Efficient readout of quantum states requires resolving low-power quantum signals in noisy environments. Linear phase-preserving amplifiers are key tools, boosting signal amplitude in both quadratures without phase alteration. This is particularly relevant for quantum microwave state tomography, essential for protocols such as entanglement generation, secure quantum remote state preparation, quantum teleportation, quantum illumination, and quantum state transfer. Conventional dispersive readout of superconducting qubits also relies on amplifying microwave signals carrying qubit information, often consisting of only a few photons. While successful, quantum physics dictates that any phase-preserving amplifier adds at least half a noise photon (the standard quantum limit or SQL), limiting quantum efficiency to 0.5. Superconducting Josephson parametric amplifiers (JPAs) and Josephson traveling-wave parametric amplifiers (JTWPAs) have achieved quantum-limited amplification, but alternative methods are crucial for applications needing efficient signal amplitude detection, like parity measurements in multi-qubit systems, quantum amplitude sensing, dark matter axion detection, and cosmic microwave background detection. This work explores nondegenerate parametric amplification of broadband microwave signals, aiming to identify conditions for noiseless amplification. This broadband nondegenerate regime, using a flux-driven JPA, contrasts with the conventional phase-preserving nondegenerate (narrowband, quantum-limited) or phase-sensitive degenerate (narrowband, potentially noiseless) regimes. In the nondegenerate regime, signal and idler are distinct frequency modes, with potentially correlated or uncorrelated phases, leading to phase-dependent or phase-independent amplification. The nondegenerate amplifier can also perform two-mode squeezing. The experiment demonstrates a JPA for phase-independent linear amplification of broadband signals with performance exceeding the SQL.

The paper extensively reviews existing literature on quantum noise, measurement, and amplification, citing key works on quantum limits on phase-preserving linear amplifiers, quantum state tomography of microwave fields, and various quantum protocols involving microwave manipulation. It highlights the importance of quantum-limited amplification in superconducting qubit readout and the limitations imposed by the standard quantum limit. Previous work on superconducting JPAs and JTWPAs achieving quantum-limited amplification is discussed, motivating the need for alternative approaches to achieve noiseless amplification. The authors also refer to prior research on parametric amplification techniques, including degenerate and nondegenerate regimes, phase-sensitive and phase-preserving amplification, and two-mode squeezing, establishing the context and novelty of their work within the broader field.

The quantum efficiency (η) is defined as the ratio of input signal vacuum fluctuations to output signal fluctuations, expressed as η = 1/(1 + 2nf), where nf is the number of noise photons added by the amplifier. For narrowband amplification, the idler mode adds noise photons, leading to η ≤ 2Gn − 1, approaching 0.5 at high gain. However, for broadband amplification, where the signal bandwidth covers both signal and idler modes, the idler contributes to the amplified signal, potentially reaching η = 1. The study uses a flux-driven JPA, where parametric amplification is achieved by driving a nonlinear electromagnetic resonator with a strong pump field (ωp = 2ω0), creating signal and idler photons (ωs = ωp/2 + Δ and ωi = ωp/2 − Δ). The nondegenerate regime (Δ ≠ 0) leads to spectrally separated signal and idler modes. The experimental setup includes a flux-driven JPA connected to a cryogenic HEMT amplifier, a circulator to separate input/output, and a reference-state reconstruction method to analyze the output signal within a measurement bandwidth. Broadband thermal states are generated using a heatable attenuator. The quantum efficiency is determined by Planck spectroscopy for broadband signals, varying the attenuator temperature to measure the photon number at different gains. For narrowband signals, a coherent input tone is used, varying its photon number to extract noise photons from the linear amplifier response. The theoretical analysis involves calculating the commutator [ĉ(ω), ĉ†(ω’)] from the linear combination of amplified incoming signals, phase-conjugated idler signals, and additive noise, using bosonic commutation relations and the Heisenberg uncertainty principle to derive the quantum limit for additive noise photons. A Lorentzian JPA gain function is used to solve for the quantum efficiency limit, differentiating between broadband and narrowband regimes. Finally, the deviation from the ideal quantum efficiency is attributed to pump signal noise, modeled by a noisy pump signal described by A(t) = (a0 + fp(t))e−iωpt, using the Wiener-Khinchin theorem to calculate power fluctuation variance. The measured quantum efficiencies are fit using a model incorporating pump-induced noise.

The paper theoretically demonstrates the possibility of achieving unity quantum efficiency (η = 1) for nondegenerate parametric amplification of broadband signals. This contrasts with the standard quantum limit (SQL) of η = 0.5 for narrowband, phase-preserving amplification. The experimental results using a flux-driven JPA and broadband thermal signals show a quantum efficiency of 0.69 ± 0.02, significantly exceeding the SQL. This demonstrates that, in the broadband regime, the idler mode does not add extra noise but contributes to the amplified signal, carrying signal information. The experimentally observed broadband gain (Gb) is consistent with the theoretical prediction Gb = 2Gn − 1, where Gn is the narrowband gain. The gain dependence of quantum efficiency exhibits a maximum, then decreases at higher gains. At low gains, HEMT noise dominates, while at high gains, pump signal fluctuations are the primary noise source. The observed deviation from unity quantum efficiency is successfully explained by modeling the pump-induced noise. This model incorporates both the gain-dependent JPA noise and the HEMT noise using a Friis equation, successfully fitting the experimental data.

The results validate the theoretical prediction that nondegenerate parametric amplification of broadband signals can surpass the standard quantum limit. The experimental quantum efficiency of 0.69 ± 0.02 significantly exceeds the SQL, indicating that signal information is encoded in both the signal and idler modes, with the idler not acting as a pure noise source. The key to exceeding the SQL is not phase interference (as in phase-sensitive amplification) but rather the information encoded in the average amplitudes of both modes. This is because it is difficult to define a phase for broadband signals, and thus encoding information in photon numbers is more natural. This contrasts with the conventional phase-sensitive amplification where SQL violation relies on phase interference, which is not possible here due to the uncorrelated phases in the broadband thermal signals. While relative phase between signal and idler has no impact on the quantum limit, it can influence the amplifier output. Pump photon number fluctuations limit the achievable quantum efficiency, a significant factor at high gains. This surpasses the quantum efficiency (0.32) previously reported for phase-preserving JTWPAs. This technology has significant implications for improving low-noise amplification in various quantum applications.

This research demonstrates that nondegenerate parametric amplification of broadband signals allows surpassing the standard quantum limit, achieving a quantum efficiency significantly higher than 0.5. The experimental results, reaching 0.69 ± 0.02, support the theoretical analysis which attributes the SQL violation to the combined information encoded in both signal and idler modes. Pump noise is identified as a limiting factor preventing reaching unity quantum efficiency. Future research could focus on mitigating pump noise to further improve quantum efficiency, and exploring applications in high-efficiency parity detection of entangled qubits and broadband dispersive qubit readout.

The primary limitation is the pump-induced noise that prevents achieving the theoretical limit of unity quantum efficiency. While the model successfully accounts for this noise, reducing it further would improve performance. The experimental setup's specific parameters (measurement bandwidth, JPA characteristics) could also affect the results. Extending the study to different bandwidths and JPA designs could provide a more comprehensive understanding. The study focused on thermal states, and investigating other input states would be valuable. Finally, the experiment’s precision might slightly affect the calculation of the quantum efficiency.