Abstract Quantum entanglement of coupled harmonic oscillators is frequently applied in quantum and non-linear physics, molecular chemistry and biophysics, which is why its study is of a great interest for modern physics. In this work a quantum entanglement of a coupled harmonic oscillator in a simple form was found. This simple form is presented as a single parameter – reflection coefficient R. All parameters of the studied system are included in the R coefficient. It is shown that the derivation of the expression can have applications in quantum optics, in particular in quantum metrology.
Abstract Quantum entanglement of coupled harmonic oscillators is frequently applied in quantum and non-linear physics, molecular chemistry and biophysics, which is why its study is of a great interest for modern physics. In this work a quantum entanglement of a coupled harmonic oscillator in a simple form was found. This simple form is presented as a single parameter – reflection coefficient R. All parameters of the studied system are included in the R coefficient. It is shown that the derivation of the expression can have applications in quantum optics, in particular in quantum metrology.
Quantum entanglement of a system of two coupled quantum harmonic oscillators with a Hamiltonian H[over ̂]=1/2(1/m_{1}p[over ̂]_{1}^{2}+1/m_{2}p[over ̂]_{2}^{2}+Ax_{1}^{2}+Bx_{2}^{2}+Cx_{1}x_{2}) can be found in many applications of quantum and nonlinear physics, molecular chemistry, and biophysics. Despite this, the quantum entanglement of such a system is still a problem under study. This is primarily due to the fact that the system is multiparametric and the quantum entanglement of such a system is not defined in a simple analytical form. This paper solves this problem and shows that quantum entanglement depends on only one parameter that has a simple physical meaning: the reflection coefficient R∈(0,1). The reflection coefficient R has a simple analytical form and includes all the parameters of the system under consideration. It is shown that for certain values of the coefficient R, the quantum entanglement can be large. The developed theory can be used not only for calculating quantum entanglement, but also for many other applications in physics, chemistry, and biophysics, where coupled harmonic oscillators are considered.
In many applications of quantum optics, nonlinear physics, molecular chemistry and biophysics, one can encounter models in which the coupled quantum harmonic oscillator provides an explanation for many physical phenomena and effects. In general, these are harmonic oscillators coupled via coordinates and momenta, which can be represented as H^=∑i=12p^i22mi+miωi22xi2+H^int, where the interaction of two oscillators H^int=ik1x1p^2+ik2x2p^1+k3x1x2−k4p^1p^2. Despite the importance of this system, there is currently no general solution to the Schrödinger equation that takes into account arbitrary initial states of the oscillators. Here, this problem is solved in analytical form, and it is shown that the probability of finding the system in any states and quantum entanglement depends only on one coefficient R∈(0,1) for the initial factorizable Fock states of the oscillator and depends on two parameters R∈(0,1) and ϕ for arbitrary initial states. These two parameters R∈(0,1) and ϕ include the entire set of variables of the system under consideration.
Quantum harmonic oscillators coupled through coordinates and momenta, represented by the Hamiltonian $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}={\ensuremath{\sum}}_{i=1}^{2}(\frac{{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}}_{i}^{2}}{2{m}_{i}}+\frac{{m}_{i}{\ensuremath{\omega}}_{i}^{2}}{2}{x}_{i}^{2})+{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{\text{int}}$, where the interaction of two oscillators ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{\text{int}}=i{k}_{1}{x}_{1}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}}_{2}+i{k}_{2}{x}_{2}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}}_{1}+{k}_{3}{x}_{1}{x}_{2}\ensuremath{-}{k}_{4}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}}_{1}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}}_{2}$, are found in many applications of quantum optics, nonlinear physics, molecular chemistry, and biophysics. Despite this, there is currently no general solution to the Schr\"odinger equation for such a system. This is especially relevant for quantum entanglement of such a system in quantum optics applications. Here this problem is solved and it is shown that quantum entanglement depends on only one coefficient, $R\ensuremath{\in}(0,1)$, which includes all the parameters of the system under consideration. It has been shown that quantum entanglement can be very large at certain values of this coefficient. The results obtained have a fairly simple analytical form, which facilitates analysis.
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.
A system of two coupled quantum harmonic oscillators with the Hamiltonian $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}=\frac{1}{2}(\frac{1}{{m}_{1}}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}}_{1}^{2}+\frac{1}{{m}_{2}}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}}_{2}^{2}+A{x}_{1}^{2}+B{x}_{2}^{2}+C{x}_{1}{x}_{2})$ can be found in many applications of quantum and nonlinear physics, molecular chemistry, and biophysics. The stationary wave function of such a system is known, but its use for the analysis of quantum entanglement is complicated because of the complexity of computing the Schmidt modes. Moreover, there is no exact analytical solution to the nonstationary Schrodinger equation $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}\mathrm{\ensuremath{\Psi}}=i\ensuremath{\hbar}\frac{\ensuremath{\partial}\mathrm{\ensuremath{\Psi}}}{\ensuremath{\partial}t}$ and Schmidt modes for such a dynamic system. In this paper we find a solution to the nonstationary Schrodinger equation; we also find in an analytical form a solution to the Schmidt mode for both stationary and dynamic problems. On the basis of the Schmidt modes, the quantum entanglement of the system under consideration is analyzed. It is shown that for certain parameters of the system, quantum entanglement can be very large.
We apply the Walborn [Walborn S P, Taketani B G, Salles A, Toscano F and de Matos Filho R L 2009 Entropic Entanglement Criteria for Continuous Variables Phys. Rev. Lett. 103 160505] Shannon Entropic Entanglement Criterion (SEEC), to simple harmonic oscillators, a system typically discussed in undergraduate quantum mechanics courses. In particular, we investigate the entanglement of a system of coupled harmonic oscillators. A simple form of entanglement criterion in terms of their interaction is found. It is shown that a pair of interacting ground state coupled oscillators are more likely to be entangled for weaker coupling strengths while coupled oscillators in excited states entangle at higher coupling strengths. Interacting oscillators are more likely to be entangled as their interaction increases.