Broadband CPW-based impedance-transformed Josephson parametric amplifier

In superconducting circuits, parametric amplification is usually achieved by Josephson parametric amplifiers (JPAs) utilizing multi-wave mixing processes in nonlinear superconducting resonators, where Josephson junctions provide the nonlinearity without introducing dissipation Roy and Devoret (2016); Manucharyan et al. (2007). The nonlinear resonator of the JPA is typically coupled to a fixed 50-$\Omega$ embedding environment Castellanos-Beltran and Lehnert (2007); Yamamoto et al. (2008); Bergeal et al. (2010); Hatridge et al. (2011), which prevents tuning its linewidth and saturation power independently, resulting in a trade-off between saturation power and bandwidthMutus et al. (2014). Hence, its bandwidth is inherently limited to 10-50 MHz, and its saturation power is typically -120 dBm for a nominal gain of 20 dB.

There are two strategies to overcome this limitation. In the first approach, the nonlinear resonator is unpacked into a nonlinear transmission line by thousands of Josephson junctions, forming a traveling-wave-parametric-amplifier (TWPA), in which the gain is optimized across a broad range of frequencies Esposito et al. (2021). Such a device must satisfy two conditions: (i) the impedance of the amplifier has to match the 50-$\Omega$ environmental impedance, and (ii) the phases of the pump, signal, and idler tones have to match Esposito et al. (2021). To fulfill these challenging requirements, 2000 nonlinear unit cells were constructed to form a TWPA with a 20-dB-gain bandwidth over 3 GHz and a saturation input power of -99 dBm Macklin et al. (2015). However, TWPA noise processes can be complex Esposito et al. (2021), and the stringent requirements on the phase matching and the large number of Josephson junctions make the design, calibration and fabrication difficult for TWPAs Ezenkova et al. (2022).

In the second approach, the environmental impedance is engineered to enhance the JPA’s coupling, which leads to enhanced bandwidth and saturation power Mutus et al. (2014). This has been accomplished by both simple narrowband impedance transformers Roy et al. (2015); Grebel et al. (2021) and more complex broadband impedance transformers Mutus et al. (2014); Lu et al. (2022); Ranzani et al. (2022). Narrowband impedance transformers constructed using $\lambda/4$ and $\lambda/2$ coplanar-waveguides (CPWs) Roy et al. (2015); Grebel et al. (2021) enhances the JPA’s bandwidth to 200 MHz with fixed operating frequency. JPAs with broadband impedance transformers following Klopfenstein Mutus et al. (2014); Lu et al. (2022) or Ruthroff Ranzani et al. (2022) taper designs have instantaneous bandwidth in the range of 600-700 MHz and allow tunable operating frequencies, but their current implementations either require multi-layer fabrication with lossy dielectrics Mutus et al. (2014); Ranzani et al. (2022) or careful simulation and calibration of finely patterned interdigitated capacitors Ranzani et al. (2022). These requirements increase the complexity of the amplifiers’ design, calibration, and fabrication.

In this Letter, we present a CPW-based broadband impedance-transformed parametric amplifier (CIMPA) design and showcase its performance. By reducing the technical requirements of the Klopfenstein taper, we can implement it with a single horn-like CPW, which keeps the broadband behavior while maintaining the simplicity of the design and fabrication. The horn-like CPW structure allows us to leverage streamlined RF design tools Simons (2004) and minimize the complexity of simulating and fabricating the impedance transformer. Notably, the CIMPA performs as well as the other IMPAs despite such simplicity. The amplifier displays an instantaneous bandwidth of 700(200) MHz for 15(20) dB gain, a 1.4 GHz flux-tunable bandwidth, a saturation input power of approximately -110 dBm, and no significant back action on the qubit.

The CIMPA operates as follows. When an RF microwave pump is applied to the nonlinear resonator via the flux line at approximately twice its resonant frequency $\omega_{p}\approx 2\omega_{0}$, we can apply the rotating wave approximation (RWA) in the frame rotating at the signal frequency $\omega_{s}\approx\omega_{0}$ and write the dressed Hamiltonian as Eddins (2017)

where $\hat{a}(\hat{a}^{\dagger})$ is the bosonic annihilation (creation) operator associated with the resonator, $\lambda$ originates from the modulation of the external flux by the pump, and $\Delta=\omega_{p}/2-\omega_{0}\sim 0$. If the circuit is coupled to the environment with strength $\kappa$, its input-output equation of motion is given by

where $\hat{a}_{\mathrm{in}}$ and $\hat{a}_{\mathrm{out}}$ are the bosonic operators associated with incoming and outgoing signals. We then proceed to define $\bar{a}[\omega]=\int_{-\infty}^{\infty}\text{d}t\ e^{i\omega t}\hat{a}(t)$ as the frequency-domain bosonic annihilation operator, and use the boundary condition $\sqrt{\kappa}\hat{a}=\hat{a}_{\text{in}}+\hat{a}_{\text{out}}$ to obtain

from which the gain is define, $G(\omega)=|\bar{a}_{\text{out }}/\bar{a}_{\text{in}}|^{2}$.

For a suitable value of $\lambda$, the denominator in Eq. 3 approaches zero, resulting in large signal gain, which remains at finite values in practice because of higher-order terms that are not included in Eq. 1. Boutin et al. (2017) In addition, large signal power may lead to an ac Stark shift and pump power depletion, thereby decreasing the effective gain Boutin et al. (2017). The incoming signal power associated with a gain drop of 1 dB is defined as the saturation input power, or alternatively, the 1-dB compression point.

Importantly, Eq. 3 shows that the amplifier’s bandwidth is determined by the coupling strength $\kappa$, and thereby can be improved by enhancing the coupling between the resonator and the signal. For the circuit shown in Fig. 1(a), this coupling rate is given as $\kappa=(CZ_{0})^{-1}$, while the saturation power is $P_{\mathrm{sat}}\sim C/Z_{0}$Mutus et al. (2014), where $Z_{0}$ is the characteristic impedance of the coupled environment. Thus, by engineering $Z_{0}$, one can increase both the bandwidth and the dynamic range of the amplifier. Since the signals must come through a nominal 50-$\Omega$ transmission line before and after interacting with the amplifier, an impedance transformer is necessary for enhanced coupling, shown on the left-hand side of Fig. 1(a).

Previously, this idea was realized using a Klopfenstein taper with a characteristic impedance ranging from $50~{}\Omega$ at one end to $15~{}\Omega$ at the other end, and a maximum return loss of -20 dB above the cut-off frequency Mutus et al. (2014); Lu et al. (2022). These stringent requirements may lead to sophisticated designs that are associated with tedious simulation procedures and complex nanofabrication processes. Alternatively, a combination of quarter-wavelength and half-wavelength CPW resonators can be used Roy et al. (2015), with the setback of having a narrower bandwidth, thereby negating the flux-tunable advantage of the SQUID.

Our design unifies the advantages of these two approaches by relaxing the condition on maximum return loss to -10 dB. This allows us to engineer a horn-like CPW transformer with characteristic impedance varying from 50 $\Omega$ to 18 $\Omega$ (Fig. 1(b)). The taper is designed by fixing its gap to 3 $\mathrm{\mu m}$ and varying the center-trace width from 7.8 $\mathrm{\mu m}$ to 218 $\mathrm{\mu m}$. The single CPW structure can be simulated using simple RF design suites such as TX-Line Simons (2004), and its dimensions are compatible with single-layer photo-lithography and reactive-ion etching techniques. As a result, this approach significantly reduces the complexity overhead in both design and fabrication, which is reflected by the simple construction of our device shown in Fig. 1(c).

In the first step, we extract the parameters of the nonlinear resonator by measuring the small-signal linear response of the device using a vector network analyzer (VNA). As the probe signal goes through $\omega_{0}/(2\pi)$, the reflected signal phase is twisted, indicating a resonant condition. When we adjust the external flux bias $\Phi_{e}$ of the SQUID, the resonant frequency changes according to

where $L_{\text{geo}}$ is the parasitic geometric inductance originating from the leads, $L_{\text{eff}}$ is the effective inductance of the SQUID, $\Phi_{0}$ is the magnetic flux quantum. Measuring the phase response at each flux point gives us the spectrum shown in Fig. 2. By fitting the measured spectrum, we extract the following parameters: $L_{\text{J,eff}}=69~{}\text{pH},~{}C=4~{}\text{pF}$, and $L_{\text{geo}}<1~{}\text{fH}$. The spectrum also allows us to estimate the asymmetry of the SQUID to be less than 1%.

We proceed to characterize the CIMPA’s gain profile, noise performance, and dynamic range. We set the external flux bias to $\Phi_{e}=0.3\Phi_{0}$, and apply a microwave pump tone at frequency $\omega_{p}/(2\pi)=13.41~{}\text{GHz}$ with 3.41 dBm power outside the fridge. we measure the microwave response using the VNA, observing over 20 (15) dB gain over a 200 (700)-MHz bandwidth (Fig. 3(a)). Such a high increase in the bandwidth cannot be explained solely by increasing the resonator linewidth Mutus et al. (2014). As discussed in Ref. Mutus et al. (2014), the weak frequency dependence of the environment’s characteristic impedance strongly assists the enhancement of the bandwidth, but hinders the comparison between the theory and experiments without the careful calibration of the environment’s characteristic impedance.

The noise performance of the device can be characterized via the signal-to-noise ratio (SNR) improvement $\text{SNR}_{\text{imprv}}$ measured by a spectrum analyzer (SA) Mutus et al. (2014); Macklin et al. (2015); Roy et al. (2015); Aumentado (2020). In an amplification chain consisting of a CIMPA and a HEMT, the SNR improvement due to the CIMPA is given as

where $T_{\text{CIMPA}}$ and $T_{\text{HEMT}}$ indicate the noise temperatures of the CIMPA and the HEMT, respectively. The extracted noise temperature with $T_{\text{HEMT}}=2.3$ K - $2.9$ K is shown in Fig. 3(b), which approaches the quantum noise limit. The uncertainty here originates from the estimated $T_{\text{HEMT}}$ range. According to the convention used in Ref. Mutus et al. (2014), we define the bandwidth of our device to be 700 MHz as the frequency range over which the device approaches the quantum noise limit. The noise increase beyond this range is mainly because the CIMPA’s gain is not large enough to compensate for the noise at the HEMT.

Presently, we characterize the saturation power of the device as follows. First, the relation between the actual signal power going into the CIMPA and the power set on the VNA must be established. By measuring the ac Stark shift and measurement-induced dephasing of the transmon qubit at various VNA powers, we can calibrate the cable attenuation Macklin et al. (2015). Then, we gradually increase the power until the gain falls off by 1 dB, and convert this using the established attenuation. The result in Fig. 3(c) shows an average 1-dB compression point of approximately -110 dBm. Compared with the saturation power of simple JPAs coupled to a $50~{}\Omega$ environment (with $Q\approx 15$), which is less than -120 dBm Yamamoto et al. (2008); Mutus et al. (2014); Ranzani et al. (2022), our saturation power is increased by around $10$ dB. This can be explained by the reduction in the environment’s characteristic impedance in our design (with $Q\approx 4$), as $P_{\mathrm{sat}}\sim C/Z_{0}$ Mutus et al. (2014).

In addition to the instantaneous bandwidth, we can flux-tune the device and operate it at other frequencies. To showcase this capability, we calibrate the CIMPA’s parameters at five different external flux points corresponding to a frequency range of 1.4 GHz, then characterize the gain performance of the device across this broad range. Figure 3 (d) shows that the combination of the engineered broadband impedance transformer and the inherent flux-tunability allows the device to amplify signals with frequency ranging from 6 GHz to 7.4 GHz and at least 15 dB of maximum gain.

Importantly, possible back-action from the parametric process can undermine the integration of an amplifier into a qubit measurement setup. For example, spurious microwave radiation from the CIMPA can populate the readout resonator connected to the qubit, leading to photon-noise dephasing Gambetta et al. (2006); Kindel et al. (2016). To investigate the possible back-action of the CIMPA prototype, we measure the echo time $T_{2,\text{echo}}$ of the qubit with and without amplification. To turn off the CIMPA, we remove the pump tone and flux-tune its frequency away from the qubit’s readout resonator, such that the amplifier acts as a passive mirror. When the CIMPA is turned on, it provides a gain of 21 dB for the readout signal frequency at 6.633 GHz. The $T_{2,\text{echo}}$ measurement is repeated continuously over 201 iterations in each setting to provide reliable statistics despite temporal fluctuations. We attribute the wider fluctuation range associated with the measurement when the CIMPA is off to the approximately ten times longer data acquisition time. As shown in Fig. 4, there is no significant degradation of the qubit’s coherence time beyond random fluctuation when the CIMPA is operated.

Supplementary Materials