Synthetic areas spread in two-dimensional Superconducting Quantum Interference Arrays

TL;DR

The paper addresses the challenge of achieving absolute magnetometry with 2D SQUID arrays without sacrificing per-loop inductive optimization. It introduces synthetic area spread by inserting bare superconducting loops, and develops a RSJ-based framework showing that the array dynamics are governed by a synthetic area vector $\underline{a}' = \underline{a}_J + C\underline{a}_B$ instead of the physical loop areas. The authors provide analytical derivations, numerical simulations, and experimental validation with Nb-based 2D SQUID arrays that exhibit a robust anti-peak VMF response near zero flux, in agreement with theory. This method enables absolute quantum sensing in large-scale SQUID arrays without degrading performance, opening pathways to ultra-high-performance integrated sensors and approaching the quantum limit of detection.

Abstract

Superconducting Quantum Interference Devices (SQUIDs), formed by incorporating Josephson junctions into loops of superconducting material, are the backbone of many modern quantum sensing systems. It has been demonstrated that, by combining multiple SQUID loops into a two-dimensional (2D) array, it is possible to fabricate ultra-high-performing Radio frequency sensors. However, to function as absolute magnetometers, current-in-use arrays require the area of each SQUID loop in the array to be incommensurate and, in turn, forbid the achievement of their full potential in terms of quantum-limited performances. This is because imposing incommensurability in the areas contrasts with optimised performance in each single SQUID loop. In this work, we report that by selectively inserting bare sections of a superconducting circuit with no Josephson junctions, 2D SQUID arrays can operate as an absolute magnetometer even when no physical area spread is applied. Based on a generalisation of current available theories, a complete analytical formulation for the one-to-one correspondence between the distribution of these bare loops and what we call a synthetic area spread is unveiled. This synthetic spread represents the equivalent physical spread of incommensurate SQUID loops that you will use to obtain the absolute Voltage-Magnetic Flux response if no bare loops were in use. Our work opens the way to a broader use of this technology for the fabrication of ultra-high-performance absolute quantum sensors. Our approach is also experimentally verified by fabricating several 2D SQUID arrays incorporating bare superconducting loops and by demonstrating that they behave in alignment with what is suggested by our theory.

Figures (9)

• Figure 1: a) Schematic of a prototypical 2D SQUID array containing both bare loops and SQUID loops. The numbering systems of the bias currents ($I^b_i$), the horizontal wires ($J_i$), the vertical wires ($I_i$), and the final row of horizontal wires ($I^F_i$) are labelled; the notation is identical to the circuit diagram of Ref. galilabariasModelingTransferFunction2022 and is also described more in detail in the Appendix \ref{['sec:2D']}. In this notation, two vertical wires i-1 and i with Josephson junctions will surround the generic $i^{th}$ SQUID and the magnetic field $\vec{B}$ is directed coming off the page. b) Illustration for a possible kind of unit-cell for the experimental array considered in this work. It consists of two bare loops surrounding a SQUID loop with the superconducting material shown in silver. Within the illustration, the red regions are vertical Josephson junctions, whilst the blue regions are made of insulating material. These junctions are normally shunted (not shown) in real circuits oppenlanderTwoDimensionalSuperconducting2003mitchell2DSQIFArrays2016kornevSQIF2015. A circuit schematic for this cell is also in the top left side.
• Figure 2: (a) The Voltage-Magnetic Flux (VMF) response of a $3\times 2$ array with equal loop areas is shown. (b) The VMF response of a $5\times 2$ array containing two rows of solely bare loops is shown with black diamonds; each loop has an equal area as depicted in the circuit diagram. The VMF response of a $3\times 2$ array where the loop area is given by the synthetic areas derived from the original circuit is overlain; this was performed by solving Eq. \ref{['eq::syntheticArea']}. The schematics for these different situations are also shown in the figure.
• Figure 3: Shown with the black empty diamonds is the Voltage-Magnetic Flux (VMF) response of a $10\times 10$ array containing a) one or b) three rows of solely bare loops on each loop has an equal area as depicted in the left circuit diagram. Shown with the black full circles is the VMF response of arrays where the loop area is given by the synthetic areas derived from the two different configurations a) and b); this was performed by solving Eq. \ref{['eq::syntheticArea']}. For both these Figures, the differences between the bare loop and synthetic area approaches are minimal.
• Figure 4: Experimental VMF curves, for different biases close to the critical one, for the type A $16\times16$ device consisting of the unit-cell depicted in Fig. \ref{['fig::Figure1']} and with two lines of bare loops between each line of SQUIDs. Although all the SQUID loops and all the bare loops in this structure have identical areas, a strong anti-peak response visible indicates the appearance of synthetic loop areas induced by the bare loops present in the circuit. The data for a device with an identical design and a demonstrated identical low temperature response are shown in Fig. 2 of Appendix \ref{['sec::Moreresults']}.
• Figure 5: Experimental VMF curves for different biases close to the critical one and for a device that has a similar design (but not identical) as the one in Figure \ref{['fig::expData']}, but no bare loops. As expected, the VMF curves of the device are not anti-peaked.