Avoided crossings, degeneracies and Berry phases in the spectrum of quantum noise of driven-dissipative bosonic systems

Avoided crossings are fundamental phenomena in quantum mechanics and photonics that originate from the interaction between coupled energy levels and have been extensively studied in linear dispersive dynamics. Their manifestation in open, driven-dissipative systems, however, where nonlinear dynamics of quantum fluctuations come into play, remains largely unexplored. In this work, we analyze the hitherto unexplored occurrence of avoided and genuine crossings in the spectrum of quantum noise. We demonstrate that avoided crossings arise naturally when a single parameter is varied, leading to hypersensitivity of the associated singular vectors and suggesting the presence of genuine crossings (diabolical points) in nearby systems. We show that these spectral features can be deliberately designed, highlighting the possibility of programming the quantum noise response of photonic systems. As a notable example, such control can be exploited to generate broad, flat-band squeezing spectra - a desirable feature for enhancing degaussification protocols. Our analysis is based on a detailed study of the Analytic Bloch-Messiah Decomposition (ABMD), which we use to characterize the parameter-dependent behavior of singular values and their corresponding vectors. This study provides new insights into the structure of multimode quantum correlations and offers a theoretical framework for the experimental exploitation of complex quantum optical systems.

💡 Research Summary

The paper investigates the emergence of avoided crossings and genuine degeneracies (diabolic points) in the quantum‑noise spectrum of driven‑dissipative bosonic systems. Starting from the linearized quantum Langevin equations around a stable classical steady state, the authors obtain a frequency‑dependent transfer matrix S(ω) that is conjugate‑symplectic (ω‑symplectic). Because S(ω) is generally non‑diagonalizable by a static symplectic transformation, they employ the Analytic Bloch‑Messiah Decomposition (ABMD) to factorize S(ω) as S(ω)=U(ω)D(ω)V†(ω), where U(ω) and V(ω) are unitary ω‑symplectic matrices and D(ω) contains the singular values dj(ω)≥1 and their inverses. These singular values directly encode the frequency‑dependent amplification (anti‑squeezing) and squeezing of the so‑called morphing supermodes.

The authors then explore the behavior of the singular‑value decomposition (SVD) when the system depends on external parameters λ. For a single real parameter, analytic dependence guarantees smooth SVD factors even at points where singular values coalesce. However, for merely Ck‑smooth dependence, the singular vectors can become discontinuous at degeneracies. The set of matrices with degenerate singular values has real codimension three, meaning that three independent parameters are required to generically encounter a true diabolic point. Consequently, when only one parameter is varied, one typically observes an avoided crossing: two singular values approach each other, the associated singular vectors rotate rapidly, and the system exhibits hypersensitivity to small perturbations.

The paper links this hypersensitivity to a topological Berry phase: encircling the avoided‑crossing region in parameter space leads to a non‑trivial phase accumulation in the singular‑vector basis. By deliberately engineering the system’s parameters, the authors show that the avoided crossing can be turned into a useful tool. Specifically, they design a configuration where the squeezing spectrum becomes broad and nearly flat across a wide frequency band. Such a flat‑band squeezing is highly desirable for de‑gaussification protocols, because it avoids the loss of non‑classical correlations that occurs when narrow‑band filters are used.

A practical contribution is an algorithm for computing a “joint‑minimum‑variation” ABMD, which ensures that the unitary factors U(ω) and V(ω) vary smoothly with ω and with external parameters, rather than being obtained independently at each point. This smoothness is essential for tracking avoided crossings and locating nearby diabolic points.

Through several realistic examples—such as two coupled modes with asymmetric losses and Kerr‑type parametric interactions—the paper demonstrates the appearance of avoided crossings in the quantum‑noise spectrum, the associated rapid change of singular vectors, and the possibility of flattening the squeezing response. The work thus provides a comprehensive theoretical framework that connects linear algebraic properties of the transfer matrix (singular values, co‑dimension of degeneracies, Berry phases) with concrete quantum‑optical applications, opening new avenues for programmable quantum‑noise engineering in photonic platforms.