Double-Pumped Kerr Parametric Amplifier Beyond the Gain-Bandwidth Limit

Abstract

Superconducting standing$-$wave parametric amplifiers are crucial for the readout of microwave quantum devices. Despite significant improvements in recent years, the need to operate near an instability point imposes a fundamental constraint: the instantaneous bandwidth decreases with increasing amplifier gain. Here we show that it is possible to obtain parametric amplification without instability by using two simultaneous drives that activate phase-preserving gain and frequency conversion. Realized in a granular aluminum dimer with Kerr nonlinearity, our method demonstrates a sixfold bandwidth increase at 20 dB gain, surpasses the conventional gain$-$bandwidth scaling up to 25 dB, and remains near the quantum limit.

Figures (12)

• Figure 1: Device design and modes of operation.(a) Circuit diagram of the granular aluminum parametric amplifier (grAlPA). The Bose-Hubbard dimer design Eichler2014DimerWinkel2020DJA is similar to Ref. Zapata2024Dec and consists of a pair of capacitively coupled grAl resonators with frequencies and self-Kerr coefficients denoted by $\omega\mathrm{_i}$ and $\mathcal{K}\mathrm{_{i}}$ (i = L, R), respectively. The nonlinearity of grAl resonators can be modeled as an effective Josephson junction array Maleeva2018. Resonator L is coupled with rate $\kappa$ to an input port, through which two pump tones are applied to activate gain (brown) and frequency conversion (purple). (b) Dressed mode structure of the dimer. The hopping interaction $J$ gives rise to hybridized modes $\omega \mathrm{_{a/b}}$. When driven, both modes are red shifted by $|\alpha\mathrm{^{in}_{tot}}|^{2} \mathcal{K}\mathrm{_{a/b}}$ to $\tilde{\omega}\mathrm{_{a/b}}$, where $|\alpha\mathrm{^{in}_{tot}}|$ is proportional to the total input power and $\mathcal{K}\mathrm{_{a/b}}$ are effective Kerr coefficients (see \ref{['A_Theory']}). (c) Configuration for double-pumping experiments. A pump applied at $\omega\mathrm{_g}$ = $(\bar{\omega}\mathrm{_a}+\bar{\omega}\mathrm{_b})/2$, produce phase-preserving gain by transferring two pump photons into one signal and one idler photon split between the dimer modes. A second pump, applied at $\omega\mathrm{_c}$ = $(\omega\mathrm{_g}+\bar{\omega}\mathrm{_a}-\bar{\omega}\mathrm{_b})$, converts a photons between $\bar{\omega}\mathrm{_a}$ and $\bar{\omega}\mathrm{_b}$, mediated by the creation of a gain pump photon. (d) Dynamics of the system following \ref{['eq_linearized_H']}. Each mode has a damping rate $\kappa\mathrm{_j}$, a frequency detuning $\Delta\mathrm{_j}$ relative to the rotating frame at $\omega\mathrm{_g}$, and single-mode squeezing interactions $\Lambda\mathrm{_{S_{j}}}$ (j = a, b). The modes are coupled through beam-splitter $\Lambda\mathrm{_{BS}}$ and two-mode squeezing $\Lambda\mathrm{_{TMS}}$ interactions, activated by the parametric drives.
• Figure 2: Eigenvalues and operational modes of a symmetric grAlPA with balanced single-mode squeezing interactions ($|\Delta\mathrm{_{a,b}}|=2|\Lambda\mathrm{_{S}}|$).(a) Real and imaginary parts of the eigenvalues $\epsilon\mathrm{_{\pm,\pm}}$ (cf. \ref{['Eigenvalues_MXP']}) vs. cooperativity difference $\mathcal{C\mathrm{_{TMS}}}-\mathcal{C\mathrm{_{BS}}}$. When the two cooperativities are equal the eigenvalues become degenerate and the system shows an exceptional point (EP). For optimally imbalance $\mathcal{C\mathrm{_{TMS}}}-\mathcal{C\mathrm{_{BS}}}=-1$, the system reaches the Bogoliubov point (BP), where the gain profile exhibits a flattened maximum. The grAlPA surpasses the GBW limit when operated at any point between the EP and BP. Below the BP, the gain profiles split into two peaks. Above the EP, the system provides gain as it approaches the instability region $\mathcal{C\mathrm{_{TMS}}}-\mathcal{C\mathrm{_{BS}}}>1$, exhibiting a conventional GBW scaling. Moving along the black arrow, the amplifier BW gradually increases. (b) Gain profiles for three possible operational modes of the grAlPA: single-pump (SP), double-pump at exceptional point (EP), and double-pump at the Bogoliubov point (BP). All curves are calculated in the quadrature representation of the hybridized basis (see \ref{['A_Theory']}). (c) Bandwidth scaling vs maximum gain $G\mathrm{_0}$ for the modes of operation in panel (b). Both the EP and BP regimes overcome the SP GBW product.
• Figure 3: Improved gain-bandwidth scaling close to the BP. In panel (a) we show the gain performance under a single pump, in panel (b) we show the two-pump protocol that enables bandwidth optimization, and in panel (c) we demonstrate the optimized operation of the amplifier close to the BP. The schematics above each panel depict the corresponding pump configurations. As visible in panel (a), the gain pump with power $P\mathrm{_g}$ applied at frequency $\omega\mathrm{_g}$ = $\tilde{\omega}\mathrm{_b}$ - $\Delta$ = ($\tilde{\omega}\mathrm{_a}$ + $\tilde{\omega}\mathrm{_b}$)$/2$ gives rise to phase-preserving gain close to $\tilde{\omega}\mathrm{_b}$ (and $\tilde{\omega}\mathrm{_a}$, not shown). The upward frequency shift of the gain curve is due to the Kerr nonlinearity of the hybridized modes (see \ref{['A_Gain Fits']}). As illustrated in panel (b), applying the conversion pump with power $P\mathrm{_c}$ at frequency $\omega\mathrm{_c}$ = $\omega\mathrm{_g}$ + $\tilde{\omega}\mathrm{_a}$ - $\tilde{\omega}\mathrm{_b}$ in addition to the gain pump, activates beam-splitter interactions between the hybridized modes, resulting in the appearance of a new idler tone and a second peak in the gain profile close to $\tilde{\omega}\mathrm{_b}$. The optimal bandwidth is achieved when the two peaks around $\tilde{\omega}\mathrm{_b}$ coalesce near the BP operational point, as illustrated in panel (c). For all gain curves in (c), the ratio $P\mathrm{_c}$/$P\mathrm{_g}$ remains approximately constant. The black dashed lines in (a) and (c) depict fits obtained with the Bose-Hubbard dimer model (see \ref{['A_Gain Fits']}). (d) Comparison of the measured GBW product for single-pumped and double-pumped grAlPA. The green dashed line represents the upper limit, given by the linewidth of the hybridized mode $\kappa\mathrm{_a}$, which can be independently optimized, for example by impedance engineering Naaman2017Naaman2022May. BWs extracted from the fits in (a) and (c) are depicted by the black dashed lines. For the single-pumped grAlPA, the BW scaling is consistent with the measured equivalent damping rate $\kappa\mathrm{_{eq}}/2\pi$ = 19 $\pm$ 4 MHz (see \ref{['A_circuitParameters']}).
• Figure 4: Phase-dependent gain close to the BP. The measurements are taken at $\tilde{\omega}\mathrm{_b}$, for a pump configuration giving a maximum gain $G\mathrm{_0}$ in the range of 20 dB. The left panel shows phase-dependent gain as a function of the input phases of the gain pump $\phi\mathrm{_g}$ and conversion pump $\phi\mathrm{_c}$. In the right panel we plot linecuts taken for fixed $\phi\mathrm{_g}$ (purple) and fixed $\phi\mathrm{_c}$ (brown).
• Figure 5: Noise performance in the phase-preserving regime close to the BP.(a) Noise visibility of the grAlPA when driven by a single pump (brown) and two pumps (red), at a maximum gain $G\mathrm{_0}$ = 22.5 dB. (b) Input-referred noise temperature as a function of detuning $\mathrm{\Delta} \omega$ from a power-calibrated tone at $\omega/2\pi$ = 8.412 GHz. The green and red solid lines correspond to measurements with both pumps off and on, respectively. When both driving tones are on, the noise approaches the standard quantum limit for phase-preserving amplification depicted by the solid blue line. (c) Added noise of the grAlPA as a function of gain. The quantum limit is defined as half a photon of noise from vacuum fluctuations at the idler frequency. Errorbars represent the uncertainty propagated from the power calibration (see \ref{['A_PowerCal']}). For comparison, we show the brown point, measured under single-pump operation.