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Superconductor Science and Technology

Distribution of flux states in a hysteretic dc SQUID

Kaiyong He, Liangliang Yang, Genting Dai, Jianshe Liu, Wei Chen

Published Jan 22, 2025• 0 citations

Abstract

Abstract This work investigates the distribution of flux states of hysteretic dc superconducting quantum interference devices (SQUIDs) obtained from standard current ramp measurement at 2.50 and 0.35 K. Each flux state corresponds to a specific number of flux quanta trapped in the superconducting loop, associated with a metastable state in the two-dimensional potential energy of the hysteretic SQUID. The flux states can be distinguished by the multiple switching currents. The observed distribution is different from the distribution caused only by thermal excitation at a specific temperature. There are notable systematic deviations between the measurement results of three samples and the standard Boltzmann distribution, which can be clearly identified from the ratios between the proportions of different flux states. A formula resembling the Boltzmann distribution, involving multiple effective temperatures, is proposed to provide a better fit to the measured results. The finding suggests that the observed distribution of flux states is established during the SQUID’s transition from the dissipative state to the zero-voltage state, and multiple equivalent temperatures may be involved in the process.

DOI: 10.1088/1361-6668/adad4b

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Abstract This work investigates the distribution of flux states of hysteretic dc superconducting quantum interference devices (SQUIDs) obtained from standard current ramp measurement at 2.50 and 0.35 K. Each flux state corresponds to a specific number of flux quanta trapped in the superconducting loop, associated with a metastable state in the two-dimensional potential energy of the hysteretic SQUID. The flux states can be distinguished by the multiple switching currents. The observed distribution is different from the distribution caused only by thermal excitation at a specific temperature. There are notable systematic deviations between the measurement results of three samples and the standard Boltzmann distribution, which can be clearly identified from the ratios between the proportions of different flux states. A formula resembling the Boltzmann distribution, involving multiple effective temperatures, is proposed to provide a better fit to the measured results. The finding suggests that the observed distribution of flux states is established during the SQUID’s transition from the dissipative state to the zero-voltage state, and multiple equivalent temperatures may be involved in the process.

We have measured the thermally-activated escape rate out of the zero-voltage state of hysteretic dc superconducting quantum interference devices (SQUID's) at 4.2 K. We found evidence that the zero-voltage state of these SQUID's is in general not unique, but can correspond instead to several metastable states. Each of these substates is associated with a given number of flux quanta trapped in the super-conducting loop of the SQUID's. The existence of this multiplicity is a direct consequence of the two-dimensional character of the dc-SQUID dynamics. For a fixed external magnetic flux, each zero-voltage substate is characterized by a particular value of the critical current, the expression of which can be derived from the double-cosine tilted two-dimensional SQUID potential. Experimentally, we observed multiple peaks in the switching-current distribution. Each peak was assigned to a given zero-voltage substate and we measured the dependence of the associated critical currents on the external magnetic flux. A procedure was developed in order to select a given substate among the multiplicity of the zero-voltage state. This allowed a precise measurement of the lifetime of one preselected zero-voltage substate as a function of the barrier height. Good agreement with two-dimensional transition-state theory was obtained.

The escape probability out of the superconducting state of a hysteretic dc SQUID has been measured at different values of the applied magnetic flux. At low temperatures, the escape current and the width of the probability distribution are temperature independent but they depend on flux. Experimental results do not fit the usual one-dimensional macroscopic quantum tunneling (MQT) law but are perfectly accounted for by the two-dimensional MQT behavior as we propose here. Near zero flux, our data confirms the recent MQT observation in a dc SQUID [Phys. Rev. Lett. 89, 98301 (2002)]].

A hysteretic dc-superconducting quantum interference device (SQUID) is used to trace the flux characteristic of a tunable rf-SQUID, the basic element for the realization of superconducting flux qubits. This allows important simplifications of circuitry and electronics in developing devices for quantum computing, by eliminating the necessity of more complex magnetometers. A hysteretic dc-SQUID is usually operated as a comparator, distinguishing only which one of two adjacent flux states is occupied by the rf-SQUID. The necessary sensitivity, moreover, is usually reached only at temperatures in the mK range. However, by exploiting the statistical properties of the current–voltage curve in the region where the switching from the zero-voltage state occurs, it is possible to obtain an accurate tracing of the input flux, even at a relatively high temperature (a few Kelvin). In our case, the input signal is given by the internal flux of a tunable rf-SQUID, the building block of a flux qubit.

We have realized a hysteretic dc SQUID to measure the flux states of a rf SQUID inductively coupled to the hysteretic dc SQUID. This type of measurement is required whenever is necessary to make a non invasive measure of the state of the rf SQUID. This is the case of measurements of macroscopic quantum coherence or measurements on q-bits systems. With the proposed set up the hysteretic dc SQUID can switch from the superconducting to the normal state depending on the total applied magnetic flux. On this system we measured at 4.2 K the change on the switching probability of the dc SQUID by varying the flux state of the rf SQUID. The experimental results agree very well with the theoretical predictions showing that this device can be used to perform non invasive measurements of the rf SQUID flux state.

We studied the escape process from the zero-voltage state to the running state in hysteretic dc-SQUIDs made with trilayer Nb-AlOx-Nb junctions. We analyze the experimental results, according to the thermal activation model, for devices having similar parameters except for the dimensionless inductance β L , which however is always very small. The interest in hysteretic dc-SQUIDs is in the possibility of using them to perform non-invasive measurements of the flux states in rf-SQUIDs: this achievement would be very important both in tests of macroscopic quantum properties of SQUIDs and in measurements on a class of quantum bits that has been recently proposed, based on magnetic flux states in SQUIDs.

We have measured and propose a model for switching rates in a hysteretic dc superconducting quantum interference device (SQUID) in the regime where phase-diffusion processes start to occur. We show that the switching rates in this regime are smaller than the rates given by Kramers' formula due to retrapping of the Josephson phase. The retrapping process, which is affected by the frequency dependent impedance of the environment of the dc SQUID, leads to a peaked second moment of the switching distribution as a function of temperature. The temperatures where the peaks occur are proportional to the critical current of the dc SQUID.