[Paper Review] Quantum-limited amplification without instability

A. Metelmann, O. Lanes+4 more|arXiv (Cornell University)|Jul 29, 2022

Quantum Information and Cryptography8 citations

TL;DR

This paper proposes a novel class of quantum parametric amplifiers that achieve quantum-limited gain without relying on dynamical instability, using number-conserving Bogoliubov modes in a stable system. The key contribution is a fundamental gain-bandwidth independence enabled by squeezed photon dynamics, demonstrated experimentally in a superconducting circuit with zero reflection and ideal single-mode squeezing in transmission.

ABSTRACT

Quantum parametric amplifiers typically generate by operating in proximity to a point of dynamical instability. We consider an alternate general strategy where quantum-limited, large-gain amplification is achieved without any proximity to a dynamical instability. Our basic mechanism (involving dynamics that conserves the number of squeezed photons) enables the design of a variety of one and two mode amplifiers that are not limited by any fundamental gain-bandwidth constraint. We focus on a particular realization that allows us to realize an ideal single-mode squeezing operation in transmission, and which has zero reflection. We present both a thorough theoretical analysis of this system (including pump-depletion effects), and also discuss results of an experimental superconducting quantum circuit implementation.

Motivation & Objective

• To overcome the fundamental gain-bandwidth trade-off inherent in conventional parametric amplifiers that rely on dynamical instability.
• To develop a stable, quantum-limited amplifier architecture that avoids critical slowing down and instability-related limitations.
• To realize ideal single-mode squeezing with zero reflection and enhanced bandwidth flatness via intrinsic parametric design.
• To demonstrate experimentally a superconducting quantum circuit implementation achieving quantum-limited performance without instability.

Proposed method

• The amplifier uses a two-mode system with parametric coupling that conserves the number of Bogoliubov quasiparticles, ensuring dynamical stability without external dissipation.
• Photons entering the amplifier are converted into Bogoliubov quasiparticles via an input squeezing transformation $\hat{S}_{\text{in}}$, which undergoes number-conserving dynamics.
• The output is formed by an inverse squeezing transformation $\hat{S}_{\text{out}}$, where the asymmetry $\hat{S}_{\text{out}} \neq \hat{S}_{\text{in}}^{-1}$ leads to net amplification.
• The scattering matrix in the quadrature basis is derived as $\textbf{s}[\omega]$, with frequency-dependent gain $\mathcal{G}[\omega] = \mathcal{G}_Q / \left[1 + 4\omega^2/\kappa^2\right]^2$, showing flat gain near resonance.
• The system is designed to achieve impedance matching via optimal imbalance of parametric gains, with $\widetilde{G} = \sqrt{\kappa_+^2 + \kappa_-^2}/2$, ensuring maximum gain and minimal reflection.
• Theoretical analysis includes pump-depletion effects and noise spectral density, showing added noise $\bar{n}_{\text{add}} = \frac{1}{\mathcal{G}_Q}\left(\frac{1}{2} + \bar{n}^T\right)$, reaching the quantum limit at high gain.

Experimental results

Research questions

• RQ1 Can quantum-limited amplification be achieved without proximity to a dynamical instability?
• RQ2 Does a number-conserving Bogoliubov mode system enable gain-bandwidth independence in parametric amplifiers?
• RQ3 Can a stable parametric amplifier achieve zero reflection and ideal single-mode squeezing in transmission?
• RQ4 How does asymmetry in cavity decay rates affect amplifier performance and optimal gain?
• RQ5 Can the intrinsic bandwidth enhancement of Roy *et al.* (2015) be realized without external impedance engineering?

Key findings

• The proposed amplifier achieves quantum-limited noise performance with added noise $\bar{n}_{\text{add}} = \frac{1}{\mathcal{G}_Q}\left(\frac{1}{2} + \bar{n}^T\right)$, approaching the quantum limit at high gain.
• The system exhibits no fundamental gain-bandwidth constraint due to the conservation of Bogoliubov mode number, enabling flat, wideband gain $\mathcal{G}[\omega] = \mathcal{G}_Q / \left[1 + 4\omega^2/\kappa^2\right]^2$.
• The amplifier achieves zero reflection at the impedance-matching condition $\Delta_{\mathcal{C}} = 0$, corresponding to $\widetilde{G} = \sqrt{\kappa_1\kappa_2}/2$.
• The optimal imbalance condition $\widetilde{G} = \sqrt{\kappa_+^2 + \kappa_-^2}/2$ maximizes gain while preserving stability and minimizing reflection.
• The experimental superconducting quantum circuit implementation confirms the theoretical predictions, demonstrating stable, high-gain amplification without instability.
• The QND-type variant achieves a bandwidth of $D_{QND} = \sqrt{\sqrt{2}-1}\kappa$, half that of the OIBA, but still enables quantum-limited operation.