Nonreciprocal Microwave Signal Processing with a Field-Programmable Josephson Amplifier

We report on the design and implementation of a field-programmable Josephson amplifier (FPJA)—a compact and lossless superconducting circuit that can be programmed in situ by a set of microwave drives to perform reciprocal and nonreciprocal frequency conversion and amplification. In this work, we demonstrate four modes of operation: frequency conversion (transmission of $-0.5$ dB, reflection of $-30$ dB), circulation (transmission of $-0.5$ dB, reflection of $-30$ dB, isolation of 30 dB), phase-preserving amplification (gain $>20\mathrm{dB}$, one photon of added noise) and directional phase-preserving amplification (reflection of $-10$ dB, forward gain of 18 dB, reverse isolation of 8 dB, one photon of added noise). The system exhibits quantitative agreement with the theoretical prediction. Based on a gradiometric superconducting quantum-interference device with $\mathrm{Nb}/\mathrm{Al}\text{−}\mathrm{Al}{\mathrm{O}}_x/\mathrm{Nb}$ Josephson junctions, the FPJA is first-order insensitive to flux noise and can be operated without magnetic shielding at low temperature. Owing to its flexible design and compatibility with existing superconducting fabrication techniques, the FPJA offers a straightforward route toward on-chip integration with superconducting quantum circuits such as qubits and microwave optomechanical systems.

Figure 1

Device and experimental setup. (a) A superconducting resonant circuit is measured in reflection in a cryostat (LPF: low-pass filter). It exhibits three resonances whose frequencies are tunable by the flux applied to a single SQUID. (b) Measured resonance frequencies ${\omega }_a$, ${\omega }_b$, and ${\omega }_c$, as a function of the flux bias to the SQUID. (c) Table of device parameters for $\mathrm{\Phi }/{\mathrm{\Phi }}_0\approx 0.29$, showing the resonance frequency ${\omega }_j$, the total linewidth ${\kappa }_j$, and the coupling efficiency ${\eta }_j={\kappa }_j^{\mathrm{ext}}/{\kappa }_j$ of each mode $j$.

Figure 2

Frequency converter. (a) Frequency-space diagram. A strong pump of frequency ${\omega }_{ab}^p=|{\omega }_b-{\omega }_a|$ converts an input signal at the frequency ${\omega }_a^s={\omega }_a+{\delta }_a$ to a output signal frequency ${\omega }_b^s={\omega }_b+{\delta }_b$, and vice versa. (b) Graph representation of the EOM (2). (c) Ideal signal-flow diagram. (d) Measured scattering parameters (the dots) and fits to Eq. (2) (the lines), for a fixed pump strength $|{\beta }_{ab}|\approx 0.5$, as a function of the detuning ${\delta }_j$. The device exhibits good impedance matching (low return loss) and near-unity transmission.

Figure 3

Two-mode amplifier. (a) Frequency-space diagram. A strong pump of frequency ${\omega }_{ab}^p={\omega }_b+{\omega }_a$ amplifies an input signal at the frequency ${\omega }_a^s={\omega }_a+{\delta }_a$ and generates an idler at the frequency $-{\omega }_b^s=-{\omega }_b+{\delta }_b$, and vice versa. (b) Graph representation of the EOM (4). (c) Ideal signal-flow diagram. (d) Measured scattering parameters (the dots) and fits to Eq. (4) (the lines), as a function of the detuning ${\delta }_j$, for the increasing pump strength ${\beta }_{ab}$ (from dark to light color). (e) Measured system-added noise (the dots) and theoretical predictions (the lines) referred to the input of the FPJA, as a function of the detuning ${\delta }_j$.

Figure 4

Circulator. (a) Frequency-space diagram. Three pumps, with the frequencies ${\omega }_{ab}^p\approx |{\omega }_b-{\omega }_a|$, ${\omega }_{bc}^p\approx |{\omega }_c-{\omega }_b|$, ${\omega }_{ac}^p\approx |{\omega }_c-{\omega }_a|$ and the respective phases ${\varphi }_{ab}$, ${\varphi }_{bc}$, and ${\varphi }_{ac}$, allow a conversion between signals at the frequencies ${\omega }_j^s={\omega }_j-{\delta }_j$ ($j\in \{a,b,c\}$), closing a loop in frequency space. Interference between paths enables nonreciprocal signal circulation. The circulation direction is set by the total loop phase, ${\varphi }_{\mathrm{loop}}={\varphi }_{ab}+{\varphi }_{bc}-{\varphi }_{ac}$. (b) Graph representation of the EOM (6). (c) Ideal signal-flow diagram. (d) Measured scattering parameters (the dots) and fits to Eq. (6) (the lines), as a function of the detuning ${\delta }_j$, for fixed pump powers ($|{\beta }_{jk}|=1/2$) and loop phase (${\varphi }_{\mathrm{loop}}\approx -\pi /2$).

Figure 5

Directional phase-preserving amplifier. (a) Frequency-space diagram. Three pumps, with the frequencies ${\omega }_{ac}^p\approx |{\omega }_c-{\omega }_a|$, ${\omega }_{ab}^p\approx {\omega }_b+{\omega }_a$, and ${\omega }_{bc}^p\approx {\omega }_c+{\omega }_b$ and the respective phases ${\varphi }_{ab}$, ${\varphi }_{bc}$, and ${\varphi }_{ac}$, allow a conversion between signals at the frequencies ${\omega }_a^s$, $-{\omega }_b^s$ and ${\omega }_c^s$, closing a loop in frequency space. Interferences between paths enable nonreciprocal signal amplification. The amplification direction is set by the total loop phase, ${\varphi }_{\mathrm{loop}}={\varphi }_{ab}+{\varphi }_{bc}+{\varphi }_{ac}$. (b) Graph representation of the EOM (8). (c) Ideal signal-flow diagram. (d) Measured scattering parameters (the dots) and fits to Eq. (8) (the lines) as a function of the detuning ${\delta }_j$, for fixed pump powers ($|{\beta }_{jk}|\approx 1/2$) and loop phase (${\varphi }_{\mathrm{loop}}\approx -\pi /2$). (e) Return noise of the amplifier. Ratio of the measured noise power out of mode $a$ for the pumps on and off, showing that no extraneous noise is added by the amplifier. (f),(g) Measured system-added noise (the dots) and theoretical predictions (the lines) referred to the input of the FPJA, as a function of the detuning ${\delta }_j$, for modes $b$ and $c$, respectively.

Figure 6

Device fabrication and layout. (a) False colored optical micrograph. The silicon substrate is in gray, the bottom Nb layer in red, and the top Nb layer in blue. Overlap between the two Nb layers forms parallel-plate capacitors, in purple. (b) Circuit equivalent of the device. (c) Fabrication process, along the cross sections $z_1$ and $z_2$ shown in (d) and (e). In step 1, the $\mathrm{Nb}/\mathrm{Al}\text{−}\mathrm{Al}{\mathrm{O}}_x/\mathrm{Nb}$ trilayer is prepared on an intrinsic silicon wafer. In step 2, the trilayer is patterned top down in three steps to define Josephson junctions and the bottom Nb wiring layer. In step 3, the $a$-Si dielectric layer is deposited and patterned to define vias. In step 4, the Nb layer is deposited over the $a$-Si and vias, and it is patterned to define the top wiring layer. Any uncovered $a$-Si is also removed in this step. (d),(e) Scanning electron micrograph of one of the coil inductors, and of the gradiometric SQUID, using the same color scheme, with, additionally, the Josephson junction in yellow.

Figure 7

Process spectroscopy. (a) Mode frequencies ${\omega }_j$, and first-order modulation frequencies $|{\omega }_j\pm {\omega }_k|$ ($j$, $k\in \{a,b,c\}$) as a function of flux. The shaded areas represent a bandwidth of 180 MHz, necessary to ensure a good rotating-wave approximation. Graph representation of the EOM for (b) parametric amplification between two modes, (c) a single harmonic oscillator, (d) frequency conversion between two modes, (e) circulation, and (f) directional phase-preserving amplification.

Figure 8

Circulator. Magnitude of the (a) measured and (b) simulated scattering parameters as a function of the loop phase ${\varphi }_{\mathrm{loop}}$ for near-ideal pump strengths.

Figure 9

Directional phase-preserving amplifier. Magnitude of the (a) measured and (b) simulated scattering parameters as a function of the loop phase ${\varphi }_{\mathrm{loop}}$ for near-ideal pump strengths. In practice, the device undergoes free oscillations for $|{\varphi }_{\mathrm{loop}}|\lesssim 80$ deg (see Appendix pp2-s3).

Figure 10

Noise calibration. (a) The noise emitted by a biased metallic tunnel junction is measured using the same setup as for the FPJA (LPF: low-pass filter). (b)–(d) Measured power spectral density in photon unit as a function of the voltage across the shot-noise junction, measured at the frequencies ${\omega }_1/2\pi =4.155\mathrm{GHz}$, ${\omega }_2/2\pi =5.756\mathrm{GHz}$, and ${\omega }_3/2\pi =7.915\mathrm{GHz}$, corresponding to the resonances of the FPJA shown in Fig. 1. The solid black lines are a fit to Eq. (c2). The dashed lines are linear extrapolations of the noise at high voltage, and their crossing point corresponds to the system-added noise. (e) Measured system-added noise as a function of frequency.