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Physical review. B, Solid state

Measurement of the probability distribution of thermally excited fluxoid quantum transitions in a superconducting ring closed by a Josephson junction

L. D. Jackel, Watt W. Webb, James E. Lukens, S. S. Pei

Published Jan 1, 1974• 72 citations
Publisher

Abstract

The thermally induced distribution of transitions between fluxoid quantum states of a superconducting ring closed by a Josephson junction has been measured. The observed dependence of its width $\ensuremath{\sigma}$ on temperature, ring inductance, and weak-link critical current, as well as its shape, confirms the recent theoretical results of Kurkij\"arvi, $\ensuremath{\sigma}\ensuremath{\propto}L{({i}_{c})}^{\frac{1}{3}}{T}^{\frac{2}{3}}$.

DOI
10.1103/physrevb.9.115

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Quantum States and Transitions in Weakly Connected Superconducting Rings

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This paper reports the results of an experimental and phenomenological investigation of the static and dynamic behavior of weakly connected superconducting rings. The configuration is essentially a macroscopic superconducting ring of inductance $L$ incorporating a point contact as a weak link which determines the critical supercurrent in the ring, ${i}_{c}$. A phenomenological model for the stationary quantum states of the system as a function of an applied field is developed. The dynamic behavior is obtained directly from the time dependence of the applied fields. Experiments demonstrating both the stationary and the time-dependent magnetic behavior are described. The stationary behavior was obtained with a magnetometer incorporating a weakly connected ring as a sensor to measure the flux through the ring under test. The experimental results confirm the phenomenological model if the critical curr\`ent ${i}_{c}$ is greater than $\frac{{\ensuremath{\Phi}}_{0}}{2L(1+\ensuremath{\gamma})}$, where ${\ensuremath{\Phi}}_{0}=\frac{h}{2e}$ is the flux quantum and $\ensuremath{\gamma}$ is a material and geometric parameter which is usually small compared to unity. In the regime $L{i}_{c}>\frac{{\ensuremath{\Phi}}_{0}}{2(1+\ensuremath{\gamma})}$, the quantum states are discrete, and the transitions between states are well defined and irreversible. If the critical current is not too large, the transitions generally occur only between adjacent states; that is, $\ensuremath{\Delta}k=\ifmmode\pm\else\textpm\fi{}1$. At large critical current, multiple quantum jumps are observed. On the other hand, if $L{i}_{c}<\frac{{\ensuremath{\Phi}}_{0}}{2(1+\ensuremath{\gamma})}$, the quantum states merge into one another continuously and reversibly. In this case the magnetic behavior inthe neighborhood of the half-quantum points is related to the depairing or gapless regime in superconductivity. Measurements of the ac properties of the weakly connected ring at 30 MHz are interpreted directly in terms of the static properties under the influence of a time-varying applied field. In fact, no qualitative corrections to the theory are expected up to frequencies of the order of the superconducting energy gap.

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arXiv:0801.3761•Low Temperature Physics•Jun 1, 2008•2 citations
The thermally induced distribution of transitions between fluxoid quantum states of a superconducting ring closed by a Josephson junction has been measured. The observed dependence of its width \ensuremathσ\ensuremath{\sigma}\ensuremathσ on temperature, ring inductance, and weak-link critical current, as well as its shape, confirms the recent theoretical results of Kurkij"arvi, \ensuremathσ\ensuremath∝L(ic)13T23\ensuremath{\sigma}\ensuremath{\propto}L{({i}_{c})}^{\frac{1}{3}}{T}^{\frac{2}{3}}\ensuremathσ\ensuremath∝L(ic​)31​T32​.
This paper reports the results of an experimental and phenomenological investigation of the static and dynamic behavior of weakly connected superconducting rings. The configuration is essentially a macroscopic superconducting ring of inductance LLL incorporating a point contact as a weak link which determines the critical supercurrent in the ring, ic{i}_{c}ic​. A phenomenological model for the stationary quantum states of the system as a function of an applied field is developed. The dynamic behavior is obtained directly from the time dependence of the applied fields. Experiments demonstrating both the stationary and the time-dependent magnetic behavior are described. The stationary behavior was obtained with a magnetometer incorporating a weakly connected ring as a sensor to measure the flux through the ring under test. The experimental results confirm the phenomenological model if the critical curr\`ent ic{i}_{c}ic​ is greater than $\frac{{\ensuremath{\Phi}}_{0}}{2L(1+\ensuremath{\gamma})}$, where ${\ensuremath{\Phi}}_{0}=\frac{h}{2e}$ is the flux quantum and $\ensuremath{\gamma}$ is a material and geometric parameter which is usually small compared to unity. In the regime $L{i}_{c}>\frac{{\ensuremath{\Phi}}_{0}}{2(1+\ensuremath{\gamma})}$, the quantum states are discrete, and the transitions between states are well defined and irreversible. If the critical current is not too large, the transitions generally occur only between adjacent states; that is, $\ensuremath{\Delta}k=\ifmmode\pm\else\textpm\fi{}1$. At large critical current, multiple quantum jumps are observed. On the other hand, if $L{i}_{c}<\frac{{\ensuremath{\Phi}}_{0}}{2(1+\ensuremath{\gamma})}$, the quantum states merge into one another continuously and reversibly. In this case the magnetic behavior inthe neighborhood of the half-quantum points is related to the depairing or gapless regime in superconductivity. Measurements of the ac properties of the weakly connected ring at 30 MHz are interpreted directly in terms of the static properties under the influence of a time-varying applied field. In fact, no qualitative corrections to the theory are expected up to frequencies of the order of the superconducting energy gap.
We report on measurements of the switching current probability both in the thermal and quantum regime on moderately damped NbN/MgO/NbN Josephson junctions with very low critical current density (’ 2A/cm2). We observed three distinct regimes: below 90 mK the width of the switching distributions is constant within the experimental error, while from 90 mK to 1.6 K the width increases with temperature, this change of behavior signaling the standard transition from macroscopic quantum tunneling (MQT) to thermal activation (TA) regime respectively. Above 1.6 K we find effects of anti-correlation between the bath temperature and thermal fluctuations since the width of the histograms starts to collapse, and indications of a transition to a third regime called Phase Diffusion (PD) are evident, such as the change of the switching distributions symmetry. Experimental data have been analyzed through numerical fitting of the switching probability and Monte Carlo simulations, and we found a good agreement with theoretical expectations based on multiple re-trapping processes, which strongly depend on the junction damping. The parameters which completely characterize the dynamics of such moderately damped Josephson junctions have been also extracted.
Superconducting closed loops composed of N series Josephson junctions are discussed. The conditions to achieve the n quantum state of a fluxoid in the loop are analyzed. It is found that the n quantum state exists in Josephson loops having more than 4n pieces of junctions even if it is assumed that the inductance L in the loop is negligibly small. This n quantum state is achieved by an asymmetric current feed to the Josephson loop. The properties of the loop are easily explained by using a pendulum analog.
We have studied the temperature dependence of escape phenomena in various underdamped Josephson junctions (JJs). The junctions had different Josephson coupling energies E J which were relatively small, but larger than the charging energy E C . Upon increasing the temperature T , we first observe the usual cross-over between macroscopic quantum tunnelling and thermally activated (TA) behaviour at temperatures k B T ∼ ℏω p , where ω p is the plasma frequency of the junction. Increasing T further, the width of the switching current distribution has, counterintuitively, a non-monotonic temperature dependence. This can be explained by the novel cross-over from TA behaviour to underdamped phase diffusion. We show that this cross-over is expected to occur at temperatures such that k B T ∼ E J (1 − 4/π Q ) 3 /2, where Q is the quality factor of the junction at the plasma frequency, in agreement with experiment. Our findings can be compared with detailed model calculations which take into account dissipation and level quantization in a metastable well. Particular attention is paid to the sample with the smallest E J , which shows extensive phase diffusion even at the lowest temperatures. This sample consists of a dc-SQUID and a single JJ close to each other, such that the SQUID acts as a tunable inductive protection for the single junction from fluctuations of a dissipative environment. By varying the flux through the dc-SQUID, we present, for the first time, experimental evidence of the escape of a JJ from the phase diffusion regime to the free running state in a tunable environment. We also show that in the zero voltage state the losses mainly occur at frequencies near the plasma resonance.
Superconducting quantum circuits (SQCs) are being explored as model systems for scalable quantum computing architectures. Josephson junctions are extensively used in superconducting quantum interference devices (SQUIDs) and in persistent-current qubit systems. Noise excitations, however, have a critical influence on their dynamics. Thus, the primary focus of this research was to investigate the effects of thermal ac-tivation on the superconducting properties of Josephson junctions. Specifically, ther-mal noise tends to result in a range of switching currents, values less than the critical current at which a junction switches from the superconducting to the normal state. First, a general review of superconductivity concepts is given, including a treatment of the Josephson phenomena. Next, I describe some of my work on characterizing the current-voltage traces of Josephson junctions tested at 4 K with a Multi-Chip Probe (MCP). Then, I describe thermal activation theory and examine the equations useful for modeling switching current distributions. The Josephson junctions of a SQUID with a ramped bias current were tested for numerous temperatures T < 4.5 K (and
We analyze the influence of noise on magnetic properties of a superconducting loop which contains three Josephson junctions. This circuit is a classical analog of a persistent current (flux) qubit. A loop supercurrent induced by external magnetic field in the presence of thermal fluctuations is calculated. To connect with experiment, we calculate the impedance of a low-frequency tank circuit which is inductively coupled with the loop of interest. We compare the results with the results obtained in the quantum mode — when the three junction loop exhibits quantum tunneling of the magnetic flux. We demonstrate that the tank–loop impedance in the classical and quantum modes have different temperature dependence and can be easily distinguished experimentally.