---
title: One-Dimensional SQUID Arrays
url: https://www.emergentmind.com/topics/one-dimensional-squid-arrays
type: topic
---

# One-Dimensional SQUID Arrays

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 [2108.11059] [1304.7371] [1411.5885] [1409.1297]. 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 [2108.11059].

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 [1304.7371].

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 [1411.5885]. 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 [1409.1297].

| Realization | Elementary cell | Reported focus |
|---|---|---|
| \((1,N_p)\) dc-SQUID/SQIF array | One row of parallel SQUID cells | Transfer function, coupling radius, noise [2108.11059] [2409.19837] |
| rf-SQUID chain in CPW | Single-junction rf-SQUID | Collective resonance, tunable \(\mu_{\rm eff}\) [1304.7371] |
| Asymmetric SQUID ring | Three-junction SQUID cell | Fluxon mobility, IVIs, depinning [1411.5885] |
| nSQUID chain | Two-junction SQUID with negative mutual inductance | Moving localized qubit states [1409.1297] |

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 \(k\),
\[
I_k(t) = I_c \sin \varphi_k(t) + \frac{\Phi_0}{2\pi R}\frac{d \varphi_k(t)}{dt} + I^n_k(t),
\]
with \(I_c\) the critical current, \(R\) the normal-state resistance, \(\varphi_k(t)\) the gauge-invariant phase difference, and \(I_k^n(t)\) Johnson thermal noise. The cited formulation explicitly includes all circulating currents in the array, the fluxes generated by those currents, and thermal-noise currents [2108.11059]. A more general vector-phase RSJ formulation for 1D parallel arrays writes the dynamics as
\[
\overrightarrow{\frac{d \varphi}{d \tau}} = \widehat{r}\left[\vec{i}_n - \widehat{i_c}\,\overrightarrow{\sin(\varphi)} + \frac{\Phi_0}{2\pi I_c}\widehat{K}\widehat{L}^{-1}\widehat{D}\vec{\varphi} + \vec{C}\right],
\]
with diagonal matrices for normalized resistances and critical currents, explicit inductive-coupling matrices, and junction-specific thermal noise strengths \(\Gamma_k\) [2211.13833].

The principal normalized variables for these arrays are
\[
i_b=\frac{I_b}{N_p I_c}, \qquad \phi_a=\frac{\Phi_a}{\Phi_0},
\]
and the time-averaged voltage \(\bar v\) in units of \(RI_c\) [2108.11059] [2211.13833]. The screening parameter is
\[
\beta_L=\frac{2L_s I_c}{\Phi_0},
\]
which controls the role of SQUID-cell inductance in both transfer-function and disorder studies [2211.13833] [2508.07685].

In rf-SQUID metamaterials, the relevant description is **RCSJ** plus magnetic coupling. For the \(n\)-th SQUID, the total flux is
\[
\Phi_n = \Phi_{\text{ext}} + L I_n + L \sum_{m\neq n}\lambda_{|m-n|} I_m,
\]
and after normalization the dynamics become
\[
\ddot{\phi}_n + \gamma \dot{\phi}_n + \beta \sin(2\pi \phi_n) = \sum_{m=1}^N (\hat{\Lambda}^{-1})_{nm}(\phi_{\text{ext}}-\phi_m),
\]
with nonlocal dipole-dipole coupling \(\lambda_{|m-n|}=\lambda_0 |m-n|^{-3}\) [1408.6072]. This nonlocality is structurally different from the finite-range current redistribution in \((1,N_p)\) sensor arrays.

For fluxon transport in asymmetric SQUID arrays, the collective coordinate is a lattice phase \(\phi_n\) governed by the **discrete double sine-Gordon equation**
\[
\ddot{\phi}_n - \kappa\, \hat{\Delta} \phi_n +\frac{2}{1+2\eta}\left ( \eta\sin \phi_n + \sin {\phi_n \over 2} \right )+ \alpha \dot{\phi}_n=\gamma,
\]
where \(\eta=I_c^{(r)}/I_c^{(l)}\) is the junction-asymmetry parameter, \(\kappa\) measures intercell coupling, and \(\gamma\) is normalized dc bias current [1411.5885].

In nSQUID arrays, the basic variables are the **common-mode** phase \(\chi\) and the **differential-mode** phase \(\phi\). 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 [1409.1297].

## 3. Transfer function, coupling radius, and field response

The standard figure of merit for 1D SQUID sensor arrays is the **maximum transfer function**
\[
\bar{v}_\phi^{\max}=\max\left(\frac{\partial \bar{v}}{\partial \phi_a}\right),
\]
evaluated at an optimal applied flux \(\phi_a^*\). For SQIF field response, the analogous quantity is the maximum slope \(\bar{v}_B^{\max}/N_s\) of \(\bar v/N_s\) versus \(B_a\) [2108.11059]. Simulations at \(77\) K used typical YBCO parameters \(I_c=20\,\mu\text{A}\), \(d=0.2\,\mu\text{m}\), \(w=2\,\mu\text{m}\), \(\lambda=0.4\,\mu\text{m}\), and an optimal bias current for uniformly biased arrays \(i_b^{\mathrm{opt}}\approx 0.75\) [2108.11059].

A central result for \((1,N_p)\) arrays is the existence of a **coupling radius** \(N_p^*\), also called the interaction radius. The reported behavior is that \(\bar{v}_\phi^{\max}\) increases with \(N_p\) at first and then **plateaus**; the onset of that plateau defines \(N_p^*\). For \(N_p>N_p^*\), the array sensitivity no longer improves appreciably. This plateauing occurs in both \((1,N_p)\) and \((3,N_p)\) geometries, and the coupling radius is reported to be **independent of the number of junctions in series** \(N_s\) for the geometries studied [2108.11059].

The coupling radius depends on the **normalised impedance** of the SQUID-loop inductance, represented by the product \(\omega l\), where
\[
l = \frac{2\pi I_c L_s}{\Phi_0}, \qquad \omega=\frac{\bar v^*}{N_s}.
\]
The reported trend is that \(N_p^*\) **increases when \(\omega l\) decreases** and decreases as \(\omega l\) increases [2108.11059]. 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
\[
B_a^*=\frac{\Phi_a^*}{a^2}
\]
is reported to be **independent of \(N_s\)**, to depend strongly on \(N_p\), and to decrease rapidly as \(N_p\) increases [2108.11059]. 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 \(\Phi_0\)-periodicity while preserving a strong central dip around \(B_a=0\). 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 \(N_s\)-normalized maximum transfer function [2108.11059].

## 4. Noise, disorder, and junction nonuniformity

The noise properties of one-dimensional SQUID arrays do not follow the simplest independent-junction picture. For high-\(T_c\) commensurate 1D arrays with \(N_s=1\), the voltage noise spectral density does **not** obey the expected scaling \(S_v(0)\propto 1/N_p\). Instead, the reported low-frequency behavior is approximately
\[
S_v(0)\sim \frac{N_s}{N_p^{0.3}},
\]
so that for \(N_s=1\), \(S_v(0)\sim N_p^{-0.3}\). By contrast, 2D arrays with larger \(N_s\) follow the expected \(S_v(0)\propto N_s/N_p\) scaling much more closely [2409.19837]. 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 [2409.19837].

Because flux noise is defined through
\[
S_{\phi}^{1/2}(f) = \frac{S_v^{1/2}(f)}{\bar{v}_{\phi}},
\]
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 \(\bar v_\phi\) do not necessarily minimize \(S_\phi^{1/2}(0)\) [2409.19837]. 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
\[
\eta (N_p, \beta_L,\Gamma, \sigma) = \frac{\langle \Delta\overline{v}(N_p,\beta_L,\Gamma,\sigma) \rangle} {\Delta\overline{v}(N_p,\beta_L,\Gamma,0)}
\]
**decreases as \(\sigma\) increases**. The reduction becomes faster when \(N_p\) increases and is more pronounced at larger \(\beta_L\), while variation of \(\Gamma\) does not significantly change the trend of \(\eta\) versus \(\sigma\) [2508.07685]. The reported design conclusion is that one should prioritize low junction-parameter spread, smaller \(N_p\), and smaller \(\beta_L\) [2508.07685].

The effect of nonuniformity depends on which junction parameter varies. When **critical currents** vary through
\[
i_{c_k}=1-\alpha+\frac{2\alpha}{N_p-1}(k-1),
\]
the maximum transfer function eventually decreases with increasing \(N_p\). When **shunt resistances** vary through
\[
\frac{1}{r_k}=1-\rho+\frac{2\rho}{N_p-1}(k-1),
\]
the reported behavior is qualitatively different: \(\bar v_\phi^{\max}\) can continue to increase with \(N_p\) beyond the plateau seen for identical-junction arrays, and voltage-response linearity increases [2211.13833]. 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 [2211.13833].

## 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 \(I_c = 1.8\,\mu\text{A}\), \(L_J \approx 183\,\text{pH}\), \(L_{\text{geo}} = 82.5\,\text{pH}\), and \(C_{\text{shunt}} = 2.0\,\text{pF}\), the resonance frequency was tunable from approximately \(9~\text{GHz}\) to \(15~\text{GHz}\), with reliable metamaterial behavior over about \(10~\text{GHz}\) to \(14.5~\text{GHz}\). The measured collective mode had \(Q_{\text{collective}} = 100\) [1304.7371].

A central methodological result was the extraction of an effective relative permeability \(\mu_{\rm r,eff}\) from the complex transmission coefficient \(S_{21}\) alone. The retrieved response had the expected resonant form: \(\operatorname{Re}(\mu_{\rm eff})\) rose from near \(1\), became **negative** near resonance, and returned toward \(1\) at higher frequency. The reported tuning range was
\[
\operatorname{Re}(\mu_{\rm eff})_{\min} = -2,\qquad \operatorname{Re}(\mu_{\rm eff})_{\max} = 3,
\]
and at \(13.83\) GHz the real part was tunable over approximately \(-1.5\) to \(+2\) [1304.7371]. 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,
\[
\lambda_{|m-n|} = \lambda_0 |m-n|^{-3},
\]
numerical integration with random initial fluxes and alternating magnetic drive produced long-lived **chimera states**, i.e. coexistence of coherent and incoherent clusters [1408.6072]. Synchronization was quantified by a Kuramoto-type order parameter
\[
\Psi(\tau)=\frac{1}{M}\sum_{m=1}^M e^{i[2\pi \phi_m(\tau)]},
\]
with \(|\Psi|=1\) for complete synchronization and \(|\Psi|=0\) for complete desynchronization [1408.6072].

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 [1408.6072]. 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.

## 6. Fluxon transport, quantum-information proposals, and related Josephson-network analogues

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**
\[
v=\{v_0\equiv 0, v_1,v_2,\ldots,v_k\},
\]
at which a fluxon propagates with constant shape and without radiation [1411.5885]. These velocities appear experimentally through **inaccessible voltage intervals** in the current-voltage characteristics, because for uniform fluxon motion
\[
\bar{V}=\frac{4\pi v}{N}.
\]
The same work reports that the critical depinning current is nonmonotonic in the asymmetry parameter \(\eta\) and has a clear minimum that coincides with the minimum of the **Peierls–Nabarro barrier** [1411.5885]. 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
\[
\chi(x,t) =4 \tan^{-1} \left[\exp\left(\frac{x-vt}{\lambda_0}\right)\right],
\]
and a **differential mode**, which in the fluxon background experiences a Pöschl–Teller-type potential well and therefore supports localized bound states [1409.1297]. After quantization, the lowest localized mode defines a two-state subspace
\[
\{ |0\rangle\, ,a^{\dagger}_0 |0\rangle \},
\]
so that a moving fluxon carries a qubit-like internal excitation [1409.1297].

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 \(1/v\) in the rapid-motion regime. For the representative estimate quoted in the paper, a stationary-qubit dephasing time \(t_d\sim 200\) ns together with propagation frequency \(\nu\sim 1\) GHz yields a moving-qubit dephasing time \(t_d^{(m)}\sim 10\,\mu\)s [1409.1297]. 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
\[
I_c(d)=I_0\left(\frac{d_0}{d}\right)^2, \qquad E_J=\frac{I_c\Phi_0}{2\pi},
\]
in the low-temperature approximation [2208.09084]. 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 [2208.09084]. 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.

Source: https://www.emergentmind.com/topics/one-dimensional-squid-arrays