High-gain weakly nonlinear flux-modulated Josephson parametric amplifier using a SQUID array

We have developed and measured a high-gain quantum-limited microwave parametric amplifier based on a superconducting lumped LC resonator with the inductor L including an array of eight superconducting quantum interference devices (SQUIDs). This amplifier is parametrically pumped by modulating the flux threading the SQUIDs at twice the resonator frequency. Around 5 GHz, a maximum gain of 31 dB, a product amplitude gain $\times$ bandwidth above 60 MHz, and a 1 dB compression point of $-123$ dBm at 20 dB gain are obtained in the nondegenerate mode of operation. Phase-sensitive amplification-deamplification is also measured in the degenerate mode and yields a maximum gain of 37 dB. The compression point obtained is 18 dB above what would be obtained with a single SQUID of the same inductance, due to the smaller nonlinearity of the SQUID array.

Figure 1

Experimental setup. (a) Optical micrograph of the tested parametric amplifier showing its 50 $\Omega$ coplanar waveguide (CPW) signal input port (top), its coupling capacitor $C_c$, its capacitor $C_R$ (left and right), its inductor $L_g$ (middle) terminated by an eight-SQUID array with total inductance $L_A\left(\Phi \right)$, and its split magnetic flux line coupled to a $50\Omega$ CPW (bottom). The dc current in the flux line sets the dc flux ${\Phi }_{DC}$ and the resonance frequency $f_R$ of the resonator, whereas the ac current parametrically pumps the resonator at ${\omega }_P\simeq 2{\omega }_R$. The black arrow points to the equivalent circuit. (b) Electrical circuit diagram showing (from left to right) the pump line, the dc flux line added to the pump with a bias tee, an additional flux line feeding a coil for compensating any flux offsets, the signal line, and a circulator routing the reflected and amplified signal to a measurement line through an isolator protecting the sample from the noise of the first amplifier placed at 4 K. Feeding lines are attenuated and filtered. The output signal is split after amplification and analyzed both with a spectrum analyzer and by homodyne demodulation.

Figure 2

DC flux modulation. Experimental (dots) and calculated (line) resonator frequency $f_R$ as a function of the dc current in the on-chip flux line [see Fig. 1] measured by fitting the phase of a weak signal reflected on the resonator in the absence of parametric pumping, as shown in the inset for zero flux bias (point A). We attribute the two shoulders on the sides of the measured resonance to multiple wave interferences due to an imperfect impedance matching somewhere in the setup. Parameters used for calculation of the modulation curve are $f_{R0}$, $p=0.45$, and $\beta =0.1$ . Full characterization in the next figures are done at working point B.

Figure 3

Signal power gain $|G_S{|}^2$ at working point B of Fig. 2 for a nominal input power $P_{S,n}=-142.5\mathrm{dBm}$. Left: Nondegenerate gain as a function of the signal frequency $f_S$ at different nominal pumping powers $P_{P,n}$ between $-58.7$ and $-54.4\mathrm{dB}\mathrm{m}$ (top dashed curve just before the onset of parametric oscillation in the absence of incident signal). (b) Phase-sensitive degenerate gain for $P_{P,n}=-54.4$ and $-55.4\mathrm{dBm}$. Inset: Demodulated signal in the IQ plane at maximum gain $(P_{P,n}=-54.4\mathrm{dBm})$ filtered at 1 MHz. I and Q voltages are digitized at 1 MSample/s during 2 s, and the color encodes the density of samples from 0 (dark blue) to maximum (red).

Figure 4

Amplifier characterization at the working point B of Fig. 2 for nondegenerate pumping. (a) Signal power gain as a function of the nominal input power $P_{S,n}$ showing the saturation at the same nominal pumping powers $P_{P,n}$ as in Fig. 3 (top dashed line corresponds again to the onset of parametric oscillation). Vertical dashed and dotted lines correspond to the input powers where gain was measured in Fig. 3 and where reference gain for saturation was defined, respectively. (b) Power gain $|G_S{|}^2$, bandwidth $BW$, and product $|G_S|\times BW$ deduced from measurements (dots) of Fig. 3 at $P_{S,n}=-142.0\mathrm{dBm}$, and calculated (solid lines) from the model with the parameters indicated in the text. (c) 1 dB compression point deduced from (a) (dots), calculated from the model (solid line), and calculated with the same parameters but only one SQUID (dashed line). Note that given the $\pm 1\mathrm{dB}$ precision on the calibration of the pumping and signal lines, $+0.8$ and $-1.0\mathrm{dB}$ were added to the nominal $P_{P,n}$ and $P_{S,n}$ values to match the data to the theoretical curves at low pumping strength. The vertical dotted line indicates the frontier between parametric amplification and parametric oscillation (infinite gain) for the linear model.