Koopman Operator and Path Integral of Quantum Free-Electron Laser Model

Abstract

A quantum model of a free-electron laser (FEL) is considered. Two different approaches for the exploration of the the FEL system are considered. In the first case, the Heisenberg equations of motion are mapped on the basis of the initial wave functions, which consists of the photon coherent states and many-dimensional electron coherent states. This mapping is an exact procedure, which makes it possible to obtain an exact equation of motion for the intensity of the laser field in a closed form. The obtained equation is controlled by a Koopman operator. The analytical expression for the evolution of the FEL intensity is obtained in the framework of a perturbation theory, which is constructed for a small time scale. The second way of the consideration is based on the construction of the many-dimensional path integrals for the evolution of the wave function. This method also makes it possible to estimate the time evolution and the gain of the FEL intensity.

1. Introduction

In this paper, we discuss a quantum model of a free-electron laser (FEL). Experimental implementation and theoretical description of the FEL is a long-lasting problem that started in the seventies of the last century. This extensively studied phenomenon is well described and reviewed in [1,2,3,4], to mention a few. Contemporary studies are also reflected in recent publications and related to both classical and quantum descriptions [5,6], including application of fractional calculus to the FEL model [7,8]. In the present study, our interest to the model relates to a quantum description of the quantum FEL model with the quantum Hamiltonian of an electromagnetic field interacting with a system of N electrons [5,9,10,11]

$$\widehat{\mathcal{H}}=\sum _{j=1}^N\frac{{\widehat{p}}_j^2}{2m}+\hslash g\left(\widehat{a}\sum _{j=1}^N{e}^{2ik{\widehat{z}}_j}+{\widehat{a}}^{†}\sum _{j=1}^N{e}^{-2ik{\widehat{z}}_j}\right).$$

(1)

Here, summation relates to the positions $z_j$ and momenta ${\widehat{p}}_j$ of electrons with the mass m and the wave number k. The position–momentum commutation rule is $[{\widehat{z}}_k,{\widehat{p}}_j]=i\hslash {\delta }_{k,j}$. The laser mode is described by the photon annihilation and creation operators $\widehat{a}$ and ${\widehat{a}}^{†}$, respectively, with the commutation rule $[\widehat{a},{\widehat{a}}^{†}]=1$. The interaction parameter g couples the electron dynamics to the photon laser field. The Hamiltonian (1) has been obtained from the consideration of a non-relativistic electron in an electromagnetic field [9,10]; see also ref. [5]. Note also that the starting point of inferring the Hamiltonian (1) is the classical relativistic Hamiltonian [5,10]; see also Appendix A, where some grounds of the applicability of the model are explained.

It is worth stressing that, discussing the dynamics of the system, we also set the notation for the operators considered here. Namely, for any time-dependent operator $\widehat{x}\left(t\right)\equiv {\widehat{x}}_t$, its initial value is denoted by $\widehat{x}=\widehat{x}(t=0)$. The same is concerned with its average values denoting $x\left(t\right)\equiv x_t$ and $x=x(t=0)$, where $x=(a^{*},a,\{p_j,z_j\})$. The only exclusion is for the Hamiltonian, since it is the integral of motion and, correspondingly, $\widehat{\mathcal{H}}\left(t\right)=\widehat{\mathcal{H}}$.

The paper relates to mathematical aspects of the quantum mechanical treatment of the system (1) in the framework of two different approaches. The first one is the Koopman operator construction by mapping the Heisenberg equation of motion on the basis of the coherent states. The second approach is the Schrödinger–Feynman consideration of the Green function in the framework of the path integral. The main issue in the task is the laser field intensity. It is shown that both methods lead to the consistent results for the laser field output.

The dynamics of the intensity of the laser field in the operator form $\widehat{\mathcal{A}}\left(t\right)={\widehat{a}}^{†}\left(t\right)\widehat{a}\left(t\right)$ is described by the Heisenberg equation of motion

$$\dot{\widehat{\mathcal{A}}}\left(t\right)=\frac{1}{i\hslash }[\widehat{\mathcal{A}}\left(t\right),\widehat{\mathcal{H}}].$$

(2)

In this approach, our main concern is a technical consideration of the Heisenberg Equation (2) and finding the evolution of the averaged value of the laser field intensity in the form

$$⟨\widehat{\mathcal{A}}\left(t\right)⟩=e^{\mathcal{K}t}⟨\widehat{\mathcal{A}}(t=0)⟩,$$

where $\mathcal{K}$ is a so-called Koopman operator [12]; see also, e.g., [13,14]. Another approach developed in the paper is the investigation of the evolution of the initial wave function, which relates to the construction of the evolution operator by means of the path integrals [15,16]. For the latter example, the initial wave functions are the direct product of the photon and electron wave functions

$$|{\Psi }_0⟩=|\alpha ⟩\otimes |\mathbf{z}⟩,$$

(3)

where the coherent states $|\alpha ⟩$ are chosen for the photon wave function [17], while the electron wave function is $|\mathbf{z}⟩={\prod }_{j=1}^N|z_j⟩\equiv |z_1,\cdots ,z_N⟩$, where $⟨z_k|{z^{\prime }}_j⟩={\delta }_{k,j}\delta (z-z^{\prime })$.

The paper is organized as follows. In Section 2, the Koopman operator is constructed by mapping the Heisenberg equations of motion for the intensity of the photon laser field on the basis of the coherent states, both photon and electron. The obtained equation is exact and determines the dynamics of the laser field intensity. The second approach to the system is the description in the framework of the Schrödinger–Feynman consideration, which is presented in Section 3 and Section 4. The path integral is constructed to describe both the evolution of the initial wave function and the intensity of the laser field. The obtained results and the consistency of the two suggested methods are discussed in Section 5. Concluding remarks are presented in Section 6. Appendix A is devoted to some details of the construction and the applicability of the Hamiltonian (1). The path integral construction and its evaluation in the framework of the stationary-phase approximation are presented in Appendix B and Appendix C, respectively.

2. Quantum Evolution according to Koopman Operator

For the quantum mechanical analysis of the Heisenberg equations of motion, we use a technique of mapping the Heisenberg equations on a basis of the coherent states [18,19,20,21,22]. Note that the Hamiltonian is the time-independent integral of motion, $\widehat{\mathcal{H}}\left(t\right)=\widehat{\mathcal{H}}(t=0)$. Therefore, the Hamiltonian in the Heisenberg equations can be mapped on a basis of the coherent states $|\alpha ⟩$ and $|\overrightarrow{\beta }⟩$, constructed at the initial moment $t=0$. First, we construct the basis of the photon coherent states. That is, at the initial moment $t=0$, one introduces the coherent states vector $|\alpha ⟩$ as the eigenfunction of the annihilation operators $\widehat{a}=\widehat{a}(t=0)$, such that $\widehat{a}(t=0)|\alpha ⟩\equiv \widehat{a}|\alpha ⟩=\alpha |\alpha ⟩$ and, correspondingly, ${⟨\alpha |{\widehat{a}}^{†}=\left[\widehat{a}\right|\alpha ⟩]}^{†}={\alpha }^{*}⟨\alpha |$. The coherent state can also be constructed from a vacuum state $|0⟩$ as follows [23]

$$|\alpha ⟩=exp[\alpha {\widehat{a}}^{†}-{\alpha }^{*}\widehat{a}]|0⟩,\widehat{a}|0⟩=0.$$

(4)

Then, defining the c-number function as an average value of the photon field operator

$$\mathcal{A}\left(t\right)=⟨\alpha |\widehat{\mathcal{A}}\left(t\right)|\alpha ⟩,$$

(5)

one maps the Heisenberg equation of motion (2) on the basis of the coherent states $|\alpha ⟩$ as follows

$$i\hslash \frac{d}{dt}\mathcal{A}\left(t\right)=⟨\alpha |\left[\widehat{\mathcal{A}}\widehat{\mathcal{H}}-\widehat{\mathcal{H}}\widehat{\mathcal{A}}\left(t\right)\right]|\alpha ⟩.$$

(6)

Accounting for Equation (4) and (5), one obtains the mapping rules

$$\begin{array}{cc}\hfill & ⟨\alpha |\widehat{\mathcal{A}}\left(t\right){\widehat{a}}^{†}|\alpha ⟩=e^{-{\left|\alpha \right|}^2}\frac{\partial }{\partial \alpha }{e}^{{\left|\alpha \right|}^2}\mathcal{A}\left(t\right),\hfill \end{array}$$

(7a)

$$\begin{array}{cc}\hfill & ⟨\alpha |\widehat{a}\widehat{\mathcal{A}}\left(t\right)|\alpha ⟩=e^{-{\left|\alpha \right|}^2}\frac{\partial }{\partial {\alpha }^{*}}{e}^{{\left|\alpha \right|}^2}\mathcal{A}\left(t\right).\hfill \end{array}$$

(7b)

As admitted, the Hamiltonian is the integral of motion $\widehat{\mathcal{H}}\left({\widehat{a}}^{†}\left(t\right),\widehat{a}\left(t\right)\right)=\widehat{\mathcal{H}}({\widehat{a}}^{†},\widehat{a})$, therefore the mapping rules (7a) and (7b) yield the equation of motion (6) for $\mathcal{A}\left(t\right)$ in the form

$$\frac{d}{dt}\mathcal{A}\left(t\right)={\widehat{L}}_1\mathcal{A}\left(t\right)+{\widehat{L}}_2\mathcal{A},$$

(8)

where the photon–electron operator reads

$${\widehat{L}}_1=\frac{1}{i\hslash }{e}^{-{\left|\alpha \right|}^2}\left[\widehat{H}({\partial }_{\alpha },\alpha )-\widehat{H}({\alpha }^{*},{\partial }_{{\alpha }^{*}})\right]e^{{\left|\alpha \right|}^2}=ig\sum _{j=1}^N\left(e^{2ik{z}_j}{\partial }_{{\alpha }^{*}}-e^{-2ik{z}_j}{\partial }_{\alpha }\right),$$

(9)

while the commutator related to the electron part of the Hamiltonian (defined on the test function $|\mathbf{z}⟩$) reads

$${\widehat{L}}_2=\frac{1}{2m\hslash i}[\mathcal{A}\left(t\right),{\widehat{\mathbf{p}}}^2].$$

(10)

Note that both operators ${\widehat{L}}_1$ and ${\widehat{L}}_2$ are Hermitian.

The next step is the averaging procedure for the electron part of the Hilbert space. To this end, we construct a basis of the electron coherent states at the initial moment $t=0$. We admit that the electron system in the Hamiltonian (1) is considered as a system of free spinless particles, and their commutation rules corresponds to a so-called Heisenberg–Weyl group [23]. Therefore, introducing creation ${\widehat{b}}^{†}$ and annihilation $\widehat{b}$ operators:

$$\begin{array}{cc}\hfill & {\widehat{b}}_j=\frac{z_j+i{\widehat{p}}_j}{\sqrt{2\hslash }},{\widehat{b}}^{†}=\frac{z_j-i{\widehat{p}}_j}{\sqrt{2\hslash }},\hfill \end{array}$$

(11a)

$$\begin{array}{cc}\hfill & z_j=\sqrt{\frac{\hslash }{2}}({\widehat{b}}_j+{\widehat{b}}_j^{†}),{\widehat{p}}_j=i\sqrt{\frac{\hslash }{2}}({\widehat{b}}_j^{†}-{\widehat{b}}_j),\hfill \end{array}$$

(11b)

where $[{\widehat{b}}_j,{\widehat{b}}_{j^{\prime }}^{†}]={\delta }_{j,j^{\prime }}$, we introduce the coherent state basis as follows:

$$|\overrightarrow{\beta }⟩=\prod _{j=1}^N|{\beta }_j⟩=\prod _{j=1}^{N}exp[{\beta }_j{\widehat{b}}_j^{†}-{\beta }_j^{*}{\widehat{b}}_j]|0⟩,{\widehat{b}}_j|0⟩=0,$$

(12)

which belongs to the Hilbert space of electrons.

Introducing the double average

$$⟨\mathcal{A}⟩\equiv ⟨\widehat{\mathcal{A}}\left(t\right)⟩=⟨⟨\widehat{\mathcal{A}}\left(t\right){⟩}_{\alpha }{⟩}_{\beta }$$

(13)

and using properties (7a) and (7b) for both $\widehat{a},{\widehat{a}}^{†}$ and $\widehat{b},{\widehat{b}}^{†}$ operators, and taking into account the commutator (10), we obtain the “Koopman equation”

$$\frac{d}{dt}⟨\mathcal{A}⟩=\mathcal{K}⟨\mathcal{A}⟩,$$

(14)

where the Hermitian Koopman operator reads

$$\begin{array}{cc}\hfill \mathcal{K}=& \frac{i}{4m}\sum _{j=1}^N\left({\partial }_{{\beta }_j^{*}}^2-{\partial }_{{\beta }_j}^2+2{\beta }_j{\partial }_{{\beta }_j^{*}}-2{\beta }_j^{*}{\partial }_{{\beta }_j}+2{\beta }_j{\partial }_{{\beta }_j}-2{\beta }_j^{*}{\partial }_{{\beta }_j^{*}}\right)\hfill \end{array}$$

(15a)

$$\begin{array}{cc}\hfill & +ig\sum _{j=1}^N\left[{\partial }_{{\alpha }^{*}}{e}^{2ik({\beta }_j^{*}+{\beta }_j)}{e}^{2ik{\partial }_{{\beta }_j^{*}}}-{\partial }_{\alpha }{e}^{-2ik({\beta }_j^{*}+{\beta }_j)}{e}^{-2ik{\partial }_{{\beta }_j}}\right].\hfill \end{array}$$

(15b)

The Koopman operator consists of two parts: $\mathcal{K}={\mathcal{K}}_e+{\mathcal{K}}_{e-ph}$. The first part ${\mathcal{K}}_e$, described by Equation (15a), corresponds to the kinetic part of free electrons, while the second part describes the electron–photon interaction, as in Equation (15b).

The solution of the Koopman Equation (14) for the initial condition $⟨\mathcal{A}(t=0)⟩={\alpha }^{*}\alpha$ is presented in the exponential form

$$⟨\mathcal{A}⟩=e^{t\mathcal{K}}{\alpha }^{*}\alpha =\sum _{n=0}^{\infty }\frac{t^n}{n!}{\mathcal{K}}^n{\alpha }^{*}\alpha .$$

(16)

We take into account that electrons spend a very short time inside the laser size, namely $t\ll 1$, which results from the fact that the laser size is finite and electron velocity is close to the speed of light. Then, we should take into account only a few first terms in the expansion (16), and we limit ourselves by the second order of $t^2$. Obviously, for $t^0$, the zero-order term corresponds to the initial condition: ${⟨\mathcal{A}⟩}^{\left(0\right)}={\left|\alpha \right|}^2$. Correspondingly, the first-order term for $t^1$ reads

$$\begin{array}{c}{⟨\mathcal{A}⟩}^{\left(1\right)}=t\mathcal{K}{\alpha }^{*}\alpha =igt\sum _{j=1}^N\left[{\partial }_{{\alpha }^{*}}{e}^{2ik({\beta }_j^{*}+{\beta }_j)}{e}^{2ik{\partial }_{{\beta }_j^{*}}}-{\partial }_{\alpha }{e}^{-2ik({\beta }_j^{*}+{\beta }_j)}{e}^{-2ik{\partial }_{{\beta }_j}}\right]{\alpha }^{*}\alpha \hfill \\ \hfill =igt\sum _{j=1}^N\left[\alpha e^{2ik({\beta }_j^{*}+{\beta }_j)}-{\alpha }^{*}{e}^{-2ik({\beta }_j^{*}+{\beta }_j)}\right].\end{array}$$

(17)

At this step, the action of the Koopman operator reduces to differentiation with respect to $\alpha$ and ${\alpha }^{*}$ only. The derivatives ${\partial }_{{\beta }_j}$ and ${\partial }_{{\beta }_j^{*}}$ do not “work” here. The situation changes for the second-order term, for which we obtain

$$\begin{array}{c}{⟨\mathcal{A}⟩}^{\left(2\right)}=\frac{t}{2}\mathcal{K}{⟨\mathcal{A}⟩}^{\left(1\right)}={\mathcal{K}}_e{⟨\mathcal{A}⟩}^{\left(1\right)}-\frac{g^2{t}^2}{2}\hfill \\ \times \sum _{j=1}^N\left[{\partial }_{{\alpha }^{*}}{e}^{2ik({\beta }_j^{*}+{\beta }_j)}{e}^{2ik{\partial }_{{\beta }_j^{*}}}-{\partial }_{\alpha }{e}^{-2ik({\beta }_j^{*}+{\beta }_j)}{e}^{-2ik{\partial }_{{\beta }_j}}\right]·\sum _{j=1}^N\left[\alpha e^{2ik({\beta }_j^{*}+{\beta }_j)}-{\alpha }^{*}{e}^{-2ik({\beta }_j^{*}+{\beta }_j)}\right]\\ \hfill =\frac{t}{2}{\mathcal{K}}_e{⟨\mathcal{A}⟩}^{\left(1\right)}+e^{4k^2}\frac{g^2{t}^2}{2}\sum _{j^{\prime },j=1}^N{e}^{2ik({\beta }_j^{*}+{\beta }_j-{\beta }_{j^{\prime }}^{*}+{\beta }_{j^{\prime }})}.\end{array}$$

(18)

We restrict the expansion by the second order in Equation (18). The last term is the main contribution to the laser field intensity

$$⟨\mathcal{A}⟩\approx e^{4k^2}\frac{g^2{t}^2}{2}\sum _{j^{\prime },j=1}^N{e}^{2ik({\beta }_j^{*}+{\beta }_j-{\beta }_{j^{\prime }}^{*}+{\beta }_{j^{\prime }})},$$

(19)

and it is the first main result, which is discussed in Section 5. It is also interesting to admit that in the coherent states, the averaged energy of electrons is conserved that immediately follows from the specific form of the Koopman operator (15) and the transformations (11). Namely, $\mathcal{K}⟨p_j^2⟩=0$, then $⟨p_j^2\left(t\right)⟩=⟨p_j^2⟩$. This situation is a pure quantum effect. It can be explained by coherent Cherenkov radiation, which is described just by the same Hamiltonian as in Equation (1) [10]; see also Appendix A.

3. Evolution of Initial Wave Function

It is also instructive to obtain an analytical expression for the evolution operator $U\left(t\right)=e^{-i\widehat{\mathcal{H}}t/\hslash }$, which determines the evolution of the initial wave function ${\Psi }_0={\psi }_0\left(\mathbf{z}\right)|\alpha ⟩$, where ${\psi }_0$ can be an arbitrary suitable function. We consider ${\psi }_0\left(\mathbf{z}\right)=|\mathbf{z}⟩$ defined in Equation (3). The evolution operator can be presented in the form of the path integral related to the electron Hilbert space. In this case, the interaction term in the Hamiltonian (1) plays the role of the potential, which is however the operator valued the function of the operators $\widehat{a}$ and ${\widehat{a}}^{†}$.

We follow the standard procedure [15,16] partitioning the time interval $t=R\Delta t$ at the condition $\Delta t\to 0$, while $R\to \infty$, where the boundary (initial and finite) conditions for the path integral defined at $r=0$ and $r=R$ are $\mathbf{z}(r=0)=\mathbf{z}$ and $\mathbf{z}\left(R\right)=\mathbf{z}\left(t\right)$. The procedure is presented in Appendix B. Then, the path integral arises from this partition as follows (A9):

$$\begin{array}{c}{e}^{-it(\widehat{K}+\widehat{V})/\hslash }=\underset{\begin{array}{c}\mathbf{z}\left(0\right)=\mathbf{z}\\ \mathbf{z}\left(t\right)=\mathbf{z}\left(t\right)\end{array}}{\int }D\left[\mathbf{z}\left(\tau \right)\right]\hfill \\ \hfill \times exp\left\{\frac{i}{\hslash }{\int }_0^{t}d\tau \left[\sum _{j=1}^N\frac{m}{2}{\left({\dot{z}}_j\right)}^2-\hslash g\sum _{j=1}^N\left(e^{2ik{z}_j}\widehat{a}+e^{-2ik{z}_j}{\widehat{a}}^{†}\right)\right]\right\}\end{array}$$

(20)

Therefore, the wave function at time t is defined by Equation (20) as follows:

$$\begin{array}{c}|\Psi (t,\alpha ,\mathbf{z})⟩=e^{-i\widehat{\mathcal{H}}t/\hslash }{\psi }_0\left(\mathbf{z}\right)|\alpha ⟩={\int }_{\mathbf{z}}^{\mathbf{z}\left(t\right)}D\left[\mathbf{z}\left(\tau \right)\right]\hfill \\ \times exp\left[\frac{i}{\hslash }{\int }_0^{t}d\tau \sum _{j=1}^N\frac{m}{2}{\left({\dot{z}}_j\left(\tau \right)\right)}^2\right]\\ \hfill \times exp\left[-ig{\int }_0^{t}d\tau \sum _{j=1}^N\left(e^{2ik{z}_j\left(\tau \right)}\widehat{a}+e^{-2ik{z}_j\left(\tau \right)}{\widehat{a}}^{†}\right)\right]|\alpha ⟩|\mathbf{z}⟩.\end{array}$$

(21)

Here, ${\psi }_0\left(\mathbf{z}\right)=|\mathbf{z}⟩$. The last line in Equation (21) describes the shift of the coherent states $|\alpha ⟩$ by means of the shift operator $\widehat{D}\left(\mu \left(t\right)\right)=e^{\mu \left(t\right){\widehat{a}}^{†}-{\mu }^{*}\left(t\right)\widehat{a}}$, where $\mu \left(t\right)=-ig{\int }_0^t{\sum }_{j=1}^N{e}^{-2ik{z}_j\left(\tau \right)}d\tau$. It transforms one coherent state to another [23,24]:

$$\widehat{D}\left(\mu \left(t\right)\right)|\alpha ⟩=e^{iIm\left[\mu \left(t\right){\alpha }^{*}\right]}|\alpha +\mu \left(t\right)⟩.$$

(22)

In Equations (20)–(22), the shift operator in the photon Hilbert space is

$$\widehat{D}\left(\mu \left(t\right)\right)=e^{\mu \left(t\right){\widehat{a}}^{†}-{\mu }^{*}\left(t\right)\widehat{a}}=exp\left[-ig{\int }_0^{t}d\tau \sum _{j=1}^N\left(e^{2ik{z}_j\left(\tau \right)}\widehat{a}+e^{-2ik{z}_j\left(\tau \right)}{\widehat{a}}^{†}\right)\right].$$

(23)

Correspondingly, the shifted coherent state in Equation (22) at time t reads

$$\begin{array}{c}|\alpha (t,\mathbf{z})⟩=e^{iIm\left[\mu \left(t\right){\alpha }^{*}\right]}|\alpha +\mu \left(t\right)⟩=exp\left[\frac{-ig}{2}{\int }_0^{t}d\tau \sum _{j=1}^N\left({\alpha }^{*}{e}^{-2ik{z}_j\left(\tau \right)}+\alpha e^{2ik{z}_j\left(\tau \right)}\right)\right]\hfill \\ \hfill \times \left|\alpha -ig{\int }_0^{t}d\tau \sum _{j=1}^N{e}^{-2ik{z}_j\left(\tau \right)}\right.⟩.\end{array}$$

(24)

Note that the wave function in Equation (24) reflects the electron–photon interaction. Substituting Equation (24) in Equation (21), we obtain the wave function in the formal form. Eventually, the exact expression for the evolution of the wave function reads as follows:

$$\begin{array}{c}|\Psi (t,\alpha ,\mathbf{z})⟩={\int }_{\mathbf{z}}^{\mathbf{z}\left(t\right)}D\left[\mathbf{z}\left(\tau \right)\right]exp\left[\frac{i}{\hslash }{\int }_0^{t}d\tau \sum _{j=1}^N\frac{m}{2}{\left({\dot{z}}_j\left(\tau \right)\right)}^2\right]\hfill \\ \hfill \times exp\left[\frac{-ig}{2}{\int }_0^{t}d\tau \sum _{j=1}^N\left({\alpha }^{*}{e}^{-2ik{z}_j\left(\tau \right)}+\alpha e^{2ik{z}_j\left(\tau \right)}\right)\right]\times \left|\alpha -ig{\int }_0^{t}d\tau \sum _{j=1}^N{e}^{-2ik{z}_j\left(\tau \right)}\right.⟩|\mathbf{z}⟩,\end{array}$$

(25)

where the initial wave function is defined in Equation (3). The path integration is performed in the semiclassical or the stationary-phase approximation, defined by the path integral as follows:

$$\begin{array}{c}G\left[\mathbf{z};\mathbf{z}\left(t\right)\right]|\alpha +\mu \left(t\right)⟩|\mathbf{z}⟩=\prod _{j=1}^N{\int }_{z_j}^{z_j\left(t\right)}D\left[z_j\left(\tau \right)\right]exp\left\{\frac{i}{\hslash }\sum _{j=1}^N\right.\hfill \\ \left.\times {\int }_0^{t}d\tau \left[\frac{m}{2}{\left({\dot{z}}_j\left(\tau \right)\right)}^2+\hslash g\left|\alpha \right|cos\left({\varphi }_{\alpha }+2k{z}_j\left(\tau \right)\right)\right]\right\}\\ \hfill \times \left|\alpha +ig{\int }_0^{t}d\tau \sum _{j=1}^N{e}^{-2ik{z}_j\left(\tau \right)}\right.⟩|\mathbf{z}⟩,\end{array}$$

(26)

where $\alpha =\left|\alpha \right|e^{i{\varphi }_{\alpha }}$. Note that it is instructive to apply the stationary-phase approximation for the averages, namely for the intensity of the laser field.

4. Evolution of the FEL Intensity

Let us estimate the evolution of the intensity of the laser field $⟨{\widehat{a}}^{†}\left(t\right)\widehat{a}\left(t\right)⟩$, which is

$$⟨{\widehat{a}}^{†}\left(t\right)\widehat{a}\left(t\right)⟩=⟨\Psi (t,\alpha ,\mathbf{z})|{\widehat{a}}^{†}\widehat{a}|\Psi (t,\alpha ,\mathbf{z})⟩.$$

(27)

To perform this calculation, we first map the photon operator ${\widehat{a}}^{†}\left(t\right)\widehat{a}\left(t\right)$ on the shifted coherent states in Equation (26) that yields the averaged value

$$[{\alpha }^{*}+{\mu }^{*}\left(t\right)][\alpha +{\mu }^{\prime }\left(t\right)]⟨\alpha +\mu \left(t\right)|\alpha +{\mu }^{\prime }\left(t\right)⟩=[{\alpha }^{*}+{\mu }^{*}\left(t\right)][\alpha +{\mu }^{\prime }\left(t\right)]e^{-\frac{1}{2}{|\mu \left(t\right)-{\mu }^{\prime }\left(t\right)|}^2}.$$

The term with $\alpha$ is estimated by the stationary phase approximation, which is $\alpha G\left[{\mathbf{z}}_e\left(\tau \right)\right]$, where

$$G\left[\mathbf{z};\mathbf{z}\left(t\right)\right]=\prod _{j=1}^N\int D\left[z_j\left(\tau \right)\right]e^{\frac{i}{\hslash }S[z_j;z_j\left(t\right)]}\approx G\left[{\mathbf{z}}_e\left(\tau \right)\right]=\prod _{j=1}^N{A}_j{e}^{\frac{i}{\hslash }{S}_e\left[z_{j,e}\right]}.$$

(28)

The dynamics of $z_{e,j}\left(\tau \right)$ are controlled by a physical pendulum with the principle Hamiltonian function $S_e\left(z_{j,e}\right)$ of a j-th electron and the constant energy $E_j=\frac{p_j^2}{2m}-2k\hslash \left|\alpha \right|cos({\varphi }_{\alpha }+2k{z}_j)$, determined at the initial time $\tau =0$, while $A_j$ is a constant resulted from the Gaussian integration, which corresponds to the second variation ${\delta }^{\left(2\right)}S$ at the stationary point. Note that the principle Hamiltonian function $S\left[z_{e,j}\left(\tau \right)\right]$ results from the first variation $\delta S[z_j;z_j\left(t\right)]=0$. Details of the calculation are shown in Appendix C.

The dynamics are described by the elliptic functions with a separatrix, which separates unlimited flight trajectories from the confined elliptic ones. Therefore, the stationary points are determined by the elliptic amplitude function $\mathrm{am}(\tau ,\kappa )$, [25], where $\kappa =\kappa \left(E_j\right)\le 1$ is defined in Equation (A23). Therefore, $2k{z}_j=\mathrm{am}(\tau ,\kappa )-{\varphi }_{\alpha }$.

Our main concern, however, is the path integration of the interaction term

$$ig{\int }_0^{t}d{t}^{\prime }\sum _{j=1}^N{e}^{-2ik{z}_j\left(t^{\prime }\right)}.$$

This part of the FEL intensity is presented in the form of the $2N$ dimensional functional integration. Therefore, the main part (M.P.) of the averaged value is defined as follows:

$$\begin{array}{c}\mathrm{M}.\mathrm{P}.⟨{\widehat{a}}^{†}\left(t\right)\widehat{a}\left(t\right)⟩=g^2\prod _{n=1}^N\int D\left[z_n\left(\tau \right)\right]\prod _{n^{\prime }=1}^N\int D\left[z_{n^{\prime }}^{\prime }\left(\tau \right)\right]\sum _{j=1}^N\sum _{j^{\prime }=1}^N{\int }_0^{t}d{t}_1{\int }_0^{t}d{t}_2{e}^{-2ik[z_j\left(t_1\right)-z_{j^{\prime }}^{\prime }\left(t_2\right)]}\hfill \\ \times exp\left\{\frac{i}{\hslash }\sum _{j=1}^N{\int }_0^{t}d\tau \left[\frac{m}{2}{\left({\dot{z}}_j\left(\tau \right)\right)}^2-\hslash g\left|\alpha \right|cos\left({\varphi }_{\alpha }+2k{z}_j\left(\tau \right)\right)\right]\right\}\\ \times exp\left\{-\frac{i}{\hslash }\sum _{j=1}^N{\int }_0^{t}d\tau \left[\frac{m}{2}{\left({\dot{z^{\prime }}}_j\left(\tau \right)\right)}^2-\hslash g\left|\alpha \right|cos\left({\varphi }_{\alpha }+2k{z}_j^{\prime }\left(\tau \right)\right)\right]\right\}\\ \hfill \times ⟨\alpha -ig{\int }_0^{t}d\tau \sum _{j=1}^N{e}^{-2ik{z}_j\left(\tau \right)}\left|\alpha -ig{\int }_0^{t}d\tau \sum _{j=1}^N{e}^{-2ik{z}_j\left(\tau \right)}\right.⟩.\end{array}$$

(29)

The stationary-phase approximation in Equation (29) is performed according to Equation (28). It is reasonable to take the stationary solution for $\kappa =1$ at the the asymptotic times, according to Figure A1 in Appendix C. Therefore, we obtain that $2k{z}_{j,e}\left(\tau \right)=2\pi -{\varphi }_{\alpha }$. Eventually, we obtain

$$⟨{\widehat{a}}^{†}\left(t\right)\widehat{a}\left(t\right)⟩\approx g^2{N}^2{t}^2\prod _{j=1}^N{\left|A_j\right|}^2.$$

(30)

5. Discussion

We collect some physical consequences of the calculations of the laser intensity and discuss them in this section. The main contribution to the intensity of the laser photon field is due to the interaction term in the Hamiltonian (1), that is

$$⟨\mathcal{A}⟩\sim g^2⟨{\left[\widehat{a}\sum _{j=1}^N{e}^{2ik{z}_j}+{\widehat{a}}^{†}\sum _{j=1}^N{e}^{-2ik{z}_j}\right]}^2⟩.$$

(31)

Two different approaches for the evaluation of the laser intensity are suggested in the framework of the Heisenberg and the Schrödinger representations of quantum mechanics.

In the first approach, the Heisenberg equations of motion are mapped on the basis of the coherent states, which are constructed at the initial moment of time $t=0$ for both the photon laser field and electrons. Since the algebras of the photon and electron operators belong to the same Heisenberg–Weyl algebras, the analytical forms of the coherent states (as the eigenstates of the annihilation operators) are the same, although the wave functions $|\alpha ⟩$ and $|\overrightarrow{\beta }⟩$ belong to the different Hilbert spaces. The Heisenberg equations of motions for the laser photon field intensity are exactly mapped on the basis of the coherent states and the obtained “Koopman equation”. The work in (14) describes the gain process, and it is controlled by the Koopman operator $\mathcal{K}$. The main advantage of this approach is that the gain intensity can be easily obtained by the small-time perturbation theory, (19), which yields

$$⟨\mathcal{A}⟩\approx e^{4k^2}\frac{g^2{t}^2}{2}\sum _{j^{\prime },j=1}^N{e}^{2ik({\beta }_j^{*}+{\beta }_j-{\beta }_{j^{\prime }}^{*}+{\beta }_{j^{\prime }})}.$$

If initially all coherent states are just the same, that is ${\beta }_j=\beta ,\forall j$, then according to Equation (11), the distributions of the initial conditions $p_j$ and $z_j$ are such that $(p_j^2+z_j^2)/2=\hslash {\left|\beta \right|}^2\forall j$, which is defined as $N^2$ points on the hyper-sphere in the $2N$-dimensional phase space. In other words, all $p_j$ and $z_j$ are independent initial values defined on the hyper-sphere, which is just the direct product of N circles for the non-interacting electrons. In this case, the gained intensity is

$$⟨\mathcal{A}⟩\sim g^2{N}^2{t}^2,$$

(32)

which is superradiance. Note that super-radiance supposes that the intensity gain is of the order of $N^2$.

The second approach relates to the construction of the path integral for the evolution operator in Equation (20). This makes it possible to estimate the FEL intensity according to the interaction term. This procedure is performed in the stationary-phase approximation. A detailed analysis is presented in Appendix C. The maximum gain effect is estimated for the large-time asymptotics of the stationary-phase solution for $E_{\gamma }{t}^2\gg 1$. This result is described by Equation (30) and eventually coincides with Equation (32). This corresponds to super-radiance as well.

Note also that the scalar product of the wave function in Equation (25) is

$$⟨\Psi (t,\alpha ,\mathbf{z})\left|\Psi \right(t,\alpha ,\mathbf{z})⟩=1.$$

Therefore, in the stationary-phase approximation, the scalar product

$$⟨\alpha -ig{\int }_0^{t}d\tau \sum _{j=1}^N{e}^{-2ik{z}_j\left(\tau \right)}\left|\alpha -ig{\int }_0^{t}d\tau \sum _{j=1}^N{e}^{-2ik{z}_j\left(\tau \right)}\right.⟩=1$$

for the stationary-phase solution $2k{z}_{j,e}\left(\tau \right)=\mathrm{am}(\sqrt{E_{\gamma }}\tau ,\kappa )-{\varphi }_{\alpha }$, which is valid for all physical values of $\kappa$ and $E_{\gamma }$ and time $\tau \in [0,t]$.

6. Conclusions

A quantum model of a free-electron laser (FEL) is considered. Two different approaches for the evaluation of the laser field intensity are suggested, and these are the Heisenberg and the Schrödinger representations of quantum mechanics. In the first case, the Heisenberg equations of motion are mapped on the many-dimensional basis, which consists of the photon coherent states and many-dimensional electron coherent states. This mapping is an exact procedure, which makes it possible to obtain an equation of motion for the laser intensity in a closed form. The obtained equation is controlled by a Koopman operator. The analytical expression for the evolution of the FEL intensity is obtained in the framework of a perturbation theory, which is constructed for a small time scale.

The second way of the solution is based on the construction of the many-dimensional path integral for the evolution of the wave function. This method also makes it possible to estimate the time evolution and the gain of the FEL intensity.

It should be admitted that the laser field outputs presented in Equations (30) and (32) obtained by these different methods are mathematically consistent. However, the description of the electron dynamics underlying the FEL is completely different. In particular, in the path integral consideration, the laser field intensity is estimated by the stationary-phase approximation, where the electron dynamics are controlled by a physical pendulum and described semiclassically by the elliptic functions. Contrary to that, the Koopman operator describes the electron dynamics exactly, and the latter is a free-particle dynamic. This phenomenon is pure quantum and can be explained by the Cherenkov radiation, which is another scenario of the FEL [10]. It should be admitted that this behavior cannot be explained by the wiggler mechanism of interaction. Citing from the Tamm and Frank’s paper [26], we note that “if one takes in account the fact that an electron moving in a medium does radiate light even if it is moving uniformly provided that its velocity is greater than the velocity of light in the medium”. (See also [27]).

Appendix A.1. Quantization

In the quantization procedure of the classical Hamiltonian (A5), both the electron dynamics and the laser field are quantized. Therefore, the electron coordinate and the momentum are operators $(z,p)\to (\widehat{z},\widehat{p})$, where their commutation relation is $[\widehat{z},\widehat{p}]=i\hslash$. Then, treating the laser field amplitude ${\tilde{A}}_L$ as the quantized field, the photon annihilation $\widehat{a}$ and creation ${\widehat{a}}^{dag}$ operators are introduced, and the commutation relation is $[\widehat{a},{\widehat{a}}^{†}]=1$. Then, the quantized amplitudes of the laser field are defined by the substitutions

$$\begin{array}{cc}\hfill & {\tilde{A}}_L\to A_L\widehat{a}\hfill \\ \hfill & {\tilde{A}}_L^{*}\to A_L{\widehat{a}}^{†},\hfill \end{array}$$

where $A_L$ is the amplitude of the quantized laser field. However, the wiggler field is considered as an external classical parameter ${\tilde{A}}_W^{*}={\tilde{A}}_W=\mathrm{const}$ [5,10].

Thus, the quantized Hamiltonian reads

$$\widehat{H}=\frac{{\widehat{p}}^2}{2m}+\hslash g\left(\widehat{a}{e}^{2ik\widehat{z}}+{\widehat{a}}^{†}{e}^{-2ik\widehat{z}}\right),$$

(A6)

where the coupling constant is

$$g=\frac{e^2}{\hslash m}{A}_L{\tilde{A}}_W,$$

(A7)

which relates to the wiggler scenario of the FEL. Eventually, the straightforward generalization of the Hamiltonian (A6) for the N electrons leads to the Hamiltonian (1).

Appendix A.2. Cherenkov Radiation Scenario of the FEL

A slightly different consideration has been suggested in ref. [10], which is also suitable for the Cherenkov radiation (CR) scenario of the FEL. In the case of stimulated CR, the wiggler is replaced by a medium with a refractive index $n>1$. Since ${\mathbf{A}}_W=0$, the interaction term is due to the transverse canonical momentum of the electron ${\mathbf{p}}_{\perp }\ne 0$, while linear polarization $\overrightarrow{ϵ}={\overrightarrow{ϵ}}^{*}$ is supposed in Equation (A2). The relativistic Hamiltonian now reads

$$H={[c^2{\mathbf{p}}^2-2ce\overrightarrow{ϵ}{\mathbf{A}}_L+m_0^2{c}^4]}^{\frac{1}{2}}.$$

(A8)

As shown in ref. [10], quantization of the Hamiltonian (A8) for the N electrons systems yields

$$\widehat{H}=\hslash \omega {\widehat{a}}^{†}\widehat{a}+\sum _j\frac{p_j^2}{2m}+\hslash g\sum _j\left[\widehat{a}{e}^{i(k{z}_j+\omega t)}+{\widehat{a}}^{†}{e}^{-i(k{z}_j+\omega t)}\right],$$

(A9)

where $\omega =ck/\sqrt{n^2-1}$ is the frequency of the laser field. The coupling constant now is $g=-\frac{e}{\hslash mc}\overrightarrow{ϵ}·{\mathbf{p}}_{\perp }{A}_L$, while $m^2=m_0^2+{\mathbf{p}}_{\perp }^2$,10].

In the interaction representation

$$\widehat{H}\to e^{-i\omega {\widehat{a}}^{†}\widehat{a}t}\widehat{H}{e}^{i\omega {\widehat{a}}^{†}\widehat{a}t}$$

and re-scaling $k\to 2k$, one arrives at the Hamiltonian $\widehat{\mathcal{H}}$ in Equation (1), obtained by means of the commutation rule [31],

$$e^{-\eta \widehat{b}}\widehat{c}{e}^{\eta \widehat{b}}=\sum _{n=0}^{\infty }\frac{{(-\eta )}^n}{n!}\left[\stackrel{n}{\overbrace{\widehat{b},[\widehat{b}\cdots [\widehat{b},\widehat{c}]\cdots }}\right],$$

(A10)

where over-brace contains n operators $\widehat{b}$. Taking into account that $\widehat{b}\equiv {\widehat{a}}^{†}\widehat{a}$, while $\widehat{c}=(\widehat{a},{\widehat{a}}^{†})$, we obtain

$$\begin{array}{cc}\hfill & e^{-i\omega {\widehat{a}}^{†}\widehat{a}t}\widehat{a}{e}^{i\omega {\widehat{a}}^{†}\widehat{a}t}=\widehat{a}{e}^{-i\omega t},\hfill \end{array}$$

(A11)

$$\begin{array}{cc}\hfill & e^{-i\omega {\widehat{a}}^{†}\widehat{a}t}{\widehat{a}}^{dag}{e}^{i\omega {\widehat{a}}^{†}\widehat{a}t}={\widehat{a}}^{dag}{e}^{i\omega t}.\hfill \end{array}$$

(A12)