Nonlinear Scattering Matrix in Quantum Optics

Abstract

It is well known that the scattering matrix plays an important role in quantum optics. This matrix converts the incoming characteristics of scattered radiation into output characteristics. Currently, only special cases of such a scattering matrix are known, which are determined by a specific problem. In this work, a general form of the scattering matrix is obtained, which can be applied to a wide range of problems. It is shown that previously well-known and widely used scattering matrices can be obtained from the resulting scattering matrix. The results obtained can be used to describe new quantum states, the scattering matrices of which have not yet been studied.

1. Introduction

It is well known in quantum optics that the relationship between the characteristics of the electromagnetic field before and after scattering is determined by the scattering matrix S [1,2,3,4,5]. In general, this matrix is square and consists of a certain number of rows and columns, depending on the problem being solved. The main characteristic of scattering is the quantum state before scattering $|{\mathsf{\Psi }}_{in}\rangle$ and after $|{\mathsf{\Psi }}_{out}\rangle =e^{i\widehat{H}t}|{\mathsf{\Psi }}_{in}\rangle$, where $\widehat{H}$ is the Hamiltonian that includes the interaction of the incident radiation in the scattering system, and t is the interaction time. Also an important characteristic of scattering is the operators of the creation and annihilation of mode i photons before and after their interaction ${\widehat{b}}_i=e^{i\widehat{H}t}{\widehat{a}}_i{e}^{-i\widehat{H}t},{\widehat{b}}_i^{†}=e^{i\widehat{H}t}{\widehat{a}}_i^{†}{e}^{-i\widehat{H}t}$, where ${\widehat{a}}_i$ and ${\widehat{a}}_i^{†}$ are the annihilation and creation operators before interaction, respectively, and ${\widehat{b}}_i$ and ${\widehat{b}}_i^{†}$ after interaction. Of greatest interest, there is the transformation with two modes, i.e., $i=1,2$. Indeed, such a transformation is associated with many applications in quantum optics [5,6,7,8,9,10], namely with two-port beam splitters (BS), parametric amplifier, phase-conjugating mirror, squeezed states, etc. The simplest and most accurate definition of this matrix can be considered [5,6]

$$\begin{array}{c}\left(\begin{array}{c}{\widehat{b}}_1\\ {\widehat{b}}_2\\ {\widehat{b}}_1^{†}\\ {\widehat{b}}_2^{†}\end{array}\right)=S\left(\begin{array}{c}{\widehat{a}}_1\\ {\widehat{a}}_2\\ {\widehat{a}}_1^{†}\\ {\widehat{a}}_2^{†}\end{array}\right).\hfill \end{array}$$

(1)

This type of Equation (1) scattering matrix is nonlinear, since it connects the output characteristics of the field with the input ones through combinations of annihilation and creation operators, which leads to an increase or decrease in the number of photons in the system. It is well known that, if in the scattering matrix, the creation operators at the output are only connected with the creation operators at the input, and the same with the annihilation operators, then in such a system, the number of photons is conserved and the scattering matrix is linear [5]. For example, in the case of a linear beam splitter, the matrix S will be [5,8,11,12,13,14,15]

$$\begin{array}{c}\hfill S=\left(\begin{array}{cc}Q& 0\\ 0& Q^{*}\end{array}\right),Q=\left(\begin{array}{cc}\sqrt{1-R}& \sqrt{R}{e}^{i\varphi }\\ -\sqrt{R}{e}^{-i\varphi }& \sqrt{1-R}\end{array}\right),\end{array}$$

(2)

where R is the reflection coefficient of the beam splitter, and $\varphi$ is the phase shift. The beam splitter is known to play an important role in quantum optics and is used in many optical circuits. For example, BS is used in linear optical quantum computing, including its use in the creation of a quantum computer [16]. Using BS, it is possible to create quantum entanglement between the input modes of electromagnetic fields [17,18,19,20,21], simulate quantum transport [22], determine the degree of photon identity [17,23], as well as use quantum metrology [24] and quantum information [25].

There are quite a lot of examples wherein the matrix S has its own form [5,6,26]. Some of these will be discussed in this work. One way or another, in general, the matrix S contains 16 elements $S_{ij}$ which have their own form depending on the problem being solved. It should be added that not all 16 elements are independent coefficients; some can be repeated or be complexly conjugate to other coefficients. It is possible to determine a specific type of matrix S when solving a specific problem such as: a two-port beam splitter, two-mode compressed states, parametric amplifier, phase-conjugating mirror, etc. [5,6]. Currently, there is no matrix S in a general form that is applicable to many physical systems. Although such a solution would make it possible to analyze quantum states that could be implemented in various devices. Indeed, at the output ports of a linear beam splitter, quantum entangled states arise that have high quantum entanglement. It is obvious that, in the case of a nonlinear scattering matrix, the quantum states at the output will also be quantum entangled. Since the study of quantum entanglement is currently one of the most promising areas for physics, the scattering matrix obtained in general form will provide a good resource for studying quantum entanglement for many physical systems.

In this work, the matrix S is found in general form, not based on a specific physical problem. It is shown that this matrix has certain properties and only has 6 independent elements out of 16 different options. Some special cases are considered and it is shown when they can be obtained from the general scattering matrix.

Next, the atomic system of units is used: ℏ = 1; $\left|e\right|$ = 1; $m_e$ = 1, where ℏ is the Dirac constant, e is the electron charge, and $m_e$ is the electron mass.

2. Nonlinear Scattering Matrix

Usually, in order to find the scattering matrix, it is necessary to consider a specific physical problem. In the general case of the interaction between a quantized two-mode electromagnetic field and matter, to find the scattering matrix, it is necessary to consider a Hamiltonian of the form

$$\widehat{H}=\sum _{i=1}^2{\omega }_i{\widehat{a}}_i^{†}{\widehat{a}}_i+U\left(\{{\widehat{a}}_i^{†},{\widehat{a}}_i\}\right),$$

(3)

where ${\omega }_i$ is the frequency of the i-th mode ($i=\{1,2\}$), and $U\left(\{{\widehat{a}}_i^{†},{\widehat{a}}_i\}\right)$ is the energy with which electromagnetic field modes interact with matter. Of course, the representation (3) does not reflect the entirety of the interaction of the field with matter, where the field interacts with all charged particles in the matter and where each charged particle is also described in a quantum way. However, we choose the Hamiltonian in the form (3), since it reflects many known models of the interaction of photons in matter and the change in field characteristics [27].

Furthermore, it is convenient to present the Hamiltonian $\widehat{H}$, not in the form of the operators ${\widehat{a}}_i$ and ${\widehat{a}}_i^{†}$, but in the coordinate representation [1,2], those ${\widehat{a}}_i=\frac{1}{\sqrt{2}}\left(q_i+\frac{\partial }{\partial q_i}\right)$ and ${\widehat{a}}_i^{†}=\frac{1}{\sqrt{2}}\left(q_i-\frac{\partial }{\partial q_i}\right)$. We will assume that we have such an interaction that the Hamiltonian $\widehat{H}$ can be represented in the form

$$\begin{array}{c}\hfill \widehat{H}=\sum _{i=1}^2\frac{{\Omega }_i}{2}\left({\widehat{p}}_i^2+y_i^2\right)=\sum _{i=1}^2\frac{{\Omega }_i}{2}\left(-\frac{{\partial }^2}{\partial y_i^2}+y_i^2\right),\\ \hfill y_1=a_1{q}_{1}cos\alpha -a_2{q}_{2}sin\alpha ,y_2=a_3{q}_{1}sin\alpha +a_4{q}_{2}cos\alpha ,a_1{a}_4=a_2{a}_3,\end{array}$$

(4)

where $a_i$ ($i=\{1,2,3,4\}$) are arbitrary coefficients that determine the degree of change (compression or stretching) of the coordinates $q_1$ and $q_2$, the angle $\alpha$ can take any value on the interval $\alpha \in (0,2\pi )$, as determined by the angle of rotation of the coordinate planes $y_1$ and $y_2$ relative to the coordinate plane $q_1$ and $q_2$; frequency ${\Omega }_i$ is now a parameter responsible for changing the frequency of incident photons. We choose the condition $a_1{a}_4=a_2{a}_3$ for convenience so that the linear transformations $y_i\to \{q_1,q_2\}$ are similar to the inverse transformation $q_i\to \{y_1,y_2\}$. In other words, representing Hamiltonian (3) as Equation (4), we specify the change in all parameters of incident photons when interacting with matter, i.e., the frequency, compression, and stretching of their coordinates and impulses, as well as coordinate rotation. As will be shown below, such a choice of Hamiltonian (4) can describe many real physical systems and known scattering matrices.

Next, we set ourselves the task of finding an explicit form of the scattering matrix S using the definition of such a matrix, as can be seen in Equation (1) and above.

2.1. Operator Decomposition

Let us represent the operators ${\widehat{a}}_i$ and ${\widehat{a}}_i^{†}$ in coordinate representation, but through the variables $y_1$ and $y_2$, we ultimately obtain

$$\begin{array}{c}\hfill {\widehat{a}}_1=\frac{1}{\sqrt{2}}\left(q_1+\frac{\partial }{\partial q_1}\right)=\frac{1}{\sqrt{2}}\left[\frac{cos\alpha }{a_1}{y}_1+\frac{sin\alpha }{a_3}{y}_2+a_{1}cos\alpha \frac{\partial }{\partial y_1}+a_{3}sin\alpha \frac{\partial }{\partial y_2}\right],\end{array}$$

$$\begin{array}{c}\hfill {\widehat{a}}_2=\frac{1}{\sqrt{2}}\left(q_2+\frac{\partial }{\partial q_2}\right)=\frac{1}{\sqrt{2}}\left[-\frac{sin\alpha }{a_2}{y}_1+\frac{cos\alpha }{a_4}{y}_2-a_{2}sin\alpha \frac{\partial }{\partial y_1}+a_{4}cos\alpha \frac{\partial }{\partial y_2}\right].\end{array}$$

Then, we divide the Hamiltonian into two parts $\widehat{H}={\widehat{H}}_1+{\widehat{H}}_2$, and the following expressions are valid

$$\begin{array}{c}\hfill \left[{\widehat{H}}_i,y_i\right]=-{\Omega }_i\frac{\partial }{\partial y_i},\left[{\widehat{H}}_i,\frac{\partial }{\partial y_i}\right]=-{\Omega }_i{y}_i.\end{array}$$

Next, we consider four operators, the first of which is ${\widehat{f}}_1=e^{i\widehat{H}t}{y}_1{e}^{-i\widehat{H}t}$. Let us make replacements $\widehat{X}=i\widehat{H}t$ and $\widehat{Y}=y_1$. Then, using the well-known expansion (BCH formula), we obtain

$${\widehat{f}}_1=e^{\widehat{X}}\widehat{Y}{e}^{-\widehat{X}}=\widehat{Y}+\left[\widehat{X},\widehat{Y}\right]+\frac{1}{2!}\left[\widehat{X},\left[\widehat{X},\widehat{Y}\right]\right]+\frac{1}{3!}\left[\widehat{X},\left[\widehat{X},\left[\widehat{X},\widehat{Y}\right]\right]\right]+\dots ,$$

where

$$\left[\widehat{X},\widehat{Y}\right]=it\left[\widehat{H},y_1\right]=-i{\Omega }_{1}t\frac{\partial }{\partial y_1},$$

$$\left[\widehat{X},\left[\widehat{X},\widehat{Y}\right]\right]=it\left(-i{\Omega }_{1}t\right)\left[\widehat{H},\frac{\partial }{\partial y_1}\right]={\left(-i{\Omega }_{1}t\right)}^2{y}_1,$$

$$\left[\widehat{X},\left[\widehat{X},\left[\widehat{X},\widehat{Y}\right]\right]\right]=it{\left(-i{\Omega }_{1}t\right)}^2\left[\widehat{H},y_1\right]={\left(-i{\Omega }_{1}t\right)}^3\frac{\partial }{\partial y_1},$$

$$\left[\widehat{X},\left[\widehat{X},\left[\widehat{X},\left[\widehat{X},\widehat{Y}\right]\right]\right]\right]=it{\left(-i{\Omega }_{1}t\right)}^3\left[\widehat{H},\frac{\partial }{\partial y_1}\right]={\left(-i{\Omega }_{1}t\right)}^4{y}_1.$$

Continuing this sequence, we obtain

$$\begin{array}{c}\hfill {\widehat{f}}_1=y_1-i{\Omega }_{1}t\frac{\partial }{\partial y_1}+\frac{1}{2!}{\left(-i{\Omega }_{1}t\right)}^2{y}_1+\frac{1}{3!}{\left(-i{\Omega }_{1}t\right)}^3\frac{\partial }{\partial y_1}+\dots =\\ \hfill =\left(1+\frac{1}{2!}{\left(-i{\Omega }_{1}t\right)}^2+\cdots \right)y_1+\left(-i{\Omega }_{1}t+\frac{1}{3!}{\left(-i{\Omega }_{1}t\right)}^3+\dots \right)\frac{\partial }{\partial y_1}=\\ \hfill =cos\left({\Omega }_{1}t\right)y_1-isin\left({\Omega }_{1}t\right)\frac{\partial }{\partial y_1}.\end{array}$$

(5)

Carrying out similar reasoning for ${\widehat{f}}_2=e^{i\widehat{H}t}{y}_2{e}^{-i\widehat{H}t}$, we ultimately obtain

$${\widehat{f}}_2=cos\left({\Omega }_{2}t\right)y_2-isin\left({\Omega }_{2}t\right)\frac{\partial }{\partial y_2}.$$

(6)

Similarly, we can consider the operator ${\widehat{g}}_1=e^{i\widehat{H}t}\frac{\partial }{\partial y_1}{e}^{-i\widehat{H}t}$. As a result, we obtain

$$\begin{array}{c}\hfill {\widehat{g}}_1=\frac{\partial }{\partial y_1}-i{\Omega }_{1}t{y}_1+\frac{1}{2!}{\left(-i{\Omega }_{1}t\right)}^2\frac{\partial }{\partial y_1}+\frac{1}{3!}{\left(-i{\Omega }_{1}t\right)}^3{y}_1+\dots =\\ \hfill =\left(1+\frac{1}{2!}{\left(-i{\Omega }_{1}t\right)}^2+\cdots \right)\frac{\partial }{\partial y_1}+\left(-i{\Omega }_{1}t+\frac{1}{3!}{\left(-i{\Omega }_{1}t\right)}^3+\dots \right)y_1=\\ \hfill =-isin\left({\Omega }_{1}t\right)y_1+cos\left({\Omega }_{1}t\right)\frac{\partial }{\partial y_1}.\end{array}$$

(7)

With similar reasoning for ${\widehat{g}}_2=e^{i\widehat{H}t}\frac{\partial }{\partial y_2}{e}^{-i\widehat{H}t}$, we ultimately obtain

$${\widehat{g}}_2=-isin\left({\Omega }_{2}t\right)y_2+cos\left({\Omega }_{2}t\right)\frac{\partial }{\partial y_2}.$$

(8)

2.2. Finding the Operators ${\widehat{b}}_1$ and ${\widehat{b}}_1^{†}$

As a result, the operator ${\widehat{b}}_1$ can be represented

$$\begin{array}{c}\hfill {\widehat{b}}_1=e^{i\widehat{H}t}\frac{1}{\sqrt{2}}\left[\frac{cos\alpha }{a_1}{y}_1+\frac{sin\alpha }{a_3}{y}_2+a_{1}cos\alpha \frac{\partial }{\partial y_1}+a_{3}sin\alpha \frac{\partial }{\partial y_2}\right]e^{-i\widehat{H}t}=\\ \hfill =\frac{1}{\sqrt{2}}\left[\frac{cos\alpha }{a_1}{\widehat{f}}_1+\frac{sin\alpha }{a_3}{\widehat{f}}_2+a_{1}cos\alpha {\widehat{g}}_1+a_{3}sin\alpha {\widehat{g}}_2\right],\end{array}$$

Expressing $y_1$, $y_2$, $\frac{\partial }{\partial y_1}$, $\frac{\partial }{\partial y_2}$ through of ${\widehat{a}}_1$, ${\widehat{a}}_1^{†}$, ${\widehat{a}}_2$, ${\widehat{a}}_2^{†}$, we obtain

$${\widehat{b}}_1=Q_{11}{\widehat{a}}_1+Q_{12}{\widehat{a}}_2+F_{11}{\widehat{a}}_1^{†}+F_{12}{\widehat{a}}_2^{†},$$

(9)

where

$$\begin{array}{c}\hfill Q_{11}={cos}^2\alpha cos\left({\Omega }_{1}t\right)+{sin}^2\alpha cos\left({\Omega }_{2}t\right)-\\ \hfill -\frac{i}{2}\left\{\left(\frac{1}{a_1^2}+a_1^2\right){cos}^2\alpha sin\left({\Omega }_{1}t\right)+\left(\frac{1}{a_3^2}+a_3^2\right){sin}^2\alpha sin\left({\Omega }_{2}t\right)\right\},\end{array}$$

(10)

$$\begin{array}{c}\hfill Q_{12}=\frac{1}{4}sin\left(2\alpha \right)\left[\left(\frac{a_2}{a_1}+\frac{a_1}{a_2}\right)\left(cos\left({\Omega }_{2}t\right)-cos\left({\Omega }_{1}t\right)\right)-\right.\\ \hfill \left.-i\left\{\left(a_3{a}_4+\frac{1}{a_3{a}_4}\right)sin\left({\Omega }_{2}t\right)-\left(a_1{a}_2+\frac{1}{a_1{a}_2}\right)sin\left({\Omega }_{1}t\right)\right\}\right],\end{array}$$

(11)

$$\begin{array}{c}\hfill F_{11}=\frac{i}{2}\left\{\left(\frac{1}{a_1^2}-a_1^2\right){cos}^2\alpha sin\left({\Omega }_{1}t\right)+\left(\frac{1}{a_3^2}-a_3^2\right){sin}^2\alpha sin\left({\Omega }_{2}t\right)\right\},\end{array}$$

(12)

$$\begin{array}{c}\hfill F_{12}=\frac{1}{4}sin\left(2\alpha \right)\left[\left(\frac{a_2}{a_1}-\frac{a_1}{a_2}\right)\left(cos\left({\Omega }_{2}t\right)-cos\left({\Omega }_{1}t\right)\right)-\right.\\ \hfill \left.-i\left\{\left(a_3{a}_4-\frac{1}{a_3{a}_4}\right)sin\left({\Omega }_{2}t\right)-\left(a_1{a}_2-\frac{1}{a_1{a}_2}\right)sin\left({\Omega }_{1}t\right)\right\}\right].\end{array}$$

(13)

Then, we obtain

$${\widehat{b}}_1^{†}=F_{11}^{*}{\widehat{a}}_1+F_{12}^{*}{\widehat{a}}_2+Q_{11}^{*}{\widehat{a}}_1^{†}+Q_{12}^{*}{\widehat{a}}_2^{†}.$$

2.3. Finding the Operators ${\widehat{b}}_2$ and ${\widehat{b}}_2^{†}$

As a result, the operator ${\widehat{b}}_2$ can be represented

$$\begin{array}{c}\hfill {\widehat{b}}_2=e^{i\widehat{H}t}\frac{1}{\sqrt{2}}\left[\frac{cos\alpha }{a_4}{y}_2-\frac{sin\alpha }{a_2}{y}_1+a_{4}cos\alpha \frac{\partial }{\partial y_2}-a_{2}sin\alpha \frac{\partial }{\partial y_1}\right]e^{-i\widehat{H}t}=\\ \hfill =\frac{1}{\sqrt{2}}\left[-\frac{sin\alpha }{a_2}{\widehat{f}}_1+\frac{cos\alpha }{a_4}{\widehat{f}}_2-a_{2}sin\alpha {\widehat{g}}_1+a_{4}cos\alpha {\widehat{g}}_2\right],\end{array}$$

Expressing $y_1$, $y_2$, $\frac{\partial }{\partial y_1}$, $\frac{\partial }{\partial y_2}$ through ${\widehat{a}}_1$, ${\widehat{a}}_1^{†}$, ${\widehat{a}}_2$, ${\widehat{a}}_2^{†}$, we obtain

$${\widehat{b}}_2=Q_{12}{\widehat{a}}_1+Q_{22}{\widehat{a}}_2-F_{12}^{*}{\widehat{a}}_1^{†}+F_{22}{\widehat{a}}_2^{†},$$

(14)

where

$$\begin{array}{c}\hfill Q_{22}={sin}^2\alpha cos\left({\Omega }_{1}t\right)+{cos}^2\alpha cos\left({\Omega }_{2}t\right)-\\ \hfill -\frac{i}{2}\left\{\left(\frac{1}{a_2^2}+a_2^2\right){sin}^2\alpha sin\left({\Omega }_{1}t\right)+\left(\frac{1}{a_4^2}+a_4^2\right){cos}^2\alpha sin\left({\Omega }_{2}t\right)\right\},\end{array}$$

(15)

$$\begin{array}{c}\hfill F_{22}=\frac{i}{2}\left\{\left(\frac{1}{a_2^2}-a_2^2\right){sin}^2\alpha sin\left({\Omega }_{1}t\right)+\left(\frac{1}{a_4^2}-a_4^2\right){cos}^2\alpha sin\left({\Omega }_{2}t\right)\right\}.\end{array}$$

(16)

Then, we obtain

$${\widehat{b}}_2^{†}=-F_{12}{\widehat{a}}_1+F_{22}^{*}{\widehat{a}}_2+Q_{12}^{*}{\widehat{a}}_1^{†}+Q_{22}^{*}{\widehat{a}}_2^{†}.$$

(17)

2.4. Scattering Matrix and Its Properties

As a result, we can write

$$\left(\begin{array}{c}\widehat{B}\\ {\widehat{B}}^{†}\end{array}\right)=\left(\begin{array}{cc}Q& F\\ F^{*}& Q^{*}\end{array}\right)·\left(\begin{array}{c}\widehat{A}\\ {\widehat{A}}^{†}\end{array}\right),$$

(18)

where the matrices are entered

$$\widehat{A}=\left(\begin{array}{c}{\widehat{a}}_1\\ {\widehat{a}}_2\end{array}\right),{\widehat{A}}^{†}=\left(\begin{array}{c}{\widehat{a}}_1^{†}\\ {\widehat{a}}_2^{†}\end{array}\right),$$

$$\widehat{B}=\left(\begin{array}{c}{\widehat{b}}_1\\ {\widehat{b}}_2\end{array}\right),{\widehat{B}}^{†}=\left(\begin{array}{c}{\widehat{b}}_1^{†}\\ {\widehat{b}}_2^{†}\end{array}\right),$$

$$Q=\left(\begin{array}{cc}{Q}_{11}& Q_{12}\\ Q_{12}& Q_{22}\end{array}\right),F=\left(\begin{array}{cc}{F}_{11}& F_{12}\\ -F_{12}^{*}& F_{22}\end{array}\right),$$

(19)

and it turns out that $Q^T=Q,F^{†}=-F,$ that is, Q is a symmetric matrix and F is an anti-Hermitian matrix. As a result, the scattering matrix can be written as

$$\begin{array}{c}\hfill S=\left(\begin{array}{cccc}{Q}_{11}& Q_{12}& F_{11}& F_{12}\\ Q_{12}& Q_{22}& -F_{12}^{*}& F_{22}\\ F_{11}^{*}& F_{12}^{*}& Q_{11}^{*}& Q_{12}^{*}\\ -F_{12}& F_{22}^{*}& Q_{12}^{*}& Q_{22}^{*}\end{array}\right).\end{array}$$

(20)

You can see that there are only six independent components here: $Q_{11},Q_{12},Q_{22}$, $F_{11},F_{12},$$F_{22}$. The remaining components of the matrix S are obtained by the complex conjugation “${}^{*}$” and multiplication by the “−” sign of these six main components.

3. Special Cases of the Scattering Matrix

Let us show how previously known cases can be obtained from a general scattering matrix (18). To do this, it is necessary to select such parameters of our general model ${\Omega }_i,a_i,\alpha$ so that the result coincides with the previously known one. It should also be added that, in the scattering matrix, each operator ${\widehat{b}}_i$ and ${\widehat{b}}_i^{†}$ is defined up to an arbitrary phase factor, so that the well-known commutation conditions are satisfied $[{\widehat{b}}_i,{\widehat{b}}_i^{†}]=1$, i.e., ${\widehat{b}}_i\to {\widehat{b}}_i{e}^{i{\varphi }_i}$ and ${\widehat{b}}_i^{†}\to {\widehat{b}}_i^{†}{e}^{-i{\varphi }_i}$. Furthermore, we will use this phase condition where necessary.

3.1. Scattering Matrix of a Linear Beam Splitter

The scattering matrix of a linear beam splitter is described by Equation (2) and the matrix B can be obtained based on general physical considerations (conservation of energy, i.e., the number of photons in the system and the unitarity of the matrix B) [11,12,14], as well as based on the conservation of angular momentum and its projection [8,13,14]. In the latter case, one can not only find the scattering matrix, but also the general form of the operator that transforms the initial quantum states $|{\mathsf{\Psi }}_{in}\rangle$ into the final $|{\mathsf{\Psi }}_{out}\rangle$, i.e., $|{\mathsf{\Psi }}_{out}\rangle =\widehat{S}|{\mathsf{\Psi }}_{in}\rangle$.

In the case of a linear beam splitter, the condition $F=0$ must be satisfied in the scattering matrix (18). In this case, it is possible with $a_i=1$. In this case, up to the phase factor $e^{i{\varphi }_i}$, the matrix Q can be represented as

$$\begin{array}{c}\hfill Q=\left(\begin{array}{cc}{Q}_{11}& Q_{12}\\ -Q_{12}^{*}& Q_{11}^{*}\end{array}\right),\\ \hfill Q_{11}={cos}^2\alpha +{sin}^2\alpha e^{-i\Delta \Omega t},\\ \hfill Q_{12}=\frac{1}{2}sin\left(2\alpha \right)\left(e^{-i\Delta \Omega t}-1\right),\end{array}$$

(21)

where $\Delta \Omega ={\Omega }_2-{\Omega }_1$. Equation (21) can be further simplified by representing $Q_{11}=\left|Q_{11}\right|e^{i{\varphi }_1}$ and $Q_{12}=\left|Q_{12}\right|e^{i{\varphi }_2}$ and taking advantage of the fact that, in the matrix Q, the first row is defined up to the phase factor $e^{i{\varphi }_1}$, and the second row is defined up to the phase factor $e^{-i{\varphi }_1}$; then, in the end, it is easy to obtain

$$\begin{array}{c}\hfill Q=\left(\begin{array}{cc}\sqrt{1-R}& \sqrt{R}{e}^{i\varphi }\\ -\sqrt{R}{e}^{-i\varphi }& \sqrt{1-R}\end{array}\right),\\ \hfill R={sin}^2\left(\frac{\Delta \Omega t}{2}\right){sin}^2\left(2\alpha \right),\varphi ={\varphi }_2-{\varphi }_1,cos\varphi =-cot\left(2\alpha \right)\sqrt{\frac{R}{1-R}}.\end{array}$$

(22)

The result obtained makes it possible to find not only the scattering matrix of the beam splitter, but also, in explicit form, the reflection coefficient R, which is important in quantum optics. One can also see how the coordinates in Equation (4) are transformed with such substitutions

$$\begin{array}{c}\hfill y_1=q_{1}cos\alpha -q_{2}sin\alpha ,y_2=q_{1}sin\alpha +q_{2}cos\alpha .\end{array}$$

(23)

The transformation (23) is responsible for rotating coordinates, without the “squeezing” of quantum states.

If we choose an arbitrary angle $\alpha =\frac{\pi }{4}$, we obtain the scattering matrix in the form

$$\begin{array}{c}\hfill Q=\left(\begin{array}{cc}cos\frac{\Delta \Omega t}{2}& \pm isin\frac{\Delta \Omega t}{2}\\ \pm isin\frac{\Delta \Omega t}{2}& cos\frac{\Delta \Omega t}{2}\end{array}\right),\\ \hfill R={sin}^2\left(\frac{\Delta \Omega t}{2}\right),\varphi =\pm \frac{\pi }{2}.\end{array}$$

(24)

If we choose $\Delta \Omega =2vs.\kappa$ and $t=z/v$, where v is the wave speed in the beam splitter, $\kappa$ is the wave vector, and z is the effective coupling length of the beam splitter, then the beam splitter matrix will coincide with the known theory of a waveguide beam splitter with reflection coefficients $R={sin}^2\left(\kappa z\right)$ [28].

It is interesting to look at the scattering matrix at $\varphi =\pm \pi$. In this case, $\frac{\Delta \Omega t}{2}=\frac{\pi }{2}+\pi n$, where $n=0,1,2,\cdots$, and the scattering matrix will be

$$\begin{array}{c}\hfill Q=\left(\begin{array}{cc}cos\left(2\alpha \right)& -sin\left(2\alpha \right)\\ sin\left(2\alpha \right)& cos\left(2\alpha \right)\end{array}\right),\\ \hfill R={sin}^2\left(2\alpha \right),\varphi =\pm \pi .\end{array}$$

(25)

In general, it is believed that, in a linear beam splitter, physically measurable results do not depend on the phase $\varphi$ and the choice of $\varphi$ can be arbitrary. It can be seen that, in this type of beam splitter, there is no dependence on time t, i.e., the reflection coefficient is constant and only depends on $\alpha$, which is a constant coefficient. This type of beam splitter can be, for example, a cubic beam splitter.

It is also noteworthy that, as a result of considering the beam splitter, we obtained not only the beam splitter matrix, but also the reflection coefficient R in explicit form Equation (22).

3.2. Scattering Matrix of a Parametric Amplifier, Phase-Conjugating Mirror, Squeezed States

Here, we will consider the scattering matrix, which describes many physical systems: a parametric amplifier or a phase-conjugating mirror, as well as squeezed two-mode states [5,6]. The scattering matrix in all these cases will be

$$\begin{array}{c}\hfill S=\left(\begin{array}{cccc}cosh\xi & 0& 0& sinh\xi \\ 0& cosh\xi & sinh\xi & 0\\ 0& sinh\xi & cosh\xi & 0\\ sinh\xi & 0& 0& cosh\xi \end{array}\right),\end{array}$$

(26)

where $\xi$ is some parameter in the problem under consideration.

Let us choose ${\Omega }_{1}t=\pi n$, and ${\Omega }_{2}t=\pi (n+1)$, where $n=0,1,2,\cdots$ Conventionally, for simplicity, we choose $n=0$ and consider this case. For the remaining n, the results will be similar. In this case, we obtain the coefficients in the scattering matrix in the form

$$\begin{array}{c}\hfill Q_{11}=cos2\alpha ,Q_{22}=-cos2\alpha ,Q_{12}=-\frac{1}{2}sin2\alpha \left(\frac{a_2}{a_1}+\frac{a_1}{a_2}\right),\\ \hfill F_{11}=0,F_{22}=0,F_{12}=-\frac{1}{2}sin2\alpha \left(\frac{a_2}{a_1}-\frac{a_1}{a_2}\right).\end{array}$$

(27)

In order to obtain the matrix (26), it is necessary that $Q_{12}=0$. To do this, we choose $\frac{a_4}{a_3}=\frac{a_2}{a_1}=i$ (furthermore, we will assume that $a_1=a_4=1,a_2=-a_3=i$), as well as $\alpha =i\xi /2$. We also take into account that the operators ${\widehat{b}}_2$ and ${\widehat{b}}_2^{†}$ are defined up to the phase factor ${\widehat{b}}_2^{†}\to {\widehat{b}}_2^{†}{e}^{i\pi }$ and ${\widehat{b}}_2^{†}\to {\widehat{b}}_2^{†}{e}^{-i\pi }$. As a result, we will obtain a scattering matrix that completely coincides with the matrix (26). You can also see how the coordinates in Equation (4) are transformed with such substitutions

$$\begin{array}{c}\hfill y_1=q_{1}cosh(\xi /2)+q_{2}sinh(\xi /2),y_2=q_{1}sinh(\xi /2)+q_{2}cosh(\xi /2).\end{array}$$

(28)

It can be seen that coordinate transformations in Equation (28) are ultimately expressed through real quantities, which are responsible for the “squeezing” of quantum states.

Thus, in this section, we considered two limiting cases of the scattering matrix: 1. rotation of the coordinate system, in which the quantum states of the system change; and 2. the squeezing of the coordinate system, in which squeezed quantum states arise.

4. Conclusions

Thus, in this work, we obtained the scattering matrix S defined by Equation (20). This matrix contains six independent components, and the remaining components of the matrix are obtained by complex conjugation or multiplication by minus these six components. Each of these six components was found in analytical form depending on the arbitrary parameters of our model: $a_1,a_2,a_3,a_4(a_1{a}_4=a_2{a}_3);{\Omega }_1,{\Omega }_2;\alpha$. In total, the six independent parameters of our model and a certain choice of these parameters can reflect some physical situation. The experimental implementation of such quantum states depends on the choice of the physical system, which is described by the Hamiltonian Equation (4). Two scattering matrices were demonstrated and choices were made regarding which components we can obtain, including the beam splitter matrix and the matrix for the parametric amplifier, phase-conjugating mirror, and squeezed states. The results obtained are important not only in the theory of nonlinear quantum optics, but also in the applied aspect. Using the scattering matrix (20), it is possible to implement such output characteristics of quantum states that are needed in applied problems, but cannot be implemented using known scattering matrices. Moreover, Equation (4) describes a coupled harmonic oscillator and obtaining such physical systems at the present stage of technology can be realized.