Entropy Flow and Exceptional-Point Structure in Two-Mode Squeezed-Bath Dynamics

TL;DR

This work analyzes two linearly coupled bosonic modes each coupled to its own squeezed reservoir within a Gaussian framework to reveal how phase-sensitive noise channels generate coherence-driven entropy flow and shape non-Hermitian spectral features. The authors derive a closed-form, covariance-based description via a Lindblad master equation with both standard thermal damping and squeezing-induced dissipators, uncovering an exceptional-point fan in the squeezing-parameter plane and a PT-symmetry structure that requires opposite-quadrature squeezing for symmetry preservation. A key finding is that entropy generation from squeezing arises only at second order in the anomalous correlations, highlighting a nonlinear, information-driven mechanism for irreversibility even in the absence of a thermal gradient; perturbative analysis shows cross-mode anomalous correlations and a second-order Rényi-2 entropy shift. Collectively, the results connect coherence, non-Hermitian spectral transitions, and information flow in structured quantum environments, and point to experimentally accessible probes of entropy flow and critical mode behavior in platforms such as superconducting circuits and driven optical systems.

Abstract

Squeezed reservoirs provide a powerful means of engineering nonclassical noise and controlling irreversible dynamics in open quantum systems. Here we develop a comprehensive analysis of two coupled harmonic oscillators driven by independent squeezed baths, focusing on the emergence of coherence-driven entropy flow and the structure of exceptional points (EPs) in the corresponding Lindblad dynamics. Working entirely within the Gaussian formalism, we derive closed-form evolution equations for the covariance matrix and show that squeezing induces entropy generation only at *second order* in the anomalous correlations, a nonlinear mechanism absent in thermal environments. This entropy flow is accompanied by a rich non-Hermitian structure: by scanning the squeezing parameters we uncover a characteristic "exceptional-point fan" in the (M1, M2) plane, which separates a narrow PT-unbroken region of oscillatory dynamics from broad PT-broken sectors in which one normal mode becomes purely overdamped. This geometric organization of EPs reveals that PT symmetry survives only when the two reservoirs squeeze opposite quadratures, and is generically broken for in-phase squeezing. Our analysis establishes squeezed reservoirs as a natural setting where information-bearing noise drives irreversible behavior through coherent pathways, and lays the groundwork for experimentally accessible probes of entropy flow and critical mode behavior in more complex open systems.

Figures (5)

• Figure 1: Conceptual illustration of a quantum system coupled to two local squeezed reservoirs. Each reservoir acts as a phase-sensitive environment characterized by squeezing parameters $(r_1, \varphi_1)$ and $(r_2, \varphi_2)$, shown here as mechanical clamps exerting quadrature-specific noise. The central system consists of interacting bosonic modes. Asymmetry in the squeezing profiles leads to a directional flow of entropy and coherence, even in the absence of a thermal gradient. The cartoon demon highlights the role of information in structuring quantum noise and emphasizes the analogy to feedback-driven thermodynamic processes.
• Figure 2: Wigner representation of thermal (gray circle) and squeezed (red ellipse) states for a single-mode bosonic system. The horizontal and vertical axes correspond to the dimensionless position and momentum quadratures $x$ and $p$, respectively. The unit lengths along each axis are set by the thermal variances $\sigma_x = \sqrt{\langle \hat{x}^2 \rangle}$ and $\sigma_p = \sqrt{\langle \hat{p}^2 \rangle}$, such that the thermal (unsqueezed) state appears as a unit circle. Squeezing distorts this distribution into an ellipse with principal axes oriented at angle $\phi$, corresponding to the squeezing phase. The arc marks this squeezing angle relative to the $x$-axis.
• Figure 3: Steady-state purity $\mu = \mathrm{Tr}[\rho^2]$ versus squeezing strength $R = \Re(M)$ for a single harmonic oscillator coupled to a squeezed thermal bath, with $N = 0.5$ and $\omega = 1$. Each curve corresponds to a different value of the damping rate $\gamma$. The vertical dashed line marks the uncertainty-limited maximum value of $R = \sqrt{N(N+1)} \approx 0.866$, beyond which the bath state becomes unphysical. As $R$ increases, the system is driven to a lower-entropy, more coherent steady state. As the squeezing strength increases, the oscillator approaches a minimum-uncertainty state, demonstrating how phase-sensitive dissipation can purify steady-state coherence.
• Figure 4: Schematic diagram of the two-mode system. Each harmonic oscillator ($\mathbf{a}_1, \mathbf{a}_2$) is coupled to an independent squeezed thermal bath characterized by occupation $N_i$ and squeezing parameter $M_i$. The two oscillators interact via a coherent hopping term $J$.
• Figure 5: PT–phase structure of the two–oscillator squeezed–bath model. Shown are the exceptional–point (EP) contours in the $(M_{1},M_{2})$–plane for several values of the dissipation rate $\gamma$ (here $\gamma = 0.1,\,0.5,\,0.9$ from blue to red). Each curve marks the locus where the drift matrix $A$ (Eq. (45)) becomes non-diagonalizable and two eigenvalues coalesce. Outside the colored "fan’’ (first and third quadrants), the system lies in the broken PT–symmetric phase, where one normal mode acquires purely real eigenvalues (overdamped, non-oscillatory dynamics). Inside the narrow region spanning the second and fourth quadrants, PT symmetry is unbroken: both normal modes have complex-conjugate eigenvalues and oscillatory dynamics survive. The boundaries sharpen and contract as $\gamma$ decreases, showing that strong dissipation is required for EPs to occur within the physically allowed squeezing amplitudes.