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.
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−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 ϑ2, the frequency parameter β0 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 ω1=ω2=1 and strong couplings ε≈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 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 ) that scales with the tunneling coherence 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%.
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.
We investigate the quantum-classical border, the entanglement and decoherence of an analytically solvable model, comprising a first subsystem (a harmonic oscillator) coupled to a driven and damped second subsystem (another harmonic oscillator). We choose initial states whose dynamics is confined to a couple of two-level systems, and show that the maximum value of entanglement between the two subsystems, as measured by concurrence, depends on the dissipation rate to the coupling-constant ratio and the initial state. While in a related model the entropy of the first subsystem (a two-level system) never grows appreciably (for large dissipation rates), in our model it reaches a maximum before decreasing. Although both models predict small values of entanglement and dissipation, for fixed times of the order of the inverse of the coupling constant and large dissipation rates, these quantities decrease faster, as a function of the ratio of the dissipation rate to the coupling constant, in our model.
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.
The time evolution of a quantum particle’s product of uncertainties in position and momentum is calculated when it is coupled with an external source. We have used a simple toy model where the particle is subject to a harmonic potential and coupled with an equivalent harmonic oscillator via a linear term. It is found that the long-time-averaged product is an increasing function of the coupling strength. It diverges when one of the eigenmodes of the coupled system goes soft, with the singular term twice of that for the stationary state. Generally, there is a jump of finite size for this quantity when a small coupling is turned on, compared to the uncoupled case. Similar behaviors have also been found for the von Neumann entanglement entropy, which is calculated exactly using a covariance matrix formalism. We find that the mode-interference plays an important role in the main features of this work.