Nonlinearity and Parametric Amplification of Superconducting Nanowire Resonators in Magnetic Field

The measured transmission spectrum near resonance is fitted to the model in Refs. [76, 77, 74], which allows us to extract the low-power resonance frequency $\omega_{0}$, the intrinsic quality factor $Q_{i}$, the coupling quality factor $Q_{c}$ and the dimensionless power-dependent nonlinearity parameter $\xi$, as explained in Appendix A.

The resonators are first characterized in the linear regime (i.e. $\xi\approx 0$) to determine the dominant source of intrinsic loss. At zero magnetic field, $Q_{i}$ exhibits the typical behavior for a resonator dominated by two-level-system (TLS) loss [78, 79, 80], with $Q_{i}\approx 4,000$ for unsaturated TLS and $30,000$ for totally saturated TLS for the 0.1 $\mu$m resonator (see appendix B).

By ramping up the magnetic field we see two effects. Figure 1(d) shows $\omega_{0}$ and $Q_{i}$ as a function of magnetic field $B$ at an excitation power that is high enough to saturate TLS but not too high to maintain the linear behavior. The resonance frequency decreases quadratically with the field strength according to $\Delta\omega_{0}/\omega_{0}=-(\pi/48)[De^{2}t^{2}B^{2}/(\hbar k_{B}T_{c})]$ [66, 67, 68], where $D$ is the electronic diffusion constant and $e$ is the electron charge constant. By fitting the data to this relation we get $D\approx 5$ cm²s^-1, which is the same order of magnitude as the previously reported value for NbTiN [66]. The dip in $Q_{i}$ at $B\approx 0.2$ T, also observed before[66] is because of the coupling to the magnetic centers in the Si substrate. For magnetic fields above this dip, $Q_{i}$ increases with $B$ up to 2 T. A similar behavior has been reported before for narrow NbTiN resonators [66, 69]. This indicates that the resonator losses are dominated by the magnetic centers in the substrate even at zero magnetic field [69], and that losses coming from magnetic vortices are negligible. The observed enhancement in the internal quality factor improves the performance of linear devices, and as we will see, can improve the performance of nonlinear KI devices in magnetic field.

In our resonators, the dominant nonlinearity is 4WM (the Kerr effect). Three-wave mixing (3WM) is anticipated to be negligible because no DC current or flux is applied, so it is not explored. Different applications have different requirements on the strength of the nonlinearity. For example, in KI detectors the nonlinearity is a side effect and the Kerr coefficient should be as low as possible, while for qubits, large Kerr coefficients are preferable since they isolate the computation space, e.g., Kerr coefficients of 100-400 MHz are typical for transmons [2]. Parametric amplification falls between these two limits, where Kerr coefficient has to be higher than the nonlinear loss to maximize gain, but should be lower than the coupling loss to the feedline to maximize the dynamic range[49], as discussed in next section.

The nonlinearity of the KI resonator was studied in detail in Ref. [81]. The Hamiltonian of the resonator is

where $\omega_{n}$ and $K_{n}$ are the resonance frequency and self-Kerr coefficient of mode $n$, respectively, $K_{mn}$ is the cross-Kerr coefficient between modes $m$ and $n$, and $a_{n}$ ($a_{n}^{\dagger}$) is the annihilation (creation) operator of the intra-resonator field of mode $n$.

We characterize the nonlinearity of our resonators by experimentally measuring their self- and cross-Kerr coefficients. This requires estimating the average number of photons in the resonator corresponding to a certain driving power at the feedline using the extracted fitting parameters. Figure 2(a) displays the transmission response at three different powers. At low power ($\xi\approx 0$) the resonance frequency is $\omega_{r}=\omega_{0}$ and the resonator operates in the linear regime (top curve in Fig.2(a)). By driving at a higher power and at a frequency closer to $\omega_{r}$, more photons couple to the resonator driving it into the nonlinear regime. As the average number of photons in the resonator increases, $\omega_{r}$ red-shifts resulting in the asymmetric resonance lineshape (middle curve in Fig.2(a)). Above a certain power, corresponding to the onset of bifurcation ($\xi_{b}=-2/\sqrt{27}$ ignoring nonlinear loss [77]), the resonator goes into the bistability regime, where the resonator has three valid states at the same frequency with different amplitudes; two of which are stable and one is unstable (bottom curve in Fig.2(a)). This bistability feature has been used for digital threshold quantum detection in Josephson bifurcation amplifier, where the sensitivity is only limited by quantum fluctuations [82, 83, 84].

The red-shift in $\omega_{r}$ at increasing power is proportional to the average number of photons in the resonator, $n_{\text{ph}}$, such that $\Delta\omega_{r}=K_{0}n_{\text{ph}}/2$, where $K_{0}$ is self-Kerr coefficient depending on the critical current and kinetic inductance of the resonator. The value of $n_{\text{ph}}$ is determined for each input power as explained in Appendix A. We do the measurement at different powers and extract $K_{0}$ from the slope of the linear fit of $\Delta\omega_{r}$ versus $n_{\text{ph}}$. In this work, we focus on the trends with magnetic field and NW width, rather than the exact values of Kerr coefficients, so that uncertainties in $n_{\text{ph}}$, which is difficult to precisely determine, do not impact interpretation of the data. Figure 2(b) presents $\Delta\omega_{r}$ versus $n_{\text{ph}}$ for three resonators of widths 0.1, 0.3 and 1 $\mu$m. The decrease in self-Kerr parameter $|K_{0}|$ as the resonator width $w_{r}$ increases comes from the dependence of $K_{0}$ on the resonator critical current $I_{c}=J_{c}w_{r}t$ and kinetic inductance $L_{k}=L_{\square}l_{r}/w_{r}$, where $|K_{0}|\propto\omega_{0}^{2}/L_{k}I_{c}^{2}$ [81, 77].

Next, we extract the cross-Kerr coefficients which describe the strength of the intermode coupling in the resonator. This coupling can be used in circuit quantum electrodynamics for nondemolition measurements [85, 86, 87, 88]. One approach is to measure the number of photons in a certain mode, called signal mode, by driving another mode, called detector mode, and detecting its dephasing [85, 87] or frequency shift [88]. The sensitivity of this scheme is determined by the magnitude of the cross-Kerr coefficient between the two modes [85], where according to (1) the resonance shift of mode $n$ induced by pumping at mode $m$ is $\Delta\omega_{r}^{(n)}=K_{mn}n_{\text{ph}}^{(m)}/2$. To extract $K_{mn}$ for $m\neq n$, we apply a two-tone technique [89, 90, 91, 68] by pumping mode $m$ with high power using a MW source ($\omega_{p}\sim\omega_{m}$) while probing mode $n$ with low power using the VNA, as depicted in Fig. 1(c). We sweep the VNA frequency around $\omega_{n}$ for different pump powers at $\omega_{m}$. The shifts in the resonance frequency of the fundamental mode (i.e. $n=0$) of a resonator of width 0.1 $\mu$m when pumping at the first excited mode (i.e. $m=1$) and vice versa (i.e. $n=1$ and $m=0$) are shown in Fig. 2(c) as a function of the number of pump photons in the resonator. A meaningful conclusion on the relation between $K_{01}$ and $K_{10}$, or between either of them and $K_{0}$, requires the power levels at $\omega_{0}$ and $\omega_{1}$ to be estimated accurately with respect to each other. The ratio between the self-Kerr coefficients can be used for this purpose since we know that $|K_{n}|\propto\omega_{n}^{2}$ [81]. We take the power level at $\omega_{0}$ as a reference and calibrate the estimated power at $\omega_{1}$ such that $K_{1}/K_{0}=(\omega_{1}/\omega_{0})^{2}=4$. The extracted values for $K_{01}$ and $K_{10}$ after power calibration are almost equal, as shown in Fig. 2(c), which agrees with previous theoretical predictions [81]. We also find that the ratio $K_{0}/K_{01}\approx 0.4$, which is close to 0.375 calculated in [45].

The nonlinearity of the NW resonators is measured as a function of applied in-plane magnetic field (Fig. 1(d)). The value of $|K_{0}|$ for two different 1 $\mu$m wide NW resonators monotonically increases by around $20\%$ and $60\%$ for magnetic fields up to $\sim 3$ T, the highest magnetic field we studied on those chips (Fig. 2d, bottom). Notably, Kerr coefficients extracted from multiple 0.1 and 0.3 $\mu$m NW resonators do not exhibit a monotonic upward trend in $|K_{0}|$. Rather, $|K_{0}|$ increases for some values of $B$, and decreases in other regions, and exhibits a larger relative scatter in best-fit values compared to the 1 $\mu$m NWs (Fig. 2(d), top and middle). Despite the presence of some scattering of data, the overall trend of for $|K_{0}|$ to increase between 0 and 2 T (2.5 T) for the 0.1 $\mu$m (0.3 $\mu$m) resonators.

We compare the measured variation of the Kerr coefficient $K_{0}$ with magnetic field $B$ to the variation calculated from the theory of superconductivity. The Kerr coefficient depends on $T_{c}$, which decreases with increasing magnetic field due to breaking the time-reversal degeneracy of Cooper pairs [92]. The relation between $K_{0}$ and $T_{c}$ originates from the dependence of $K_{0}$ on the superconducting parameters $I_{c}$ and $L_{\square}$, which both vary with $T_{c}$ [75], and it is given by (see Appendix C)

According to this relation, the expected change in $K_{0}$ at $B=2$ T is only $\sim$0.1%. Therefore, the observed increase in $|K_{0}|$ for the $1$ $\mu$m NWs (Fig. 2(d), bottom) is more than 100 times larger than the theoretically predicted increase in $|K_{0}|$. It is difficult to pinpoint the cause for this enhancement compared to theoretical predictions. We speculate that it may reflect inhomogeneity present in the sputtered NbTiN film. For example, weak links, which may be viewed as sections of the NW with smaller critical currents and temperatures, could contribute to the enhancement of the Kerr coefficient. The less systematic variation in Kerr coefficients with magnetic field for the narrower wires may also be a signature of film or nanowire etching inhomogeneity.

A KI resonator can be operated as a parametric amplifier when pumped by a high-power tone. Here we consider a 4WM process, where the signal is slightly detuned from the pump within the linewidth of the same mode and the amplification occurs by the conversion of two pump photons into signal and idler photons. The performance of such parametric amplifiers was analyzed theoretically [81] and measured experimentally [93]. In the stiff-pump approximation, where the pump power at the output is assumed to be equal to the pump power at the input, the signal gain in the hanger configuration (Fig.  1(a)) is given by [77]

where $\kappa_{c}=\omega_{0}/Q_{c}$ and $\kappa_{i}=\omega_{0}/Q_{i}$ are the coupling and intrinsic loss rates, respectively, $\xi$ is the reduced power-dependent nonlinearity parameter, $\eta$ is the reduced power-dependent nonlinear loss, $n$ is the reduced number of photons in the cavity, $\tilde{\delta}=(\omega_{p}-\omega_{0})/(\kappa_{c}+\kappa_{i})$ is the reduced pump-resonance detuning, $\tilde{\Delta}=(\omega_{s}-\omega_{p})/(\kappa_{c}+\kappa_{i})$ is the reduced signal-pump detuning, $\phi$ is a phase representing the asymmetry in the feedline and $\lambda_{\pm}=1/2\pm\sqrt{(\xi n)^{2}+-(\tilde{\delta}-2\xi n)^{2}}$.

We measure the performance of the KI resonators as nondegenerate (phase insensitive) parametric amplifiers in magnetic field. We stimulate a 4WM process by supplying a pump tone detuned from the resonance frequency by $\delta=\omega_{p}-\omega_{0}$ and a low-power signal detuned from the pump tone by $\Delta=\omega_{s}-\omega_{p}$, where $|\Delta|\ll\kappa_{c}+\kappa_{i}$. Inside the resonator two pump photons at $\omega_{p}$ are converted into two photons at $\omega_{s}$ and $2\omega_{p}-\omega_{s}$, leading to the amplification of the signal and the generation of a new tone (idler) at frequency $\omega_{i}=2\omega_{p}-\omega_{s}$, as illustrated in the inset of Fig. 3(a). The amplified output power of the signal is then measured by a spectrum analyzer using a measurement bandwidth well below $|\Delta|$. We define the signal gain at $\omega_{s}=\omega_{p}+\Delta$ as the ratio of the signal output power at $\omega_{s}$ when pump is on to the signal output power at $\omega^{\prime}_{s}$ when pump is off, where $\omega^{\prime}_{s}$ is far from resonance. The signal output power at $\omega^{\prime}_{s}$ when pump is off is used as an approximation for the signal input power at $\omega_{s}$ (the signal output power at $\omega_{s}$ when pump is off does not reflect the signal input power at $\omega_{s}$ when $\omega_{s}$ is close to the resonance frequency).

The gain data for $\Delta=10$ kHz for a resonator of width 0.1 $\mu$m at $B=0$ T is fitted with the expression in (3) and plotted in Fig. 3(a) at different pump powers. The gain increases with pump power until the onset of bifurcation where the system enters the bistability regime. The parametric amplifier exhibits large gain when operated at pump power and frequency near the bifurcation point. The condition for large amplification is that the system can access the bifurcation regime, which is satisfied when the nonlinear (i.e. two-photon) loss rate is smaller than $|K_{0}|/\sqrt{3}$ [81].

Figures 3(b) and (c) show the full measurement of the gain versus pump power and pump frequency for the same resonator of width 0.1 $\mu$m at 0 and 1.7 T, respectively. In both cases, gain is observed in the NW resonator. The marked points which are connected by solid lines in Fig. 3(b) and (c) indicate the frequency where the maximum gain is obtained for each pump power. The maximum gain is plotted in Fig. 3(d) versus the pump power for the $0.1$ $\mu$m and $1$ $\mu$m NW resonators. At zero magnetic field, the maximum gains of the two resonators are almost equal, $\sim$12.5 dB. However, to reach this gain, the 1 $\mu$m resonator requires higher pump power due its lower Kerr coefficient. This verifies that the maximum gain is insensitive to the Kerr nonlinearity, so that high gains can still be achieved for resonators with arbitrarily small Kerr-coefficients if pumped by sufficiently high power. However, as mentioned before, pumping with high power is not desirable since it heats up the resonator by generating thermal quasi-particles. Above a certain threshold, the quasi-particle losses will dominate the total losses of the resonator decreasing the intrinsic quality factor and increasing the noise level. On the other hand, the gain of the two resonators is quite different at high magnetic field. The maximum gain of the 0.1 $\mu$m resonator is almost not affected at 1.7 T, except for a small shift in the power at which it is observed, while the maximum gain of the 1 $\mu$m is significantly suppressed at 2 T. The measured maximum gain is plotted versus magnetic field in the inset of Fig. 3(d). The 0.1 $\mu$m resonator exhibits a stable gain over the whole measurement range of $B$, which is limited by the microwave switches in our setup, while the gain of the 1 $\mu$m resonator degrades gradually with $B$. To confirm this result, another 1 $\mu$m resonator was measured and similar degradation for its gain versus $B$ was observed (not shown). The shift in the power at which the maximum gain is observed at high field, (see Fig. 3(d) main panel), is due to the fact that the resonators have higher $Q_{i}$ and slightly higher $|K_{0}|$ in magnetic field, as illustrated in Fig. 1(d) and Fig. 2(d), respectively. The higher $Q_{i}$ translates into a larger average number of photons in the resonator given a certain pump power, while the higher $|K_{0}|$ translates into a larger $\xi$ given a certain number of photons in the resonator. Most importantly, these results demonstrate the sensitivity of the gain levels obtained at high magnetic field to the resonator width, where narrower KI devices can maintain their high gains up to higher values of magnetic field.

For operating points near the bifurcation, gain can additionally be limited by pump depletion, where as the signal output power approaches the pump power, the gain begins to saturate. This defines the dynamic range of the device and it is limited by the pump power at which the bifurcation occurs. For a lossless device ($\kappa_{i}\ll\kappa_{c}$), the bifurcation power is inversely proportional to $|K_{0}|Q_{c}^{2}$ (see Appendix A). Therefore lower $|K_{0}|$ and $Q_{c}$ gives larger dynamic range. However, this requires operating at higher pump power leading to the excitation of thermal quasi-particles and lowering $Q_{i}$ [49]. We determine the dynamic range of our devices by measuring their gain as a function of input signal power $P_{s}$, and observing the signal power at which 1-dB compression of the gain occurs. In Fig. 4(a) we show the data for the 1 $\mu$m resonator for two different gains of 11 and 6 dB, determined by pump power and frequency, at 0 T, and for a gain of 6 dB at 2 T. The 11 dB gain gives lower 1-dB compression point, i.e. smaller dynamic range, $\sim-110$ dBm compared to $-105$ dBm for the 6 dB gain, which agrees with both theory and experiment in literature [49, 54]. The presence of magnetic field does not affect the 1-dB compression point as long as the device operates at a similar gain, as shown in Fig. 4(a) for the case of 6 dB gain with and without field. The 0.1 $\mu$m KI PA also exhibits a dynamic range that is insensitive to magnetic field, where the 1-dB compression point occurs at $\sim-118$ dBm at both 0 and 1.7 T, as shown in Fig. 4(b) for a gain of 11 dB. The smaller dynamic range of the 0.1 $\mu$m KI PA compared to the 1 $\mu$m one is attributed to the larger Kerr coefficient of the 0.1 $\mu$m resonator.

We also characterize the noise properties of our KI PAs. Figure 4(c) shows the noise power referred to the input of the 0.1 $\mu$m KI PA in the presence of a coherent input signal at $\sim-120$ dBm. The input referred noise decreases by roughly the same amount of the gain when the resonator is pumped for amplification, improving the signal-to-noise ratio (SNR) by the same factor. Similar results are obtained at high magnetic field and for the 1 $\mu$m resonator. In addition, we measure the noise figure as a function of gain. We define the noise figure as the ratio between the SNR when pump is off, SNR${}_{\text{off}}$, and the SNR when pump is on at a certain gain, SNR${}_{\text{on}}(G)$, where in this definition the SNR${}_{\text{off}}$ is used as an indication of the SNR at the input of the KI PA. The results for the 1 $\mu$m and the 0.1 $\mu$m KI PAs are shown in Figs. 4(d) and (e), respectively, at zero and high magnetic fields. The noise figure decreases linearly with gain for both KI PAs over the whole range of amplification, even with the existence of the magnetic field. This means that the gains of these resonators are not high enough to reach the saturation point of SNR${}_{\text{on}}(G)$, where the noise at the mixing chamber, either from quantum fluctuations or added noise by the KI PA, dominates over the noise added by the HEMT or thermal noise at higher stages. Despite this limitation in our devices, we can conclude that the magnetic field did not significantly enhance the added noise of the 0.1 $\mu$m KI PA up to 12 dB gain.

These results show that, other than the gain suppression of the 1 $\mu$m resonator, the high magnetic field does not affect the characteristics of our KI PAs. Therefore, KI devices of sufficiently narrow widths, $\sim$0.1 $\mu$m, are capable of operating efficiently as PAs at high magnetic fields up to $\sim$2 T.

Finally we explore the phase-sensitive amplification of our NW resonators in magnetic field. Here the device operates in the degenerate mode, where the pump and signal tones are set to the same frequency ($\Delta=0$), so the idler is generated at the same frequency. This results in squeezing the output phase space, with squeezing level and direction determined by the power and phase of the pump tone, respectively, as illustrated by the schematic in Fig. 5(a). The magnitude of the output signal is then either amplified or deamplified depending on its phase relative to the pump phase. Similarly the variance of the output signal undergoes the same effect, which can be used for squeezing thermal and vacuum noise. Squeezing level up to 26 dB has been recently achieved using a KI-based degenerate parametric amplifier [54] operating in zero magnetic field.

In order to probe the gain of our NW resonators in the degenerate regime, we amplitude-modulate the signal with a sinusoidal wave of small frequency $\omega_{M}$ and measure the gain at the sideband with measurement bandwidth much less than $\omega_{M}$. We use $\omega_{M}=20$ kHz, and we adjust the pump frequency for maximum gain. Similar to the case of nondegenerate amplification, the gain here is obtained by comparing the output sideband power when the pump is on to that when the pump is off at a reference point far from resonance. Figures 5(b) and (c) show the measured gain versus $\Delta\phi=\phi_{s}-\phi_{p}$ for a resonator of width 0.1 $\mu$m at 0 and 2 T with estimated pump power at the resonator $-81$ and $-83$ dBm, respectively. In both figures the gain oscillates at phase period $\pi$, such that the gain varies from a maximum value of 7.5 dB at $\Delta\phi=0$ to a minimum value of -7.5 dB at $\Delta\phi=\pm\pi/2$. In Fig. 5(d) the maximum gain per phase difference and pump frequency is plotted as a function of pump power at 0 and 2 T. The gain peaks at the bifurcation power with almost the same value of $\sim 16$ dB at 0 and 2 T. Similar to the nondegenerate case, there is a power shift between the measurements at 0 and 2 T, due to the combination of higher $Q_{i}$ and slightly higher $|K_{0}|$ at 2 T. These results demonstrate the ability of KI resonators to operate as phase-sensitive parametric amplifiers at high magnetic field without compromising their amplification or squeezing capabilities.