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arXiv:2606.29617

Dynamics of quantum entanglement in two time-dependent coupled harmonic oscillators

Ayoub Ghaba, Radouan Hab-arrih, Elhoussine Atmani, Abdallah Slaoui

quant-phSubmitted Jun 28, 2026• Announced Jun 30, 2026• 0 citations
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Abstract

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 $S_L=1-\mathcal{P}$ for arbitrary excitation numbers $(n,m)$, 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 $θ$ and $\vartheta_2$, the frequency parameter $β_0$ and the coupling strength $ε$: 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 $ω_1=ω_2=1$ and strong couplings $ε\approx 0.99$, 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.

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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.
Seeing its simplicity of processing, the system of two coupled harmonic oscillators have witnessed a quick growth of research conducted to quantum information. This approach is explicitly investigated in this paper, in particularly entanglement concept is proved to be intimately related to the tunneling effect. We examine analytically entanglement dynamics by introducing the Lewis and Riesenfeld invariant operator in the Heisenberg picture approach to compute the density matrix on the based of an exact treatment. We use Wigner function of the mixed state as an essential tool to move to linear entanglement entropies and we compute the corresponding correction factor. We follow numerically the evolution of entanglement dynamics without and under tunneling through the quantum potential barrier by introducing two particular models between simple and damped coupled harmonic oscillators. Entanglement dynamics is considered to be average without tunneling, it grows upon encountering the potential barrier and remains moderately constant inside barrier. This specificity is extended for both models but with higher values for damped coupled harmonic oscillators consequently the damping effect rapidly increases entanglement. Correction factor is also considered for both models, it show that; low temperature and high potential barriers make the system more disruptive. An increase of the coupling parameter of the system increase correction factor. Damping make the correction factor more important so damping disturbs more the system. Interference effect increase correction factor and it shows an interference between the values of the barrier penetration integral.
In the dynamic description of physical systems, the two coupled harmonic oscillators’ time-dependent mass, angular frequency, and coupling parameter are recognized as a good working example. We present in this work an analytical treatment with a numerical evaluation of the entanglement and the nonadiabatic Berry phases in the vacuum state. On the basis of an exact resolution of the wave function solution of the time-dependent Schrödinger equation (TDSE) using the Heisenberg picture approach, we derive the wave function of the two coupled harmonic oscillators. At the logarithmic scale, we derive the entanglement entropies and the temperature. We discuss the existence of the cyclical initial state (CIS) based on an instant Hamiltonian and we obtain the corresponding nonadiabatic Berry phases through a period T. Moreover, we extend the result to the case of N coupled harmonic oscillators. We use the numerical calculation to follow the dynamic evolution of the entanglement in comparison to the time dependance of the nonadiabatic Berry phases and the time dependance of the temperature. For two coupled harmonic oscillators with time-independent mass and angular frequency, the nonadiabatic Berry phases present very slight oscillations with the equivalent period as the period of the entanglement. A second model is composed of two coupled harmonic oscillators with angular frequency, which change initially as well as later. Herein, the entanglement and the temperature exhibit the same oscillatory behavior with exponential increase in temperature.
In this work, we explore the dynamics of entanglement of an isolated quantum system consisting of two time-dependent, coupled harmonic oscillators. Through the use of a numerical method that relies on the estimation of the system's Wigner representation by a specific Gaussian function, we investigate the time evolution of the entanglement entropy after an instant quench in the inherent parameters of the system. Besides, from the comparison of the results obtained from the analytical expression for the time-dependent von Neumann entropy with the numerically computed entropy data, the effectiveness of the numerical method is tested for a variety of angular frequency combinations. Also, we analyze how the entropy of entanglement change as a function of time.
Quantum entanglement serves as a key phenomenon in understanding correlations in many-body systems, but analytical results remain scarce for coupled three-body oscillators. In this work, we address this gap by introducing a geometrical diagonalization approach that constrains Euler angles, thereby reducing the degrees of freedom in the entanglement analysis. It consists of deriving analytical expressions for linear entropy and purity under the bipartitions (x∣yz)(x|yz)(x∣yz), (y∣xz)(y|xz)(y∣xz), and (xy∣z)(xy|z)(xy∣z) using the Wigner function framework. Our results indicate that excitations in any oscillator basically enhance the redistribution of correlations across the system. The mixing angle θθθ governs entanglement intensity, ranging from separability to maximal correlation. Moreover, we reveal the symmetry relations, notably SLy[(n,m,l),θ]=SLz[(n,m,l),−θ]S_{Ly}[(n,m,l),θ]=S_{Lz}[(n,m,l),-θ]SLy​[(n,m,l),θ]=SLz​[(n,m,l),−θ] and an intrinsic symmetry within (x∣yz)(x|yz)(x∣yz). Hence, we clarify how excitation levels and mixing angles create and enhance entanglement in the three coupled harmonic oscillators.
Abstract We present an exact analytical solution of the Hu–Paz–Zhang master equation in a precise Markovian limit for a system of two harmonically coupled harmonic oscillators interacting with a common thermal bath of harmonic oscillators. The thermal bath is initially considered to be at arbitrary temperatures and characterized by an Ohmic Lorentz–Drude spectral density. In the examined system, couplings between the two harmonic oscillators and the environment ensure a complete decoupling of the center-of-mass and relative degrees of freedom, resulting in undamped dynamics in the relative coordinate. The exact time evolution is used to analyze the system’s entanglement dynamics, quantified through logarithmic negativity and quantum mutual information, while ensuring the positivity of the density operator to confirm the physical validity of the results. We demonstrate that, under certain parameter regimes and initial conditions, the asymptotic dynamics can give rise to periodic entanglement-disentanglement behavior. Furthermore, numerical simulations reveal that for negative values of the direct coupling between the oscillators, which are sufficiently close to a critical lower bound beyond which the system becomes unstable, the system can maintain entanglement across a broad temperature range and for arbitrarily long durations.
We study the relationship between entanglement and parametric resonance in a system of two coupled time-dependent oscillators. As a measure of bipartite entanglement, we calculate the linear entropy for the reduced density operator, from which we study the entanglement dynamics. In particular, we find that the bipartite entanglement increases in time up to a maximal mixing scenario, when the set of auxiliary dynamical parameters are under parametric resonance. Moreover, we obtain a closed relationship between the correlations in the ground state, the localization of the Wigner function in phase space, and the localization of the wave function of the total system.