We study the parametric resonance in a system of two coupled quantum harmonic oscillators when the frequency of one of them (describing the field mode) is modulated at exactly twice the unperturbed eigenfrequency (the case of 'unshifted resonance'). The coordinate–momentum coupling between the two oscillators is chosen in order to simulate the dynamical Casimir effect in a cavity containing a harmonic oscillator detector (LC-contour). Analytical solutions are obtained with the aid of the multiple scales method. For the field mode and detector initially prepared in thermal squeezed states, we calculate the mean numbers of quanta in both modes, as well as the invariant squeezing coefficients, purity and photon distribution functions, analyzing the influence of coupling strength and frequency modulation depth on the dynamics of these quantities. We show that the system goes to strongly squeezed states even for high temperature initial states of the detector. Depending on the relative strength of coupling coefficient and the parametric modulation amplitude, initially separable states go to permanently entangled states or they show periodic appearance and disappearance of entanglement.
We study the parametric resonance in a system of two coupled quantum harmonic oscillators when the frequency of one of them (describing the field mode) is modulated at exactly twice the unperturbed eigenfrequency (the case of 'unshifted resonance'). The coordinate–momentum coupling between the two oscillators is chosen in order to simulate the dynamical Casimir effect in a cavity containing a harmonic oscillator detector (LC-contour). Analytical solutions are obtained with the aid of the multiple scales method. For the field mode and detector initially prepared in thermal squeezed states, we calculate the mean numbers of quanta in both modes, as well as the invariant squeezing coefficients, purity and photon distribution functions, analyzing the influence of coupling strength and frequency modulation depth on the dynamics of these quantities. We show that the system goes to strongly squeezed states even for high temperature initial states of the detector. Depending on the relative strength of coupling coefficient and the parametric modulation amplitude, initially separable states go to permanently entangled states or they show periodic appearance and disappearance of entanglement.
We study a system of two quadratically-coupled harmonic oscillators, prepared respectively in the ground state and in a coherent state. This toy model can describe for instance the dynamics of one mode of a quantum field evolving in a harmonically-oscillating classical background. In such situations, it is well-known that, depending on the background oscillation amplitude and coupling strength, the system may exhibit parametric resonance. Using the Feynman-Vernon influence functional, we analyze the interplay between parametric resonance and quantum entanglement by perturbatively computing the purity of the mixed state resulting from tracing over one of the oscillators. We find a runaway behavior of the quantum entanglement between the two oscillators, as well as a rapid breakdown of the validity of perturbation theory, in regions of parametric instability. This provides a quantitative confirmation of various statements in the literature and is highly relevant to questions pertaining to the validity of the semiclassical approximation in coupled quantum-classical systems.
We study the dynamics of the coefficient of quantum separability between resonantly coupled modes of the massless scalar field in a one-dimensional ideal cavity, whose boundary performs small harmonic oscillations at the frequency ω w = p ω 1 (where ω 1 is the fundamental field eigenfrequency). A qualitative difference between the cases p = 1 and p = 2 is discovered. In the first case, initially factorized thermal states remain separable in the process of evolution, whereas the dynamics of entanglement of initial entangled states (modelled by two-mode squeezed states ) turns out to be more intricate, showing sudden changes from ‘classical’ to ‘nonclassical’ regimes. In contrast, in the ‘principal resonance case’ p = 2 the degree of entanglement changes with time monotonically, and any initially factorized thermal states go asymptotically to mixed quantum states with maximal degree of entanglement. Moreover, for highly thermalized states, the time dependence of the separability coefficient exhibits sharp jumps, which are, nonetheless, not correlated with time dependences of the energies of the field modes.
Abstract In the present paper, we study the influence of non-commutativity on entanglement in a system of two oscillators-modes in interaction with its environment. The considered system is a two-dimensional harmonic oscillator in non-commuting spatial coordinates coupled to its environment. The dynamics of the covariance matrix, the separability criteria for two Gaussian states in non-commutative space coordinates, and the logarithmic negativity are used to evaluate the quantum entanglement in the system, which is compared to the commutative space coordinates case. The result is applied for two initially entangled states, namely the squeezed vacuum and squeezed thermal states. It can be observed that the phenomenon of entanglement sudden death appears more early in the system for the case of squeezed vacuum state than in the case of squeezed thermal state. Thereafter, it is also observed that non-commutativity effects lead to an increasing of entanglement of initially entangled quantum states, and reduce the separability in the open quantum system. It turns out that a separable state in the usual commutative quantum mechanics might be entangled in non-commutative extension.
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.
Abstract We investigate the exact dynamics of a system of two independent harmonic oscillators coupled through their angular momentum. The exact analytic solution of the equations of motion for the field operators is derived, and the conditions for dynamical stability are obtained. As for the application, we examine the emergence of squeezing and mode entanglement for an arbitrary separable coherent initial state. It is shown that close to instability, the system develops considerable entanglement, which is accompanied with simultaneous squeezing in the coordinate of one oscillator and the momentum of the other oscillator. In contrast, for weak coupling away from instability, the generated entanglement is small, with weak alternating squeezing in the coordinate and momentum of each oscillator. Approximate expressions describing these regimes are also provided.
The quantum oscillator model plays a significant role in quantum optics and quantum information and has been one of the hot topics in related research fields. Inspired by the single-mode linear harmonic oscillator and the two-mode entangled state representation, we construct a two-mode coupled harmonic oscillator in this work. Different from the quantum transformation method used in previous literature, the entangled state representation is directly used in this work to solve its energy eigenvalues and eigenfunctions easily. The energy eigenvalues and eigenfunctions of this two-mode coupled harmonic oscillator are continuous compared with those of the one-mode harmonic oscillator. Using the matrix theory of quantum operators, we derive the transformation and inverse transformation of the time evolution operator corresponding to the two-mode coupled harmonic oscillator. In addition, using the entangled state representation, the specific form of the time evolution of the two-mode vacuum state under the action of the oscillator is obtained. Through the analysis of quantum fidelity, it is found that the fidelity of the output quantum state decreases with the oscillator frequency increasing, and the fidelity eventually tends to zero with the increase of time. When analyzing the orthogonal squeezing properties of the output quantum state, this type of two-mode oscillator does not have the orthogonal squeezing effect, but it has a strong quantum dissipation effect instead. This conclusion is further verified by the quasi-probability distribution Q function of the quantum state phase space. Therefore, the two-mode coupled harmonic oscillator has a major reference value in quantum control such as quantum decoherence and quantum information transmission. Like the two-mode squeezed vacuum state, the photon distribution of the output quantum light field corresponding to the two-mode harmonic oscillator presents a super-Poisson distribution, and the photons exhibit a strong anti-bunching effect. Using the three-dimensional discrete plot of the photon number distribution, the super-Poisson distribution and quantum dissipation effect of the output quantum state are intuitively demonstrated. Finally, the SV, which is an entanglement criterion, is used to determine that the output quantum state has a high degree of entanglement. Further numerical analysis shows that the degree of entanglement increases with the action time and the oscillator frequency. In summary, the two-mode coupled harmonic oscillator constructed in this work can be used to prepare highly entangled quantum states through a complete quantum dissipation process. This provides theoretical support for experimental preparing quantum entangled states based on dissipative mechanisms.