Hidden quantum correlations in cavity-based quantum optics

In multimode optical systems, the spectral covariance matrix encodes all the information about quantum correlations between the quadratures of Gaussian states. Recent research has revealed that, in scenarios that are more common than previously thought, part of these correlations remain inaccessible to standard homodyne detection scheme. Formally, this effect can be attributed to a non-real spectral covariance matrix. In this work, we provide a systematic framework and explicit criteria for identifying experimental configurations leading to such a behavior. This study will facilitate the proper exploitation and optimal engineering of CV quantum resources.

💡 Research Summary

The paper investigates a subtle but important limitation of standard homodyne detection (HD) in multimode continuous‑variable (CV) quantum optics. In Gaussian states the full quantum information is encoded in the covariance matrix of the quadrature operators. For stationary processes this matrix can be expressed in the frequency domain as the spectral covariance matrix σ(ω). While most previous works assumed σ(ω) to be real, recent studies have shown that σ(ω) can acquire an imaginary part when the sideband modes at +ω and –ω are not in the same quantum state. In such cases the imaginary component σ_I(ω) is antisymmetric (σ_I(ω)=–σ_I(–ω)) and cannot be accessed by conventional HD, which only measures symmetric combinations of sidebands. Consequently, a portion of the quantum correlations—often termed “hidden squeezing” or “hidden quantum correlations”—remains invisible to HD.

The authors develop a systematic theoretical framework to predict when σ(ω) will be real or complex, based solely on the physical parameters of a cavity‑based multimode system. Starting from a general quadratic Hamiltonian
H = ℏ∑{m,n} G{mn} a†m a_n + (ℏ/2)∑{m,n} (F_{mn} a†_m a†_n + h.c.),
they derive linear Langevin equations for the quadrature vector R(t). The damping matrix Γ (diagonal, possibly mode‑dependent) and the interaction matrix M (constructed from G and F) fully determine the input‑output transfer function S(ω). By decomposing S(ω) = S_R(ω) + i S_I(ω) and inserting it into the expression for the output spectral covariance σ_out(ω) = (1/2π) S(ω) S^T(–ω), they obtain explicit formulas for the real and imaginary parts of σ_out(ω).

The key result is that σ_out(ω) is real (i.e., σ_I(ω)=0 for all ω) if and only if two conditions are simultaneously satisfied:

1. Commutation condition: