blockade of Cooper pair tunneling, influence the transport properties. In order to observe charging effects in a single Josephson junction, the impedance of the electromagnetic environment surrounding the junction has to be larger than the quantum resistance (RQ=h/4e2≈6.45kΩ). In this work the high impedance environment is obtained by biasing the sample under test (single Josephson junction or CPT) with four one-dimensional Josephson junction arrays having SQUID geometry. The advantage of this configuration is the possibility of tuning in situ the effective impedance of the electromagnetic environment. By applying a magnetic field perpendicular to the SQUID loops, the Josephson energy EJ of the SQUIDs is suppressed, resulting in an increase of the measured zero bias resistance of the arrays of several orders of magnitude (104< R0 (Ω) <109). This bias method enables the measurement of the same sample in environments with different impedance. As the impedance of the environment is increased, the current-voltage characteristics (IVCs) of the single Josephson junction and of the CPT show a well defined Coulomb blockade feature with a region of negative differential resistance, signature of the coherent tunneling of single Cooper pairs. The measured IVCs of a single Josephson junction with SQUID geometry in the high impedance environment show a qualitative agreement with the Bloch band theory as the EJ/EC ratio of the junction is tuned with the magnetic field. We also studied a single nontunable Josephson junction with strong coupling (EJ/EC > 1), where the exact dual of the overdamped Josephson effect is realized, resulting in a dual shape of the IVC, where the roles of current and voltage are exchanged. Here, we make for the first time a detailed quantitative comparison with a theory which includes the effect of fluctuations due to the finite temperature of the environment. The measurements on CPTs in the high impedance environment showed that the Coulomb blockade voltage is modulated periodically by the gate-induced charge. The gate-voltage dependence of the CPT changes from e-periodic to 2e-periodic as the impedance of the environment is increased. The high impedance environment reduces quasiparticle tunneling rates, thereby restoring the even parity of the CPT island. This behavior suggests that high impedance leads can be used to effectively suppress quasiparticle poisoning.
We have developed a quantitative theory of Cooper pair pumping in gated one-dimensional arrays of Josephson junctions. The pumping accuracy is limited by quantum tunneling of Cooper pairs out of the propagating potential well and by direct supercurrent flow through the array. Both corrections decrease exponentially with the number N of junctions in the array, but give a serious limitation of accuracy for any practical array. The supercurrent at resonant gate voltages decreases with N only as $\mathrm{sin}(\ensuremath{\varphi}/N)/N,$ where $\ensuremath{\varphi}$ is the Josephson phase difference across the array.
This thesis describes fabrication, measurements and analysisof two different Josephson junction circuits; the Cooper pairtransistor (CPT) and the one dimensional Josephson junctionarray (JJA). All experiments are conducted in a dilutionrefrigerator with a base temperature of 15{30 mK.In the experiments on CPTs the maximum supercurrent,called the switch- ing current, is measured versus the gatecharge. The importance of filtering of the measurement leads inorder to suppress quasi-particle excitations and to get2e-periodic switching currents is demonstrated. Normal metalgold leads close to the CPTs are introduced to further reducethe effect of quasi-particles. To prove that switching currentis 2e-periodic, magnetic fields and increased temperatures areused to induce a transition to e-periodicity. CPTs where eachJosephson junction is replaced by a Superconducting QuantumInterference de- vice (SQUID) have also been fabricated andmeasured. The SQUIDs make it possible to tune the effectiveJosephson energy with an external magnetic field. The measuredswitching current is tunable in magnitude and 2e-periodic. Theswitching currents are compared with theory. Good agreementbetween the- ory and experiments is found using a model of theCPT in an electrodynamic environment with a high effectivetemperature.The JJAs are also fabricated with a SQUID geometry toenable tuning of the effective Josephson energy. The currentvoltage characteristics (IVCs) of the measured JJAs exhibit aCoulomb blockade of Cooper pair tunneling which is tunable withmagnetic field and periodic with the ux quanta. The IVCs of theJJA show a region of negative differential resistance, alsocalled back- bending. This is interpreted as evidence of Blochoscillations in the JJAs. Two different models are discussed toexplain the measured IVC.The thesis ends by presenting a theoretical proposal ona Josephson junction charge qubit, called the array-qubit. Thearray-qubit consists of an array with N SQUIDs and N +1islands. The two basis states of the qubit corresponds to anexcess Cooper pair on either island 1 or N +1. These two statesare coupled through an Nth-order tunneling process which makesit possible to effectively suppress the Josephson coupling.This facilitate a long relaxation time so that the state of thequbit can be read with a slow detector.
Properties of weak links between two superconductors, or Josephson junctions, make them interesting for fundamental physics research. Since their discovery over four decades ago, they have provided a unique way to study the behavior of the superconducting quantum phase. More recently, ultra small, or mesoscopic, Josephson junctions with substantial single Cooper pair charging energy have gained interest due to their behavior as macroscopic quantum objects. In addition to the theoretical interest, Josephson junctions can be used as active elements in circuit applications. Particularly, in this Thesis we study two different devices. We develop the required theoretical treatments, derive device properties, and compare the results with experimental data. The first application is a Josephson voltage standard based on externally damped Superconductor - Insulator - Superconductor junctions. It consists of an array of large Josephson junctions connected in series and irradiated with a 70 GHz microwave signal. Phase locking the Josephson dynamics into the signal leads to the quantization of the voltage. This is utilized in metrology. We introduce a new circuit solution based on frequency dependent damping of the junctions. Optimization and some designs for practical arrays are presented. The purpose is to find such a design that the array is fast, has low power consumption and is as stable as possible. Arrays able to generate DC voltages of order 1 volt with metrological accuracy are demonstrated experimentally and their applicability in AC voltage calibrations is analyzed. The second application is the Bloch Oscillating Transistor (BOT). The BOT is based on controlling the Cooper pair current in an ultra small Josephson junction by means of quasiparticles tunneling through a normal junction. As part of the thesis work, the principle of operation is first demonstrated computationally. The model is then refined to yield quantitative predictions of the characteristics. Finally, an analytic theory for the device is developed and the properties as an amplifier are derived.
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.