Practical Limits to Single-Mode Vacuum Squeezing in a SNAIL Parametric Amplifier

Abstract

We characterize single-mode vacuum squeezing generated by a SNAIL Parametric Amplifier (SPA) operated under conditions representative of practical sensing and qubit-readout experiments. Motivated by prior expectations that Kerr-induced distortion limits squeezing in degenerate parametric amplifiers, we varied external flux and pump power to explore operating points where Kerr nonlinearity is theoretically minimized. We find that for practical applications where the squeezing frequency is fixed, the Kerr was variable by about a factor of two and the achievable squeezing showed no significant dependence on Kerr. Theoretical modeling supports this observation and indicates that baseline Kerr values in state-of-the-art SPAs are already too small to impose a practical limitation. Instead, squeezing was dominated by internal resonator loss and insertion loss in the microwave chain. These results indicate that, in practical SPAs, reducing loss, rather than suppressing Kerr, is the primary route to improved squeezing performance.

Figures (12)

• Figure 1: \ref{['fig:circuit']}. Simplified circuit diagram of the measurement setup. During the squeezing measurement, no signal is applied to the left port of the first circulator. A 3D cavity-qubit system is connected to the output of the squeezer SPA to calibrate the output line. The output state is detected at room temperature with a heterodyne receiver. \ref{['fig:freqs']}. Measured SPA parameters (dots) compared with theory (solid lines) over the experimental flux range. The resonant frequency $\omega_0$ is obtained from fits of the SPA linear response measured with a vector network analyzer (VNA). The Kerr nonlinearity $g_4$ is extracted from IMD measurements with the pump off, performed at resonance (green) and detuned, at $f_s$ (red). $g_3$ is calculated from the SPA gain equation with $f_p=14.5$ GHz and pump power $P_p$ set to yield 10 dB of gain.
• Figure 2: \ref{['fig2:a']}. Histogram showing the difference in counts between measured vacuum state and squeezed vacuum states. A flux of counts towards large $|I|$ values and a reduction at extreme $|Q|$ values, indicate phase-sensitive amplification along $I$ and squeezing along $Q$. Insets show the vacuum (top right) and squeezed vacuum (bottom left) histograms. \ref{['fig2:b']}. Line cuts of the histograms along the $I$ and $Q$ with Gaussian fits. \ref{['fig2:c']}. Distributions at the qubit inferred from $\eta$, assuming Gaussian statistics. The squeezed vacuum state exhibits a narrower $Q$ quadrature compared to the vacuum state. \ref{['fig2:d']}. Gain of the SPA as measured with a VNA. At each flux, the pump power was tuned to achieve target gains of 10, 12.5, 15, and 17.5 dB at $f_s$, defining the operating points for squeezing and IMD measurements. The pump frequency is fixed at $f_p=14.5$ GHz. \ref{['fig2:e']}. Squeezing inferred from measurements at each operating point after efficiency calibration. Points with $\Phi_{\mathrm{ext}}>0.375\Phi_0$ tend to show the highest squeezing.
• Figure 3: \ref{['fig:iip3']}. Input-referred third order intercept point ($IIP_3$) measured for the SPA versus pump power and flux (bottom axis)/detuning (top axis). \ref{['fig:s_kerr']}. Comparison of squeezing and $|K|$ extracted from $IIP_3$ (circles), for several gain settings (colors: 10 dB--blue, 12.5 dB--orange, 15 dB--green, 17.5 dB--red). No clear correlation between $|K|$ and $S$ is observed. This absence of dependence is reinforced by theoretical calculations of $S$ versus $|K|$ for each gain (colored lines). Instead, the observed variation in $S$ is attributed to changes in the uncalibrated measurement efficiency, represented by clusters of colored lines corresponding to five values of $\eta_{\mathrm{int}}\eta_{\mathrm{cold}}$.
• Figure 4: Measured squeezing versus amplifier detuning, controlled via external flux, for several gain settings (colors: 10 dB--blue, 12.5 dB--orange, 15 dB--green, 17.5 dB--red). Squeezing degrades as $|\Delta|$ increases. This degradation is attributed to changes in the uncalibrated measurement efficiency $\eta_{\mathrm{int}}\eta_{\mathrm{cold}}$ with flux. Clusters of colored lines show theoretical predictions of $S$ versus $\Delta$ for five representative values of $\eta_{\mathrm{int}}\eta_{\mathrm{cold}}$.
• Figure 5: Hardware setup used in this experiment. The Quantum Machines OPX, Octave, and OPT are grouped into a single symbol. Inset is the symbol key for the experimental setup. Not shown in this legend, the VNA is a Copper Mountain Technologies M5180, and the Spectrum Analyzer is a Keysight N9000B; a Stanford Research Systems FS725/3 provides the reference clock signal for the experiment.