Josephson parametric amplifiers are an important part of a modern superconducting quantum computing platform and squeezed quantum states generation devices. Traveling wave and impedance-matched parametric amplifiers provide broad bandwidth for high-fidelity single-shot readout of multiple qubit superconducting circuits. Here, we present a quantum-limited 3-wave-mixing parametric amplifier based on superconducting nonlinear asymmetric inductive elements (SNAILs), whose useful bandwidth is enhanced with an on-chip two-section impedance-matching circuit based on microstrip transmission lines. The amplifier dynamic range is increased using an array of 67 SNAILs with 268 Josephson junctions, forming a nonlinear quarter-wave resonator. Operating in a current-pumped mode, we experimentally demonstrate an average gain of 17 dB across 300 MHz bandwidth, along with an average saturation power of –100 dBm, which can go as high as −97 dBm with quantum-limited noise performance. Moreover, the amplifier can be fabricated using a simple technology with just one e-beam lithography step.

Parametric amplifiers have become key components in quantum information processing due to their near quantum-limited noise performance. Those devices are used as the first stage of the low-noise amplification schemes defining primarily the overall system noise and signal-to-noise ratio.11. D. M. Pozar, Microwave Engineering, 4th ed. ( John Wiley and Sons, Inc., 2012). Broadband Josephson parametric amplifiers are the only tools to perform single-photon power measurements of both microwave2–42. S. J. Asztalos, G. Carosi, C. Hagmann, D. Kinion, K. van Bibber, M. Hotz, L. J. Rosenberg, G. Rybka, J. Hoskins, J. Hwang, P. Sikivie, D. B. Tanner, R. Bradley, and J. Clarke, Phys. Rev. Lett. 104, 041301 (2010). https://doi.org/10.1103/PhysRevLett.104.0413013. A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, J. Majer, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, Phys. Rev. Lett. 95(6), 060501 (2005). https://doi.org/10.1103/PhysRevLett.95.0605014. G. P. Fedorov, S. V. Remizov, D. S. 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Walter, P. Kurpiers, S. Gasparinetti, P. Magnard, A. Potočnik, Y. Salathé, M. Pechal, M. Mondal, M. Oppliger, C. Eichler, and A. Wallraff, Phys. Rev. Appl. 7, 054020 (2017). https://doi.org/10.1103/PhysRevApplied.7.054020 real-time quantum feedback control,13–1513. R. Vijay, C. Macklin, D. H. Slichter, S. J. Weber, K. W. Murch, R. Naik, A. N. Korotkov, and I. Siddiqi, Nature 490, 77–80 (2012). https://doi.org/10.1038/nature1150514. D. Riste, M. Dukalski, C. A. Watson, G. de Lange, M. J. Tiggelman, Y. M. Blanter, K. W. Lehnert, R. N. Schouten, and L. DiCarlo, Nature 502, 350–354 (2013). https://doi.org/10.1038/nature1251315. H. A. Bachor and T. C. Ralph, A Guide to Experiments in Quantum Optics, 3rd ed. ( John Wiley and Sons, 2019). and generating squeezed quantum states.16,1716. M. A. Castellanos-Beltran, K. D. Irwin, G. C. Hilton, L. R. Vale, and K. W. Lehnert, Nat. Phys. 4, 929–931 (2008). https://doi.org/10.1038/nphys109017. M. Malnou, D. A. Palken, L. R. Vale, G. C. Hilton, and K. W. Lehnert, Phys. Rev. Appl. 9, 044023 (2018). https://doi.org/10.1103/PhysRevApplied.9.044023There are several types of parametric amplifiers that have been demonstrated in the last few years for multiplexed qubits readout.18–2018. C. Macklin, K. O'brien, D. Hover, M. E. Schwartz, V. Bolkhovsky, X. Zhang, W. D. Oliver, and I. Siddiqi, Science 350(6258), 307–310 (2015). https://doi.org/10.1126/science.aaa852519. T. C. White, J. Y. Mutus, I. C. Hoi, R. Barends, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey, J. Kelly, A. Megrant, C. Neill, P. J. J. O'Malley, P. Roushan, D. Sank, A. Vainsencher, J. Wenner, S. Chaudhuri, J. Gao, and J. M. Martinis, Appl. Phys. Lett. 106, 242601 (2015). https://doi.org/10.1063/1.492234820. S. Kundu, N. Gheeraert, S. Hazra, T. Roy, K. V. Salunkhe, M. P. Patankar, and R. Vijay, Appl. Phys. Lett. 114, 172601 (2019). https://doi.org/10.1063/1.5089729 The first one is an impedance-matched Josephson parametric amplifier (IMPA). It allows us to increase gain-bandwidth product due to the impedance-matching of the amplifier to its input. This idea was first demonstrated using the Klopfenstein taper structure to make a stronger coupling between a Josephson Parametric Amplifier (JPA) and its environment.21,2221. J. Y. Mutus, T. C. White, R. Barends, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey, J. Kelly, A. Megrant, C. Neill, P. J. J. O'Malley, P. Roushan, D. Sank, A. Vainsencher, J. Wenner, K. M. Sundqvist, A. N. Cleland, and J. M. Martinis, Appl. Phys. Lett. 104, 263513 (2014). https://doi.org/10.1063/1.488640822. R. Yang and H. Deng, IEEE Trans. Appl. Supercond. 30(6), 1100306 (2020). https://doi.org/10.1109/TASC.2020.2985650 Later, a two-section impedance transformer was proposed2323. T. Roy, S. Kundu, M. Chand, A. M. Vadiraj, A. Ranadive, N. Nehra, M. P. Patankar, J. Aumentado, A. A. Clerk, and R. Vijay, Appl. Phys. Lett. 107, 262601 (2015). https://doi.org/10.1063/1.4939148 to broaden amplifiers' bandwidth and was realized by several scientific groups.24–2624. J. Grebel, A. Bienfait, É. Dumur, H. S. Chang, M. H. Chou, C. R. Conner, G. A. Peairs, R. G. Povey, Y. P. Zhong, and A. N. Cleland, Appl. Phys. Lett. 118, 142601 (2021). https://doi.org/10.1063/5.003594525. P. Duan, Z. Jia, C. Zhang, L. Du, H. Tao, X. Yang, L. Guo, Y. Chen, H. Zhang, and Z. Peng, Appl. Phys. Express 14, 042011 (2021). https://doi.org/10.35848/1882-0786/abf02926. S. Wu, D. Zhang, R. Wang, Y. Liu, S. P. Wang, Q. Liu, J. S. Tsai, and T. Li, Chin. Phys. B 31, 010306 (2022). https://doi.org/10.1088/1674-1056/ac280c A maximum gain of 15–20 dB, a bandwidth in the range of 150–600 MHz, a saturation power of –120 to –100 dBm, and a noise near the quantum limit have been demonstrated for these devices. However, the saturation power of those devices is limited as its parametric interaction is described by Kerr nonlinearity. Moreover, these devices require a complex multi-stage fabrication process.To enhance both bandwidth and dynamic range, the Josephson traveling-wave parametric amplifier (JTWPA) was introduced. These devices are based on a nonlinear transmission line demonstrating a bandwidth up to several gigahertz at 10–20 dB gain, which allows reading out dozens of superconducting qubits in a single shot.27–2927. M. Esposito, A. Ranadive, L. Planat, and N. Roch, Appl. Phys. Lett. 119, 120501 (2021). https://doi.org/10.1063/5.006489228. M. Esposito, A. Ranadive, L. Planat, S. Leger, D. Fraudet, V. Jouanny, O. Buisson, W. Guichard, C. Naud, J. Aumentado, F. Lecocq, and N. Roch, Phys. Rev. Lett. 128, 153603 (2022). https://doi.org/10.1103/PhysRevLett.128.15360329. C. Kow, V. Podolskiy, and A. Kamal, arXiv:2201.04660 (2022). Usually, it requires the fabrication of several thousands of nearly identical nonlinear elements (Josephson junctions and parallel plate capacitors), demanding sophisticated fabrication facilities which are not widely available. Furthermore, the noise temperature of such devices is 2–4 times higher the quantum limit due to the losses in the circuit.3030. L. Planat, “ Resonant and traveling-wave parametric amplification near the quantum limit,” Doctoral dissertation ( University Grenoble Alpes, 2020). Recent advances in this direction are associated with the development of JTWPA based on superconducting nonlinear asymmetric inductive elements (SNAIL).3131. A. Ranadive, M. Esposito, L. Planat, E. Bonet, C. Naud, O. Buisson, W. Guichard, and N. Roch, Nat. Commun. 13, 1737 (2022). https://doi.org/10.1038/s41467-022-29375-5 This approach ensures eliminating the gap in the dispersion relation, but the noise temperature is still too high.In this work, we present a quantum-limited 3-wave mixing impedance-matched JPA consisting of a SNAIL array with an on-chip two-section microstrip impedance transformer. It operates in reflection mode when the input signal reflects off, generating the amplified output signal with a gain of more than 17 dB and idler tone. To improve the dynamic range of our degenerate parametric amplifier, we use the SNAILs, which provide the flexibility in optimizing a 3-wave-mixing amplification process, while simultaneously minimizing a 4-wave-mixing Kerr nonlinearity suspected to cause amplifier saturation.31–3531. A. Ranadive, M. Esposito, L. Planat, E. Bonet, C. Naud, O. Buisson, W. Guichard, and N. Roch, Nat. Commun. 13, 1737 (2022). https://doi.org/10.1038/s41467-022-29375-532. N. E. Frattinia, U. Vool, S. Shankar, A. Narla, K. M. Sliwa, and M. H. Devoret, Appl. Phys. Lett. 110(22), 222603 (2017). https://doi.org/10.1063/1.498414233. N. E. Frattini, V. V. Sivak, A. Lingenfelter, S. Shankar, and M. H. Devoret, Phys. Rev. Appl. 10, 054020 (2018). https://doi.org/10.1103/PhysRevApplied.10.05402034. V. V. Sivak, N. E. Frattini, V. R. Joshi, A. Lingenfelter, S. Shankar, and M. H. Devoret, Phys. Rev. Appl. 11, 054060 (2019). https://doi.org/10.1103/PhysRevApplied.11.05406035. V. V. Sivak, S. Shankar, G. Liu, J. Aumentado, and M. H. Devoret, Phys. Rev. Appl. 13, 024014 (2020). https://doi.org/10.1103/PhysRevApplied.13.024014 The substitution of a SNAIL array for each of the junctions in the JPA SQUID increases the saturation power of the amplifier (scales as Ic2/Q3, where Ic is the critical current of large Josephson junction in a SNAIL loop and Q is the coupled Q-factor) while keeping the total inductance of the device roughly the same. The operation principle of the amplifier is described in detail in previous works.36,3736. W. J. Getsinger, IEEE Trans. Microwave Theory Tech. 11, 486–497 (1963). https://doi.org/10.1109/TMTT.1963.112571537. O. Naaman and J. Aumentado, PRX Quantum 3, 020201 (2022). https://doi.org/10.1103/PRXQuantum.3.020201 Based on this, we have engineered the IMPA with 1 dB compression power P−1dB ∈ [−97, −100] dBm at 17 dB gain and quantum-limited noise performance.The impedance-matched parametric amplifier consists of JPA and an on-chip impedance transformer. In this case, JPA is the LC oscillator realized by the series connection of the SNAIL array with inductance Larray and microstrip parallel plate capacitor with C capacitance. JPA resonance frequency is defined as ω0=1/LarrayC, where Larray is defined as follows:3131. A. Ranadive, M. Esposito, L. Planat, E. Bonet, C. Naud, O. Buisson, W. Guichard, and N. Roch, Nat. Commun. 13, 1737 (2022). https://doi.org/10.1038/s41467-022-29375-5 where M is the number of SNAILs in the array, Lj is the Josephson inductance of large junction in the loop, c2=α cos φmin+13 cos(φmin−φext3) is the flux-tunable constant, α=Lj(large)Lj(small) is the cell asymmetry, φext=2πΦ/Φ0 is the magnetic flux quantum, and φmin is defined as c1≡α sin φmin+sinφmin−φext3=0.IMPA tunable range is directly defined by SNAIL asymmetry α, which depends on the junction's critical current ratio. Maximum asymmetry value is defined as 1/n, where n is the number of large Josephson junctions in the loop. Figure 1(a) shows the simulation results of the amplifier tunable range with decreasing SNAIL asymmetry. The number of SNAILs M and asymmetry coefficient α characterize nonlinear processes in the device. SNAIL has both types of nonlinearities, where a third-order nonlinearity (g3) depicts a 3-wave-mixing process and maximum gain, and a fourth-order nonlinearity or Kerr nonlinearity (g4) describes a 4-wave-mixing process and saturation power of the parametric amplifier. The ratio between third- and fourth-order nonlinearities defines the IMPA working point.Figure 1(b) shows the simulation results of the SNAIL third-order nonlinearity vs normalized magnetic flux for various asymmetry coefficients α at a fixed value of Josephson inductance Lj and number of nonlinear elements M. One can clearly notice that it is non-zero in the whole range of magnetic flux values except Φ/Φ0 = 0 and Φ/Φ0 = 0.5 points, where the 3-wave mixing term is forbidden. At small asymmetry values α = 0.09 (black), third-order nonlinearity g3 has a flat profile, which can provide a stable 20 dB gain in the whole tunable range of a narrow-banded Josephson parametric amplifier. As α increases, the third-order nonlinearity value is increased, providing higher gain at the central resonance frequency [see Fig. 1(c)]. For a greater part of the flux range, the fourth-order nonlinearity g4 is positive, but the SNAILs have a region with the negative values of g4 (Kerr-free region), where g3 is dominant and the three-wave mixing process takes place [see Fig. 1(d)].For impedance transformer modeling, the negative-resistance prototypes method proposed in Ref. 3636. W. J. Getsinger, IEEE Trans. Microwave Theory Tech. 11, 486–497 (1963). https://doi.org/10.1109/TMTT.1963.1125715 was utilized. We used the second-order Chebyshev (equal-ripple) bandpass filter, provided a multi-peak gain for the impedance transformer circuit. We chose prototype3636. W. J. Getsinger, IEEE Trans. Microwave Theory Tech. 11, 486–497 (1963). https://doi.org/10.1109/TMTT.1963.1125715 with a minimum gain of Gmin = 20 dB, a passband ripple of 0.5 dB, an equivalent capacitance of g1 = 0.5, an inductance of g2 = 0.24, and a conductance of g3= 1.22. The gain is defined as the ratio of the reflected power, dissipated in the load resistance, to the power available from the generator and can be described as follows:3737. O. Naaman and J. Aumentado, PRX Quantum 3, 020201 (2022). https://doi.org/10.1103/PRXQuantum.3.020201 G=Г(ω)2=PLω−1PLω,(3)where PLω=1+k2TN2(ω) is the Chebyshev power loss function and TN=2ω2−1 is the Chebyshev polynomial for 2nd-order circuit (n = 2). On the other hand, the reflection coefficient Г(ω) depends on the input impedance of IMPA circuit (Zin), Гω=(Zin −1)/(Zin +1), and, hence, resulting power loss function can be described as follows: PLω=Zin+122(Zin+Zin*),(4)where Zin=R+jωLarray−jωC2 is the input impedance of IMPA and C2=b2ω0 is the capacitance of the λ/2 resonator. The bandwidth improvement of this IMPA is defined by fractional bandwidth w, which determines as w=g1Rx1=ω2−ω1ω0, where ω0=ω1ω2 is the central resonance frequency and R is the negative resistance of parametric amplifier [see Fig. 2(d)].The key task of the modeling is to achieve a reasonable correlation between the prototype and the actual circuitry of the parametric amplifier. It requires creating a lumped-element circuit with the prototype gain and ripple as well as designed center-frequency, impedance, and bandwidth. Prototype parameters and parametric amplifier circuit characteristics are defined by impedance or admittance slope parameters. An actual negative-resistance device (parametric amplifier) determines the minimum value of the slope parameter of the λ/2 resonator. For the electrical prototype circuit proposed in Fig. 2(c), impedance of λ/4 and λ/2 transformers can be defined as where Z0 = 50 Ω is the characteristic impedance of the circuit, R0=R/r0 is the source impedance, b=g2/wR=ω0C2 is the admittance slope parameter, and x=g1R/w is the impedance slope parameter. It is an important condition that each transformers resonance frequency has to be equal to the parametric amplifier resonance frequency (ωλ/4 = ωλ/2 = ωJPA = ω0). The impedance transformer provides an increase in the amplifier bandwidth lowering the quality factor of the IMPA resonance. The proposed method results in a multi-peak gain profile [see Fig. 2(d)]. The ripples of the gain are defined by negative-resistance prototype parameters: number of sections, section impedance values, and negative-resistance value. We first carefully matched circuit design and prototype parameters, and then we simulated a full IMPA circuit in Ansys HFSS.Figures 2(a) and 2(b) show scanning electron microscopy images of the impedance-matched Josephson parametric amplifier and its equivalent circuit. It consists of an array of M = 67 SNAILs connected in series with a 30 fF microstrip parallel plate capacitor. The number of SNAILs defined the full nonlinear inductance of IMPA (Larray) and depends on resonance frequency value [Eqs. (1) and (2)]. Each SNAIL consists of an array of three large Josephson junctions (with Josephson inductance LJ = 80 pH) in a loop with one smaller junction (with inductance LJ/α), which has an asymmetry α = 0.18 [see Fig. 2(b)]. The impedance transformer is implemented by a series connection of λ/4 resonator with 87Ω characteristic impedance, lowering the quality factor of the JPA resonance, and λ/2 resonator with 59Ω characteristic impedance, which increases the amplification bandwidth [see Fig. 2(c)]. Both microstrip transmission lines are centered at 6.4 GHz corresponding to JPA central resonance frequency (ωλ/4 = ωλ/2 = ωJPA = 6.4 GHz). As we used a series connection of the inductance and the capacitance in the circuit, we need a smaller capacitance to obtain desired resonance frequency, which allows us to use the substrate as a dielectric layer of a parallel plate capacitor.

The IMPA fabrication includes a two-step process: (I) SNAIL and impedance transformer pattering with double-angle evaporation and liftoff, and (II) wafer backside metallization. The above described processes start on a high-resistivity silicon substrate (525 μm thick). Prior to exposure, the substrate is cleaned in Piranha solution at 80 °C. Then, a bilayer mask is spin-coated on the substrate, which consists of 500 nm MMA (methyl methacrylate) and 150 nm AR-P (CSAR). The impedance transformer input line and SNAIL array are patterned using direct 50 kV e-beam lithography. Al/AlOx/Al Josephson junction and microstrip impedance transformer are e-beam shadow-evaporated in a single vacuum cycle. Liftoff is performed in a bath of N-methyl-2-pyrrolidone with sonication at 80 °C for 3 h and rinsed in IPA with ultrasonication. The microstrip transmission line is finally formed using the substrate backside silver ground plane deposited.

Finally, we experimentally measured our amplifier in a dilution refrigerator with a base temperature near 10 mK. The cryogenic characterization was done in reflection mode with a circulator connected in series. We used the current pump for uniform pumping of the SNAIL array. The flux bias of the SNAIL loops was controlled by the superconducting coil. Here, we define the operational bandwidth as the range of frequencies where the gain is greater than 17 dB, and the noise temperature corresponds to the standard quantum limit. First, we determine the IMPA characteristic frequency by applying flux to the SNAIL loop through an external coil. As shown in Fig. 3(a), the resonant frequency can be tuned from 4.9 to 8.2 GHz. Second, we chose the flux-bias operating point ФDC = 0.42Ф0 and measure the gain response. For SNAIL IMPA characterization, we work at the area with negative Kerr. It is well known that the IMPA gain profile depends on the components of the cryogenic setup. In our model, we assume an ideal 50 Ω impedance connected to the amplifier circuit. The real measurement circuit includes non-ideal cables, circulator, and wire-bond connections, which also affect the gain profile.Figure 3(b) shows the best gain profile obtained at central frequency ωJPA = 6.4 GHz with a bandwidth over 300 MHz. This profile was obtained at the source pump power of Ppump = 11, 5 dBm and a DC coil current of 2 mA. The measuring device operates in a three-wave mixing mode. The saturation power of the amplifier is defined as a value of the input signal power at which the gain is decreased by 1 dB. The saturation power was also measured at the center frequency ωJPA = 6.4 GHz. We measured the saturation power of −97 to −100 dBm in the bandwidth of 300 MHz with a gain above 17 dB [see Fig. 3(b)].For noise temperature measurement, we used the method described in Ref. [2121. J. Y. Mutus, T. C. White, R. Barends, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey, J. Kelly, A. Megrant, C. Neill, P. J. J. O'Malley, P. Roushan, D. Sank, A. Vainsencher, J. Wenner, K. M. Sundqvist, A. N. Cleland, and J. M. Martinis, Appl. Phys. Lett. 104, 263513 (2014). https://doi.org/10.1063/1.4886408]. We calibrated the noise of the HEMT following our IMPA and then calculate system noise using the SNR improvement method. The estimated IMPA noise temperature is consistent with near-quantum-limited operation [see Fig. 3(b)].In summary, we have designed, fabricated, and characterized the impedance-matched Josephson parametric amplifier based on the SNAIL array and microstrip impedance transformer connected in series. The proposed device does not require a complicated multi-step process and can be fabricated with just one single e-beam lithography. We demonstrated an average gain of 17 dB with a bandwidth over 300 MHz at the central resonance frequency of 6.4 GHz, which corresponds to the design frequency. The noise temperature was estimated to be close to the standard quantum limit with the saturation power in the range of [−97, −100] dBm. We experimentally tested the proposed device for ultrahigh efficient multi-resonator quantum memory readout, where a broadband amplifier is required to perform quantum tomography of the quantum states storing process with a low number of photons.3838. A. R. Matanin, K. I. Gerasimov, E. S. Moiseev, N. S. Smirnov, A. I. Ivanov, E. I. Malevannaya, V. I. Polozov, E. V. Zikiy, A. A. Samoilov, I. A. Rodionov, and S. A. Moiseev, arXiv:2207.14092 (2022).See the supplementary material for the amplifier equivalent circuit and gain simulations as well as cryogenic experimental setup description.

The device was fabricated at the BMSTU Nanofabrication Facility (Functional Micro/Nanosystems, FMN REC, ID 74300).

Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

Daria Ezenkova: Conceptualization (equal); Formal analysis (lead); Methodology (equal); Investigation (lead); Writing – original draft (lead). Dmitry Moskalev: Conceptualization (equal); Formal analysis (lead); Methodology (equal); Investigation (lead); Writing – review and editing (equal). Nikita Smirnov: Formal analysis (equal); Investigation (equal); Writing – review and editing (equal). Anton Ivanov: Formal analysis (equal); Investigation (equal); Writing – review and editing (equal). Alexey Matanin: Formal analysis (supporting); Investigation (supporting); Writing – review and editing (supporting). Victor Polozov: Investigation (supporting); Writing – review and editing (supporting). Vladimir Echeistov: Formal analysis (supporting); Investigation (supporting). Elizaveta Malevannaya: Formal analysis (supporting); Investigation (supporting). Andrey Samoilov: Formal analysis (supporting); Writing – review and editing (supporting). Evgeniy Zikiy: Formal analysis (supporting); Investigation (supporting). Ilya Rodionov: Project administration (lead); Conceptualization (lead); Formal analysis (equal); Writing – review and editing (equal).

The data that support the findings of this study are available within the article.

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