We investigate the entanglement dynamics of two harmonic oscillators with a weak time-dependent coupling. We find strong connections between the dynamical behaviors of the entanglement and the stability of the normal modes. An abrupt transition of the entanglement dynamics from periodic and bounded to monotonically increasing occurs, when one of the normal modes becomes dynamically unstable as the modulation amplitude crosses the critical value. In particular, when the modulation frequency is roughly twice the oscillator frequency, an explicit critical value for the modulation strength is derived to specify the transition. Moreover, it is found that one can significantly change the dynamical behavior of the entanglement by adding an extremely weak time modulation with specific frequency to a constant coupling.
We investigate the entanglement dynamics of two harmonic oscillators with a weak time-dependent coupling. We find strong connections between the dynamical behaviors of the entanglement and the stability of the normal modes. An abrupt transition of the entanglement dynamics from periodic and bounded to monotonically increasing occurs, when one of the normal modes becomes dynamically unstable as the modulation amplitude crosses the critical value. In particular, when the modulation frequency is roughly twice the oscillator frequency, an explicit critical value for the modulation strength is derived to specify the transition. Moreover, it is found that one can significantly change the dynamical behavior of the entanglement by adding an extremely weak time modulation with specific frequency to a constant coupling.
We study the entanglement dynamics of two coupled mechanical oscillators, within a modulated optomechanical system. We find that, depending on the strength of the mechanical coupling, one could observe either a stationary or a dynamical behavior of the mechanical entanglement, which is extremely robust against the oscillator temperature. Moreover, we have shown that this entanglement dynamics is strongly related to the stability of the normal modes. Taking mechanical damping effects into account, an analytical expression corresponding to the critical mechanical coupling strength, where the transition from stationary to dynamical entanglement occurs, is also reported. The proposed scheme is analyzed with experimentally realistic parameters, making it a promising means to realize macroscopic quantum entanglement within current state-of-the-art experimental setups.
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.
The exact dynamics of the entanglement between two harmonic modes generated by an angular momentum coupling is examined. Such a system arises when considering a particle in a rotating anisotropic harmonic trap or a charged particle in a fixed harmonic potential in a magnetic field, and it exhibits a rich dynamical structure, with stable, unstable, and critical regimes according to the values of the rotational frequency or field and trap parameters. Consequently, it is shown that the entanglement generated from an initially separable Gaussian state can exhibit quite distinct evolutions, ranging from quasiperiodic behavior in stable sectors to different types of unbounded increase in critical and unstable regions. The latter lead, respectively, to logarithmic and linear growth of the entanglement entropy with time. It is also shown that entanglement can be controlled by tuning the frequency, such that it can be increased, kept constant, or returned to a vanishing value with just stepwise frequency variations. Exact asymptotic expressions for the entanglement entropy in the different dynamical regimes are provided.
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.
The time-dependent behavior of a two-level system interacting with a quantum oscillator system is analyzed in the case of a coupling larger than both the energy separation between the two levels and the energy of the quantum oscillator ($\ensuremath{\Omega}<\ensuremath{\omega}<\ensuremath{\lambda}$, where $\ensuremath{\Omega}$ is the frequency of the transition between the two levels, $\ensuremath{\omega}$ is the frequency of the oscillator, and $\ensuremath{\lambda}$ is the coupling between the two-level system and the oscillator). Our calculations show that the amplitude of the expectation value of the oscillator coordinate decreases as the two-level system undergoes the transition from one level to the other, while the transfer probability between the levels is staircaselike. This behavior is explained by the interplay between the adiabatic and nonadiabatic regimes encountered during the dynamics with the system acting as a quantum counterpart of the Landau-Zener model. The transition between the two levels occurs as long as the expectation value of the oscillator coordinate is driven close to zero. On the contrary, if the initial conditions are set such that the expectation values of the oscillator coordinate are far from zero, the system will remain locked on one level.
Abstract The time evolution of the quantum kicked harmonic oscillator (KHO) is described by the Floquet operator which maps the state of the system immediately before one kick onto the state at a time immediately after the next. Quantum KHO is characterized by three parameters: the coupling strength V 0 , the so-called Lamb–Dicke parameter η whose square is proportional to the effective Planck constant ℏ eff , and the ratio T of the natural frequency of the oscillator and the kick frequency. To a given coupling strength and depending on T being a natural or irrational number, the phase space of the classical kicked oscillator can display different behaviors, as for example, stochastic webs or quasicrystal structures, thus showing a chaotic or localized behavior that is mirrored in the quantum phase space. On the other hand, the classical limit is studied letting ℏ eff become negligible. In this paper we investigate how the ratio T , considered as integer, rational or irrational, influences the entanglement dynamics of the quantum KHO and study how the entanglement dynamics behaves when varying either V 0 or ℏ eff parameters.