Three-wave mixing traveling-wave parametric amplifier with periodic variation of the circuit parameters

Over six decades have passed since the first proposals on traveling-wave parametric amplifiers (TWPAs).^1–3 Since then, TWPAs have been widely used in optics;^4,5 however, in electronics and microwave circuits, they gave way to transistor-based amplifiers. Recent progress in superconducting quantum computing and the demand for quantum-limited amplifiers has brought the attention back to the TWPAs. Being composed of passive, low-loss elements such as capacitors and inductors, TWPAs are inherently low noise. This makes them ideal for millikelvin measurements. Furthermore, embedding the elements in a transmission-line frame liberates TWPAs from the constraint of a fixed gain-bandwidth product inherent in parametric amplifiers based on lumped element oscillators and resonators.^6,7

Parametric amplification results from frequency mixing in a nonlinear element, where a strong pump is mixed with the weak signal, causing energy transfer from the pump to the signal and, thus, amplification. There are two main schemes of frequency mixing, namely, three-wave mixing (3WM) and four-wave mixing (4WM). 4WM-TWPAs appeared earlier and progressed both in the form of kinetic-inductance-based transmission lines^8–12 and of lumped-element circuits composed of Josephson junctions.^13–18 There are fewer implementations of 3WM-TWPAs,^19–25 despite the abundance of the theoretical work.^2,19,26–30

The quadratic nonlinearity in a 3WM-TWPA leads to a greater amplification at lower pump powers compared with a 4WM-TWPA, which relies on cubic nonlinearity. This translates into a shorter chain for the 3WM-TWPA to achieve a target amplification,²⁶ which in turn introduces less loss to the system. Moreover, in a 3WM-TWPA, maximum amplification is close to half of the pump frequency, in contrast to a 4WM-TWPA, where the gain is maximum near the pump frequency. Therefore, the strong pump that is detuned from the signal can be filtered out easily. Furthermore, the 3WM-TWPA is not greatly affected by the intrinsic phase mismatch induced by the Kerr effect, which was believed to eliminate the need for dispersion engineering. In practice, however, the linear dispersion facilitates the generation of higher pump harmonics and up-converted signal modes. Therefore, the desired gain may not be achieved unless one increases the number of device unit cells.^29,31 This problem has so far been addressed by adding features to the TWPA chain to deliberately distort the linear dispersion and increase the phase mismatch at higher frequencies while maintaining finely tuned phase matching for a particular pump frequency. Although this approach results in gain enhancement, it suffers from the limited flexibility of the amplification band.^20,24

Our proposed solution to the problem of high harmonic generation is creating a stopband for the second harmonic of the pump, $2 ω p$⁠, thus suppressing the generation and propagation of all pump harmonics. This is similar to an approach for the 4WM-kinetic-inductance TWPAs,^8–10 where a stopband is used to suppress the third harmonic of the pump. A strong nonlinear dispersion in the vicinity of the stopband would also prevent upconversion of the signal and idler by the pump. Therefore, we anticipate a better amplifier performance as well as improved flexibility in adjusting the pump frequency compared to other proposals.^20,24,27

We use the periodic loading method³² and modify the chain inductance to create a distributed filter, resulting in a stopband at the frequency of $2 ω p$⁠. Such a filter is in the form of nonlinear inductor-capacitor (LC) cells, and besides frequency elimination, it contributes to the gain. Moreover, the design is free from additional sources of reflection and loss present in resonator-based dispersion engineering. With this design, we demonstrate up to $20 dB$ gain with a TWPA that contains only 440 LC cells.

The TWPA is illustrated in Fig. 1(a). The nonlinearity results from the Josephson junctions, arranged in the form of a superconducting nonlinear asymmetric inductive element (SNAIL) loop³³ to provide 3WM. The equivalent SNAIL inductance, L, is periodically modulated such that the three consecutive cells form a supercell, $L 1 − L 1 − L 2$⁠, with $L 2 = ( 3 / 2 ) L 1$ being the loaded feature. We choose the capacitance C such that the supercell, as a whole, is impedance matched.³² (Variation in the TWPA impedance with magnetic flux is insignificant at the frequencies of interest well below the stopband.) In general, one can place the loaded feature periodically after any number of L₁-cells and use a different capacitance, $C 2 ≠ C$⁠.

The dispersion relation is obtained by calculating the supercell transmission (ABCD) matrix, $M ̂$⁠, and solving the equation, $2 cosh ( γ a ) = Tr ( M ̂ )$⁠, where γ is the complex propagation constant and a is the length of the supercell.³²  Figure 1(b) shows the dispersion relation where the wave number k is the imaginary part of γ. The plot consists of three propagation bands separated by two stop-bands. The stop-bands variation with magnetic flux is illustrated in Fig. 1(c). The stopbands shift downwards in frequency as the magnetic flux approaches $0.5 Φ 0$⁠, as the SNAIL inductance increases.

We fabricated the device on a high resistivity silicon wafer (⁠ $ρ ≥ 10 k Ω cm$⁠) with evaporated aluminum as the superconducting material, aluminum/aluminum oxide/aluminum (Al/AlO_x/Al) Josephson junctions, and parallel plate capacitors with amorphous aluminum nitride dielectric. We mounted a $5 × 10 mm 2$ TWPA chip, containing 440 cells or $≈ 147$ supercells, in a copper box and attached the assembly to the mixing chamber of a Bluefors dilution refrigerator to measure the device at temperatures around $10 mK$⁠. Figure 2 shows the wiring diagram of the cryogenic measurement setup. To apply the dc magnetic flux, we placed a superconducting coil under the sample. To calibrate the measurement, we used a $50 Ω$ SMA cable as the reference, marked with REF in the figure. The TWPA and the reference were connected to RF switches to share the input/output lines. The signal and the pump were fed through line 1 and line 2, respectively. They combined in a directional coupler before entering the TWPA. This was made to protect a qubit, accessible via the signal line, from being perturbed by the strong pump. We used this qubit to calibrate the power, allowing us to measure the noise temperature, as discussed later. During gain measurements, we combined the signal and the pump at room temperature and fed both through line 2.

We present the gain for pump frequencies between 7 and $8.4 GHz$ in Fig. 3 (blue, green, and purple traces). For each pump frequency, we kept the pump second harmonic within the stopband; thus, we changed the flux bias from $0.34 Φ 0$ down to $0.21 Φ 0$⁠. Simultaneously, we ramped up the pump power not to decrease the ratio of the pumping current to the critical current. The gain increases with the growing pump frequency, following the theory prediction,^2,31 as the gain coefficient is proportional to the wave number of the pump. The gain maximum increases from $≈ 15 dB$ with the pump frequency of $7 GHz$ to about $20 dB$ within the pump frequency interval 7.8– $8.4 GHz$⁠. At the same time, the shape of the gain traces does not change, indicating that 3WM remains dominant within the range of the applied magnetic bias, despite the increasing 4WM admixture. The frequency shift of the gain peak from half of the pump frequency is due to phase mismatch.^2,24,31 In Fig. 3, we also present the gain of a SNAIL-TWPA that contains no dispersion-engineering features (yellow trace). The latter was made of the same number of 440 SNAILs with similar configurations and measured at the same operational point. The clear difference in the frequency dependence and the level of the gain of the loaded TWPA indicate the effect of the suppression of high harmonics.

To investigate how effective the suppression by the stopband was, we probed the generation of the second harmonic of an injected mode. We biased the TWPA at $≈ 0.38 Φ 0$⁠, where the stopband was accessible by our measurement setup and 4WM was negligible. We sent in a tone at frequency ω and power $− 111 dBm$ and measured the output signals at ω and $2 ω$ using a spectrum analyzer. The experimentally measured transmission results are shown in Fig. 4. Each result is plotted after subtracting the measurement taken at zero flux instead of the reference cable to exclude the effect of the insertion loss. The response of the tone itself (blue) reveals the stopband starting at approximately $11.5 GHz$⁠. However, the second harmonic (green) does not change significantly in the stopband. This surprising observation, the absence of the extinction of the second harmonic, is confirmed by numerical simulation (yellow and purple dashed–dotted lines). The amplitude of the second harmonic in the stopband is approximately 10 dB smaller than the maximum below the stopband edge, whereas the main tone decreases by orders of magnitude. The simulation was performed using the harmonic-balance method implemented in the advanced design system simulator.³⁴ This method accounts for the full nonlinear dynamics of the Josephson inductance in response to the applied ac current and magnetic flux and returns all inter-mixing products generated in the circuit.

The difference in the behavior of the main tone and its second harmonic in the stopband can be understood by considering the non-linear wave interactions. The second harmonic continuously interacts, even within the stopband, with the main tone, which freely propagates at smaller frequencies within the propagating band and provides energy to the second harmonic. This understanding is supported by an analytical study of a simplified model for a TWPA with a stopband. The model reveals that the second harmonic (with frequency $2 ω$⁠) contains a propagating part with a wave vector $2 i k ( ω )$⁠, in addition to an evanescent part with complex wave vector of the second harmonic $γ ( 2 ω )$⁠. Once the evanescent term vanishes, the amplitude of the propagating wave becomes independent of the position in the TWPA chain. The amplitude of the wave is inversely proportional to the width of the stopband.

The observed small suppression of the second harmonic in the stopband illustrates a general property of the nonlinear media—the absence of the full extinction of a wave in a stopband due to nonlinear wave interactions.

We further investigated the presence of the (down-converted) idler, $ω i = ω p − ω s$⁠, and the up-converted modes (⁠ $ω p + ω s , i$⁠), for a signal within the interval 0.1– $6.2 GHz$⁠. We biased the device at $Φ ≈ 0.38 Φ 0$ and applied the pump at $6.2 GHz$ so that its second harmonic was in the stopband. We used a spectrum analyzer and measured the response between 3.5– $12.5 GHz$⁠, as shown in Fig. 5(a). The blue trace indicates the response at the signal frequency, ω_s, while the yellow depicts the idler, ω_i, when the signal is applied below $2.7 GHz$⁠. The green and purple traces represent the upconversion of the signal and idler, respectively, $ω p + ω s , i$⁠. The peak at frequencies between 6.2 and $9 GHz$ indicates efficient upconversion from the quasi-linear dispersion region below $3 GHz$⁠. The significantly weaker response at frequencies above $9 GHz$ reveals the increased phase mismatch close to the stopband. Therefore, we conclude that the shape of the gain in Fig. 5(a) is mainly formed by an interplay of the three modes, ω_s, ω_i, and $ω p + ω s ( i )$⁠, Fig. 5(b).

We fit the data in Fig. 6(a) using Eqs. (4) and (5), solid lines in the plots. Assuming vacuum input noise, we extract the parameters, D = 0.73 and $A H = 24.85$⁠. Note that D is comparable with the value of 0.79 that we obtain from the insertion loss measurement. We use D to compute the added noise of the TWPA, Eq. (3), as shown in Fig. 6(b). The saturated value of the added noise at large pump power is found, $A ∞ = 0.86$ photons. Thus, the added noise by TWPAs is only 0.36 photons above the standard quantum limit of A_SQL = 0.5.³⁸ 

In summary, we demonstrated an enhanced gain in a Josephson junction-based TWPA in the three-wave-mixing regime, once we reduced the undesired effect of higher mode generation. Using periodic loading, we engineered a stopband in the frequency spectrum of the TWPA. We demonstrated that placing the stopband at twice the pump frequency suppressed the propagation of the second harmonic of the pump and the up-converted signal and idler. However, due to nonlinear interactions, the stopband did not fully eliminate these modes. Nevertheless, even with $10 dB$ suppression of the second harmonic of the pump, the maximum gain of the loaded TWPA is about $10 dB$ larger than that of a device with no dispersion engineering. A TWPA with a small number of 147 supercells shows a maximum gain of $20 dB$⁠, a 3-dB bandwidth of $≈ 1 GHz$ between 4 and $8 GHz$⁠, and an added noise below one photon. The width of the stopband provides the flexibility of tuning the pump frequency over a range of $1.4 GHz$⁠, while the maximum gain remains above $15 dB$⁠.

This research was funded by the Knut and Alice Wallenberg Foundation through the Wallenberg Center for Quantum Technology (No. WACQT KAW 2021.0009) and by the EU Flagship on Quantum Technology (No. H2020-FETFLAG-2018-03) Project 820363 OpenSuperQ. The authors are grateful to Christian Fager for valuable advice on device modeling. The authors acknowledge the use of the Nanofabrication Laboratory (NFL) at Chalmers University of Technology. The authors are thankful for the support from Marco Scigliuzzo, Andreas Bengtsson, Yong Lu, Liangyu Chen, Hangxi Li, Christopher Warren, David Niepce, Eleftherios Moschandreou, Philip Krantz, and Lars Jönsson.