Kerr nonlinearity in a superconducting Josephson metamaterial

We present a detailed experimental and theoretical analysis of the dispersion and nonlinear Kerr frequency shifts of plasma modes in a one-dimensional Josephson junction chain containing 500 superconducting quantum interfence devices in the regime of weak nonlinearity. The measured low-power dispersion curve agrees perfectly with the theoretical model if we take into account the Kerr renormalization of the bare frequencies and the long-range nature of the island charge screening by a remote ground plane. We measured the self- and cross-Kerr shifts for the frequencies of the eight lowest modes in the chain. We compare the measured Kerr coefficients with theory and find good agreement.

Figure 1

(a) SEM image of one part of the 500-SQUID chain. (b) Optical image showing the SQUID chain connected at both ends to a microstrip transmission line. (c) Equivalent electrical scheme of the chain and transmission line for the local screening model.

Figure 2

One-tone spectroscopy of propagation modes in the chain of 500 SQUIDs. (a) Fifteen modes are resolved within the bandwidth of the measuring circuit 2–18 GHz. (b) Zoom of mode 3, up: amplitude, bottom: phase of the signal. Continuous line is the theoretical fit using Eq. (16) with $Q_i=2540, Q_c=535$, and ${\omega }_r/\left(2\pi \right)=6.717$ GHz.

Figure 3

Power-dependent two-tone spectroscopy measurement of propagation modes in the chain of 500 SQUIDs. The measurement tone for this two-tone measurement was mode $n=4$ at $8.41\mathrm{GHz}$. The first 43 modes are clearly resolved. There is a cutoff frequency for the transmission at 22 GHz, above which no signal is transmitted.

Figure 4

(a) Mode dispersion for the 500-SQUID chain. Red stars: measured data, extracted from low-power, two-tone spectroscopy (see Fig. 3); green dots: theoretical fit using the local screening model; blue dots: theoretical fit using the long-range screening model; dashed black line: low-frequency slope. (b) Deviation of the mode frequencies obtained using the two models including the Kerr shifts from the measured frequencies. While the local screening model shows a maximum deviation of 25%, the long-range screening model agrees within 0.1% with the measured mode frequencies.

Figure 5

Kerr shifts of the mode frequencies: (a) the self-Kerr frequency shift of mode $k=2$, the regimes of weak and strong nonlinearity are indicated. The inset shows the resonance at the power where bistable behavior starts to appear. From the fit of the Lorentzian, we deduced $Q_{\text{tot}}=149$ and (b) cross-Kerr frequency shifts for mode $k=3$ in the weakly nonlinear regime as a function of pumping power to a different mode (modes $k^{\prime }=2,4,5,6,7$, and 8). For each curve, the corresponding values of the slope $X_{k{k}^{\prime }}$ are extracted.

Figure 6

The principal scheme of the experimental setup for one-tone- and two-tone-driven transmission measurements, performed at a temperature of 10 mK. The transmission amplitude and phase through the SQUID-chain is measured by a Vector Network Analyzer (VNA). For mode frequencies larger than 16 GHz, we use a two-tone measurement where a second microwave tone is swept over frequency by a second microwave generator while the frequency of the VNA is kept constant.

Figure 7

Experimental dispersions for two identical chains of 500 SQUIDS having different coupling to the transmission line. The inset shows the equivalent scheme for the capacitively coupled chain.

Figure 8

Amplitude and phase of the resonance for the second mode. The continuous line is the fit of the resonance with Eqs. (B1). Fitting parameters are: $Q_i=9988, Q_c=3318, Z_0=50\mathrm{\Omega }, X_e=10\mathrm{\Omega }$, and $\omega /\left(2\pi \right)=4.909$ GHz.

Figure 9

Frequency dependance of the internal quality factor: blue points are experimental data, continous line is the theoretical fit with a $1/{\omega }^2$-dependance.

Figure 10

Two different models describing the effective screening of the charge $Q_n$. (a) In the local screening model, an island charge is fully screened by the ground capacitance $C_g$ and there is no long-range interaction between different island charges.(b) The long-range screening model takes into account long range interactions between different island charges in the chain. For the calculation of the potential of island $n$, multiple image charges are introduced to satisfy the boundary conditions of the electrical field at the interface silicon/air and at the ground plane where $V=0$. A zoom on the single island is presented as an inset for each model.

Figure 11

Dispersion relation of an infinite chain (black line) compared to the propagating quantized modes (black dots) of the 500-SQUID chain without the Kerr nonlinearity. Blue dots show the dispersion of the propagating modes of the 500-SQUID chain by taking into account the Kerr nonlinearity.