Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Mathematics

Coupled Harmonic Oscillator in a System of Free Particles

Dmitry N. Makarov

Published Jan 18, 2022• 2 citations
Publisher

Abstract

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.

DOI
10.3390/math10030294

Related papers

Coupled harmonic oscillators and their quantum entanglement

Dmitry N. Makarov

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.

arXiv:1710.01158•Physical review. E•Apr 5, 2018•70 citations

Linearly Coupled Quantum Harmonic Oscillators and Their Quantum Entanglement

Dmitry N. Makarov, Ksenia Makarova

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.

arXiv:2402.00806•Mathematics•Apr 28, 2025•3 citations

Quantum entanglement of linearly coupled quantum harmonic oscillators

Dmitry N. Makarov, K. A. Makarova

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.

Physical review. E•Apr 28, 2025•0 citations

Quantum entanglement of a harmonic oscillator with an electromagnetic field

Dmitry N. Makarov

At present, there are many methods for obtaining quantum entanglement of particles with an electromagnetic field. Most methods have a low probability of quantum entanglement and not an exact theoretical apparatus based on an approximate solution of the Schrodinger equation. There is a need for new methods for obtaining quantum-entangled particles and mathematically accurate studies of such methods. In this paper, a quantum harmonic oscillator (for example, an electron in a magnetic field) interacting with a quantized electromagnetic field is considered. Based on the exact solution of the Schrodinger equation for this system, it is shown that for certain parameters there can be a large quantum entanglement between the electron and the electromagnetic field. Quantum entanglement is analyzed on the basis of a mathematically exact expression for the Schmidt modes and the Von Neumann entropy.

arXiv:1709.04716•Scientific Reports•May 22, 2018•30 citations

Coupled harmonic oscillators and their application in the dynamics of entanglement and the nonadiabatic Berry phases

Adel Abidi, A. Trabelsi, S. Krichene

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.

Canadian Journal of Physics•Jun 15, 2021•4 citations

Exact Solution for a Non-Markovian Dissipative Quantum Dynamics

Luca Ferialdi, Angelo Bassi

We provide the exact analytic solution of the stochastic Schrödinger equation describing a harmonic oscillator interacting with a non-Markovian and dissipative environment. This result represents an arrival point in the study of non-Markovian dynamics via stochastic differential equations. It is also one of the few exactly solvable models for infinite-dimensional systems. We compute the Green's function; in the case of a free particle and with an exponentially correlated noise, we discuss the evolution of Gaussian wave functions.

arXiv:1204.4348•Physical Review Letters•Apr 26, 2012•36 citations
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.
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.
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.
At present, there are many methods for obtaining quantum entanglement of particles with an electromagnetic field. Most methods have a low probability of quantum entanglement and not an exact theoretical apparatus based on an approximate solution of the Schrodinger equation. There is a need for new methods for obtaining quantum-entangled particles and mathematically accurate studies of such methods. In this paper, a quantum harmonic oscillator (for example, an electron in a magnetic field) interacting with a quantized electromagnetic field is considered. Based on the exact solution of the Schrodinger equation for this system, it is shown that for certain parameters there can be a large quantum entanglement between the electron and the electromagnetic field. Quantum entanglement is analyzed on the basis of a mathematically exact expression for the Schmidt modes and the Von Neumann entropy.
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 provide the exact analytic solution of the stochastic Schrödinger equation describing a harmonic oscillator interacting with a non-Markovian and dissipative environment. This result represents an arrival point in the study of non-Markovian dynamics via stochastic differential equations. It is also one of the few exactly solvable models for infinite-dimensional systems. We compute the Green's function; in the case of a free particle and with an exponentially correlated noise, we discuss the evolution of Gaussian wave functions.