Quantum Reports

Complete Lewis–Riesenfeld Theory of Gaussian Entanglement in Coupled Position-Dependent-Mass Oscillators with Microscopic Tunneling and Dynamical Phase Accrual

Chadia Qotni, Taoufik Hachimi

Published Oct 28, 2026• 0 citations

Abstract

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 + g$\sqrt{C_T}$ ) that scales with the tunneling coherence $\sqrt{C_T}$ 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%.

DOI: 10.3390/quantum8040114

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American Journal of Physics and ApplicationsOct 31, 20240 citations

Coupled harmonic oscillators and their application in the dynamics of entanglement and the nonadiabatic Berry phases

Adel Abidi, A. Trabelsi, S. Krichene

Canadian Journal of PhysicsJun 15, 20214 citations

Quantum Reports

Complete Lewis–Riesenfeld Theory of Gaussian Entanglement in Coupled Position-Dependent-Mass Oscillators with Microscopic Tunneling and Dynamical Phase Accrual

Chadia Qotni, Taoufik Hachimi

Published Oct 28, 2026• 0 citations

Publisher

Abstract

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 + g(\sqrt{C_T}) ) that scales with the tunneling coherence (\sqrt{C_T}) 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%.

DOI: 10.3390/quantum8040114

Related papers

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

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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 $E_{N}=\frac{1}{2} \operatorname{arcosh}(X/2)$ , 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.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.

quant-pharXiv:2609.32709Sep 29, 20260 citations

Entanglement of Coupled Harmonic Oscillators without and with Tunneling Effect and Correction Factor

Ahlem Abidi

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.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.

European Journal of Applied PhysicsOct 13, 20240 citations

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

Ayoub Ghaba, Radouan Hab-arrih, Elhoussine Atmani +1 more

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.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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Field-Space Entanglement Dynamics Between Tunnel-Coupled Luttinger Liquids

Léonce Dupays, Taufiq Murtadho, Bi Hong Tiang +2 more

Entanglement dynamics depend not only on how a quantum system is partitioned, but critically on how interactions across that partition are structured. For a spatial bipartition of a locally interacting system, entanglement is generated near the boundary and then propagates into the bulk. By contrast, when two extended quantum fields are coupled locally along their entire length, the interaction crosses the field-space partition everywhere, and this generates correlations throughout the system. Here, we study the entanglement dynamics between two gapless one-dimensional quantum many-body systems described by Luttinger liquid theory. The systems are initially decoupled and prepared at zero or finite temperature, after which a time-dependent tunneling interaction is activated uniformly along their length. Within a Gaussian approximation, we derive general analytical expressions for the logarithmic negativity, mutual information, and Rényi entropies under arbitrary coupling protocols. At zero temperature, entanglement displays an early-time power-law growth whose exponent is fixed solely by the first non-null derivative of the tunneling protocol. Once the coupling saturates, we obtain exact long-time averages of the information-theoretic quantities and characterise how the correlations scale with temperature and the final coupling strength. We also analyse how mutual information and logarithmic negativity approach the adiabatic limit for a very slow protocol with respect to intrinsic system timescale. This work extends the study of entanglement dynamics in nonequilibrium field theory to field-space partitions and mixed initial states.Entanglement dynamics depend not only on how a quantum system is partitioned, but critically on how interactions across that partition are structured. For a spatial bipartition of a locally interacting system, entanglement is generated near the boundary and then propagates into the bulk. By contrast, when two extended quantum fields are coupled locally along their entire length, the interaction crosses the field-space partition everywhere, and this generates correlations throughout the system. Here, we study the entanglement dynamics between two gapless one-dimensional quantum many-body systems described by Luttinger liquid theory. The systems are initially decoupled and prepared at zero or finite temperature, after which a time-dependent tunneling interaction is activated uniformly along their length. Within a Gaussian approximation, we derive general analytical expressions for the logarithmic negativity, mutual information, and Rényi entropies under arbitrary coupling protocols. At zero temperature, entanglement displays an early-time power-law growth whose exponent is fixed solely by the first non-null derivative of the tunneling protocol. Once the coupling saturates, we obtain exact long-time averages of the information-theoretic quantities and characterise how the correlations scale with temperature and the final coupling strength. We also analyse how mutual information and logarithmic negativity approach the adiabatic limit for a very slow protocol with respect to intrinsic system timescale. This work extends the study of entanglement dynamics in nonequilibrium field theory to field-space partitions and mixed initial states.

quant-pharXiv:2608.05968Aug 7, 20260 citations

The Schmidt Decomposition for Entangled System and Nonadiabatic Berry Phases

Ahlem Abidi, A. Trabelsi

The time-dependent Hamiltonians are a very important portion in the modeling of real systems. In fact, the dynamic description of an entangled quantum systems is reflected in full coherence with the resolution of a wave function, solution of the Schrödinger equation throughout the entire study path. In this regard, we specify in this paper the system of two-site Bose-Hubbard model that obeys tunnel behavior, as two coupled harmonic oscillators, to examine quantum entanglement. The dynamics of such a system is described by the Schrödinger equation have introduced to the solution, the non-linear Ermakov equations as well as through a passage to the Heisenberg picture approach and the general Lewis and Riesenfeld invariant method compute between coupled harmonic oscillators and the coupled Caldirola Kanai oscillators. We prove that a time exponential increase in the mass of the system brings back to an exponential increase of entanglement and the Heisenberg picture approach is the most stable method to quantum entanglement because, this last has reached very large values. Also, we specify a cyclic time evolution, we find analytically the nonadiabatic Berry phases. In a particular case, such an entangled system acquired a nonadiabatic Berry phases that exhibits the same behavior as the Schmidt parameter.The time-dependent Hamiltonians are a very important portion in the modeling of real systems. In fact, the dynamic description of an entangled quantum systems is reflected in full coherence with the resolution of a wave function, solution of the Schrödinger equation throughout the entire study path. In this regard, we specify in this paper the system of two-site Bose-Hubbard model that obeys tunnel behavior, as two coupled harmonic oscillators, to examine quantum entanglement. The dynamics of such a system is described by the Schrödinger equation have introduced to the solution, the non-linear Ermakov equations as well as through a passage to the Heisenberg picture approach and the general Lewis and Riesenfeld invariant method compute between coupled harmonic oscillators and the coupled Caldirola Kanai oscillators. We prove that a time exponential increase in the mass of the system brings back to an exponential increase of entanglement and the Heisenberg picture approach is the most stable method to quantum entanglement because, this last has reached very large values. Also, we specify a cyclic time evolution, we find analytically the nonadiabatic Berry phases. In a particular case, such an entangled system acquired a nonadiabatic Berry phases that exhibits the same behavior as the Schmidt parameter.

American Journal of Physics and ApplicationsOct 31, 20240 citations

Coupled harmonic oscillators and their application in the dynamics of entanglement and the nonadiabatic Berry phases

Adel Abidi, A. Trabelsi, S. Krichene

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 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.

Canadian Journal of PhysicsJun 15, 20214 citations

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.

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.