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Physical review. E

Quantum entanglement of linearly coupled quantum harmonic oscillators

Dmitry N. Makarov, K. A. Makarova

Published Apr 28, 2025• 0 citations
Publisher

Abstract

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.

DOI
10.1103/physreve.111.044140

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Quantum harmonic oscillators coupled through coordinates and momenta, represented by the Hamiltonian H\ifmmode^\elseˆ\fi=\ensuremath∑i=12(p\ifmmode^\elseˆ\fii22mi+mi\ensuremathωi22xi2)+H\ifmmode^\elseˆ\fiint\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}}H\ifmmode^\elseˆ\fi​=\ensuremath∑i=12​(2mi​p\ifmmode^\elseˆ\fi​i2​​+2mi​\ensuremathωi2​​xi2​)+H\ifmmode^\elseˆ\fi​int​, where the interaction of two oscillators H\ifmmode^\elseˆ\fiint=ik1x1p\ifmmode^\elseˆ\fi2+ik2x2p\ifmmode^\elseˆ\fi1+k3x1x2\ensuremath−k4p\ifmmode^\elseˆ\fi1p\ifmmode^\elseˆ\fi2{\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}H\ifmmode^\elseˆ\fi​int​=ik1​x1​p\ifmmode^\elseˆ\fi​2​+ik2​x2​p\ifmmode^\elseˆ\fi​1​+k3​x1​x2​\ensuremath−k4​p\ifmmode^\elseˆ\fi​1​p\ifmmode^\elseˆ\fi​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∈(0,1)R\ensuremath{\in}(0,1)R\ensuremath∈(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.
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.
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.
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.
The coupled quantum harmonic oscillator is one of the most researched and important model systems in quantum optics and quantum informatics. This system is often investigated for quantum entanglement in the environment. As a result, such studies are complex and can only be carried out using numerical methods that do not reveal the general pattern of such systems. In this work, the external environment is considered to be two independent particles interacting with coupled harmonic oscillators. It is shown that such a system has an exact analytical solution to the dynamic Schrödinger equation. The analysis of this solution is carried out, and the main parameters of this system are revealed. The solutions obtained can be used to study more complex systems and their quantum entanglement.
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−PS_L=1-\mathcal{P}SL​=1−P for arbitrary excitation numbers (n,m)(n,m)(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 θθθ and ϑ2\vartheta_2ϑ2​, the frequency parameter β0β_0β0​ and the coupling strength εεε: 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=1ω_1=ω_2=1ω1​=ω2​=1 and strong couplings ε≈0.99ε\approx 0.99ε≈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.
Other than scattering problems where perturbation theory is applicable, there are basically two ways to solve problems in physics. One is to reduce the problem to harmonic oscillators, and the other is to formulate the problem in terms of two-by-two matrices. If two oscillators are coupled, the problem combines both two-by-two matrices and harmonic oscillators. This method then becomes a powerful research tool to cover many different branches of physics. Indeed, the concept and methodology in one branch of physics can be translated into another through the common mathematical formalism. Coupled oscillators provide clear illustrative examples for some of the current issues in physics, including entanglement and Feynman's rest of the universe. In addition, it is noted that the present form of quantum mechanics is largely a physics of harmonic oscillators. Special relativity is the physics of the Lorentz group which can be represented by the group of two-by-two matrices commonly called SL(2,c)SL(2,c)SL(2,c). Thus the coupled harmonic oscillators can play the role of combining quantum mechanics with special relativity. It is therefore possible to relate the current issues of physics to the Lorentz-covariant formulation of quantum mechanics.