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arXiv:2609.32709

Exact Gaussian Entanglement Dynamics and Initial-State Control in Coupled Parametric Oscillators

Roberto Bernal-Jaquez, Hector H. Hernandez-Hernandez, Alexander Schaum, Guillermo Chacon-Acosta

quant-phSubmitted Sep 26, 2026• Announced Sep 29, 2026• 0 citations
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Abstract

We study two bilinearly coupled harmonic oscillators driven by a common time-dependent spring. A fixed normal-mode transformation reduces the pair to two parametric oscillators, each solved exactly by its Lewis-Riesenfeld invariant and Ermakov-Pinney amplitude, including the ladder-operator construction and the Lewis-Riesenfeld phase. For invariant-vacuum inputs the physical two-mode covariance matrix is obtained in closed form, and the logarithmic negativity collapses to $E_{N}=\frac{1}{2} \operatorname{arcosh}(X/2)$, where a single dimensionless invariant $X \geq 2$ measures the relative phase-space deformation of the two normal modes. We give a normalization-explicit account of the Duan--Simon comparison: after local symplectic optimization the Duan test and the positive partial transpose share the entanglement threshold for this pure exchange-symmetric Gaussian family, but the optimized Duan quantity is a witness, not an independent measure, with $D_{\rm opt}/D_{\rm sep}=e^{-E_N}$. We carefully separate a genuinely separable product preparation from the already-entangled ground state of the coupled Hamiltonian, and use only the former for claims of entanglement generation. Under sinusoidal stiffness modulation a separable input shows bounded correlations away from resonance and sustained growth near parametric resonance, and we demonstrate numerically that the initial width and chirp act as two independent, experimentally accessible controls of the entanglement at a chosen target time. An $SU(1,1)$ formulation identifies $2 E_{N}$ with the hyperbolic separation of the two normal-mode squeezing trajectories on the $SU(1,1)/U(1)$ disk and shows that entanglement is governed by both the relative squeezing magnitude and the relative squeezing angle; ; we exhibit regimes in which nearly equal magnitudes still produce strong entanglement through the angle alone.

Comments
7 figures, 16 pages

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We study two bilinearly coupled harmonic oscillators driven by a common time-dependent spring. A fixed normal-mode transformation reduces the pair to two parametric oscillators, each solved exactly by its Lewis-Riesenfeld invariant and Ermakov-Pinney amplitude, including the ladder-operator construction and the Lewis-Riesenfeld phase. For invariant-vacuum inputs the physical two-mode covariance matrix is obtained in closed form, and the logarithmic negativity collapses to EN=12arcosh⁡(X/2)E_{N}=\frac{1}{2} \operatorname{arcosh}(X/2)EN​=21​arcosh(X/2), where a single dimensionless invariant X≥2X \geq 2X≥2 measures the relative phase-space deformation of the two normal modes. We give a normalization-explicit account of the Duan--Simon comparison: after local symplectic optimization the Duan test and the positive partial transpose share the entanglement threshold for this pure exchange-symmetric Gaussian family, but the optimized Duan quantity is a witness, not an independent measure, with Dopt/Dsep=e−END_{\rm opt}/D_{\rm sep}=e^{-E_N}Dopt​/Dsep​=e−EN​. We carefully separate a genuinely separable product preparation from the already-entangled ground state of the coupled Hamiltonian, and use only the former for claims of entanglement generation. Under sinusoidal stiffness modulation a separable input shows bounded correlations away from resonance and sustained growth near parametric resonance, and we demonstrate numerically that the initial width and chirp act as two independent, experimentally accessible controls of the entanglement at a chosen target time. An SU(1,1)SU(1,1)SU(1,1) formulation identifies 2EN2 E_{N}2EN​ with the hyperbolic separation of the two normal-mode squeezing trajectories on the SU(1,1)/U(1)SU(1,1)/U(1)SU(1,1)/U(1) disk and shows that entanglement is governed by both the relative squeezing magnitude and the relative squeezing angle; ; we exhibit regimes in which nearly equal magnitudes still produce strong entanglement through the angle alone.
We develop a non-perturbative treatment of entanglement dynamics in two coupled harmonic oscillators with position-dependent mass (PDM) and arbitrary Hermitian operator ordering. A time-dependent symplectic transformation diagonalizes the coupled Hamiltonian at every instant (verified symplectic and diagonal to machine precision at all sampled t and μ), reducing the dynamics, up to a small non-adiabatic term, to two Ermakov–Pinney equations with closed-form Gaussian solutions. The formalism is validated against two independent methods, truncated Fock-space propagation and direct Heisenberg-picture integration of the linear response matrix, yielding a maximum relative error of 0.339% on the covariance matrix in both cases (mean Heisenberg-picture error 0.113%; the error on the logarithmic negativity itself reaches 3.5% at t = T), which shows that the Gaussian/Lewis–Riesenfeld reduction reproduces the second moments of the full quantum dynamics to within a fraction of a percent. The bilinear coupling μ is shown to be a threshold-free entanglement resource: the logarithmic negativity at t = T rises monotonically from EN = 0 at μ = 0 to EN = 0.158 at μ = 0.55, saturating as the mixing angle approaches, starting from the decoupled regime (μ = 0), but never reaches the maximal 45° hybridization. A microscopic double-well tunneling model, derived from Wannier-state overlap integrals, yields an effective coupling μeff = μ(1 + gCT\sqrt{C_T}CT​​ ) that scales with the tunneling coherence CT\sqrt{C_T}CT​​ rather than linearly with CT (the squared coherence), enhancing EN by 72.4% relative to the zero-tunneling baseline at strong tunneling (CT = 0.91). Self-consistent PDM renormalization mildly suppresses entanglement by 7.8% at ε = 0.5, via a 9.8% effective-mass increase, while weak local dissipation, treated with the full Gaussian master equation, reduces the final entanglement only moderately (by 8.6% at κ = 0.2) and the purity by about 1%.
Abstract Generating and controlling continuous-variable entanglement is a central requirement for quantum information processing protocols. In the present contribution, we investigate the bipartite Gaussian entanglement of the two-mode Gaussian state produced by a parametric converter, with the two cavity modes initially prepared as arbitrary, non-classical single-mode Gaussian states of tunable purity and non-classicality. The entanglement is witnessed using the Simon's separability criterion, which is necessary and sufficient for Gaussian states, along with the Duan's inseparability criterion. We solve its dynamics as functions of the coherent pump-field phase, the cavity decay rates, and the non-classicality and purity of the initial modes. The entanglement evolves as a damped oscillation and emerges only once there is a difference between the non-classicalities of the two input states, becoming stronger as this difference grows. Likewise, the phase of the pump-field controls both the strength and time duration of entanglement. Higher input purity moderately strengthens and prolongs the correlations, whereas the cavity decay rates are the main source of decoherence. Finally, we compare the two criteria and find that the Simon's condition shows nearly continuous entanglement, whereas Duan's criterion responds only where the correlations are strong, leaving transient windows in which the state is separable under Duan yet still inseparable under Simon. These theoretical results suggest that, the non-classicality of the input field and the phase of the pump field can serve as effective control parameters for achieving tunable bipartite Gaussian entanglement, offering guidance for future practical realizations in cavity-based systems.
We investigate the quantum entanglement dynamics of two coupled harmonic oscillators with a time-dependent interaction. Using the Lewis-Riesenfeld invariant method, we derive the exact analytical wave functions without any perturbative or adiabatic approximations and combine this with a phase-space analysis using the Wigner function to provide a complete description of the system's quantum state evolution. We obtain general expression form for the purity and the linear entropy SL=1−PS_L=1-\mathcal{P}SL​=1−P for arbitrary excitation numbers (n,m)(n,m)(n,m), allows a systematic study of entanglement for a large class of quantum states. We show that the entanglement dynamics is very sensitive to the interplay between the detuning parameters θθθ and ϑ2\vartheta_2ϑ2​, the frequency parameter β0β_0β0​ and the coupling strength εεε: the increase of detuning takes the system from slow irregular oscillations to fast and regular periodic behavior, and the stronger coupling systematically enhances both the amplitude and the average value of the linear entropy. Most importantly, for the resonance case ω1=ω2=1ω_1=ω_2=1ω1​=ω2​=1 and strong couplings ε≈0.99ε\approx 0.99ε≈0.99, the system shows robust undamped synchronized periodic oscillations of the linear entropy for all quantum states considered, indicating preserved quantum coherence without saturation. Our findings demonstrate that linear entropy is a sensitive and practical entanglement witness, and we establish explicit analytical relations between the coupling parameters of the system and its entanglement properties, which are directly relevant to quantum information processing and the control of quantum correlations in continuous-variable systems.
In the framework of the theory of open systems based on completely positive quantum dynamical semigroups, we give a description of the continuousvariable entanglement for a system consisting of two independent harmonic oscillators interacting with a general environment.Using Peres-Simon necessary and sufficient criterion for separability of two-mode Gaussian states, we describe the generation and evolution of entanglement in terms of the covariance matrix for an arbitrary Gaussian input state.For some values of diffusion and dissipation coefficients describing the environment, the state keeps for all times its initial type: separable or entangled.In other cases, entanglement generation or entanglement collapse (entanglement sudden death) take place or even a periodic collapse and revival of entanglement.We show that for certain classes of environments the initial state evolves asymptotically to an entangled equilibrium bipartite state, while for other values of the coefficients describing the environment, the asymptotic state is separable.We calculate also the logarithmic negativity characterizing the degree of entanglement of the asymptotic state.
In the framework of the theory of open systems based on completely positive quantum dynamical semigroups, we give a description of the continuous-variable entanglement for a system consisting of two independent harmonic oscillators interacting with a general environment. Using Peres-Simon necessary and sufficient criterion for separability of two-mode Gaussian states, we describe the generation and evolution of entanglement in terms of the covariance matrix for an arbitrary Gaussian input state. For some values of diffusion and dissipation coefficients describing the environment, the state keeps for all times its initial type: separable or entangled. In other cases, entanglement generation or entanglement collapse (entanglement sudden death) take place or even a periodic collapse and revival of entanglement. We show that for certain classes of environments the initial state evolves asymptotically to an entangled equilibrium bipartite state, while for other values of the coefficients describing the environment, the asymptotic state is separable. We calculate also the logarithmic negativity characterizing the degree of entanglement of the asymptotic state.
We consider a system of two bosonic modes each subject to the dynamics induced by a thermal Markovian environment and we identify instantaneous, local symplectic controls that minimize the loss of entanglement in the Gaussian regime. By minimizing the decrease of the logarithmic negativity at every instant in time, it will be shown that a nontrivial, finite amount of local squeezing helps to counter the effect of decoherence during the evolution. We also determine optimal control routines in the more restrictive scenario where the control operations are applied on only one of the two modes. We find that applying an instantaneous control only at the beginning of the dynamics, i.e., preparing an appropriate initial state, is the optimal strategy for states with symmetric correlations and when the dynamics is the same on both modes. More generally, even in asymmetric cases, the delayed decay of entanglement resulting from the optimal preparation of the initial state with no further action turns out to be always very close to the optimized control where multiple operations are applied during the evolution. Our study extends directly to ``monosymmetric'' systems of any number of modes, i.e., to systems that are invariant under any local permutation of the modes within any one partition, as they are locally equivalent to two-mode systems.