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Topics in the physics of underdamped Josephson systems

I. Tornes

Published Jan 1, 2006• 0 citations
OpenAlex

Abstract

We describe a variety of models to characterize the behavior of several types of underdamped Josephson systems.We also present numerical results based on these models.We first discuss a stack of small junctions in a single mode resonant cavity.Such a stack is believed to be appropriate to model some high temperature cuprate superconductors such as Bi 2 Sr 2 CaCu 2 O 8+x .Our results show self-induced-resonant-steps in the current voltage (IV) curve.These steps represent a coherent phase locking between the junctions in the stack and the resonant cavity.Next we generalize these results to treat a single long junction coupled to a single mode resonant cavity.We find a number of features not seen in stacks of small junctions.Our numerical results show that if the electromagnetic fields of the cavity mode are spatially periodic, we again see steps in the IV curve and furthermore, for sufficiently large coupling, the cavity frequency is shifted by coupling to the long junction.We also discuss how a stack of long junctions could be used to model a layered high temperature superconductor.We then turn to some properties of single junctions in the "quantum" regime where the phase/number non-commutation is important.This quantum regime is crucial to all proposals to use such junctions as solid state quantum bits (qubits).We show that such qubits, when subjected to external a.c.perturbations, exhibit not only . . . . . . . . . .

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Dynamics of a Josephson array in a resonant cavity

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We derive dynamical equations for a Josephson array coupled to a resonant cavity by applying the Heisenberg equations of motion to a model Hamiltonian described by us earlier [Phys. Rev. B 63, 144522 (2001); 64, 179902(E) (2001)]. By means of a canonical transformation, we also show that, in the absence of an applied current and dissipation, our model reduces to one described by Shnirman et al [Phys. Rev. Lett. 79, 2371 (1997)] for coupled qubits, and that it corresponds to a capacitive coupling between the array and the cavity mode. From extensive numerical solutions of the model in one dimension, we find that the array locks into a coherent, periodic state above a critical number of active junctions; that the current-voltage characteristics of the array have self-induced resonant steps (SIRS's); that when ${N}_{a}$ active junctions are synchronized on a SIRS, the energy emitted into the resonant cavity is quadratic in ${N}_{a};$ and that when a fixed number of junctions is biased on a SIRS, the energy is linear in the input power. All these results are in agreement with recent experiments. By choosing the initial conditions carefully, we can drive the array into any of a variety of different integer SIRS's. We tentatively identify terms in the equations of motion which give rise to both the SIRS's and the coherence threshold. We also find higher-order integer SIRS's and fractional SIRS's in some simulations. We conclude that a resonant cavity can produce threshold behavior and SIRS's even in a one-dimensional array with appropriate experimental parameters, and that the experimental data, including the coherent emission, can be understood from classical equations of motion.

arXiv:cond-mat/0111028•Physical review. B, Condensed matter•Mar 19, 2002•40 citations

Electrical properties of stacks of many long Josephson junctions

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We investigated numerically IV-characteristics and power of emission from stacks with various quantities of long Josephson junctions (up to 6 junctions) which interacted inductively with each other. Parameters of junctions were chosen close to those for MoRe-Si(W)-MoRe heterostructures. We set Gaussian spread of about 0.01% of critical currents along junctions. Electrical properties of a stack consisted of three junctions was investigated in details. Zero-fi eld steps at voltages corresponding to frequencies of various modes of electromagnetic waves in the stack were found in the IV-characteristic. We showed that positions of zero-fi eld steps in IV-curves were in good agreement with predictions of the theory. The highest maximum of power of emission corresponded to the so-called in-phase mode at which all voltages over junctions in the stack oscillate in-phase. Considering stacks with many junctions, we showed that power of emission at the voltage of the resonance which corresponds to the in-phase mode is proportional to the square of quantity of long junctions in the stack that is the characteristic of coherent emission.

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Josephson junctions made from high temperature superconductors (HTS) are active high frequency devices which can be used as oscillators or frequency mixers in the submillimeter-wave region. The frequencies of these electromagnetic waves range from 300 GHz up to 3 THz. Up to now, this wave band is a relatively unexploited region of the electromagnetic spectrum because it is very difficult to produce and detect submillimeter-waves. However, submillimeter-wave technologies are becoming important in many practical scientific applications such as radio astronomy, atmospheric research, plasma diagnostics and THz-imaging, recently. In this work, the behaviour HTS Josephson junctions have been investigated experimentally and theoretically with the aim to THz-applications. In the first part the behaviour of intrinsic Josephson junction stacks are measured, and the possibility of phase locking is discussed by the theoretical analysis of two different shunt technologies depending on the parameter spread of the junction and the shunt parameters. It could be shown that stacks made from sputtered BSCCO layers does not act as intrinsic Josephson junction arrays because of their many micro shorts resulting from the layer growth. However, the stacks made from sputtered TBCCO show clear intrinsic Josephson junction behaviour. But their quasiparticle branches in the current-voltage characteristic deviate from the behaviour predicted by the common used RCSJ model. These branches can be discussed within a tunneling model using d-wave superconductor density of states. A temperature dependent current distribution and a zero bias anomaly was found. Furthermore, the coherent behaviour is studied for intrinsic arrays with additional side-wall shunts. The existence of thresholds of phase locking at especial junction and shunt parameters are demonstrated and discussed in regard to THz-oscillator applications. It was found that the Josephson junctions in a stack can be phase locked even if they have a parameter spread in the critical current up to 15% using a resonant shunt. In the second part of this work the mixing properties of HTS bicrystal Josephson junctions as wave guide mixers for 115 and 345 GHz have been investigated. (orig.)

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During last decade, considerable efforts were made to achieve coherent emission from stacks of many Josephson junctions. It is known that strong emission from a junction in the presence of external magnetic field appears at the so-called Fiske steps in the IV-characteristic at voltages which correspond to frequencies of geometrical resonances. However, it is possible to obtain resonant steps in long junctions without external magnetic field. The periodical movement of fluxons is excited due to some disorder in the distribution of critical currents along junctions. The so-called zero-field steps are formed in the IV-curve due to the interaction of fluxons with oscillations of voltage at Josephson frequencies. We investigated numerically IV-characteristics and the dependence of the average square of ac voltage at the end of the stack of two long Josephson junctions on the average voltage. Junctions interacted inductively with each other. We introduced not only the Gaussian distribution of critical currents along junctions but also the Gaussian distribution of coefficients of the interaction between junctions (mutual inductances). Zero-field steps in the IV-characteristic were found at voltages which corresponded to frequencies of in-phase collective modes in the stack as well as to frequencies of uncoupled junctions. Zero-field steps appeared in the hysteretic region of the IV-curve. There appeared also jumps of voltage from the resistive branch to the zero-field step. We showed that there existed distributions of mutual inductances along junctions which provided jumps to voltages at which the average square of ac voltage at the end of the stack (which is proportional to power of emission) was larger than that for the stack with the uniform distribution of mutual inductances.

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Dynamics of underdamped Josephson junctions with nonsinusoidal current-phase relation

Victor K. Kornev, T. Yu. Karminskaya, Yu. V. Kislinskiĭ +3 more

Results on analytical and computational investigations of high frequency dynamics of Josephson junctions, characterized by nonzero capacitance and the second harmonic in the current-phase relation are presented. These attributes each influence on behaviour of integer Shapiro steps and lead to formation of non-integer Shapiro steps. Analytic theory of the integer and non-integer Shapiro steps has been developed for so-called high frequency limit. The analytical and numerical results are compared with experimental data for hybrid heterostructures YBCO/Au/Nb. Detector response for the case of high fluctuation level has been considered as well.

Journal of Physics Conference Series•Jun 1, 2006•10 citations
We describe a variety of models to characterize the behavior of several types of underdamped Josephson systems.We also present numerical results based on these models.We first discuss a stack of small junctions in a single mode resonant cavity.Such a stack is believed to be appropriate to model some high temperature cuprate superconductors such as Bi 2 Sr 2 CaCu 2 O 8+x .Our results show self-induced-resonant-steps in the current voltage (IV) curve.These steps represent a coherent phase locking between the junctions in the stack and the resonant cavity.Next we generalize these results to treat a single long junction coupled to a single mode resonant cavity.We find a number of features not seen in stacks of small junctions.Our numerical results show that if the electromagnetic fields of the cavity mode are spatially periodic, we again see steps in the IV curve and furthermore, for sufficiently large coupling, the cavity frequency is shifted by coupling to the long junction.We also discuss how a stack of long junctions could be used to model a layered high temperature superconductor.We then turn to some properties of single junctions in the "quantum" regime where the phase/number non-commutation is important.This quantum regime is crucial to all proposals to use such junctions as solid state quantum bits (qubits).We show that such qubits, when subjected to external a.c.perturbations, exhibit not only . . . . . . . . . .
We derive dynamical equations for a Josephson array coupled to a resonant cavity by applying the Heisenberg equations of motion to a model Hamiltonian described by us earlier [Phys. Rev. B 63, 144522 (2001); 64, 179902(E) (2001)]. By means of a canonical transformation, we also show that, in the absence of an applied current and dissipation, our model reduces to one described by Shnirman et al [Phys. Rev. Lett. 79, 2371 (1997)] for coupled qubits, and that it corresponds to a capacitive coupling between the array and the cavity mode. From extensive numerical solutions of the model in one dimension, we find that the array locks into a coherent, periodic state above a critical number of active junctions; that the current-voltage characteristics of the array have self-induced resonant steps (SIRS's); that when Na{N}_{a}Na​ active junctions are synchronized on a SIRS, the energy emitted into the resonant cavity is quadratic in Na;{N}_{a};Na​; and that when a fixed number of junctions is biased on a SIRS, the energy is linear in the input power. All these results are in agreement with recent experiments. By choosing the initial conditions carefully, we can drive the array into any of a variety of different integer SIRS's. We tentatively identify terms in the equations of motion which give rise to both the SIRS's and the coherence threshold. We also find higher-order integer SIRS's and fractional SIRS's in some simulations. We conclude that a resonant cavity can produce threshold behavior and SIRS's even in a one-dimensional array with appropriate experimental parameters, and that the experimental data, including the coherent emission, can be understood from classical equations of motion.
We investigated numerically IV-characteristics and power of emission from stacks with various quantities of long Josephson junctions (up to 6 junctions) which interacted inductively with each other. Parameters of junctions were chosen close to those for MoRe-Si(W)-MoRe heterostructures. We set Gaussian spread of about 0.01% of critical currents along junctions. Electrical properties of a stack consisted of three junctions was investigated in details. Zero-fi eld steps at voltages corresponding to frequencies of various modes of electromagnetic waves in the stack were found in the IV-characteristic. We showed that positions of zero-fi eld steps in IV-curves were in good agreement with predictions of the theory. The highest maximum of power of emission corresponded to the so-called in-phase mode at which all voltages over junctions in the stack oscillate in-phase. Considering stacks with many junctions, we showed that power of emission at the voltage of the resonance which corresponds to the in-phase mode is proportional to the square of quantity of long junctions in the stack that is the characteristic of coherent emission.
Josephson junctions made from high temperature superconductors (HTS) are active high frequency devices which can be used as oscillators or frequency mixers in the submillimeter-wave region. The frequencies of these electromagnetic waves range from 300 GHz up to 3 THz. Up to now, this wave band is a relatively unexploited region of the electromagnetic spectrum because it is very difficult to produce and detect submillimeter-waves. However, submillimeter-wave technologies are becoming important in many practical scientific applications such as radio astronomy, atmospheric research, plasma diagnostics and THz-imaging, recently. In this work, the behaviour HTS Josephson junctions have been investigated experimentally and theoretically with the aim to THz-applications. In the first part the behaviour of intrinsic Josephson junction stacks are measured, and the possibility of phase locking is discussed by the theoretical analysis of two different shunt technologies depending on the parameter spread of the junction and the shunt parameters. It could be shown that stacks made from sputtered BSCCO layers does not act as intrinsic Josephson junction arrays because of their many micro shorts resulting from the layer growth. However, the stacks made from sputtered TBCCO show clear intrinsic Josephson junction behaviour. But their quasiparticle branches in the current-voltage characteristic deviate from the behaviour predicted by the common used RCSJ model. These branches can be discussed within a tunneling model using d-wave superconductor density of states. A temperature dependent current distribution and a zero bias anomaly was found. Furthermore, the coherent behaviour is studied for intrinsic arrays with additional side-wall shunts. The existence of thresholds of phase locking at especial junction and shunt parameters are demonstrated and discussed in regard to THz-oscillator applications. It was found that the Josephson junctions in a stack can be phase locked even if they have a parameter spread in the critical current up to 15% using a resonant shunt. In the second part of this work the mixing properties of HTS bicrystal Josephson junctions as wave guide mixers for 115 and 345 GHz have been investigated. (orig.)
During last decade, considerable efforts were made to achieve coherent emission from stacks of many Josephson junctions. It is known that strong emission from a junction in the presence of external magnetic field appears at the so-called Fiske steps in the IV-characteristic at voltages which correspond to frequencies of geometrical resonances. However, it is possible to obtain resonant steps in long junctions without external magnetic field. The periodical movement of fluxons is excited due to some disorder in the distribution of critical currents along junctions. The so-called zero-field steps are formed in the IV-curve due to the interaction of fluxons with oscillations of voltage at Josephson frequencies. We investigated numerically IV-characteristics and the dependence of the average square of ac voltage at the end of the stack of two long Josephson junctions on the average voltage. Junctions interacted inductively with each other. We introduced not only the Gaussian distribution of critical currents along junctions but also the Gaussian distribution of coefficients of the interaction between junctions (mutual inductances). Zero-field steps in the IV-characteristic were found at voltages which corresponded to frequencies of in-phase collective modes in the stack as well as to frequencies of uncoupled junctions. Zero-field steps appeared in the hysteretic region of the IV-curve. There appeared also jumps of voltage from the resistive branch to the zero-field step. We showed that there existed distributions of mutual inductances along junctions which provided jumps to voltages at which the average square of ac voltage at the end of the stack (which is proportional to power of emission) was larger than that for the stack with the uniform distribution of mutual inductances.
With the recent technical breakthroughs, physicists fully play with the beauty of quantum mechanics. In this theoretical work on the mesoscopic Josephson effect, we take advantage of the collaborations with experimentalists as well as the exchanges between the cold atoms and condensed matter communities. We consider various systems based on the Josephson junction, beginning with its quantum description in the underdamped regime. Using the Keldysh formalism, we obtain the current-voltage characteristics from the classical to the zero temperature regime and the quantum Smoluchowski equation in the quasi-classical limit. We then study the quantum dynamics of a phase qubit realized with a SQUID in a novel configuration where tunneling events occur through two quartic barriers. The escape rate in this camel-back potential, calculated with the instantons technique, allows us to describe the experiments. Circuit quantum electrodynamics predicts that a lasing effect appears when a qubit is coupled to a resonant cavity. We consider the case of a charge qubit and also that of a transmon which exploits the Purcell effect. With the Lindbladian we obtain the density matrix from which we derive the spectrum of the output field. Finally, we study a gas of cold atoms in a ring shape trap comprising a barrier, thus creating a Bose Josephson junction. The low energy physics is described through the correlation functions with the Luttinger liquid theory. We show that the quantum fluctuations in the ring induce a renormalization of the Josephson energy.
Results on analytical and computational investigations of high frequency dynamics of Josephson junctions, characterized by nonzero capacitance and the second harmonic in the current-phase relation are presented. These attributes each influence on behaviour of integer Shapiro steps and lead to formation of non-integer Shapiro steps. Analytic theory of the integer and non-integer Shapiro steps has been developed for so-called high frequency limit. The analytical and numerical results are compared with experimental data for hybrid heterostructures YBCO/Au/Nb. Detector response for the case of high fluctuation level has been considered as well.