Quantum Regime of a Free-Electron Laser: Relativistic Approach

Abstract

In the quantum regime of the free-electron laser, the dynamics of the electrons is not governed by continuous trajectories but by discrete jumps in momentum. In this article, we rederive the two crucial conditions to enter this quantum regime: (1) a large quantum mechanical recoil of the electron caused by the scattering with the laser and the wiggler field and (2) a small energy spread of the electron beam. In contrast to our recent approach based on nonrelativistic quantum mechanics in a co-moving frame of reference, we now pursue a model in the laboratory frame employing relativistic quantum electrodynamics.

This article is part of the topical collection “Enlightening the World with the Laser” - Honoring T. W. Hänsch guest edited by Tilman Esslinger, Nathalie Picqué, and Thomas Udem.

It is a great honor and pleasure for us to dedicate this article to Theodor W. Hänsch on the occasion of his $75\mathrm{th}$ birthday. On first sight, there seems to be no connection between his research interests and the field of free-electron lasers (FELs). However, a closer examination reveals once more his crucial influence in shaping a new field. Indeed, in one [1] of the pioneering articles on FEL theory, W. B. Colson acknowledges ‘fruitful discussions,’ he had with Hänsch on this topic. Therefore, we find it appropriate to contribute an article devoted to the Quantum FEL to this Special Issue celebrating the scientific achievements of Theodor W. Hänsch. Our warmest wishes, and Happy Birthday Ted.

Appendices

Appendix A: Furry Picture for Scalar quantum Electrodynamics

In this appendix, we derive, starting from basic principles, the model introduced in Sect. 2, namely the Furry or bound interaction picture [21] for a scalar field [19]. Here we first discuss the classical Klein–Gordon and Maxwell fields, as well as their interaction. Then we perform second quantization in the Schrödinger picture, before we finally investigate the transformation to the Furry picture. We conclude by presenting the Hamiltonian in the Furry picture.

1.1 Lagrangian Formulation

We start from the classical Lagrangian density [39]

$$ \mathcal{L}\equiv {\mathcal{L}}_{\mathrm{K}-\mathrm{G}}+{\mathcal{L}}_{\mathrm{M}}+{\mathcal{L}}_{\mathrm{Int}} $$

(48)

for the Klein–Gordon field $\varphi =\varphi (x)$ interacting with an electromagnetic field described by the four-potential ${A}^{\mu }$.

The dynamics of the free Klein–Gordon field is determined by

$$ {\mathcal{L}}_{\mathrm{K}-\mathrm{G}}\equiv c\left[{\hbar}^2{\partial}_{\mu }{\varphi}^{\ast }(x){\partial}^{\mu}\varphi (x)-{m}_0^2{c}^2{\varphi}^{\ast }(x)\varphi (x)\right] $$

(49)

with the electron mass ${m}_0$, the speed of light c and the reduced Planck constant $\hbar \equiv h/2\pi$.

The free Maxwell field evolves in time according to the Lagrangian density

$$ {\mathcal{L}}_{\mathrm{M}}\equiv -\frac{1}{4{\mu}_0}{F}_{\mu \nu}(x){F}^{\mu \nu}(x) $$

(50)

where we have introduced the field tensor

$$ {F}^{\mu \nu}(x)\equiv {\partial}^{\mu }{A}^{\nu }(x)-{\partial}^{\nu }{A}^{\mu }(x) $$

and the vacuum permeability ${\mu}_0$.

The interaction Lagrangian

$$ {\displaystyle \begin{array}{cc}\hfill {\mathcal{L}}_{\mathrm{Int}}& \equiv -\mathrm{e}c{A}_{\mu }(x)\left[{\varphi}^{\ast }(x)i\hbar {\partial}^{\mu}\varphi (x)-\varphi (x)i\hbar {\partial}^{\mu }{\varphi}^{\ast }(x)\right]\hfill \\ {}\hfill & \kern1em +{\mathrm{e}}^2c\kern0.277778em {A}^2(x){\varphi}^{\ast }(x)\varphi (x)\hfill \end{array}} $$

(51)

is chosen such that the Euler–Lagrange equation with respect to ${\varphi}^{\ast }(x)$, that is,

$$ {\partial}^{\mu}\frac{\partial \mathcal{L}}{\partial \left({\partial}^{\mu }{\varphi}^{\ast }(x)\right)}-\frac{\partial \mathcal{L}}{\partial {\varphi}^{\ast }(x)}=0, $$

(52)

leads to the Klein–Gordon equation

$$ \left[{\left(i\hbar {\partial}^{\mu }-\mathrm{e}{A}^{\mu }(x)\right)}^2-{m}_0^2{c}^2\right]\varphi (x)=0 $$

(53)

with minimal coupling ${\widehat{p}}^{\mu}\to {\widehat{p}}^{\mu }-\mathrm{e}{A}^{\mu }(x)$ between the operator ${\widehat{p}}^{\mu}\equiv i\mathit{\hbar}{\partial}^{\mu }$ for the four-momentum and the potential ${A}^{\mu }(x)$.

When we compute the corresponding Euler–Lagrange equations for ${A}^{\mu }$, we obtain the Maxwell equations

$$ {\partial}^{\nu }{\partial}_{\nu }{A}^{\mu }(x)={\mu}_0\mathrm{e}{j}^{\mu }(x), $$

(54)

where ${A}^{\mu }$ is driven by the four current

$$ {\displaystyle \begin{array}{cc}\hfill {j}^{\mu }(x)& \equiv c\left[{\varphi}^{\ast }(x)i\hbar {\partial}^{\mu}\varphi (x)-\varphi (x)i\hbar {\partial}^{\mu }{\varphi}^{\ast }(x)\right]\hfill \\ {}\hfill & \kern1em +2\mathrm{e}{A}^{\mu }(x){\varphi}^{\ast }(x)\varphi (x),\hfill \end{array}} $$

(55)

which represents a conserved quantity for Eq. (53), in the sense that ${\partial}_{\mu }{j}^{\mu }(x)=0$.

1.2 Hamiltonian Formulation

In order to find a quantized theory for this interaction, we change to the Hamiltonian formalism of field theory. To simplify the calculations, we choose from the beginning the gauge

$$ \phi (x)\equiv \frac{1}{c}{A}^{(0)}(x)=0 $$

(56)

and

$$ \nabla \cdotp \boldsymbol{A}(x)=0 $$

(57)

since we are only interested in the transverse degrees of freedom. In this way, we avoid that the momentum density

$$ \pi (x)\equiv \partial \mathcal{L}/\partial \dot{\varphi}(x) $$

(58)

of the Klein–Gordon field is coupled to $\phi$. The fact that we recover the same results as if we would have retained $\phi$ and set it to zero at the end of the calculations [19] justifies this procedure.

With the help of Eq. (49), we obtain

$$ \pi (x)=\left({\hbar}^2/c\right){\dot{\varphi}}^{\ast }(x) $$

(59)

and thus arrive at the Hamiltonian density

$$ \mathcal{H}\equiv {\mathcal{H}}_{\mathrm{K}-\mathrm{G}}+{\mathcal{H}}_{\mathrm{M}}+{\mathcal{H}}_{\mathrm{Int}} $$

(60)

with

$$ {\displaystyle \begin{array}{cc}\hfill {\mathcal{H}}_{\mathrm{K}-\mathrm{G}}& \equiv c\kern0.277778em \left[\frac{1}{\hbar^2}{\pi}^{\ast }(x)\pi (x)+{\hbar}^2\nabla {\varphi}^{\ast }(x)\cdotp \nabla \varphi (x)\right.\hfill \\ {}\hfill & \kern1em \left.+\kern0.166667em {m}_0^2{c}^2{\varphi}^{\ast }(x)\varphi (x)\right],\hfill \end{array}} $$

(61)

for the free Klein–Gordon field,

$$ {\displaystyle \begin{array}{cc}\hfill {\mathcal{H}}_{\mathrm{Int}}& \equiv \mathrm{e}c{A}^{\mu }(x)\kern0.277778em \left[{\varphi}^{\ast }(x)i\hbar {\partial}_{\mu}\varphi (x)-\left(i\hbar {\partial}_{\mu }{\varphi}^{\ast }(x)\right)\varphi (x)\right]\hfill \\ {}\hfill & \kern1em -{\mathrm{e}}^2c{A}^2(x){\varphi}^{\ast }(x)\varphi (x)\hfill \end{array}} $$

(62)

for the interaction and ${\mathcal{H}}_{\mathrm{M}}$ as the Hamiltonian density [39] of the free Maxwell field.

1.3 Quantization in Schrödinger Picture

Next we perform second quantization in the Schrödinger picture where the Klein–Gordon field operator ${\widehat{\varphi}}_{\mathrm{S}}\left(\boldsymbol{r}\right)$ and its conjugate momentum ${\widehat{\pi}}_{\mathrm{S}}\left(\boldsymbol{r}\right)$ fulfill the commutation relations

$$ \left[{\widehat{\pi}}_{\mathrm{S}}\left(\boldsymbol{r}\right),{\widehat{\varphi}}_{\mathrm{S}}\left({\boldsymbol{r}}^{\prime}\right)\right]=\left[{\widehat{\pi}}_{\mathrm{S}}^{\dagger}\left(\boldsymbol{r}\right),{\widehat{\varphi}}_{\mathrm{S}}^{\dagger}\left({\boldsymbol{r}}^{\prime}\right)\right]=- i\mathit{\hbar}\updelta \left(\boldsymbol{r}-{\boldsymbol{r}}^{\prime}\right) $$

(63)

while all other possible combinations of commutators vanish.

We emphasize that the operators ${\widehat{\varphi}}_{\mathrm{S}}\left(\boldsymbol{r}\right)$ and ${\widehat{\pi}}_{\mathrm{S}}\left(\boldsymbol{r}\right)$ do not depend on time in contrast to their classical counterparts $\varphi (x)\equiv \varphi \left( ct,\boldsymbol{r}\right)$ and $\pi (x)\equiv \pi \left( ct,\boldsymbol{r}\right)$. In the Schrödinger picture, all dynamics is contained in the state vector $\mid {\varPsi}_{\mathrm{S}}(t)\Big\rangle$ for the electron and the electromagnetic field and follows from the Schrödinger equation

$$ i\mathit{\hbar}\frac{\mathrm{d}}{\mathrm{d}t}\mid {\varPsi}_{\mathrm{S}}(t)\left\rangle ={\widehat{H}}_{\mathrm{S}}\mid {\varPsi}_{\mathrm{S}}(t)\right\rangle . $$

(64)

Here we have replaced all fields and momenta by their operators in the Hamiltonian

$$ {\widehat{H}}_{\mathrm{S}}\equiv \int {\mathrm{d}}^3r\kern0.277778em {\widehat{\mathcal{H}}}_{\mathrm{S}}\left(\boldsymbol{r}\right). $$

(65)

The Furry picture is most convenient when the interaction with the electromagnetic field consists of two parts: one due to an external field which is treated as classical and fixed, which is its depletion is neglected, and one with a quantized field.

In our case, we identify these two components as the classical wiggler field ${A}_{\mathrm{W}}^{\mu }$ and the quantized laser field ${\widehat{A}}_{\mathrm{L}}^{\mu }$, that is,

$$ {\widehat{A}}^{\mu }(x)={\widehat{A}}_{\mathrm{L}}^{\mu}\left(\boldsymbol{r}\right)+{A}_{\mathrm{W}}^{\mu }(x). $$

(66)

Here we have fixed the dynamics of the wiggler field in the time-dependent potential ${A}_{\mathrm{W}}(x)\equiv {A}_{\mathrm{W}}\left( ct,\boldsymbol{r}\right)$, while the operator ${\widehat{A}}_{\mathrm{L}}\left(\boldsymbol{r}\right)$ is independent of time.

We can make a similar distinction of Eq. (66) for the Hamiltonian

$$ {\widehat{H}}_{\mathrm{S}}(t)\equiv {\widehat{H}}_{\mathrm{W}}(t)+{\widehat{H}}_{\mathrm{L}}\equiv \int {\mathrm{d}}^3r\kern0.277778em \left[{\widehat{\mathcal{H}}}_{\mathrm{W}}(t)+{\widehat{\mathcal{H}}}_{\mathrm{L}}\right] $$

(67)

with

$$ {\displaystyle \begin{array}{cc}\hfill {\widehat{\mathcal{H}}}_{\mathrm{W}}(t)& \equiv c\left[\frac{1}{\hbar^2}{\widehat{\pi}}_{\mathrm{S}}^{\dagger}\left(\boldsymbol{r}\right){\widehat{\pi}}_{\mathrm{S}}\left(\boldsymbol{r}\right)+{\hbar}^2\nabla {\widehat{\varphi}}_{\mathrm{S}}^{\dagger}\left(\boldsymbol{r}\right)\cdotp \nabla {\widehat{\varphi}}_{\mathrm{S}}\left(\boldsymbol{r}\right)\right.\hfill \\ {}\hfill & \kern1em +\left({m}_0^2{c}^2-{\mathrm{e}}^2{A}_{\mathrm{W}}^2(x)\right){\widehat{\varphi}}_{\mathrm{S}}^{\dagger}\left(\boldsymbol{r}\right){\widehat{\varphi}}_{\mathrm{S}}\left(\boldsymbol{r}\right)+{\widehat{\mathcal{H}}}_{\mathrm{M}}\left(\boldsymbol{r}\right)\hfill \\ {}\hfill & \kern1em +\mathrm{e}{A}_{\mathrm{W}}^{\mu }(x)\left({\widehat{\varphi}}_{\mathrm{S}}^{\dagger}\left(\boldsymbol{r}\right)i\hbar {\partial}_{\mu }{\widehat{\varphi}}_{\mathrm{S}}\left(\boldsymbol{r}\right)\left.-\left(i\hbar {\partial}_{\mu }{\widehat{\varphi}}_{\mathrm{S}}^{\dagger}\left(\boldsymbol{r}\right)\right){\widehat{\varphi}}_{\mathrm{S}}\left(\boldsymbol{r}\right)\right)\right]\hfill \end{array}} $$

(68)

and

$$ {\displaystyle \begin{array}{c}\hfill {\widehat{\mathcal{H}}}_{\mathrm{L}}=\mathrm{e}c{\widehat{A}}_{\mathrm{L}}^{\mu}\left(\boldsymbol{r}\right)\left[{\widehat{\varphi}}_{\mathrm{S}}^{\dagger}\left(\boldsymbol{r}\right)i\hbar {\partial}_{\mu }{\widehat{\varphi}}_{\mathrm{S}}\left(\boldsymbol{r}\right)-\left(i\hbar {\partial}_{\mu }{\widehat{\varphi}}_{\mathrm{S}}^{\dagger}\left(\boldsymbol{r}\right)\right){\widehat{\varphi}}_{\mathrm{S}}\left(\boldsymbol{r}\right)\right]\\ {}\hfill \kern1em -c\left[{\mathrm{e}}^2{\widehat{A}}_{\mathrm{L}}^2\left(\boldsymbol{r}\right)+2{e}^2{A}_{\mathrm{W}}(x)\cdotp {\widehat{A}}_{\mathrm{L}}\left(\boldsymbol{x}\right)\right]{\widehat{\varphi}}_{\mathrm{S}}^{\dagger}\left(\boldsymbol{r}\right){\widehat{\varphi}}_{\mathrm{S}}\left(\boldsymbol{r}\right).\end{array}} $$

(69)

From these expressions, we observe two important aspects: (1) Although we use the Schrödinger picture, the Hamiltonian ${\widehat{H}}_{\mathrm{W}}$ is explicitly time dependent. This feature originates from the fact that we have used for the wiggler a time-dependent classical field instead of a field operator in the Schrödinger picture. By prescribing the dynamics of the wiggler, we have already incorporated the free motion of this field mode.

(2) We realize that ${\widehat{H}}_{\mathrm{L}}$ still contains a term linear in ${A}_{\mathrm{W}}^{\mu }$. This linear contribution arises as the cross-term to the square ${A}^2$ in Eq. (62) of the sum in Eq. (66). Later on, we shall recognize that this term is crucial for the FEL.

1.4 Transformation to the Furry Picture

Similar to the transition into the familiar interaction picture, we obtain the Furry picture by transforming the original state $\mid {\varPsi}_{\mathrm{S}}(t)\Big\rangle$ by a unitary operator V(t) such that

$$ \mid {\varPsi}_{\mathrm{F}}(t)\left\rangle ={V}^{-1}(t)\mid {\varPsi}_{\mathrm{S}}(t)\right\rangle . $$

(70)

With the help of the Schrödinger equation, Eq. (64), the time derivative of the new state $\mid {\varPsi}_{\mathrm{F}}(t)\Big\rangle$ reads

$$ {\displaystyle \begin{array}{cc}\hfill i\mathit{\hbar}\frac{\mathrm{d}}{\mathrm{d}t}\mid {\varPsi}_{\mathrm{F}}(t)\Big\rangle & ={V}^{-1}(t)\left[-i\hbar \dot{V}(t){V}^{-1}(t)\right.\hfill \\ {}\hfill & \left.+\kern0.166667em \left({\widehat{H}}_{\mathrm{W}}(t)+{\widehat{H}}_{\mathrm{L}}\right)\right]V(t)\mid {\varPsi}_{\mathrm{F}}(t)\Big\rangle .\hfill \end{array}} $$

(71)

If we demand the Schrödinger-like equation

$$ i\mathit{\hbar}\dot{V}(t)={\widehat{H}}_{\mathrm{W}}(t)V(t) $$

(72)

for V(t), we arrive at the time evolution

$$ i\mathit{\hbar}\frac{\mathrm{d}}{\mathrm{d}t}\mid {\varPsi}_{\mathrm{F}}(t)\left\rangle ={\widehat{H}}_{\mathrm{F}}(t)\mid {\varPsi}_{\mathrm{F}}(t)\right\rangle $$

(73)

with

$$ {\widehat{H}}_{\mathrm{F}}(t)\equiv {V}^{-1}(t){\widehat{H}}_{\mathrm{L}}V(t) $$

(74)

for the state $\mid {\varPsi}_{\mathrm{F}}\Big\rangle$ which is focused on the interaction with the laser field. Indeed, the Hamiltonian ${\widehat{H}}_{\mathrm{F}}$ in this picture is a function of the transformed field operators

$$ {\widehat{\varphi}}_{\mathrm{F}}(x)\equiv {V}^{-1}(t){\widehat{\varphi}}_{\mathrm{S}}\left(\boldsymbol{r}\right)V(t), $$

(75)

and

$$ {\widehat{\pi}}_{\mathrm{F}}(x)\equiv {V}^{-1}(t){\widehat{\pi}}_{\mathrm{S}}\left(\boldsymbol{r}\right)V(t) $$

(76)

with

$$ {\widehat{A}}_{\mathrm{L}}^{\mu }(x)\equiv {V}^{-1}(t){\widehat{A}}_{\mathrm{L}}^{\mu}\left(\boldsymbol{r}\right)V(t). $$

(77)

When we examine Eq. (72), we notice the difficulties to specify the explicit form of the transformation matrix V for the Furry picture, since we are dealing with the time-dependent Hamiltonian ${\widehat{H}}_{\mathrm{W}}$. Indeed, we find the time-ordered exponential [36]

$$ V(t)=\mathcal{T}\exp \left[-\frac{i}{\hbar}\int \kern-0.33em \mathrm{d}t{\widehat{H}}_{\mathrm{W}}(t)\right] $$

(78)

for V(t), in contrast to the simple exponential in the ordinary Dirac picture.

Only the operator of the electromagnetic field can be written in the same form as in the familiar interaction picture, because the Hamiltonian of the Maxwell field commutes with the other contributions in ${\widehat{H}}_{\mathrm{W}}$.

Nevertheless, we can formulate the equation of motion

$$ \frac{\partial }{\partial t}{\widehat{\varphi}}_{\mathrm{F}}(x)=\frac{i}{\hbar }{V}^{-1}(t)\left[{\widehat{H}}_{\mathrm{W}}(t),{\widehat{\varphi}}_{\mathrm{S}}\left(\boldsymbol{r}\right)\right]V(t) $$

(79)

for the field operator of the Klein–Gordon field where we have used Eq. (72) and the fact that ${\widehat{\varphi}}_{\mathrm{S}}\left(\boldsymbol{r}\right)$ is independent of time in the Schrödinger picture.

The evaluation of the commutator in Eq. (79), with the help of Eqs. (63) and (68), yields

$$ \frac{\partial }{\partial t}{\widehat{\varphi}}_{\mathrm{F}}(x)=\frac{c}{\hbar^2}{\widehat{\pi}}_{\mathrm{F}}^{\dagger }(x). $$

(80)

In a similar way, we compute

$$ {\displaystyle \begin{array}{cc}\hfill \frac{\partial^2}{\partial {t}^2}{\widehat{\varphi}}_{\mathrm{F}}(x)& =\frac{c}{\hbar^2}\frac{\partial }{\partial t}{\widehat{\pi}}_{\mathrm{F}}^{\dagger }(x)\hfill \\ {}\hfill & =\frac{c}{\hbar^2}\frac{i}{\hbar }{V}^{-1}(t)\left[{\widehat{H}}_{\mathrm{W}}(t),{\widehat{\pi}}_{\mathrm{S}}^{\dagger}\left(\boldsymbol{r}\right)\right]V(t),\hfill \end{array}} $$

(81)

and observe that the field operator in the Furry picture fulfills the Klein–Gordon equation in the external wiggler field

$$ \left[{\left(i\hbar {\partial}^{\mu }-\mathrm{e}{A}_{\mathrm{W}}^{\mu }(x)\right)}^2-{m}_0^2{c}^2\right]{\widehat{\varphi}}_{\mathrm{F}}(x)=0. $$

(82)

Hence, we can expand the field operator in terms of solutions of this equation of motion. This procedure is analogous to the ordinary interaction picture, where the field operator is expanded in solutions of the free Klein–Gordon equation. The task is, of course, to find a solution of this external field problem, Eq. (82), which in case of a plane wave field is given by the Volkov solution [22], as discussed in Appendix B.

1.5 Hamiltonian in the Furry Picture

We conclude by noting that the interaction Hamiltonian defined by Eq. (74) can be written as

$$ {\widehat{H}}_{\mathrm{F}}=\mathrm{e}\int {\mathrm{d}}^3r\kern0.166667em {\widehat{j}}_{\mathrm{F}}(x)\cdotp {\widehat{A}}_{\mathrm{L}}(x) $$

(83)

where the electron current $\widehat{j}$ couples to the laser field ${\widehat{A}}_{\mathrm{L}}$. Here we have neglected the term proportional to ${\widehat{A}}_L^2(x)$ and have recalled from Eq. (55) that the current

$$ {\displaystyle \begin{array}{cc}\hfill {\widehat{j}}_{\mathrm{F}}^{\mu }(x)& \equiv c\left[{\widehat{\varphi}}_{\mathrm{F}}^{\dagger }(x)i\hbar {\partial}^{\mu }{\widehat{\varphi}}_{\mathrm{F}}(x)-\left(i\hbar {\partial}^{\mu }{\widehat{\varphi}}_{\mathrm{F}}^{\dagger }(x)\right){\widehat{\varphi}}_{\mathrm{F}}(x)\right]\hfill \\ {}\hfill & \kern1em +2\mathrm{e}{A}_{\mathrm{W}}^{\mu }(x){\widehat{\varphi}}_{\mathrm{F}}^{\dagger }(x){\widehat{\varphi}}_{\mathrm{F}}(x)\hfill \end{array}} $$

(84)

is conserved.

For the sake of clarity, we omit in the main body of this article the subscript F indicating the Furry picture. Since, apart from this appendix, no quantities outside the framework of the Furry picture are considered, this procedure seems justified.

Appendix B: Volkov Solution for Klein–Gordon Equation

The term ‘Volkov solution’ usually refers to a solution of the Dirac equation coupled to an external electromagnetic field in the form of a plane wave and was derived in 1935 by D. M. Volkov [22]. However, in the present context ‘Volkov solution’ refers to a solution of the corresponding problem of the Klein–Gordon equation. In this appendix, we rederive [19] the Volkov solution in the case of an optical undulator described by the potential

$$ {A}_{\mathrm{W}}^{\mu }={\tilde{\mathcal{A}}}_{\mathrm{W}}\left({\epsilon}^{\mu }{\mathrm{e}}^{-i{k}_{\mathrm{W}}\cdotp x}+\mathrm{c}.\mathrm{c}.\right) $$

(85)

given by Eq. (4).

In order to solve the equation

$$ \left[{\left(i\hbar {\partial}^{\mu }-\mathrm{e}{A}_{\mathrm{W}}^{\mu }(x)\right)}^2-{m}_0^2{c}^2\right]\varphi (x)=0 $$

(86)

we note that ${A}_{\mathrm{W}}^{\mu }$ depends only on the phase $\xi \equiv {k}_{\mathrm{W}}\cdotp x$ and assume that the free field solution is only modified by a function $f=f\left(\xi \right)$. This assumption gives rise to the ansatz

$$ \varphi (x)=\mathcal{N}\kern0.277778em f\left(\xi \right)\kern0.277778em {\mathrm{e}}^{-i{p}_{\mathrm{f}}\cdotp x/\hbar }. $$

(87)

Here ${p}_{\mathrm{f}}^{\mu }$ describes the free four-momentum of the electron and fulfills the relation ${p}_{\mathrm{f}}\cdotp {p}_{\mathrm{f}}={m}_0^2{c}^2$, while $\mathcal{N}$ is a normalization constant.

In addition, we demand for a one-dimensional theory that before the electron enters, the wiggler its momentum is parallel to the propagation direction of the field, corresponding to the relation

$$ {p}_{\mathrm{f}}\cdotp {A}_{\mathrm{W}}^{\mu }(x)={p}_{\mathrm{f}}\cdotp {\widehat{A}}_{\mathrm{L}}^{\mu }(x)=0. $$

(88)

Since the potentials are transverse, all scalar products between the four-wave vectors and the four-potentials vanish, and in particular, we have

$$ {k}_{\mathrm{W}}\cdotp {A}_{\mathrm{W}}=0 $$

(89)

for the wiggler.

Under these assumptions, we insert the ansatz Eq. (87) into Eq. (86) and obtain the first-order differential equation

$$ {f}^{\prime}\left(\xi \right)=-\frac{i}{\hbar}\frac{{\mathrm{e}}^2{\tilde{\mathcal{A}}}_{\mathrm{W}}^2}{p_{\mathrm{f}}\cdotp {k}_{\mathrm{W}}}\kern0.277778em f\left(\xi \right) $$

(90)

for f with ${f}^{\prime}\left(\xi \right)\equiv \mathrm{d}f\left(\xi \right)/\mathrm{d}\xi$. Here we have made use of the identity ${A}_{\mathrm{W}}^2(x)=-2{\tilde{\mathcal{A}}}_{\mathrm{W}}^2$ valid for circular polarization.

Crucial for the change from a second-order differential equation, Eq. (86), to a first-order one, Eq. (90), is the fact that the only term with a second derivative, that is, the contribution $-{\hbar}^2{k}_{\mathrm{W}}\cdotp {k}_{\mathrm{W}}\kern0.277778em {f}^{{\prime\prime}}\left(\xi \right)$ vanishes due to the dispersion relation ${k}_{\mathrm{W}}\cdotp {k}_{\mathrm{W}}=0$ for the wiggler field.

Although the dispersion relation for the four-wave vector is correct only in the case of a laser wiggler the second derivative can still be neglected [19, 20] in the magnetostatic case. Indeed, the corrections due to the second derivative scale with ${a}_0^2/{\gamma}_0^2$, which is negligible for all reasonable values of the field strength of a wiggler.

The solution of the differential equation, Eq. (90), reads

$$ f\left(\xi \right)=\exp \left[-\frac{i}{\hbar}\frac{{\mathrm{e}}^2{\tilde{\mathcal{A}}}_{\mathrm{W}}^2}{p_{\mathrm{f}}\cdotp {k}_{\mathrm{W}}}\xi \right] $$

(91)

where we have paid attention to the constraint that f should reduce to $f\left(\xi \right)=1$ for a vanishing wiggler field, that is, for ${A}_{\mathrm{W}}^{\mu }(x)=0$.

When we combine all terms we obtain the Volkov solution [19]

$$ {\varphi}_p(x)=\frac{1}{\sqrt{2{p}_0V}}\kern0.277778em {\mathrm{e}}^{- ip\cdotp x/\hbar }, $$

(92)

where we have defined the effective momentum

$$ {p}^{\mu }={p}_{\mathrm{f}}^{\mu }+\frac{{\mathrm{e}}^2{\tilde{\mathcal{A}}}_{\mathrm{W}}^2}{p_{\mathrm{f}}\cdotp {k}_{\mathrm{W}}}\kern0.277778em {k}_{\mathrm{W}}^{\mu } $$

(93)

of the electron in the wiggler field which fulfills the modified energy–momentum relation

$$ p\cdotp p={p}_0^2-{\boldsymbol{p}}^2={m}_0^2{c}^2+2\kern0.277778em {\mathrm{e}}^2{\tilde{\mathcal{A}}}^2\equiv {m}^2{c}^2 $$

(94)

with the effective mass m.

The normalization constant $\mathcal{N}$ follows from the condition that the integral of the density, i.e., zeroth component of the four current, Eq. (55), divided by c, over the quantization volume V yields unity, that is,

$$ {\int}_V\kern-0.33em {\mathrm{d}}^3r\left[{\varphi}_p^{\ast }(x)\frac{i\mathit{\hbar}}{c}\frac{\partial }{\partial t}{\varphi}_p(x)-{\varphi}_p(x)\frac{i\mathit{\hbar}}{c}\frac{\partial }{\partial t}{\varphi}_p^{\ast }(x)\right]=1 $$

(95)

In conclusion, we observe that, similar to the free field case, the solution of Eq. (86) is a plane wave where the free momentum ${p}_{\mathrm{f}}$ is replaced by the effective momentum p which takes the effect of the wiggler field into account. Due to the linearity of the problem any other solution of Eq. (86) can be expanded in plane waves of this kind.

Appendix C: Two-Photon Processes

In this appendix, we show that higher-order photon processes are suppressed. For this purpose, we first analyze the influence of two-photon transitions within our perturbative approach and demonstrate that they scale with ${\alpha}^2$. We conclude by briefly analyzing the general case.

The second-order terms in the expression Eq. (34) for the time evolution operator lead to the evolved state

$$ \mid {\varPsi}^{(2)}(t)\left\rangle \equiv \underset{0}{\overset{t}{\int \limits }}\kern-0.33em \mathrm{d}{t}_2\underset{0}{\overset{t_2}{\int \limits }}\kern-0.33em \mathrm{d}{t}_1\widehat{H}\left({t}_2\right)\widehat{H}\left({t}_1\right)\mid n,{p}_z\right\rangle . $$

(96)

for the initial state $\mid n,{p}_z\Big\rangle$. Apart from the contributions with ${\widehat{\mathcal{H}}}_{\mathrm{em}}\left({t}_2\right){\widehat{\mathcal{H}}}_{\mathrm{em}}\left({t}_1\right)$ representing two-photon emission, and with ${\widehat{\mathcal{H}}}_{\mathrm{abs}}\left({t}_2\right){\widehat{\mathcal{H}}}_{\mathrm{abs}}\left({t}_1\right)$ representing two-photon absorption, there also emerge cross-terms with ${\widehat{\mathcal{H}}}_{\mathrm{abs}}\left({t}_2\right){\widehat{\mathcal{H}}}_{\mathrm{em}}\left({t}_1\right)$ and ${\widehat{\mathcal{H}}}_{\mathrm{em}}\left({t}_2\right){\widehat{\mathcal{H}}}_{\mathrm{abs}}\left({t}_1\right)$ describing processes where the number of photons remains unchanged. We omit the latter ones and discuss only the two-photon transitions.

From a calculation analogous to the single-photon case we obtain the probability density

$$ {\left|{c}_2\right|}^2={\left(\frac{gt}{\gamma}\right)}^4\left(n+1\right)\left(n+2\right)\kern0.277778em {\mathcal{E}}^{\left(+2\right)}{\left|\psi \left({p}_z\right)\right|}^2 $$

(97)

for two-photon emission, and the probability density

$$ {\left|{c}_{-2}\right|}^2={\left(\frac{gt}{\gamma}\right)}^4n\left(n-1\right)\kern0.277778em {\mathcal{E}}^{\left(-2\right)}{\left|\psi \left({p}_z\right)\right|}^2 $$

(98)

for two-photon absorption. Both expressions are weighted with the initial momentum distribution ${\left|\psi \left({p}_z\right)\right|}^2$.

Moreover, we have defined the selectivity functions

$$ {\displaystyle \begin{array}{cc}\hfill {\mathcal{E}}^{\left(\pm 2\right)}& \equiv \frac{1}{{\left(2{\phi}_{\pm 1}\right)}^2}\left[{\operatorname{sinc}}^2\left(2{\phi}_{\pm 2}\right)+{\operatorname{sinc}}^2{\phi}_{\pm 3}\right.\hfill \\ {}\hfill & \kern1em \left.-\kern0.166667em 2\cos {\phi}_{\pm 1}\operatorname{sinc}\left(2{\phi}_{\pm 2}\right)\operatorname{sinc}{\phi}_{\pm 3}\right]\hfill \end{array}} $$

(99)

for these processes with the phases

$$ {\phi}_{\pm j}\equiv \frac{\left({k}_{\mathrm{L}}+{k}_{\mathrm{W}}\right) ct}{2}\left(\beta -{\beta}_{\mathrm{BR}}\mp j\frac{\mathcal{Q}}{2}\right). $$

(100)

As shown in Fig. 2, the dominant maxima of ${\mathcal{E}}^{\left(\pm 2\right)}$ are located at $\beta ={\beta}_{\mathrm{BR}}\pm \mathcal{Q}$ as expected from energy–momentum conservation, Eq. (31). However, there are also less pronounced maxima separated from these major ones by $\mathcal{Q}/2$.

Fig. 2

Selectivity functions ${\mathcal{E}}^{\left(+2\right)}$ (red curve) or ${\mathcal{E}}^{\left(-2\right)}$ (blue curve) for two-photon emission or absorption, defined by Eq. (99), in the quantum regime for ${\omega}_{\mathrm{r}}t=5$, as functions of the scaled velocity $\beta$ of the electron. Although minor maxima exist, the dominant ones are located at $\beta ={\beta}_{\mathrm{BR}}\pm \mathcal{Q}$ as predicted by energy–momentum conservation

The probability for two-photon emission, Eq. (97), is maximized for $\beta -{\beta}_{\mathrm{BR}}=\mathcal{Q}$ where we can estimate

$$ {\mathcal{E}}^{\left(+2\right)}\cong \frac{1}{{\left(2{\omega}_{\mathrm{r}}t\right)}^2}. $$

(101)

Thus, the probability for this process scales as

$$ {\left|{c}_2\right|}^2\sim {\left(\frac{gt}{\gamma}\right)}^2\left(n+2\right)\frac{\alpha^2}{4}, $$

(102)

where we have used the definitions Eqs. (44) and Eq. (45) of the recoil frequency and the quantum parameter $\alpha$, respectively.

On the other hand, the maximum probability for single-photon emission, Eq. (38), scales with

$$ {\left|{c}_1\right|}^2\sim {\left(\frac{gt}{\gamma}\right)}^2\left(n+1\right) $$

(103)

and we arrive at the ratio

$$ \frac{{\left|{c}_2\right|}^2}{{\left|{c}_1\right|}^2}\sim \frac{\alpha^2}{4}, $$

(104)

which we have also obtained in Ref. [12] in an analogous way.

This relation demonstrates that in a Quantum FEL defined by $\alpha \ll 1$, two-photon emission is strongly suppressed in comparison with the single-photon transition. Since processes involving more than two photons occur only at even higher orders of perturbation theory, we argue that these processes are suppressed with corresponding powers of $\alpha$.