Multiparametric Amplification and Qubit Measurement with a Kerr-free Josephson Ring Modulator

TL;DR

This paper introduces a novel Josephson Ring Modulator design that nulls higher order Hamiltonian terms, enabling a bi-directional, broadband, phase-sensitive amplifier for superconducting qubit readout.

Contribution

A new linearly shunted Josephson Ring Modulator with minimized 4th-order terms and a combined parametric drive scheme for enhanced amplification and bandwidth.

Findings

Achieved nearly null 4th-order Hamiltonian terms at a specific flux bias.

Demonstrated a bi-directional, phase-sensitive amplifier with large, gain-independent bandwidth.

Successfully used the amplifier for superconducting qubit readout with high quantum efficiency.

Abstract

Josephson-junction based parametric amplifiers have become a ubiquitous component in superconducting quantum machines. Although parametric amplifiers regularly achieve near-quantum limited performance, they have many limitations, including low saturation powers, lack of directionality, and narrow bandwidth. The first is believed to stem from the higher order Hamiltonian terms endemic to Josephson junction circuits, and the latter two are direct consequences of the nature of the parametric interactions which power them. In this work, we attack both of these issues. First, we have designed a new, linearly shunted Josephson Ring Modulator (JRM) which nearly nulls all 4th-order terms at a single flux bias point. Next, we achieve gain through a pair of balanced parametric drives. When applied separately, these drives produce phase-preserving gain (G) and gainless photon conversion (C), when applied together, the resultant amplifier (which we term GC) is a bi-directional, phase-sensitive transmission-only amplifier with a large, gain-independent bandwidth. Finally, we have also demonstrated the practical utility of the GC amplifier, as well as its' quantum efficiency, by using it to read out a superconducting transmon qubit.

Equations

$$H_{\mathrm{shunt}}=\sum _{i=1}^4\frac{{\varphi }_0^2}{2L_{\mathrm{shunt}}}{\left({\phi }_i-{\phi }_E\right)}^2,$$

$$H_{\mathrm{out}}=\sum _{i=1}^4{H}_{\mathrm{seg}}\left({\phi }_{i+1}-{\phi }_i-{\varphi }_{\mathrm{ext}}/4\right).$$

$$H_{\mathrm{seg}}(\phi )=\underset{\chi }{\min }\frac{{\varphi }_0^2}{2L_{\mathrm{out}}}(\phi -\chi {)}^2-E_J\cos (\chi ),$$

$$\left\{{\phi }_1,{\phi }_2,{\phi }_3,{\phi }_4\right\}$$

$$=\left\{{\phi }_a+{\lambda }_1{\phi }_c,{\phi }_b-{\lambda }_2{\phi }_c,-{\phi }_a+{\lambda }_1{\phi }_c,-{\phi }_b-{\lambda }_2{\phi }_c\right\}.$$

$$K_{aa}$$

$$K_{ab}$$

$${\dot{\phi }}_j=\frac{1}{{\varphi }_0}{V}_j(t)\iff {\phi }_j(t)=\frac{1}{{\varphi }_0}{\int }_{-\infty }^t\left[V_j(t^{\prime })\right]d{t}^{\prime }$$

$$J_j=\frac{1}{{\varphi }_0}\frac{\partial H_{\mathrm{JRM}}}{\partial {\phi }_j},$$

$$2C_1{\varphi }_0^2{\ddot{\phi }}_1+E_J\left[\sin \left({\phi }_1-{\phi }_2+\frac{{\phi }_{\text{ext}}}{4}\right)-\sin \left({\phi }_4-{\phi }_1+\frac{{\phi }_{\text{ext}}}{4}\right)\right]+\frac{{\varphi }_0^2}{L_{\text{shunt}}}\left({\phi }_1-{\phi }_5\right)=0$$

$$2C_4{\varphi }_0^2{\ddot{\phi }}_2+E_J\left[\sin \left({\phi }_2-{\phi }_3+\frac{{\phi }_{\text{ext}}}{4}\right)-\sin \left({\phi }_1-{\phi }_2+\frac{{\phi }_{\text{ext}}}{4}\right)\right]+\frac{{\varphi }_0^2}{L_{\text{shunt}}}\left({\phi }_2-{\phi }_5\right)=0$$

$$2C_2{\varphi }_0^2{\ddot{\phi }}_3+E_J\left[\sin \left({\phi }_3-{\phi }_4+\frac{{\phi }_{\text{ext}}}{4}\right)-\sin \left({\phi }_2-{\phi }_3+\frac{{\phi }_{\text{ext}}}{4}\right)\right]+\frac{{\varphi }_0^2}{L_{\text{shunt}}}\left({\phi }_3-{\phi }_5\right)=0$$

$$2C_3{\varphi }_0^2{\ddot{\phi }}_4+E_J\left[\sin \left({\phi }_4-{\phi }_1+\frac{{\phi }_{\text{ext}}}{4}\right)-\sin \left({\phi }_3-{\phi }_4+\frac{{\phi }_{\text{ext}}}{4}\right)\right]+\frac{{\varphi }_0^2}{L_{\text{shunt}}}\left({\phi }_4-{\phi }_5\right)=0$$

$${\phi }_5=\frac{1}{4}\left({\phi }_1+{\phi }_2+{\phi }_3+{\phi }_4\right).$$

$${\ddot{\phi }}_1+\frac{E_J}{2C_1{\varphi }_0^2}\cos (\frac{{\phi }_{\mathrm{ext}}}{4})\left[2{\phi }_1-{\phi }_2-{\phi }_4\right]+\frac{1}{8C_1{L}_{\text{shunt}}}\left(3{\phi }_1-{\phi }_2-{\phi }_3-{\phi }_4\right)=0$$

$${\ddot{\phi }}_2+\frac{E_J}{2C_4{\varphi }_0^2}\cos (\frac{{\phi }_{\mathrm{ext}}}{4})\left[2{\phi }_2-{\phi }_3-{\phi }_1\right]+\frac{1}{8C_4{L}_{\text{shunt}}}\left(3{\phi }_2-{\phi }_1-{\phi }_3-{\phi }_4\right)=0$$

$${\ddot{\phi }}_3+\frac{E_J}{2C_2{\varphi }_0^2}\cos (\frac{{\phi }_{\mathrm{ext}}}{4})\left[2{\phi }_3-{\phi }_4-{\phi }_2\right]+\frac{1}{8C_2{L}_{\text{shunt}}}\left(3{\phi }_3-{\phi }_1-{\phi }_2-{\phi }_4\right)=0$$

$${\ddot{\phi }}_4+\frac{E_J}{2C_3{\varphi }_0^2}\cos (\frac{{\phi }_{\mathrm{ext}}}{4})\left[2{\phi }_4-{\phi }_1-{\phi }_3\right]+\frac{1}{8C_3{L}_{\text{shunt}}}\left(3{\phi }_4-{\phi }_1-{\phi }_2-{\phi }_3\right)=0$$

$${\omega }_0^2$$

$${\omega }_a^2$$

$${\omega }_b^2$$

$${\omega }_c^2$$

$$\frac{E_J}{{\varphi }_0}\sin \left({\phi }_j-{\delta }_j+\frac{{\phi }_{\text{ext}}}{4}\right)=\frac{{\varphi }_0}{L_{\text{out}}}({\delta }_j-{\phi }_{j+1}),$$

$$\frac{E_J}{{\varphi }_0}\sin \left(-{\delta }_j^{(0)}+\frac{{\phi }_{\text{ext}}}{4}\right))=\frac{{\varphi }_0}{L_{\text{out}}}{\delta }_j^{(0)}.$$

$${\delta }_j^{(1)}=\frac{{\phi }_j\alpha \cos (\Delta )+{\phi }_{j+1}}{1+\alpha \cos (\Delta )}.$$

$${\ddot{\delta \phi }}_1+\frac{E_J}{2C_1{\varphi }_0^2}\frac{\cos \Delta }{1+\alpha \cos \Delta }\left(2\delta {\phi }_1-\delta {\phi }_2-\delta {\phi }_4\right)+\frac{1}{8C_1{L}_{\text{shunt}}}\left(3\delta {\phi }_1-\delta {\phi }_2-\delta {\phi }_3-\delta {\phi }_4\right)=0$$

$${\ddot{\delta \phi }}_2+\frac{E_J}{2C_4{\varphi }_0^2}\frac{\cos \Delta }{1+\alpha \cos \Delta }\left(2\delta {\phi }_2-\delta {\phi }_1-\delta {\phi }_3\right)+\frac{1}{8C_4{L}_{\text{shunt}}}\left(3\delta {\phi }_2-\delta {\phi }_1-\delta {\phi }_3-\delta {\phi }_4\right)=0$$

$${\ddot{\delta \phi }}_3+\frac{E_J}{2C_2{\varphi }_0^2}\frac{\cos \Delta }{1+\alpha \cos \Delta }\left(2\delta {\phi }_3-\delta {\phi }_2-\delta {\phi }_4\right)+\frac{1}{8C_2{L}_{\text{shunt}}}\left(3\delta {\phi }_3-\delta {\phi }_1-\delta {\phi }_2-\delta {\phi }_4\right)=0$$

$${\ddot{\delta \phi }}_D+\frac{E_J}{2C_3{\varphi }_0^2}\frac{\cos \Delta }{1+\alpha \cos \Delta }\left(2\delta {\phi }_4-\delta {\phi }_1-\delta {\phi }_3\right)+\frac{1}{8C_3{L}_{\text{shunt}}}\left(3\delta {\phi }_4-\delta {\phi }_1-\delta {\phi }_2-\delta {\phi }_3\right)=0$$

Full text

††preprint: AIP/123-QED

I Introduction

In recent years, the potential of superconducting qubits as a quantum computing architecture has grown tremendously Devoret and Schoelkopf (2013); Barends et al. (2014). This is in part due to quantum limited, Josephson-junction based parametric amplifiers (JPAs), which are able to enhance qubit signals while adding the minimum amount of noise and back-action, allowed by quantum mechanics Caves (1982). In these devices, gain is achieved by intensely driving the non-linear Hamiltonian of a circuit containing one or more Josephson junctions to parametrically couple together one or more microwave modes Roy and Devoret (2016). Paramps regularly enable experiments with single-shot, high-fidelity quantum non-demolition measurements of superconducting quantum bits Vijay et al. (2011); Hatridge et. al. (2013); Walter et al. (2017).

At present, there are two families of parametric amplifiers: those based on discrete microwave resonances, and those based on nonlinear transmission lines, so-called Traveling Wave Parametric Amplifiers (TWPAs). TWPAs have superior instantaneous bandwidth and saturation power, but their typical reported noise performance is a factor of a few higher than resonant mode-based JPAs, and, although they are directional amplifiers, reflected pump and signal tones still often require operation with external circulators Macklin et al. (2015); White et al. (2015); Zorin (2016). On the other hand, resonator-based JPAs (the focus of this work) contain typically far fewer Josephson junctions, are far easier to fabricate and operate very near the quantum limit. Their limitations are a fixed gain-bandwidth product, low input saturation powers, and lack of directional amplification (that is, they amplify in reflection and so must be operated with external circulators) Castellanos-Beltran and Lehnert (2007); Bergeal et al. (2010a); Hatridge et al. (2011). These issues stem from multiple causes. Recent work has shown that higher-order terms in the amplifier Hamiltonian can limit the saturation power of the device Liu et al. (2017); Frattini et al. (2017, 2018); Sivak et al. (2019), while the lack of directionality and fixed gain-bandwidth product are inherent to the choice of parametric coupling used in the amplifier. However, there are numerous theoretical Metelmann and Clerk (2015, 2014); Ranzani and Aumentado (2015) and experimental works Sliwa et al. (2015); Lecocq et al. (2017); Peterson et al. (2017); Fleury et al. (2014); Abdo et al. (2019) showing that applying multiple, simultaneous couplings in an amplifier containing two or more modes/ports can result in devices which potentially circumvent some or all of these limitations.

In this work, we begin by developing a kinetic-inductance shunted Josephson Ring Modulator (JRM) mixing element which is free of 4th-order nonlinearities for a certain bias flux, and embed it in a Josephson Parametric Converter (JPC) Bergeal et al. (2010a, b). The JPC consists of three microwave modes, each linked to a single microwave port. The JRM provides a general third order coupling between the devices three modes. Phase preserving gain (G) using a pair of modes is typically achieved by driving one mode off resonance at the sum of the other two modes’ frequencies. Gainless photon conversion (C) between two modes, from which the JPC derives its’ name, is achieved by driving instead at the difference frequency.

Next, we follow the scheme suggested by Metelmann and Clerk (2015), in which both G and C drives are applied simultaneously to a single pair of modes with matched coupling strengths. The combination of frequency conjugating (G) and non-conjugating (C) processes results in phase-sensitive, transmission-only amplification, shown schematically in Fig. 1. The device has unity reflection gain and operates bi-directionally, and so still requires external circulators, but has a number of desirable features. First, it has a fixed, large bandwidth ($\sim$60 $\%$ of the modes’ linear bandwidths). Second, rather than having divergent gain at fixed pump power, its gain is linearly dependent on pump power, with potential advantages in gain stability and device saturation behavior.

Although this scheme has been proposed previously, efforts to implement it have been hampered by the presence of higher-order Hamiltonian terms in the JRM. Notably, unshunted Bergeal et al. (2010b) and Josephson-junction shunted JRMs Schackert (2013); Liu et al. (2017), have 4th-order self- and/or cross-Kerr terms at all bias fluxes. These terms cause the amplifier’s mode frequencies to shift with applied pump power, making biasing the GC mode of operation, which requires far more intense pumps than G or C alone, extremely difficult. By operating our kinetic-inductance shunted JRM at a bias point which nearly nulls all Kerr terms in the device simultaneously, we have demonstrated 21 dB of phase-sensitive gain (15 dB phase-preserving gain), with a greatly enhanced bandwidth. To demonstrate the viability of this mode of operation in practical quantum information experiments, we used the GC amplifier to measure a transmon qubit. By strongly measuring the qubit, we were able to perform high fidelity qubit readout. We also performed deliberately weak measurement experiments to determined the device’s quantum efficiency.

II Theory of the GC amplifier and the nulling of the 4th order terms

The block diagram describing our scheme for qubit measurement and signal amplification is shown in Fig. 1(a). The qubit resides inside the measurement cavity, which has two ports, a strong port and a weak port. The weak port (on the left) is used for driving the cavity and qubit while the strong port (on the right) is used for measuring the qubit. The measurement signal is routed via two circulators, which ensure that the reverse amplified quantum noise and reflected signals from the amplifier do not travel backwards to the qubit. The JPC’s output is further amplified by a cryogenic HEMT amplifier before being recorded at room temperature. The scattering matrix of the GC amplifier is shown in Fig. 1(b). Signals entering a port are reflected with unity gain, and transmitted with phase-sensitive gain, whose amplified quadrature is determined by the phases of the applied pumps.

The effective circuit diagram of our inductively shunted JRM is depicted in Fig. 2(a). The key ingredients of the model are (1) the Josephson junctions, (2) the shunt inductors labeled $L_{\rm shunt}$, and (3) the stray inductance on the outer arms of the JRM, labeled labeled $L_{\rm{out}}$. In this setup, the outer ring of the JRM, which contains the Josephson junctions, is the source of the non-linear couplings between the modes of the JPC. The shunt inductors are used to control the degeneracy of the ground state of the JPC at finite magnetic flux bias. They lift the energy of states in which current flows through the shunt inductors, thus preventing the device from switching to these undesired configurations. The stray inductors are an inherent property of the aluminum traces which we use to make the outer ring of our JRM.

The circuit is described by the Hamiltonian $H_{\rm JRM}=H_{\rm shunt}+H_{\rm out}$, composed of the shunt Hamiltonian

[TABLE]

and the outer ring Hamiltonian

[TABLE]

Here, $\varphi_{i}$ is the superconducting phase at the $i$-th vertex of the JRM (see Fig. 2a) and we use the notation $\varphi_{5}=\varphi_{1}$ 111We have assumed that the phase on the middle node is zero, this assumption is founded in the eigenmode analysis presented in Appendix B., $\varphi_{E}$ is the phase at the middle point which is constrained by the outer node phase by $\varphi_{E}=\sum_{j=1}^{4}\varphi_{j}/4$. The phase gain due to the externally applied flux bias in each of the four outer arms of the JRM is $\phi_{\rm ext}/4=(2e/\hbar)\,\Phi_{\textrm{ext}}/4$. The energy of an outer ring segment phase biased to $\varphi$ is

[TABLE]

and the minimization with respect to $\chi$ results in a transcendental equation that ensures that the current in the outer inductor is identical to the current in the Josephson junction.

Next, we analyze the non-linear couplings between the JPC modes. Here, we focus on the case in which the JPC has a non-degenerate ground state centered on $\varphi_{1}=\varphi_{2}=\dots=0$. In this case, the phases at the JRM vertices can be expressed in terms of the canonical variables $\varphi_{a}$, $\varphi_{b}$, and $\varphi_{c}$ that correspond to the $a$, $b$, and $c$ eigenmodes of the JPC:

[TABLE]

The coefficients $\lambda_{1}=C_{3}/\sqrt{C_{1}^{2}+C_{3}^{2}}$ and $\lambda_{2}=C_{1}/\sqrt{C_{1}^{2}+C_{3}^{2}}$ come from the JPC eigemode analysis and their values depend on the capacitors added to the JRM to make the JPC, see Appendix B. We obtain the non-linear couplings between the normal modes of the JPC by taking derivatives of $H_{\rm JRM}$ with respect to the canonical variables. For the self- and cross-Kerr terms we find:

[TABLE]

Observe that all of the Kerr terms are proportional to the fourth derivative of $H_{\rm seg}(\varphi)$ with respect to $\varphi$ evaluated at $\varphi=\varphi_{\rm ext}/4$. The cosine Hamiltonian of the junction means that the Kerr naturally passes through zero, if the desired flux configuration holds to the required flux. More, all six Kerr terms vanish identically at a single null point. In the absence of stray inductance (i.e. $L_{\rm out}\rightarrow 0$) the null occurs at $\phi_{\rm ext}=2\pi$; the null point persists and shifts towards $\phi_{\rm ext}=4\pi$ as $L_{\rm out}\rightarrow 0$ increases (see Section A).

We now introduce two dimensionless parameters that we will use to characterize our circuits. The first parameter, $\beta=L_{J}/L_{\rm shunt}=\phi_{0}^{2}/L_{\rm shunt}E_{J}$, measures the strength of the shunt inductors relative to the Josephson junction energy. The second parameter, $\alpha=L_{\rm out}/L_{J}=L_{\rm out}E_{J}/\phi_{0}^{2}$, measures the strength of the Josephson junction with respect to the stray inductance. Preferably $\alpha\ll 1$ to ensure that the Josephson energy dominates the stray inductance. At the same time, $\beta$ should be sufficiently large to ensure that $\varphi_{1}=\varphi_{2}=\dots=0$ is the global minimum of $H_{\rm JRM}$ and hence the JPC has a single, well defined ground state at the null point. For the case $\alpha=0$, the non-degeneracy of the ground state for all values of $\varphi_{\rm ext}$ is ensured by setting $\beta>4$. As $\alpha$ increases, so does the minimum required value of $\beta$ (we address these requirements by analyzing the degeneracy of the ground state in Appendix A).

A similar analysis of the JRM with Josephson-junction based shunts (as in Schackert (2013); Liu et al. (2017)), shows that the cross-Kerr terms null together at a similar point to our linearly shunted JRM, but the self-Kerr null point is shifted to a larger flux. As the behaviors of the various Kerr terms are very similar in their effect on device performance, the junction-shunted JRM ring will not realize the benefits of a linearly shunted JRM but will instead perform essentially no different from the unshunted version. We have also found, in both theory and experiment, that asymmetries in junction critical current and applied bias flux to the JRMs four loops must be minimized, as they can cause substantial difference in the flux response, and hence Kerr nulling, of the JRM.

III Experiment

We form a JPC by embedding the JRM at the intersection of two, single-ended microstrip resonators, as in Liu et al. (2017). The target parameters, junction $I_{0}=2~{}\mu$A $\beta>4$ and $\alpha\ll 1$, require either meandering shunt inductances Roch et al. (2012); Flurin (2015) or kinetic-inductance shunts; in our experiments we found the latter to produce better results.

To fabricate our JRM, we first sputter a film of 10 nm thick NbTiN onto a PMMA mask on which a cross with arm widths and lengths of 2 and 15 $\mu m$ respectively, has been patterned. The PMMA layer is subsequently lifted off with acetone. The resulting shunt inductors have $L_{\rm shunt}\simeq 75$ pH, about five times their geometric value. Next, standard double-angle aluminum deposition is used to create the outer ring of the JRM. As NbTiN does not form a native oxide, good contact to the upper (aluminum) portion of the circuit is achieved with only a standard, gentle Argon/Oxygen cleaning step prior to deposition of the first layer of aluminum.

In order to characterize the self- and cross-Kerr terms for a given mode/pair of modes, a pump tone is applied that is fixed 5 line widths away from the resonant frequency, while a second, weak tone is swept through the mode’s resonant frequency with a Vector Network Analyzer(VNA). The Kerr amplitude is seen as a shift in resonant frequency with increasing pump power Frattini et al. (2018). The cancellation points at which the resonant frequency remains fixed, is shown in Fig. 2(c). In subsequent qubit measurements, the frequency of the qubit cavity was tuned with an aluminum screw to align with the right most 4th-order free point of our JPC.

Next, we identified the signal and idler frequencies ,which are 7.4668 GHz and 4.8715 GHz respectively, and pumped at their sum, tuning the applied pump power to achieve 20 dB gain in reflection (G). We repeated the process with the pump at the difference frequency (C), tuning the applied pump power to find a 20 dB dip in reflection. These bias powers are each very close to the critical values, and tell us the room temperature ratio of applied microwave powers required to balance the G and C processes. We next fine tuned the applied pump frequencies until they linked to identical idler frequencies in transmission through the device.

Turning on both drives simultaneously achieves GC amplification. We control GC gain by increasing/decreasing both pumps simultaneously while maintaining the ratio established above. Measured GC amplification gain strengths between 8 and 15 dB, along with reflection performance, are shown in Fig. 3(a). All transmission gain curves are measured with a VNA. An external mixer at the difference frequency converts device outputs from the idler mode back to the input, signal mode, frequency. The VNA is sensitive only to a single frequency, thus all gains shown are phase-preserving, and so for large gains, the phase-preserving gain peak sits $\sim$ 6 dB below the phase-sensitive gain of the device, so that the maximum phase-sensitive gain for the 15.5 dB curve (which we will use for qubit readout) is 21.5 dB. Above 10 dB gain, the frequency of the pump tones had to be adjusted slightly to account for the imperfectly nulled higher order terms, which shifted the device’s modes, and hence, the frequency of peak gain, to lower frequencies. At the same time, the reflection curves at this higher power start to show small disturbances away from unity (see Fig. 3(a)). For clarity in these experiments, the amplifier is flux shifted slightly below the cavity frequency, which is evident as a notch to the right of the gain peaks.

The other crucial feature of these gain curves is that they demonstrate the same bandwidth regardless of gain (see Fig. 3(b)). We compare these bandwidths with the standard 20 dB for one single pump amplification process represented by the black curve. The GC amplifier shows 14 MHz of phase-preserving bandwidth. In contrast, a singly-pumped gain response of 20 dB only has a bandwidth of 2.33 MHz in the same device. This bandwidth is approximately 6 times larger, in good agreement with theory.

We also show a spectrum analyzer (received power) trace when the GC amplifier is driven with a fixed tone 5 MHz detuned from its center frequency in Fig. 3(c). The presence of a symmetrically detuned tone below the center frequency is a clear demonstration of the phase-sensitive nature of this devices’ gain. Collectively, Fig. 3 demonstrate all key predicted features of GC amplification; fixed bandwidth, 0 dB gain in reflection, and phase-sensitive amplification in transmission.

To demonstrate that our Kerr-nulled, GC pumped device is a practical, quantum-limited amplifier we also performed phase-sensitive strong/projective measurements on a superconducting transmon qubit. All further qubit measurements are performed in the configuration shown in Fig. 1(a). The qubit is first prepared in the superposition state $\ket{\Psi}=(\ket{g}+\ket{e})/\sqrt{2}$, we determined the optimal alignment of the device’s amplified quadrature by finding the largest separation between ground and excited states when projective measurement is performed, as shown in Fig. 4(a). Rotating the relative pump phase by 180 degrees from this optimal point moves the signal to the squeezed quadrature, so that the $\ket{g}$ and $\ket{e}$ states overlap, as seen in Fig. 4(b). Both histograms contain 80,000 measurements. We also measured spontaneous quantum jumps between the $\ket{g}$ and $\ket{e}$ state when the phase was rotated to the optimal alignment and the cavity was driven for 7.5 $\mu s$, as shown in Fig. 4(c).

Finally, we calculate the quantum efficiency of our amplifier via the back-action of deliberately weak measurements on the qubit’s state, as in Hatridge et. al. (2013); Sliwa (2016). The pulse sequence is detailed in section D. For phase-sensitive amplification, only one microwave quadrature can be received at room temperature. In a qubit measurement, this confines the qubit back-action to a single plane on the Bloch sphere, determined by which quadrature is amplified. To calibrate our quantum efficiency ($\eta$), we use the amplifier aligned as in Fig. 4(b), resulting in a pure back-action on the qubit’s phase. We found that our quantum efficiency was $\eta=55$ %, in good agreement with the efficiency of the device when operated as a phase-preserving amplifier at the same bias point. We also performed weak measurements with the amplifier aligned optimally (as in Fig. 4(a) (for which the back-action is only on the qubit’s z-coordinate), shown in Fig. 5.

IV Conclusion

In conclusion, we have created a JPC with a linearly shunted JRM that features flux bias points where all Kerr terms can be nulled simultaneously. We used this 4th-order free JRM to realize practical, robust GC amplification, which has phase-sensitive gain in transmission, unity gain in reflection, and a fixed bandwidth that is  6 times larger than a singly-pumped, phase-preserving 20 dB gain curve from the same device.

We have measured a transmon qubit with this device, demonstrating high fidelity projective measurements and quantum jumps. Finally, we used weak qubit measurements to find the quantum efficiency of this mode of amplification. The result, $\eta=0.55$ is comparable to the efficiency of the device when operated as a singly pumped, phase-preserving amplifier. We quote efficiency, rather than readout fidelity, as the figure of merit for the amplifier as the latter quantity involves a number of factors (such as qubit $T_{1}$ and insertion loss between cavity and amplifier) which are unrelated to virtues of GC amplification. The device used in this work was also chosen to have a modest bandwidth ($\sim 20~{}\textrm{MHz}$), to reduce the required pump powers for GC operation. Although the bandwidth was more than sufficient to amplify our qubit cavity, in future work we will open the bandwidth allow for multiple qubits to be read out in a single device.

This work suggests numerous avenues for future improvements of parameteric amplifiers. First, by extending the number of pumps to 6, while keeping the number of modes at 3, we can retain the benefits of GC pumping scheme and achieve directional amplificationMetelmann and Clerk (2015). Second, we should continue to explore the wealth of potential parametric driving schemes, which may yield devices with further improved performance. Finally, by continuing to fine-tune the engineered Hamiltonian of our JRMs we can enhance the saturation power of our deviceFrattini et al. (2018).

V Acknowledgments

C. Liu acknowledge support from a Pittsburgh Quantum Institute graduate student fellowship. This work was supported by NSF Grant No. PIRE-1743717. Research was also sponsored by the Army Research Office and was accomplished under Grant Number W911NF-18-1-0144. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Office or the U.S. Government. The U. S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein.

Appendix A The degeneracy of the ground state of the JPC

In this section, we analyze the degeneracy of the ground states of the shunted JPC for two cases: without and with the stray inductors. Specifically, to find the degeneracy of the ground state we count the number of global minima of $H_{\rm JRM}$.

We first consider the JPC without stray inductors, i.e. $L_{\textrm{out}}=0$, $\alpha=0$. We numerically find the global minima of the JRM Hamiltonian by forming a set of possible minimum points by randomly seeding a numerical minimum with initial values of $\varphi_{1}$, $\varphi_{2}$, $\varphi_{3}$, and $\varphi_{4}$. From this set, we select the minima that correspond to smallest values of $H_{\rm JRM}$, and finally we throw away repeated points. In Fig. 6(a) we plot the degeneracy of the JRM ground state as we sweep the external magnetic flux and the shunting parameter $\beta$. The blue region in the plots corresponds to conditions under which the JRM has a single global minimum (the JPC has a non-degenerate ground state), while the orange and red regions corresponds to doubly- and quadruply-degenerate ground states.

As we decrease the inductance of the shunts, states with circulating currents move up in energy, and hence the ground state degeneracy is lifted. In other words, larger $\beta$ corresponds to less degeneracy. In the absence of stray inductors, all even order terms (beyond 2nd order) in $H_{\rm JRM}$ become zero when the external magnetic flux bias is set to $2\pi$. We find that at this flux bias the ground state is non-degenerate for $\beta>1.0$ (minimum requirement for operation of the nulled JPC). Additionally, we find that the ground state is non-degenerate for all values of the magnetic flux bias when $\beta>4.0$.

Next, we take stray inductance into account. The stray inductors increase the inductance of the segments of the outer ring of the JRM, effectively decreasing $\beta$ (as compared to the case $\alpha=0$). In Fig. 6(b) we fix $\beta=4.5$ (close to our experimental parameters) and sweep the parameter $\alpha$ and external magnetic flux $\varphi_{\textrm{ext}}$. We notice that as we increase $\alpha$, the region of degenerate ground states reappears (near maximal magnetic flux bias). Further, we observe that as we increase $\alpha$ the Kerr nulling point moves from $\varphi_{\textrm{ext}}=2\pi$ towards $\varphi_{\textrm{ext}}=3.5\pi$, at which point it hits the degenerate ground state region (corresponding to $\alpha\approx 0.4$).

Appendix B The eigenmodes of the JPC

In this subsection, we discuss the JPC circuit and identify its eigenmodes. The effective circuit diagram without input-output connections is shown in Fig. 7(a). In order to simplify our analysis, we begin by making the assumption that $L_{\rm{shunt}}\rightarrow 0$. We will lift this assumption at the end.

In order to find the eigenmodes of the JPC, we construct the equations of motion (EOM) by applying Kirchhoff law to the circuit. We follow the standard prescription for superconducting circuits: relating voltage to the normalized flux (i.e. the superconducting phase) on each of the nodes via

[TABLE]

and the supercurrent to the normalized flux via

[TABLE]

where $\phi_{0}=\hbar/(2e)$ is the reduced magnetic flux quantum. The resulting equations of motion are

[TABLE]

We can linearize the EOMs by expanding around the minimum energy configuration of the JRM (that we found in previous section). We now focusing on the non-degenerate configurations of interest: $\varphi_{1}=\varphi_{2}=\varphi_{3}=\varphi_{4}=\varphi_{5}=0$. In this case $\varphi_{i}$’s can be regarded as small perturbations away from the JRM minimum point. The resulting linearized equations of motion are

[TABLE]

Making the additional assumption that $C_{1}=C_{2}$ and $C_{3}=C_{4}$, we find that the resulting eigenvalue/eigenstate pairs are

[TABLE]

where $\omega_{0}$ is a zero mode that does not couple with the rest of the modes in the Hamiltonian. The non-trivial modes $\varphi_{a}$, $\varphi_{b}$ and $\varphi_{c}$ are sketched in Fig. 7(b).

We now put the stray inductors back into the circuit. In doing so, we introduce four nodes between the stray inductors and the Josephson junctions on each arm. Because there is no kinetic terms associated with these nodes (as we neglect the capacitances to these nodes), the resulting Kirchhoff equations give us four more constrains:

[TABLE]

where $\delta_{j}$ is the phase at the node between the stray inductor and the Josephson junction on the $j$-th arm of the JRM (see Fig. 2).

In order to linearize the equations of motion, we must first find the values of $\delta_{j}$’s at the minimum energy configuration of the JRM. These correspond to the solution of the transcendental equation

[TABLE]

We set $\Delta=-\delta^{(0)}_{j}+\frac{\varphi_{\textrm{ext}}}{4}$, and use Eq. (15) to obtain an expression for the first order correction to $\delta_{j}$ in terms of $\varphi_{j}$ and $\varphi_{j+1}$

[TABLE]

We now eliminate $\delta^{(1)}_{j}$ from the linearized equations of motion to obtain

[TABLE]

Comparing equations Eq. (10) and Eq. (18) we observe that the effect of stray inductors is a renormalization of the Josephson energy $E_{J}\rightarrow E_{J}\frac{\cos\Delta}{1+\alpha\cos\Delta}$. This renormalization results in a shift of the eigenfrequencies but the eigenmodes remain identical.

Appendix C Measurement of Kerr terms

We need to quantify the strength of the Kerr terms inherent to our amplifier in order to find when they go to zero. To do this, we perform Self-Kerr and Cross-Kerr duffing sweeps across a wide range of flux. The self Kerr terms, $a^{\dagger}aa^{\dagger}a$ and $b^{\dagger}bb^{\dagger}b$, can be measured by applying a strong pump tone 5 line widths (here 100 MHz from the signal and 125 MHz from the idler) away from the respective resonant frequencies. We then sweep the flux and the pump power and measure the detuning from the resonant frequencies. Plotting this detuning visually in red, for negative detuning, and blue, for positive detuning, allows us to quickly identify where the these terms go to zero, since zero detuning, representing no 4th-order effects, is plotted in white. The cross Kerr, $a^{\dagger}ab^{\dagger}b$ is measured similarly, but this time we apply a tone 5 line widths away from the signal mode and measure the effect of the idler, as seen in Fig. 8. With a high enough pump power, it becomes easy to distinguish the red/blue regions, and thus, the crossing zone in the middle where the specific 4th-order term goes to zero. This flux bias point as shown Fig. 2 (c), is represented by a dotted black line. This JRM is designed such that all 4th order terms should be identically nulled at the same flux bias.

Appendix D Back action on the qubit for weak amplification

In addition to being an essential component to the measurement chain used to readout superconducting qubits, parametric amplification can be used to manipulate them as well. For instance, the back-action associated with parametric amplification can perform the essential function of remote entangling distant qubits. It can also be used as an accurate, self-calibrating way to determine measurement efficiency. These types of measurements will be vital in identifying and eliminating the effects which limit our ability to manipulate quantum systems.

In Fig. 9, we apply second measurement strength to be 5 times (in voltage) smaller than that of strong measurement while GC is operated at orthogonal phase. The histogram shows sinusoidal oscillation in both X and Y o highly nonclassical stochastic Ramsey processes. The expectation value of Z is affected by qubit’s relaxation and leads to the slightly red color instead of complete white. With the same procedure but squeezing the Q quadrature instead, we change’s qubit’s motion to be confined to the Y-Z plane, see Fig. 9.

The standard way to perform back-action measurement is shown in Fig.10. We first strongly read out the qubit and record the outcome, which will be used to prepare the qubit in the ground state with state selection. Then the qubit is rotated by the +y axis and measured with a variable measurement strength, and the outcome is recorded again. For the final tomography, phase measures the x, y, or z component of the qubit with a strong measurement pulse.

If we take the qubit to originally be oriented along the Y-axis and the I quadrature to be perfectly squeezed, the back-action corresponds to stochastic, trackable motion of the qubit state in the x-y plane, with the extent of the motion varying with the strength of the measurement. For weak strength, the back-action looks like a stochastic rotation in the x-y plane with the degree of rotation encoded in Qm.

Appendix E Saturation power

The amplifier’s saturation behavior was measured as shown in Fig. 11. Although operated with much higher pump powers, this dynamic range is virtually identical to singly-pumped phase-preserving and phase-sensitive gain at the same Kerr nulling point. This points to conventional pump depletion not limiting the device. In a separate work, we have concluded that still higher-order nonlinearities can contribute strongly to gain saturation and must be controlled to realize amplifiers with superior dynamic range Liu et al. (2019).