One-Dimensional SQUID Arrays

Updated 8 July 2026

One-dimensional SQUID arrays are superconducting circuits composed of a linear arrangement of SQUID or Josephson junctions that enable controlled flux-to-voltage transfer and collective resonance.
They encompass diverse designs—from dc-SQUID sensor arrays to rf-SQUID metamaterials and nSQUID chains—that are characterized by metrics such as coupling radius, noise scaling, and transfer function behavior.
These arrays are applied in high-sensitivity magnetometry and quantum information, demonstrating phenomena like tunable permeability, chimera states, and fluxon transport with engineered fluxon dynamics.

One-dimensional SQUID arrays are superconducting circuits in which SQUID-based unit cells, or closely related Josephson elements, are arranged along a single spatial direction and operated collectively. In the literature, the term spans several distinct realizations: one-row dc-SQUID and SQIF sensor arrays written as $(1,N_p)$ , chains of rf-SQUIDs embedded in microwave transmission lines, dc-biased rings of asymmetric three-junction SQUID cells that support fluxon transport, and nSQUID chains with negative mutual inductance that separate transport and information-bearing modes (Labarias et al., 2021, Butz et al., 2013, Zolotaryuk et al., 2014, Deng et al., 2014). Across these realizations, the central observables are the flux-to-voltage transfer function, voltage and flux noise, collective resonance, effective permeability, current-voltage characteristics, and the dynamics of localized excitations.

1. Architectural forms and geometric definitions

In magnetometer-oriented work, a one-dimensional SQUID array is typically a single row of $N_p$ Josephson junctions in parallel, denoted $(1,N_p)$ . The comparison two-dimensional geometry is $(N_s,N_p)$ , with $N_s=3$ series-connected rows in the cited calculations. These arrays are modeled as grid-like structures of square SQUID loops with equal loop height and width, $a=b$ , under uniform bias-current injection and with the output voltage measured between designated array terminals (Labarias et al., 2021).

In rf-SQUID metamaterials, the one-dimensionality is geometric rather than merely topological. A demonstrated implementation consisted of two one-dimensional arrays of 27 rf-SQUIDs each placed in the two gaps of a coplanar waveguide, giving a total of 54 SQUIDs along the microwave propagation path. The pitch between neighboring SQUIDs was $92~\mu\text{m}$ , about twice the SQUID width, and neighbor-to-neighbor inductive coupling was reported to be much weaker than coupling to the coplanar waveguide, so the devices were treated as weakly interacting resonators rather than a strongly synchronized array (Butz et al., 2013).

Other one-dimensional realizations change the unit cell rather than the layout. In the asymmetric-array problem, each cell is a SQUID with two junctions in the left arm and one in the right arm; because the analysis assumes small loop size and ignores mutual inductances between cells, the dynamics reduce to a nearest-neighbor lattice along a single chain (Zolotaryuk et al., 2014). In the nSQUID proposal, each cell is a two-junction SQUID with negative mutual inductance between its arms, creating distinct common and differential modes along a one-dimensional chain (Deng et al., 2014).

Realization	Elementary cell	Reported focus
$(1,N_p)$ dc-SQUID/SQIF array	One row of parallel SQUID cells	Transfer function, coupling radius, noise (Labarias et al., 2021, Nieves et al., 2024)
rf-SQUID chain in CPW	Single-junction rf-SQUID	Collective resonance, tunable $\mu_{\rm eff}$ (Butz et al., 2013)
Asymmetric SQUID ring	Three-junction SQUID cell	Fluxon mobility, IVIs, depinning (Zolotaryuk et al., 2014)
nSQUID chain	Two-junction SQUID with negative mutual inductance	Moving localized qubit states (Deng et al., 2014)

This diversity is important because “one-dimensional SQUID array” does not imply a single canonical device. It instead denotes a family of reduced geometries in which longitudinal coupling, collective response, and finite interaction range can be studied with greater control than in planar two-dimensional arrays.

2. Circuit models and collective variables

For 1D sensor arrays, the standard description is an overdamped RSJ model. For each junction $k$ ,

$N_p$ 0

with $N_p$ 1 the critical current, $N_p$ 2 the normal-state resistance, $N_p$ 3 the gauge-invariant phase difference, and $N_p$ 4 Johnson thermal noise. The cited formulation explicitly includes all circulating currents in the array, the fluxes generated by those currents, and thermal-noise currents (Labarias et al., 2021). A more general vector-phase RSJ formulation for 1D parallel arrays writes the dynamics as

$N_p$ 5

with diagonal matrices for normalized resistances and critical currents, explicit inductive-coupling matrices, and junction-specific thermal noise strengths $N_p$ 6 (Labarias et al., 2022).

The principal normalized variables for these arrays are

$N_p$ 7

and the time-averaged voltage $N_p$ 8 in units of $N_p$ 9 (Labarias et al., 2021, Labarias et al., 2022). The screening parameter is

$(1,N_p)$ 0

which controls the role of SQUID-cell inductance in both transfer-function and disorder studies (Labarias et al., 2022, Nieves et al., 11 Aug 2025).

In rf-SQUID metamaterials, the relevant description is RCSJ plus magnetic coupling. For the $(1,N_p)$ 1-th SQUID, the total flux is

$(1,N_p)$ 2

and after normalization the dynamics become

$(1,N_p)$ 3

with nonlocal dipole-dipole coupling $(1,N_p)$ 4 (Lazarides et al., 2014). This nonlocality is structurally different from the finite-range current redistribution in $(1,N_p)$ 5 sensor arrays.

For fluxon transport in asymmetric SQUID arrays, the collective coordinate is a lattice phase $(1,N_p)$ 6 governed by the discrete double sine-Gordon equation

$(1,N_p)$ 7

where $(1,N_p)$ 8 is the junction-asymmetry parameter, $(1,N_p)$ 9 measures intercell coupling, and $(N_s,N_p)$ 0 is normalized dc bias current (Zolotaryuk et al., 2014).

In nSQUID arrays, the basic variables are the common-mode phase $(N_s,N_p)$ 1 and the differential-mode phase $(N_s,N_p)$ 2. In the strong negative-coupling limit, the common mode supports fluxons, while the differential mode supports localized excitations in the fluxon background. This mode separation is the basis for the proposed dual-rail structure (Deng et al., 2014).

3. Transfer function, coupling radius, and field response

The standard figure of merit for 1D SQUID sensor arrays is the maximum transfer function

$(N_s,N_p)$ 3

evaluated at an optimal applied flux $(N_s,N_p)$ 4. For SQIF field response, the analogous quantity is the maximum slope $(N_s,N_p)$ 5 of $(N_s,N_p)$ 6 versus $(N_s,N_p)$ 7 (Labarias et al., 2021). Simulations at $(N_s,N_p)$ 8 K used typical YBCO parameters $(N_s,N_p)$ 9, $N_s=3$ 0, $N_s=3$ 1, $N_s=3$ 2, and an optimal bias current for uniformly biased arrays $N_s=3$ 3 (Labarias et al., 2021).

A central result for $N_s=3$ 4 arrays is the existence of a coupling radius $N_s=3$ 5, also called the interaction radius. The reported behavior is that $N_s=3$ 6 increases with $N_s=3$ 7 at first and then plateaus; the onset of that plateau defines $N_s=3$ 8. For $N_s=3$ 9, the array sensitivity no longer improves appreciably. This plateauing occurs in both $a=b$ 0 and $a=b$ 1 geometries, and the coupling radius is reported to be independent of the number of junctions in series $a=b$ 2 for the geometries studied (Labarias et al., 2021).

The coupling radius depends on the normalised impedance of the SQUID-loop inductance, represented by the product $a=b$ 3, where

$a=b$ 4

The reported trend is that $a=b$ 5 increases when $a=b$ 6 decreases and decreases as $a=b$ 7 increases (Labarias et al., 2021). Smaller loop size also gives a larger transfer function. These results directly constrain how far one can scale a one-row array before collective interaction ceases to produce useful gains.

The corresponding optimal applied magnetic field

$a=b$ 8

is reported to be independent of $a=b$ 9, to depend strongly on $92~\mu\text{m}$ 0, and to decrease rapidly as $92~\mu\text{m}$ 1 increases (Labarias et al., 2021). A common misconception is therefore that adding parallel junctions simply raises sensitivity at a fixed operating field. The cited calculations instead show that the operating point shifts substantially as the number of coupled parallel cells grows.

For SQIFs, intentional loop-area nonuniformity removes strict $92~\mu\text{m}$ 2-periodicity while preserving a strong central dip around $92~\mu\text{m}$ 3. In the tested arrays, the 1D SQIF and the 2D SQIF with equal row heights had similar main-dip behavior, while a 2D SQIF with varying row heights suppressed secondary oscillations more strongly and showed a slightly higher $92~\mu\text{m}$ 4-normalized maximum transfer function (Labarias et al., 2021).

4. Noise, disorder, and junction nonuniformity

The noise properties of one-dimensional SQUID arrays do not follow the simplest independent-junction picture. For high- $92~\mu\text{m}$ 5 commensurate 1D arrays with $92~\mu\text{m}$ 6, the voltage noise spectral density does not obey the expected scaling $92~\mu\text{m}$ 7. Instead, the reported low-frequency behavior is approximately

$92~\mu\text{m}$ 8

so that for $92~\mu\text{m}$ 9, $(1,N_p)$ 0. By contrast, 2D arrays with larger $(1,N_p)$ 1 follow the expected $(1,N_p)$ 2 scaling much more closely (Nieves et al., 2024). The proposed physical reason is the emergence of a finite Josephson-junction interaction radius, so that noise sources in a 1D commensurate array are not effectively independent at low frequencies (Nieves et al., 2024).

Because flux noise is defined through

$(1,N_p)$ 3

departures from ideal voltage-noise scaling combine with transfer-function changes. The cited work therefore states that transfer function and noise do not in general optimize at the same bias point; the values of bias current and applied flux that maximize $(1,N_p)$ 4 do not necessarily minimize $(1,N_p)$ 5 (Nieves et al., 2024). A plausible implication is that 1D-array optimization cannot be reduced to a single-response metric.

Fabrication disorder introduces a second nonideal mechanism. In simulations of one-dimensional SQUID arrays with log-normal critical-current spread, the performance metric

$(1,N_p)$ 6

decreases as $(1,N_p)$ 7 increases. The reduction becomes faster when $(1,N_p)$ 8 increases and is more pronounced at larger $(1,N_p)$ 9, while variation of $\mu_{\rm eff}$ 0 does not significantly change the trend of $\mu_{\rm eff}$ 1 versus $\mu_{\rm eff}$ 2 (Nieves et al., 11 Aug 2025). The reported design conclusion is that one should prioritize low junction-parameter spread, smaller $\mu_{\rm eff}$ 3, and smaller $\mu_{\rm eff}$ 4 (Nieves et al., 11 Aug 2025).

The effect of nonuniformity depends on which junction parameter varies. When critical currents vary through

$\mu_{\rm eff}$ 5

the maximum transfer function eventually decreases with increasing $\mu_{\rm eff}$ 6. When shunt resistances vary through

$\mu_{\rm eff}$ 7

the reported behavior is qualitatively different: $\mu_{\rm eff}$ 8 can continue to increase with $\mu_{\rm eff}$ 9 beyond the plateau seen for identical-junction arrays, and voltage-response linearity increases (Labarias et al., 2022). This separates two effects that are often conflated. Critical-current spread is detrimental, while deliberately engineered resistance asymmetry can be beneficial for linearity and transfer-function scaling in the studied regime (Labarias et al., 2022).

5. Metamaterial behavior and non-equilibrium collective dynamics

One-dimensional rf-SQUID arrays can function as magnetic metamaterials rather than only as magnetometers. In the demonstrated 54-element chain, the Josephson inductance of each rf-SQUID made the resonance frequency flux-tunable in situ. Using $k$ 0, $k$ 1, $k$ 2, and $k$ 3, the resonance frequency was tunable from approximately $k$ 4 to $k$ 5, with reliable metamaterial behavior over about $k$ 6 to $k$ 7. The measured collective mode had $k$ 8 (Butz et al., 2013).

A central methodological result was the extraction of an effective relative permeability $k$ 9 from the complex transmission coefficient $N_p$ 00 alone. The retrieved response had the expected resonant form: $N_p$ 01 rose from near $N_p$ 02, became negative near resonance, and returned toward $N_p$ 03 at higher frequency. The reported tuning range was

$N_p$ 04

and at $N_p$ 05 GHz the real part was tunable over approximately $N_p$ 06 to $N_p$ 07 (Butz et al., 2013). This establishes a 1D SQUID array as a homogenized magnetic medium with field-tunable effective parameters, not merely as a collection of isolated resonators.

The same rf-SQUID setting also supports strongly nontrivial driven states. In a one-dimensional SQUID metamaterial with nonlocal dipole-dipole coupling,

$N_p$ 08

numerical integration with random initial fluxes and alternating magnetic drive produced long-lived chimera states, i.e. coexistence of coherent and incoherent clusters (Lazarides et al., 2014). Synchronization was quantified by a Kuramoto-type order parameter

$N_p$ 09

with $N_p$ 10 for complete synchronization and $N_p$ 11 for complete desynchronization (Lazarides et al., 2014).

The comparison with local coupling is significant. The cited study reports that locally coupled arrays can form nonuniform states, but these are not chimera states; instead, all clusters remain internally synchronized, even if different clusters are not synchronized with one another (Lazarides et al., 2014). This suggests that in 1D SQUID metamaterials, dimensional reduction does not eliminate complex spatiotemporal dynamics; rather, the decisive distinction is between local and nonlocal coupling.

In dc-biased asymmetric SQUID arrays, the relevant nonlinear excitations are fluxons. In the Hamiltonian discrete double-sine-Gordon lattice, the continuum family of kink speeds collapses to a finite set of special sliding velocities

$N_p$ 12

at which a fluxon propagates with constant shape and without radiation (Zolotaryuk et al., 2014). These velocities appear experimentally through inaccessible voltage intervals in the current-voltage characteristics, because for uniform fluxon motion

$N_p$ 13

The same work reports that the critical depinning current is nonmonotonic in the asymmetry parameter $N_p$ 14 and has a clear minimum that coincides with the minimum of the Peierls–Nabarro barrier (Zolotaryuk et al., 2014). A common simplification is to relate depinning only to the local on-site potential; the cited result shows that the effective periodic pinning landscape for the fluxon center is the more relevant quantity.

The nSQUID array proposal extends one-dimensional SQUID physics from classical transport to moving quantum degrees of freedom. Its key construction is the separation into a common mode, which supports a sine-Gordon-like fluxon

$N_p$ 15

and a differential mode, which in the fluxon background experiences a Pöschl–Teller-type potential well and therefore supports localized bound states (Deng et al., 2014). After quantization, the lowest localized mode defines a two-state subspace

$N_p$ 16

so that a moving fluxon carries a qubit-like internal excitation (Deng et al., 2014).

The decoherence analysis in that proposal is also specifically one-dimensional. Motion spreads low-frequency noise over a larger frequency interval, giving a low-frequency noise level that scales as $N_p$ 17 in the rapid-motion regime. For the representative estimate quoted in the paper, a stationary-qubit dephasing time $N_p$ 18 ns together with propagation frequency $N_p$ 19 GHz yields a moving-qubit dephasing time $N_p$ 20s (Deng et al., 2014). This suggests a route by which one-dimensional transport may itself contribute to decoherence suppression.

A related but distinct use of one-dimensional Josephson-array language appears in engineered Nb nano-island arrays on Au thin films. There, scanning SQUID susceptometry was compared with a model that treats the system as a network of one-dimensional SNS Josephson junctions, each with

$N_p$ 21

in the low-temperature approximation (Horn et al., 2022). Although the physical sample is a square lattice rather than a one-dimensional SQUID array, the study shows that low-field magnetic response can be understood quantitatively through junction-length-dependent coupling, while higher-field nonlinearity and dissipation are associated with vortex entry and motion (Horn et al., 2022). This suggests a broader methodological connection: one-dimensional Josephson-element models remain useful even when the experimental platform is not itself a literal 1D SQUID chain.