Coherent oscillations of driven rf SQUID metamaterials

Through experiments and numerical simulations we explore the behavior of rf SQUID (radio frequency superconducting quantum interference device) metamaterials, which show extreme tunability and nonlinearity. The emergent electromagnetic properties of this metamaterial are sensitive to the degree of coherent response of the driven interacting SQUIDs. Coherence suffers in the presence of disorder, which is experimentally found to be mainly due to a dc flux gradient. We demonstrate methods to recover the coherence, specifically by varying the coupling between the SQUID meta-atoms and increasing the temperature or the amplitude of the applied rf flux.

Figure 1

(a) Experimental setup of waveguide transmission measurement with inset showing schematic of an rf SQUID including the RCSJ model for the Josephson junction. (b) Portion of the Hypres $27\times 27$ array along with SQUID design with lengths given in micrometers. (c) Design of the IREE SQUID from the $21\times 21$ array with lengths given in micrometers and $r_{\text{JJ}}$ denoting the radius of the Josephson junction.

Figure 2

(a) Measured and (b) simulated transmission for the $21\times 21$ array with $|{\kappa }_0|=0.02$ as a function of frequency and dc flux in the linearized limit of Eq. (1), ${\mathrm{\Phi }}_{\text{rf}}/{\mathrm{\Phi }}_0\ll 1$ (see Supplemental Material [33]). The simulation has a dc flux gradient such that one edge of the array experiences the ${\mathrm{\Phi }}_{\text{dc}}/{\mathrm{\Phi }}_0$ value shown and the other edge is $90\%$ of that value. Inset curves show simulated coherence $r_A$ as a function of applied dc flux with (white) and without (yellow) the flux gradient.

Figure 3

Numerical simulation for the range of frequency tunability in the real part of the effective relative permeability as a function of coherence for eight noninteracting $21\times 21$ arrays with $|{\kappa }_0|=0.02$. The coherence was varied by applying a dc flux gradient. The black portion of the curve is where the minimum effective relative permeability is negative. Inset: simulated real part of effective relative permeability as a function of frequency illustrating how the range of effective relative permeability is defined.

Figure 4

(a) Numerically simulated coherence of a $21\times 21$ rf SQUID array as a function of coupling on the primary resonance at ${\mathrm{\Phi }}_{\text{dc}}/{\mathrm{\Phi }}_0=2$ for three different flux gradients in the limit ${\mathrm{\Phi }}_{\text{rf}}/{\mathrm{\Phi }}_0\ll 1$. Insets: Simulated spatial distribution of amplitude (color) and phase (dashed contour line at ${\theta }_j=0$) of $\widehat{\delta }\left(t\right)$ with a gradient such that one edge of the array has $15\%$ of the dc flux of the other edge. (b),(c) Simulated transmission and coherence vs frequency for two different coupling values.

Figure 5

(a)–(c) Measured transmission as a function of frequency and dc flux at three values of rf flux for $27\times 27$ array $T=7$ K. (d) Local minima (normalized by the global minimum) in measured transmission as a function of dc flux $S_{21}\left({\mathrm{\Phi }}_{\text{dc}}\right)$ at the geometric resonant frequency ${\omega }_{\text{geo}}/2\pi$ for (red) ${log}_{10}({\mathrm{\Phi }}_{\text{rf}}/{\mathrm{\Phi }}_0)=-3$, (black) ${log}_{10}({\mathrm{\Phi }}_{\text{rf}}/{\mathrm{\Phi }}_0)=-1.5$, and (blue) if no flux gradient were present, independent of rf flux.

Figure 6

(a)–(c) Measured transmission as a function of frequency and dc flux at three temperatures for the $27\times 27$ array with $log{\mathrm{\Phi }}_{\text{rf}}/{\mathrm{\Phi }}_0=-2$. (d) Local minima (normalized by the global minimum) in measured transmission as a function of dc flux $S_{21}\left({\mathrm{\Phi }}_{\text{dc}}\right)$ at the geometric resonant frequency ${\omega }_{\text{geo}}/2\pi$ for (red) $T=4.5\text{K}$, (black) $T=8.0\text{K}$, and (blue) if no flux gradient were present.