A system of two coupled quantum harmonic oscillators with the Hamiltonian $\widehat{H}=\frac{1}{2}(\frac{1}{m_1}{\widehat{p}}_1^2+\frac{1}{m_2}{\widehat{p}}_2^2+A{x}_1^2+B{x}_2^2+C{x}_1{x}_2)$ 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 $\widehat{H}\mathrm{\Psi }=i\hslash \frac{\partial \mathrm{\Psi }}{\partial t}$ 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 $\widehat{H}=\frac{1}{2}\left(\frac{1}{m_1}{\widehat{p}}_1^2+\frac{1}{m_2}{\widehat{p}}_2^2+A{x}_1^2+B{x}_2^2+C{x}_1{x}_2\right)$ 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 $R\in (0,1)$. 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 theory of the beam splitter (BS) in quantum optics is well developed and based on fairly simple mathematical and physical foundations. This theory has been developed for any type of BS and is based on the constancy of the reflection coefficients R (or the transmission coefficient T, where $R+T=1$) and the phase shift $\varphi$. It has recently been shown that the constancy of these coefficients cannot always be satisfied for a waveguide BS, where R and $\varphi$ depend in a special way on photon frequencies. Based on this, this review systematizes the concept of BS in quantum optics into “Conventional” and frequency-dependent BS, and also presents the theory of such BS. It is shown that the quantum entanglement, photon statistics at the output ports, and the Hong–Ou–Mandel (HOM) effect for such BS can be very different. Taking into account the fact that the waveguide BS is currently acquiring an important role in quantum technologies due to the possibility of its miniaturization, this review will be useful not only for theoreticians, but also for experimenters.

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 theoretical description of the scattering of ultrashort electromagnetic field pulses (USP), the semiclassical approach is usually used, where the electromagnetic field is classical and the atomic system is quantum. This article shows the need to take into account the quantum properties of scattered photons, it is that if we take into account the interaction of an USP with a system of atoms, then with the scattering of the pulse it is possible to generate a given number of $n$ photons with a probability $P_n$. The main equations for the probability $P_n$ of the production of $n$ photons and their average energy $E_n$ are found in an analytical form. It is shown that only for a certain number of atoms in the system can multiphoton scattering of ultrashort electromagnetic field pulses occur, where it is necessary to take into account the obtained basic equations for $P_n$ and $E_n$. Various biomolecules, nanosystems and polyatomic structures can consist of such a number of atoms. This is especially important because experiments are currently being conducted with such structures at the present time using high-power ultrashort laser pulses. It is shown that the developed theory in limiting cases turns into well-known approaches of single-photon and multi-photon theories.

The interaction of atomic systems with high-power ultrashort electromagnetic field pulses (USPs) is currently the subject of many theoretical and experimental studies. However, a wave function has yet to be developed for the atomic electron located in such fields, including relativistic fields. In this Letter, an equation is obtained that is similar to the Schrödinger equation for the fields under consideration, but which takes into the account relativistic effects in powerful spatially inhomogeneous fields of USPs. Using the sudden disturbance approximation, an exact solution of the resulting equation is obtained in the form of an analytical wave function which is suitable for any type and form of ultrashort pulse, and which takes into account its magnetic component. It is shown that the obtained wave function satisfies the necessary completeness condition in quantum mechanics.