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.
1. Introduction
Light scattering in matter has been well studied in the past [1–6]. From the classical viewpoint, the scattering of light in matter occurs due to the accelerated motion of an atomic electron in an electromagnetic field. Thus, the energy of the incident wave is partially converted into the energy of the scattered wave, i.e. scattering occurs. From the quantum-mechanical viewpoint, light scattering processes (both elastic and inelastic scattering) are usually explained using the first or next order of perturbation theory [2,3,5,7–9]. The use of perturbation theory is fully justified, because even with a very intense electromagnetic field, the probability of light scattering on an atomic electron is not a large value. Usually, the use of the first order of the perturbation theory gives a significant predominance over its following orders, which in most cases leads to the dominance of scattering as described by classical physics. The so-called semiclassical approach is also used to describe light scattering [8,9] - this is when the atomic system is considered as quantum, and the electromagnetic field is classical. For example, the application of the first order of perturbation theory or semiclassical dipole scattering leads to a linear dependence of the scattering probability $w$ on the intensity of scattered light $P \sim E^2$ ($E$ is the electromagnetic field strength), i.e. $w \sim P$. Despite this, currently, research is mainly focused on light scattering in superpowerful electromagnetic fields, where nonlinear scattering processes occur on atomic, molecular, or free electrons [10–16], as well as the nonlinear interaction of light with matter [17–19].
Currently, with the advent of powerful ultrashort electromagnetic field sources, the problem of studying the scattering of such pulses in matter [20] is s particularly relevant. Indeed, there is a tendency to increase the power of ultrashort pulses (USP) of the electromagnetic field and to reduce their duration [21–24]. The scattering of such ultrashort electromagnetic fields is used not only to study the structure of biomolecules, nanosystems and complex polyatomic structures, but also to study various dynamic processes [25,26] and generate higher harmonics [22,23]. In the X-ray frequency range, the generation of USP is usually carried out using a free-electron laser, for example, XFEL (X-ray Free Electron Laser), and an XFEL pulse duration of less than 1 fs has already been achieved. It should be added that the development of the technique of generating USP up to the attosecond range has opened a new area of science - attosecond physics [23]. At present, it is safe to say that such pulses are generated and can be used in spectroscopic studies of matter. Theoretical methods of research in the scattering of such pulses in a substance are similar to traditional ones, i.e. This is the first and next order of perturbation theory for calculating the probability of photon production or the semiclassical approach (see, e.g., [27–34]). In addition, studies are beginning to appear where scattering (including the generation of higher harmonics) and the interaction of an electromagnetic field with a charged particle are described without using perturbation theory and semiclassical approaches [35–41]. Despite this, in most cases such approaches are based on the dipole approximation, which does not work in the X-ray frequency range and is not adapted to the use of USP. Thus, there is a need to develop the quantum theory of scattering of USP of an electromagnetic field, which operates in a wide spectral range of frequencies, including X-ray.
There is currently no USP scattering theory that does not take into account perturbation theory and the semiclassical approach. It is obvious that such a theory should be based on nonperturbative approaches with the involvement of the quantum nature of scattered light. In this paper, within the framework of quantum mechanical consideration, the theory of multiphoton scattering of USP is developed. It is shown that for a certain number of atoms in a system, multi-photon scattering can occur, which cannot be described in the framework of the traditional scattering approach. The probability $P_n$ of the production of $n$ photons and their average energy $E_n$ is found in an analytical form. For a macroscopic system and a system consisting of a smaller number of atoms, the resulting equations can be ignored and traditional approaches can be used. Since, as shown in the “Limit Cases” section, for single-photon scattering (when the number of atoms in the system is small and 1 photon is generated) and scattering on a macroscopic system (when the number of photons is infinitely many), the expressions obtained will go over to previously known approaches. The most important applied result is the application of this theory to X-ray diffraction analysis of a substance. Here, a previously unexplored area of physics is explored, that is, the case of a substance consisting of a certain number of atoms, in which multiphoton scattering of USP occurs. It is shown that such a substance can be various biomolecules, nanosystems and polyatomic structures. The fact that such structures are currently investigated using high-power ultrashort laser pulses makes it essential that the results reported here are taken into account when applied to spectroscopic studies (X-ray diffraction).
In physics, there are a lot of theoretical approaches in explaining various collective phenomena. The scattering of USP, on polyatomic structures, distinct from individual atoms, can also be attributed to phenomena that are currently little studied. The need to account for collective phenomena in scattering is well known. For example, this is a refinement of the theory of bremsstrahlung Bethe-Heitler [42] by the effect of Landau-Pomeranchuk-Migdal [43,44]. This effect takes into account the reduction of the bremsstrahlung cross sections at high energies or high densities of matter. The same applies to very intense colliding electrons, where the transmission of the transverse momentum is small, and interference can occur in transverse directions.
2. The model and its solution
Consider the Schrödinger equation for a polyatomic system, which is affected by an ultrashort (classical) electromagnetic field pulse and a quantized electromagnetic field of a scattered pulse. Let ’s present the electromagnetic field through the transverse vector potential $\textbf{A}$, in the Coulomb gauge $div \textbf{A}=0$ [8], see also [3,9] (further, the atomic system of units will be used: $\hbar = 1, | e | =1, m_e = 1$, where $\hbar$ is Dirac’s constant, $e$ is the electron charge, $m_e$ is the electron mass)
(1)$$\left\lbrace {\hat H}_{\gamma}+\frac{1}{2}\sum_a\left({\hat{\textbf p}}_a+\frac{1}{c}\hat{\textbf A}_a\right)^2 + \sum_a U(\textbf{r}_a)\right\rbrace \Psi = i\frac{\partial \Psi}{\partial t}.$$
In Eq. (1) ${\hat H}_{\gamma }$ is the Hamilton operator for a scattered field; $U(\textbf{r}_a)$ is the atomic potential acting on the electron with the number $a$; ${\hat{\textbf p}} _ a$ is the electron momentum operator with the number $a$; $\hat{\textbf A}_a = \hat{\textbf A}_{1,a} + \textbf{A}_{2,a}$, where $\hat{\textbf A}_{1, a}$ is the vector potential of the quantized scattered field; $\textbf{A}_{2,a}$ is the vector potential of the external classical field, summation $\sum _a$ in (1) is carried out on all electrons in the system under consideration.The solution of Eq. (1) will be sought in the form $\Psi = \psi (\textbf{r}, t) \Phi (\{ Q \}, \textbf{r}, t)$, where $\{ Q \}$ set of all modes of field variables. It should be added that such a representation of the solution of Eq. (1), in the form of the product of two functions $\psi (\textbf{r}, t) \Phi (\{ Q \}, \textbf{r}, t)$, is not an approximation, but is a convenient record of the desired solution. The approximation will be further used in the search for the function $\Phi (\{ Q \}, \textbf{r}, t)$. Next, we take into account the fact that the evolution of an electron is determined by the external classical electromagnetic field and the field $U(\textbf{r}_a)$, which is indeed reasonable given the weak interaction of the scattered field (powerful field USP) with the electron. With this in mind, we can distinguish two equations in Eq. (1): the first is responsible for the evolution of an electron in an external classical electromagnetic field and the field $U(\textbf{r}_a)$ (whose wave function will be $\psi$); the second is responsible for the field of the scattered USP (whose wave function will be $\Phi$). Such an approximation we will call the “approximation of a determined evolution” or abbreviated DE approximation. Since we do not take into account in this approximation the evolution of an electron in the field of a scattered photon, we should neglect the members of ${\nabla} \Phi$. This is fully justied, since the entire evolution of an electron in the USP eld is determined by the wave function $\psi$, and the coordinate component ${\textbf r}$ enters the wave function $\Phi$ as a parameter. Moreover, as will be shown in the section “Limit cases”, the results obtained in this approximation will be similar to most known scattering theories in the limiting case. As a result, the following system of differential equations requires solution
(2)$$\left\lbrace {\hat H}_{\gamma}+\sum_a\frac{\hat{\textbf A}_{1,a}}{2c}\left({\hat{\textbf p}}_a+\frac{1}{c}\textbf{A}_{2,a} \right)+\sum_a\left({\hat{\textbf p}}_a+\frac{1}{c}\textbf{A}_{2,a} \right)\frac{\hat{\textbf A}_{1,a}}{2c} + \sum_a\frac{(\hat{\textbf A}_{1,a})^2}{2c^2} \right\rbrace \Phi \psi = i\frac{\partial \Phi }{\partial t}\psi,$$
where $\psi$ satisfies equation (3)$$\left\lbrace \frac{1}{2}\sum_a\left({\hat{\textbf p}}_a+\frac{1}{c}\textbf{A}_{2,a}\right)^2 + \sum_a U(\textbf{r}_a)\right\rbrace \psi= i\frac{\partial \psi }{\partial t} .$$
We first consider the differential Eq. (3) and look for a solution in the form $\psi =exp\left ( - \frac {i}{c}\sum _a\textbf{A}_{2,a}\textbf{r}_a\right ) \psi _{L}$. We will consider the general case, not the dipole approximation, then $\textbf{A}_{2}=\textbf{A}_{2,a}(\eta )$, where $\eta =t-\textbf{n}_{0}\textbf{r}_a/c$, where $\textbf{n}_{0}$ is a unit vector directed along the propagation of a USP. It noted be said that the choice of $\psi$ in this form corresponds to the gauge transformation, where the gauge function is $f = - \sum _a \textbf{A}_{2,a}\textbf{r}_a$. In summary, it can be shown that (4)$$\left\lbrace \frac{1}{2} {\sum_a\hat{\textbf p}_a}^2+\sum_a\frac{\textbf{A}^{'}_{2,a}}{c}\hat{\textbf p}_a+\sum_a\hat{\textbf p}_a\frac{\textbf{A}^{'}_{2,a}}{2c} + \sum_a \frac{(\textbf{A}^{'}_{2,a})^2}{2c^2} - \sum_a \phi_a^{'}+\sum_a U(\textbf{r}_a) \right\rbrace \psi_{L} = i\frac{\partial \psi_{L} }{\partial t},$$
where $\textbf{A}^{'}_{2,a}=\textbf{A}_{2,a}+{\nabla}_a f(\textbf{r}_a,t)$, $\phi ^{'}_a=-\frac {1}{c}\frac {\partial f(\textbf{r}_a,t)}{\partial t}$, and it is easy to get $\textbf{A}^{'}_{2,a}=-\textbf{n}_{0}\textbf{E}\textbf{r}_a$, ${\phi }^{'}_a=-\textbf{E}\textbf{r}_a$ ($\textbf{n}_{0}=\frac {\textbf{k}_{0}}{k_{0}}$) (see, e.g., [33,34,45,46]), where $\textbf{E} = \textbf{E}(\eta )$ is the electromagnetic field strength. The second to fourth terms of Eq. (4) make a negligible contribution to the evolution of an electron in an USP field, since they are less than ${\phi }^{'}$ by approximately $c = 137$ times (of course, if we assume that an electron in an atom does not acquire a relativistic impulse under the action of a USP). Therefore, almost without loss of accuracy, it is simpler to consider equation (5)$$\left\lbrace \frac{1}{2} {\sum_a \hat{\textbf p}_a}^2-\sum_a \phi^{'}_a+\sum_a U(\textbf{r}_a) \right\rbrace \psi_{L} = i\frac{\partial \psi_{L} }{\partial t}.$$
It is known that the solution of Eq. (5) and even more so (4) can be found only by numerical calculation, although not for all parameters of the USP. We write, as a complement, the wave function in the initial gauge, i.e. when solving Eq. (3) (we denote it as $\psi _{v}$). (6)$$\psi_{v}= exp\left\lbrace -\frac{i}{c}\sum_{a}\textbf{A}_{2,a}\textbf{r}_{a} \right\rbrace \psi_L .$$
In what follows, we will restrict ourselves to the use of the wave function $\psi _L$, because it only includes the physical quantity of the USP, i.e. $\textbf{E}$. It should be noted that the use of the wave function $\psi _{v}$ will not affect the final result for calculating the main scattering characteristics. This can be shown analytically, similarly to our conclusions on the wave function $\psi _{L}$, (see Appendix E). In other words, gauge invariance will indeed work despite the separation of the original Eq. (1) into a system consisting of two differential Eqs. (2) and (3).We next proceed to the consideration of Eq. (2). If we remove the negligible last term in Eq. (2), we obtain
(7)$$\left\lbrace {\hat H}_{\gamma}+\sum_{a} \frac{\hat{\textbf A}_{1,a}}{c}\left(\psi^{{-}1}_{L}{\hat{\textbf p}}_{a}\psi_{L}+\frac{1}{c}\textbf{A}^{'}_{2,a} \right) \right\rbrace \Phi = i\frac{\partial \Phi}{\partial t}.$$
Of course, the resulting Eq. (7) contains an unknown value $\psi _L$, the search for which constitutes the main task of laser physics. In our case, it turned out that by solving Eq. (7) we reduce the problem of scattering of USP to the main problem of laser physics. Next, we turn to the solution of Eq. (7). It is known that in quantizing the electromagnetic field, in this case, the vector potential [3,8,9] can be represented as (8)$$\hat{\textbf A}_{1,a}=\sum_{k_{\sigma}}c\left(\frac{2\pi}{\omega V} \right)^{1/2}\textbf{u}_{k_{\sigma}}\left({\hat a}^{+}_{k_{\sigma}}exp\left({-}i\textbf{k}\textbf{r}_a\right)+ {\hat a}_{k_{\sigma}}exp\left(i\textbf{k}\textbf{r}_a\right) \right) .$$
In Eq. (8) ${\hat a}^{+}_{k_{\sigma }},{\hat a}_{k_{\sigma }}$ are respectively the operators of the creation and destruction of a photon with frequency $\omega$, impulse ${\textbf k}$ with polarization $\sigma$ ($\sigma =1,2$) and $\textbf{u}_{k_{\sigma }}$ are unit polarization vectors (basis of real orthogonal unit polarization vectors). Summation over $k_{\sigma }$ in Eq. (8) includes all possible states of ${\textbf k}$ and polarization $\sigma$ ($\sigma$ has two values), $V$ is the quantization volume of the electromagnetic field. The problem in question is simpler to solve if using the electromagnetic field variables $Q$ and $\hat {P}=-i\frac {\partial }{\partial Q}$ [3,8,9]. In this case, we get (9)$$\hat{\textbf A}_{1,a}=\sum_{k_{\sigma}}\sqrt{\frac{4\pi c^2}{V \omega}}\textbf{u}_{k_{\sigma}}\left( Q_{k_{\sigma}}\cos(\textbf{k} \textbf{r}_a)- \hat{P}_{k_{\sigma}}\sin(\textbf{k} \textbf{r}_a) \right) .$$
In addition, the value of ${\hat H}_{\gamma }$ is given as (10)$${\hat H}_{\gamma}=\frac{1}{2}\sum_{k_{\sigma}}\omega\left({\hat{P}}^2_{k_{\sigma}}+Q^2_{k_{\sigma}}\right) .$$
The solution to differential Eq. (7) is easiest to find using the “interaction representation” (e.g., [2,8]) in which the solution is sought in the form $|\Phi (t)\rangle =e^{-i{\hat H}_{\gamma } t}\hat {S}(t)|\Phi _0\rangle$, where $\hat {S} (t)$ is the evolution operator, and $| \Phi _0 \rangle$ is the ground state of the stray field (the Fock vacuum state). To calculate the necessary probabilities, the found wave function $| \Phi (t) \rangle$ must be decomposed into eigenvalues of a noninteracting system, i.e. (11)$$|\Phi(t)\rangle=\prod_{k_{\sigma}}\sum_{n_{k_{\sigma}}}C_{n_{k_{\sigma}}}\Phi_{n_{k_{\sigma}}}(Q_{k_{\sigma}})e^{{-}i E_{n_{k_{\sigma}}}t},$$
where $\prod _{k_{\sigma }}$ is the product over all possible states ${\textbf k}$ and polarizations $\sigma$, $C_{n_{k_{\sigma }}}$ are expansion coefficients, $\Phi _{n_{k_{\sigma }}}(Q_{k_{\sigma }})$ is an eigenfunction of the noninteracting system and $E_{n_{k_{\sigma }}}=\omega (n_{k_{\sigma }}+1/2)$ are eigenvalues of energy. The probability of detecting $n_{k_{\sigma }}$ of scattered photons from the field state $|0_{k_{\sigma }}\rangle$ when the electrons of the system under consideration move from the state $|0 \rangle$ to the state $| m \rangle$ will be $W^{0_{k_{\sigma }},0}_{n_{k_{\sigma }},m}=|C^{0_{k_{\sigma }},0}_{n_{k_{\sigma }},m}|^2$, where (12)$$C^{0_{k_{\sigma}},0}_{n_{k_{\sigma}},m}=\langle m, n_{k_{\sigma}}| \hat{S}(t=\infty)|0_{k_{\sigma}},\psi_{L} (t=\infty)\rangle .$$
Of interest is the total probability of scattering, regardless of the state of electrons in the target system. This requires the probability $W^{0_{k_{\sigma }},0}_{n_{k_{\sigma }},m}$ to sum over all electron states of the target $W^{0_{k_{\sigma }},0}_{n_{k_{\sigma }},All} =\sum _m W^{0_{k_{\sigma }},0}_{n_{k_{\sigma }},m}$. Using the condition of completeness, we obtain the probability of this process (13)$$W^{0_{k_{\sigma}},0}_{n_{k_{\sigma}},All}=\langle \psi_{L} (t=\infty) |\langle n_{k_{\sigma}}| \hat{S}(t=\infty)|0_{k_{\sigma}}\rangle |^2 \psi_{L} (t=\infty)\rangle .$$
For convenience, we rename $W^{0_{k_{\sigma }},0}_{n_{k_{\sigma }},All}=P_{n_{k_{\sigma }}}$, and $\psi _{L}(t=\infty )=\psi$. Using the “interaction representation”, the exact solution can be found (Appendix A), which has the form (14)$$P_{n_{k_{\sigma}}}=\left\langle \psi \left| \frac{|X|^{2n_{k_{\sigma}}}}{n_{k_{\sigma}}!}e^{-|X|^2}\right|\psi \right\rangle ,$$
where (15)$$X=\sum_a \sqrt{\frac{2\pi}{V \omega}} \int^{\infty}_{-\infty} e^{i(\omega t - \textbf{k}\textbf{r}_a)} \psi^{{-}1}_{L}(t,\textbf{r}_a) \textbf{u}_{k_{\sigma}} \left( \hat{\textbf p}_a + \frac{\textbf{A}^{'}_{2,a}}{c} \right) \psi_{L}(t,\textbf{r}_a) dt .$$
It should be added that $\psi ^{-1}_L (t,\textbf{r})$ may have singularity when $\psi _L(t,\textbf{r}) \to 0$. This singularity is eliminable, since $\psi _L (t,\textbf{r})$ is in the numerator of Eq. (15). If the singularity is not eliminated in the expression $\psi ^{-1}_L (t,\textbf{r}) \hat{\textbf p}_a \psi _{L}(t,\textbf{r}_a)$, then it is eliminated in the final equation for the probability $P$. Indeed, it can be seen from Eq. (14) that for large values of $X \sim \psi ^{-1}_L (t,\textbf{r})\hat{\textbf p}_a \psi _{L}(t,\textbf{r}_a)$ (this is possible for some $\textbf{r}_a$), the probability $P$ will be a finite quantity. You can also see that Eq. (14) resemble Poisson statistics, although it should be clarified that after averaging over electronic states, the statistics will be different (except for the case when $X \neq X(\textbf{r}_a)$).Using the well-known multinominal theorem [47], the total probability $P_n$ of the production of $n$ photons in a given frequency range and with averaged polarization can be found
(16)$$P_n=\left\langle \psi \left| \frac{Z^{n}}{n!}e^{{-}Z}\right| \psi \right\rangle ,$$
where (17)$$Z=\frac{1}{(2\pi)^2c^3}\int \omega \left| \sum_a \int^{\infty}_{-\infty} e^{i(\omega t - \textbf{k}\textbf{r}_a)}[\textbf{n}\times\textbf{P}(t,\textbf{r}_a)]dt \right|^2 d\Omega d\omega ,$$
where $\textbf{P}(t,\textbf{r}_a)=\psi ^{-1}_{L}(t,\textbf{r}_a) ( \hat{\textbf p}_a + \frac {\textbf{A}^{'}_{2,a}}{c} ) \psi _{L}(t,\textbf{r}_a)$ and $\textbf{n}=\textbf{k}/k$ is the direction of the scattered USP. Integration over frequency and solid angle is undertaken, depending on which frequency interval and solid angle we are interested in when calculating the probability $P_n$ in a given interval. The results of the calculation of Eq. (16) for $P_n$ are presented in Appendix B. In addition to the probability $P_n$ of the production of $n$ photons, an equally important characteristic of scattering is the average energy $E_n$ of the generated $n$ photons arising from the scattering of USP by the system under consideration. The exact equation for the energy $E_n$ (Appendix C) can be obtained in the form (18)$$E_n=\left\langle \psi \left| Z_{\omega} \frac{Z^{n-1}}{(n-1)!}e^{{-}Z}\right| \psi \right\rangle ,$$
where (19)$$Z_{\omega}=\frac{1}{(2\pi)^2c^3}\int \omega^2 \left| \sum_a \int^{\infty}_{-\infty} e^{i(\omega t - \textbf{k}\textbf{r}_a)}[\textbf{n}\times\textbf{P}(t,\textbf{r}_a)]dt \right|^2 d\Omega d\omega .$$
Eq. (18) defines the average scattering energy of $n$ photons in all directions and frequencies, and can be presented in a more convenient form (differential characteristic, see Appendix C) (20)$$\frac{d^2 E_n}{d\Omega d\omega}=\left\langle \psi \left| Z_{D} \frac{Z^{n-1}}{(n-1)!}e^{{-}Z}\right| \psi \right\rangle ,$$
where (21)$$Z_{D}=\frac{1}{(2\pi)^2c^3}\omega^2 \left| \sum_a \int^{\infty}_{-\infty} e^{i(\omega t - \textbf{k}\textbf{r}_a)}[\textbf{n}\times\textbf{P}(t,\textbf{r}_a)] dt\right|^2 .$$
It should be added that at first glance, the resulting Eqs. (16), (18) and (20) may have a singularity when in a certain neighborhood of the coordinate values $\psi$ will tend to zero $\psi \to 0$, i.e. and $\psi ^{-1} \to \infty$. There are two options why this singularity is not problematic when integrated. The first case is when ${\hat{\textbf p}} \psi$ also tends to zero and the expression for ${\textbf P}$ is finite. The second option is when ${\textbf P}$ tends to infinity ${\textbf P} \to \infty$ in the neighborhood of the singularity. In the second variant, the Eqs. (16), (18) and (20) in the vicinity of this singularity will tend to zero, which leads to the elimination of the divergence.3. Limit cases
As the basic Eqs. (16), (18), (20) have no restrictions on the number of photons $n$, it is interesting to consider the limit cases $n \gg 1$ and $n \ll 1$. Indeed, based on the obtained equations, there is a statistical distribution of the number of photons produced and it is interesting to find and analyze their limiting cases.
For further analysis, the average number of scattered photons $\bar {n} = \sum _n n P_n$ and the average value of the scattered energy $\bar {E} = \sum _{n} E_n$ are required, and are found through the following
(22)$$\bar{n}=\langle \psi | Z |\psi \rangle ,~~ \bar{E}=\langle \psi | Z_{\omega} |\psi \rangle .$$
We now consider two cases: 1. when the number of atoms in the system under consideration is so large (the transition to the case of a macroscopic system) that $\bar {n} \gg 1$ (corresponds to $n \gg 1$),
2. when the number of atoms is so small that $\bar {n} \ll 1$ (corresponds to $n \ll 1)$.
(23)$$P_n=\frac{1}{\sqrt{2\pi D}}e^{-\frac{1}{2}\frac{(n-\bar{n})^{2}}{D} }.$$
In a qualitative analysis of the macroscopic case, when $\bar {n} \to \infty$, it is clear that the dispersion of $D$ cannot give a noticeable contribution to take it into account. In other words, the distribution will be such that $n \to \bar {n}$, and it is unnecessary to take into account the specific values of $n$ for spectroscopic studies. Therefore, to study a macroscopic system, all values of $n$ should be taken into account, which means that only the mean values of $\bar {E}$ should be taken into account. It should be added that such an analysis is understandable at a qualitative level, since a macroscopic system averages all characteristics and only an average of their values can be observed, e.g., $\bar {E}$. In contrast, in the second case, $\bar {n} \ll 1$ only if $Z \ll 1$, i.e. the system comprises a relatively small number of atoms. In this case, Eq. (16), (18) show that the main contribution to the distribution of $P_n$ and $E_n$ will be at $n = 1$, and the remaining members are negligible; here $P_{n=1}=\bar {n}=\left \langle \psi \left | Z\right | \psi \right \rangle$, and the energy $E_{n=1}=\bar {E}=\left \langle \psi \left | Z_{\omega }\right | \psi \right \rangle$ meaning that for $\bar {n} \ll 1$ only one-photon processes should be considered, and the main characteristics of the scattering are determined by their average values for the energy $\bar {E}$ and the number of photons $\bar {n}$.In concluding this section, it should be pointed out that for a system consisting of a relatively small number of atoms and a macroscopic system, the equations for calculating the energy coincides and is determined by its average value $\bar {E}$. In all other cases, the average energy $\bar {E}$ is not a characteristic that can provide information about the scattering system. This is one of the main findings of this work that has practical application, meaning that in order that in order to investigate polyatomic structures, where $n \sim 1$ a counting number, knowledge of the average energy $E_n$ and the number of generated $n$ photons is necessary. In other words, the experiment should find the dependence $\frac {d^2 E_n} {d\Omega d\omega }$ and record the number of photons $n$, which are generated by scattering on a polyatomic structure. Knowing such dependence from the experiment and using the theory presented in this work, it will be possible to conduct spectroscopic studies, including X-ray structural analysis.
Let us consider one more limiting case of the equations obtained, in which the electron in the external field will be classical. It is known that there is an exact solution to the problem of the radiation probability of a quantized electromagnetic field interacting with a classical source, with a current $\textbf{j}(\textbf{r}, t)$ [48]. In order for the obtained equations to correspond to this case, it is necessary to assume that the electron is in an external electric field $\textbf{E} (\textbf{r}, t)$. To do this, in the Eq. (16) (of course, this also applies to Eqs. (18), (20)) remove $\textbf{A}^{'}_{2,a}$, since the magnetic field does not affect the electron in this case. To find the wave function $\psi _L$, it is necessary to solve the Schrödinger Eq. (5) ($U(\textbf{r}) = 0$, because in this case the electron is only in the field $\textbf{E} (\textbf{r},t)$) c ${\phi }^{'} = - \textbf{E}(\textbf{r},t)\textbf{r}$. We want the solution of this Schrödinger equation to correspond to the case under consideration, i.e. the classical current $\textbf{j}(\textbf{r},t)$. In this case, when solving the Schrödinger equation in the field $\textbf{E}(\textbf{r},t)$, you must assume that $\textbf{E}(\textbf{r},t) = \textbf{E}(t)$. In other words, this means that E(r,t) depends on $\textbf{r}$, only on the dimensions of the space much larger than the de Broglie electron wavelengths. Solving the Schrödinger equation in this case is easy; we get
(24)$$\psi_{cl}= C e^{i(\textbf{p r}-\omega_e t)}e^{{-}i\int^{t}_{-\infty}\textbf{E}(\textbf{r},t^{'})\textbf{r}dt^{'}}e^{i a(t)},$$
where $a(t)$ is an insignificant phase, $\textbf{p}$ is the electron momentum before interaction with the $\textbf{E}(\textbf{r},t)$ field, $C$ is the normalization constant, $\omega _e$ is the electron frequency. Using the well-known equation for the probability current vector, we obtain (25)$$\textbf{j}(\textbf{r},t) = \textbf{p}-\int^{t}_{-\infty}\textbf{E}(\textbf{r},t^{'})dt^{'}.$$
Eq. (25) does not contain Planck’s constant and is the classic expression for the current vector. Since we want to make the transition to the classical current vector, it is necessary to take all the quantum fluctuations of physical quantities of all order, in Eq. (16), to be equal to zero, i.e. $\langle \psi |~|f(\textbf{r})|^{2k}~|\psi \rangle - | \langle \psi | f(\textbf{r}) |\psi \rangle |^{2k} \to 0$. In this case, it is easy to get Eq. (16) in the form of $P^{cl}_n=\frac {\bar {n}^n_{cl}}{n!}$, where (26)$$\bar{n}_{cl}=\frac{1}{(2\pi)^2c^3}\int \omega \left| \int^{\infty}_{-\infty} e^{i(\omega t - \textbf{k}\textbf{r})}[\textbf{n}\times\textbf{j}(\textbf{r},t)] dt d^3\textbf{r} \right|^2 d\Omega d\omega .$$
The resulting Eq. (26) is fully consistent with the well-known [48] expression for the case under consideration (only in [48] the averaging over the polarizations of the photon was not performed). It should be added that the resulting expression for the probability in the [48] was based on boson statistics of photons. In our case, we did not use approaches based on this statistics, but received our expression on the basis of strictly mathematical reasoning and the transition from quantum physics to classical.It should be added that using the Eq. (20) is not difficult to obtain, with $n = 1, a=1$ (of course, we must assume $Z \ll 1$) and $\langle \psi |~|f(\textbf{r})|^2|\psi \rangle - | \langle \psi | f(\textbf{r}) |\psi \rangle |^2 \to 0$ expressions for classical scattering, this is Rayleigh scattering and Thomson scattering. Indeed, in the case of Rayleigh scattering, the Eq. (20) must, in view of the foregoing, be represented as
(27)$$\frac{d^2 E}{d\Omega d\omega}=\frac{1}{(2\pi)^2c^3}\omega^2 \Bigl| \int \frac{\psi^* \psi}{\psi^*_{L}(t,\textbf{r})\psi_{L}(t,\textbf{r})} \int^{\infty}_{-\infty} e^{i(\omega t - \textbf{k}\textbf{r})}[\textbf{n}\times\textbf{P}(t,\textbf{r})] dt d^3\textbf{r}\Bigr|^2 ,$$
where is $\textbf{P}(t,\textbf{r})=\psi ^{*}_{L}(t,\textbf{r}) \hat{\textbf p} \psi _{L}(t,\textbf{r})$ (taken into account that $\psi _{L}(t,\textbf{r})\approx \psi ; \to \frac {\psi ^* \psi }{\psi ^*_{L}(t,\textbf{r})\psi _{L}(t,\textbf{r})} \approx 1$, since Rayleigh scatterin, and also without taking into account the magnetic component of the electromagnetic field, because the dipole interaction, that is, $\frac {\textbf{A}^{'}_{2,a}}{c}\ll 1$ and ${\textbf{k}\textbf{r}} \ll 1$). Integrating in parts the Eq. (27) we get (28)$$\frac{d^2 E}{d\Omega d\omega}=\frac{1}{(2\pi)^2c^3}\omega^2 \Bigl| \int^{\infty}_{-\infty} e^{i\omega t}[\textbf{n}\times\textbf{p}(t)] dt\Bigr|^2 ,$$
where $\textbf{p}(t)=\langle \psi _{L}(t,\textbf{r})| \hat{\textbf p} |\psi _{L}(t,\textbf{r}) \rangle$ and makes sense momentum. If we assume that the interaction occurs in a finite period of time is $t$ (very large) and integrating (28) in parts, we get (29)$$\frac{d^2 E}{d\Omega d\omega}=\frac{1}{(2\pi)^2c^3}\omega^4 \Bigl| [\textbf{n}\times \tilde{\textbf{r}}] \Bigr|^2 ,~~\tilde{\textbf{r}}=\int^{t}_{0} e^{i\omega t^{'}}\textbf{r}(t^{'})dt^{'} ,$$
where $\textbf{r}(t)=\int ^{t}_0 \textbf{p}(t^{'})dt^{'}=\langle \psi _{L}(t,\textbf{r})| \textbf{r} | \psi _{L}(t,\textbf{r}) \rangle$ is dipole moment of the system under consideration. The Eq. (29) is standard for calculating in any form (depending on the polarizations of the scattered electromagnetic wave) of Rayleigh scattering. For example, if we assume that an electromagnetic field interacts with a classical oscillator (with a frequency of $\omega _a$) with a force of $\textbf{E}_0 \sin (\omega _0 t)$, we get $\textbf{r}(t)=\textbf{E}_0/\omega ^2_a \sin (\omega _0 t)$ (considering that $\omega _0 \ll \omega _a$ for Rayleigh scattering), and (30)$$\frac{d^2 E}{d\theta dt}=\frac{E^2_0}{8 c^3}\left( \frac{\omega_0}{\omega_a}\right)^4 (1+\cos(\theta))\sin(\theta) ,$$
where $\theta$ scattering angle. From Eq. (30) it is easy to obtain the classical cross section for Rayleigh scattering. It should be added that Eq. (29) coincides with the expressions for calculating the generation spectra of higher harmonics (HHG), see, for example, [28,49]. Of course, for this you need to take $\textbf{r} (t)$ not as a classical model, but by the formula $\textbf{r}(t) = \langle \psi _{L}(t,\textbf{r})| \textbf{r} | \psi _{L}(t,\textbf{r}) \rangle$. In addition, for USP, the integration over time in Eq. (29) is taken from $- \infty$ to $+ \infty$.To conclude this section, we show what can be obtained in the limiting case of an expression for Thomson scattering. As is known, Thomson scattering in the classical case and quantum (in the first order of the perturbation theory) coincide. To make this easier to show, it is better to use the wave function of an electron in an external electromagnetic field in the $\ psi_v$ gauge (Appendix E shows the gauge invariance of the developed theory). The solution in this calibration is well known, even in the relativistic case (Volkov’s solution) [3]. Using the nonrelativistic limit of this wave function in the form of $\psi _v (t,\textbf{r})=Ce^{-i(\omega _e t- \textbf{p r})}e^{-i\int ^{t}_0\textbf{p A}dt/c}$ you can get (for $n=1, a=1$) for (20) the equation (taking into account the finite, but long interaction time from $0$ to $t$)
(31)$$\frac{d^2 E}{d\Omega d\omega}=\frac{1}{(2\pi)^2c^3}\omega^2 \langle \psi| \left| \int^{t}_{0} e^{i(\omega t^{\prime} - \textbf{k}\textbf{r})}\left[\textbf{n}\times \frac{\textbf{A}}{c}\right] dt^{\prime}\right|^2 | \psi \rangle.$$
Integrating Eq. (31) over time, by parts and taking into account the fact that $\textbf{E} =-\frac {1}{c} \frac {\partial \textbf{A}} {\partial t}$ we get the formula for Thomson scattering.4. Attosecond pulse scattering
In order to see what a polyatomic system should be, for the number of photons produced to be a countable number, we consider a specific case in which an attosecond electromagnetic pulse interacts with a polyatomic system. In this case, it is known that the sudden perturbation approximation can be applied [50,52,53] to find an analytical solution of Eq. (5). In this approximation, the own Hamiltonian of the system can be neglected, since the electron in the atom does not have time to evolve under the action of the field of the atom due to the too rapid interaction of the pulse with the electron in the atom. As a result, the solution of Eq. (5) in the approximation of sudden disturbances will be:
(32)$$\psi_{L}=\varphi _{0}exp\left\lbrace -i\int^{t}_{-\infty}\sum_a \textbf{E}\textbf{r}_a dt \right\rbrace .$$
Using this wave function and integration by parts of the equation $\textbf{P}_{\omega }=\int ^{\infty }_{-\infty }e^{i\omega t}\textbf{P}dt$, we can see that the vector $\textbf{P}_{\omega }=-i\widetilde{\textbf E}(\omega )/\omega$, where $\widetilde{\textbf E}(\omega )= \int ^{\infty }_{-\infty }\textbf{E}e^{i\omega t}dt$. As a result, we obtain an equation for $Z$ (33)$$Z=\frac{1}{(2\pi)^2c^3}\int \frac{1}{\omega} \left| \sum_a e^{{-}i\textbf{k}\textbf{r}_a}[\textbf{n}\times \widetilde{\textbf E}(\omega )] \right|^2 d\Omega d\omega .$$
We take into account, in the general non-dipole case, that if $\textbf{E}=\textbf{E}_0 g(t-\textbf{n}_{0}\textbf{r}_a/c)$ ($g$ is a function depending on the choice of the form of a USP), then $\widetilde{\textbf E}(\omega )=\textbf{E}_0 e^{i k \textbf{n}_0\textbf{r}_a} F_{\omega }$, where $F_{\omega }=\int ^{\infty }_{-\infty }g(\eta )e^{i\omega \eta } d\eta$. With this in mind, we get (34)$$Z=\frac{[\textbf{n}\times \textbf{E}_0]^2}{(2\pi)^2c^3}\int \frac{F^2_{\omega}}{\omega} \left| \sum_a e^{{-}i\textbf{p}\textbf{r}_a} \right|^2 d\Omega d\omega ,$$
where $\textbf{p}=\omega /c(\textbf{n}-\textbf{n}_{0})$ and has the meaning of a recoil momentum when a photon is scattered.As mentioned earlier, the equation for the average number of photons in scattering coincides with the equation for the probability in a single-photon scattering process. This result makes it possible to use previously developed approaches in the case of single-photon scattering processes, e.g., in [34,50–52], and at the same time evaluate $\bar {n}$ in multiphoton processes using already known equations. We also add that in the case where the $\bar {n} \ll 1$, Eqs. (16),(18),(20) tend towards the previously known approaches of single-photon scattering of attosecond impulses [34,50–52]. We use this to find $\bar {n}$ in the case of multiphoton scattering, i.e. we use the equation $\bar {n}=\left \langle \psi | Z |\psi \right \rangle$, where $Z$ is obtained in (34) and coincides with equation obtained in [34,50]. In the cycle of these works [34,50–52], the theory of single-photon scattering on various polyatomic structures and nanosystems was developed taking into account defects and thermal vibrations of atoms, where the equations have their final and analytical form. Consider as an example the simplest polyatomic system - a linear chain consisting of identical atoms. In this case, using the studies [34,50], $\frac {d \bar {n}}{d\Omega d\omega }$ can be written in the form
(35)$$\frac{d \bar{n}}{d\Omega d\omega}=\frac{[\textbf{n}\times \textbf{E}_0]^2}{(2\pi)^2c^3}\frac{F^2_{\omega}}{\omega} \left( N \left\lbrace N_e+N_e(N_e-1)F(\omega,\textbf{p})\right\rbrace +N^2_e F(\omega,\textbf{p}) \left( \frac{\sin^2(\textbf{p}\textbf{d}N/2)}{\sin^2(\textbf{p}\textbf{d}/2)} - N \right) \right) ,$$
where the vector $\textbf{d}$ determines the interatomic distance in the linear chain in the chosen direction, $N, N_e$ is the number of atoms in the system and electrons in the atom, respectively, and the equation for $F (\omega , \textbf{p})$ found in [34] (36)$$F(\omega,\textbf{p})=\left( \sum^3_{i=1}\frac{A_i\alpha^2_i}{p^2+\alpha_i^2}\right)^2,$$
where the coefficients $A_i, \alpha _i$ are known [54] the atom in question by specifying the distribution of electron density. For the calculation, the form of a USP must be chosen and the most frequently used is the Gaussian pulse $\textbf{E}=\textbf{E}_0 exp \left (-\alpha ^2 (t-\textbf{n}_{0}\textbf{r}_a/c)^2 \right ) \cos (\omega _0 t-\textbf{k}_{0}\textbf{r}_a)$. For simplicity, let us assume that the number of oscillations in a pulse is rather large, i.e. $\omega _0/\alpha \gg 1$ (the condition $\int ^{\infty }_{-\infty }\textbf{E}dt\to 0$ and $F_{\omega }=\frac {\sqrt {\pi }}{2\alpha }exp\left ( -\frac {\omega -\omega _0}{2\alpha }\right )^2$). We perform calculations for a specific value of the orientation of the linear chain, with respect to the direction of the incident pulse. If we choose, as an example, an orientation such that the vector $\textbf{n}_0$ is parallel to $\textbf{d}$, then for sufficiently large $N$ (more precisely, for $\frac {\omega _0 d}{c}N\gg 1$) we obtain (37)$$\bar{n}=\sqrt{\frac{\pi}{2}}\frac{E^2_0}{6\omega_0 c^3 \alpha}N N_e\left(1-F\left(\frac{\omega_0}{c}\right)+N_e\frac{3\pi}{2}\frac{c}{\omega_0 d}\right),$$
where $G(\omega _0/c)=1-F\left (\frac {\omega _0}{c}\right )$ is a function (calculated in Appendix D also provides details of the calculation of $\bar {n}$ ) that is presented in Fig. 1 for the atoms $Li$ - lithium, $C$ - carbon, $Si$ - silicon, $Fe$ - iron, $Pb$ - lead. The function $G(\omega _{0}/c)$ is dimensionless and shows the transition from USP scattering by individual electrons in atoms $\omega _{0}/c \gg 1$ to scattering by a group of atoms $\omega _{0}/c \ll 1$. Moreover, it is not difficult to obtain Thomson scattering from (37) for $\omega _{0}/c \gg 1$. The figure shows that if $\frac {\omega _0}{c} \lesssim 1$ is chosen, then the last term in (37) begins to dominate $G(\omega _0/c)$, and if small $\frac {\omega _0}{c}$ and certain $N_e$, in Eq. (37) the member $G(\omega _0/c)$ can be neglected. Of course, arbitrarily reducing $\frac {\omega _0}{c}$ is impossible, since the pulse duration must be of the order of attoseconds. Neglecting $\frac {\omega _0}{c}$, i.e. small $|\textbf{p}|$ means the use of the dipole approximation. In the case of a linear chain from Eq. (37), it is simple to show that in this approximation for attosecond pulses $\bar {n}=\sqrt {\frac {\pi }{2}}\frac {E^2_0}{6\omega _0 c^3 \alpha }N N^2_e\frac {3\pi }{2}\frac {c}{\omega _0 d}$ and $\bar {n} = Z$, and that the distribution for $P_n$ and $E_n$ will be (38)$$P_n= \frac{{\bar{n}}^{n}}{n!}e^{-\bar{n}} ~,~~ E_n=\bar{E} P_{n-1} ,$$
where $\bar {E}=\sqrt {\frac {\pi }{2}}\frac {E^2_0}{6 c^3 \alpha }N N^2_e\frac {3\pi }{2}\frac {c}{\omega _0 d}$ and the condition prevails of $E_n=\hbar \omega _0 n P_n$ (in the CGS system of units) with fluctuation $\delta \bar {E} =\hbar \omega _0 \sqrt {\bar {n}}$. For such system parameters, where $\bar {n}\sim 1$, the fluctuation $\delta \bar {E}$ is large and the deviation from the mean $\bar {E}$ can no longer be neglected. It should be added that for spectroscopic studies (X-ray diffraction analysis), the dependence $\frac {d^2 E_n}{d\Omega d\omega }$ is necessary, which in this case will be (39)$$\frac{d^2 E_n}{d\Omega d\omega}=\frac{[\textbf{n}\times \textbf{E}_0]^2}{(2\pi)^2c^3}F^2_{\omega} N^2_e \frac{\sin^2(\textbf{p}\textbf{d}N/2)}{\sin^2(\textbf{p}\textbf{d}/2)} P_{n-1} .$$
We also add that in Eq. (39) $\frac {\sin ^2(\textbf{p}\textbf{d}N/2)}{\sin ^2(\textbf{p}\textbf{d}/2)}$ is responsible for the diffraction pattern in the scattering of USP.It should be recalled that Eqs. (38), (39) are obtained in the dipole approach with $\frac {\omega _0 d}{c}N\gg 1$, $\omega _0/\alpha \gg 1$ and parallelism of vectors $\textbf{n} _0$ and $\textbf{d}$. In other cases, the final equation should be calculated using Eqs. (16), (18) with $Z$ calculated using the formula (34). Distributions in the general case will be more complex and deserve separate study for specific examples of the polyatomic system and USP.
5. Discussion and conclusion
In this work we have developed a multiphoton theory of the scattering of ultrashort electromagnetic pulses, which takes into account the entire set of infinitely large modes of the scattered electromagnetic field. The resulting Eqs. (16),(18),(20) are generic and universal, and are applicable to all types of USP and polyatomic systems. It should also be noted that at present the solution to complex processes is being attempted using numerical simulation methods. This problem cannot be solved by similar methods, since the number of modes of the scattered field considered is infinitely large and the number of variables in the solved differential Eq. (1) or (7) is infinite. A noteworthy consequence of the derived exact solution, for the scattering probability $P_n$ and the average energy $E_n$ with $n$ number of photons, is the analytical result, which ultimately does not depend on the quantization volume $V$; this result is not at all obvious and confirms that Eqs. (16),(18),(20) are correct. This conclusion is quite obvious in the case of single-photon scattering, when the scattering amplitude of one mode is $\sim {\hat{\textbf A}}\sim 1/\sqrt {V}$, and the probability $\sim {\hat{\textbf A}^2}\sim 1/V$, summing over all modes, $V$ disappears. We should also add about the gauge invariance of the developed theory, which, as mentioned above, is satisfied, which can be shown exactly analytically. This fact, as well as the correct limit transitions (see “Limit Cases”), confirms the correctness of the assumptions made about the evolution of an electron in an external classical electromagnetic field and the field $U(\textbf{r}_a)$, when the general differential Eq. (1) was divided into two Eqs. (2) and (3).
We now discuss the application of the developed theory. Obviously, Eqs. (16),(18),(20) should be taken into account when $\bar {n}$ is countable. In other words, the developed theory should not be used when $\bar {n} \ll 1$ (perturbation theory) and $\bar {n} \gg 1$ (the classical case), since in this case, other older theories suffice. Let us estimate the value of the parameter $\bar {n}$ in order of magnitude, which is sufficient if it is necessary to use the developed theory. To do this, consider Eq. (17), where $Z \sim \bar {n}$, in which $\textbf{P}_{\omega }=\int ^{\infty }_{-\infty }e^{i\omega t}\textbf{P}dt$, where $\textbf{P}$ is defined above. As a rough estimate of equation $\textbf{P}$, the dimension of which is a momentum, we can assume that $\textbf{P}$ is the momentum transferred by a USP at time $t$ to an electron with coordinate $\textbf{r}_a$. This impulse will be considered equal to the impulse transmitted to the free electron $\textbf{P}=\int ^t_{-\infty }\textbf{E}(t,\textbf{r}_a)dt$, which is really justified when using a powerful USP whose fields are larger than atomic and the carrier frequency $\omega _0$ is greater than the atomic frequency $\omega _a \sim 1$ ($\omega _0$ in the X-ray frequency range). As a result, we obtain $\textbf{P}_{\omega }=-i\widetilde{\textbf E}(\omega )/\omega$, where $\widetilde{\textbf E}(\omega )= \int ^{\infty }_{-\infty }\textbf{E}e^{i\omega t}dt$, which coincides with the case considered above for attosecond pulses. With such an estimate, we obtain an equation for $Z$ in the form (34). Further, we take into account the fact that the USP scattering occurs mainly at a frequency close to $\omega _0$ (carrier frequency) with a width of $\alpha$. In this case,
(40)$$Z\sim \frac{E^2_0}{c^3\alpha \omega_0} H~, ~~ H=\begin{cases}N N_e, ~~if ~~ \frac{\omega_0}{c} \gg 1 ~- ~ inc \cr N^2 N^2_e, ~~if ~~\frac{\omega_0 d}{c} N\ll 1 ~- ~coh \end{cases},$$
where $inc$ is incoherent scattering of USP on electrons of a polyatomic system, $coh$ is coherent scattering on all electrons of a polyatomic system, and $N, N_e$ are the numbers of atoms in the system and electrons in an atom, respectively. In Eq. (40), the parameter $d$ is an estimate and characterizes the average interatomic distance in a polyatomic system. Equation (40) can be obtained in a simpler way if we analyze Eqs. (35) and (37) in the case of a linear chain for attosecond pulses. However, all components of Eq. (40) between the presented limiting cases depends on many parameters within the system under consideration. In the case of attosecond pulses, more accurate estimates can be made by using Eqs. (35) and (37), and in the case of $\frac {\omega _0 d}{c}\sim 1, N \gg 1$ the exact Eqs. (38) and (39) can be used. Choosing a USP with $E_0 \gtrsim 10$ [a.u.], with the number of atoms in the system $N \gtrsim 1000$, as well as a pulse with a duration $\tau = 1 / \alpha$ less than or of the order of femtoseconds $fs$ (in atomic units $\alpha \gtrsim 10^{- 3}$), as well as in the X-ray range $\omega _0$, we get $\bar {n} \gtrsim 1$. It is from such a number of atoms in the system that various biomolecules, proteins, nanosystems, and other promising objects of research consist. USPs with such power and duration are currently being used to study the structure and dynamics of many promising structures. This conclusion is important from the practical point of view, because for $\bar {n} \gtrsim 1$ it is necessary to use the developed theory in the case of X-ray diffraction analysis of the structures studied. It should be noted that such an assessment is valid only for powerful USP for which the fields and frequencies of $\omega _0$ are more than atomic $\omega _a \sim 1$; otherwise such an assessment will give a greater error. Despite this, at present, the fields of USP are just comparable to or larger than atomic fields in the X-ray frequency range (USP on free electrons), which in general gives a satisfactory estimate.Finally, using the developed theory, such parameters of a USP and a polyatomic system can be chosen in order to generate a certain number of $n$ photons with a given probability $P_n$. In other words, such a polyatomic system with the given USP parameters can be a source of single photons. The developed approach is applicable not only for USP scattering on a polyatomic system, but also for multiple scattering from the same electron (inside one and the same atom). Within the framework of the presented theory, multiple scatterings and scattering on a polyatomic system, including interference processes, are taken into account simultaneously. Of course, the completely quantum interference in the presented theory is not taken into account, because when solving Eq. (2), we did not take into account the term $\textbf{A}^2_{1,a}$ and also used the DE approximation, which was negligible, when scattering. For example, the presented theory does not take into account the birth of quantum-entangled photons, i.e. spontaneous parametric down-conversion (SPDC) is not taken into account. Of course, SPDC is known to have a low probability of events, compared with USP scattering.
Appendix A
Consider the differential Eq. (7). To find the probability of, we first consider this equation with the variable $Q_{k_{\sigma }}$, which will be
(41)$$\left( \frac{1}{2}\omega\left({\hat{P}}^2_{k_{\sigma}}+Q^2_{k_{\sigma}}\right)+\alpha(t)Q_{k_{\sigma}}+i \beta(t)\frac{\partial ~~}{\partial Q_{k_{\sigma}}}\right) \Phi_{k_{\sigma}}=i\frac{\partial \Phi_{k_{\sigma}}}{\partial Q_{k_{\sigma}}} ,$$
where
$\alpha (t)=\sum _a \gamma _a(t)\cos (\textbf{k}\textbf{r}_a)$,
$\beta (t)=\sum _a\gamma _a(t)\sin (\textbf{k}\textbf{r}_a)$,
(42)$$\gamma_a(t)=\sqrt{\frac{4\pi}{V\omega}}\psi^{{-}1}_{L}(t,\textbf{r}_a) ( \hat{\textbf p}_a + \frac{\textbf{A}^{'}_{2,a}}{c} ) \textbf{u}_{k_{\sigma}} \psi_{L}(t,\textbf{r}_a).$$
We need to find
$|\langle n_{k_{\sigma }}| \hat {S}(t)|0_{k_{\sigma }}\rangle |^2$, where
$\hat {S}(t)$ is an evolution operator in the interaction representation. It is known that this operator in general form can be represented as a Magnus expansion [
55,
56]
$\hat {S}(t)=exp({\hat A}_1+{\hat A}_2+{\hat A}_3+\cdots )$, where
(43)$$\begin{aligned} &{\hat W}(t)=e^{i{\hat H}_0 t} {\hat V} e^{{-}i{\hat H}_0 t} ,~{\hat A}_1 ={-}i\int^t_{-\infty}{\hat W}(t_1)dt_1 ,~ {\hat A}_2 = \frac{({-}i)^2}{2}\int^t_{-\infty}dt_1\int^{t_1}_{-\infty}dt_2 [{\hat W}(t_1),{\hat W}(t_2)]\\ &{\hat A}_3 = \frac{({-}i)^3}{6}\int^t_{-\infty}dt_1\int^{t_1}_{-\infty}dt_2\int^{t_2}_{-\infty}dt_3\left\lbrace [{\hat W}(t_1),[{\hat W}(t_2),{\hat W}(t_3)]]+ [[{\hat W}(t_1),{\hat W}(t_2)],{\hat W}(t_3)]\right\rbrace , \end{aligned}$$
where in our case the interaction operator is
${\hat V}=\alpha (t)Q_{k_{\sigma }}+i \beta (t)\frac {\partial ~~}{\partial Q_{k_{\sigma }}}$, and the free field Hamiltonian is
${\hat H}_0=\frac {1}{2}\omega \left ({\hat {P}}^2_{k_{\sigma }}+Q^2_{k_{\sigma }}\right )$. If you use commutation rules, it is easy to get that
${\hat W}(t)=Q_{k_{\sigma }}\alpha ^{'}(t)-i\beta ^{'}(t)\frac {\partial ~~}{\partial Q_{k_{\sigma }}}$, where
$\alpha ^{'}(t)=\sum _a \gamma _a(t)\cos (\omega t-\textbf{k}\textbf{r}_a)$,
$\beta ^{'}(t)=\sum _a\gamma _a(t)\sin (\omega t-\textbf{k}\textbf{r}_a)$. It is also not difficult to get an the equation for
${\hat A}_2=i\frac {(-i)^2}{2}\int ^t_{-\infty }dt_1\int ^{t_1}_{-\infty }dt_2 \sum _{a,b}\gamma _a(t_1)\gamma _b(t_2)\times\sin \left ( \omega (t_2-t_1)-\textbf{k}(\textbf{r}_a-\textbf{r}_b)\right )$. You can see that the equation for
${\hat A}_2$ does not depend on the field variables, which means that the other terms in the Magnus decomposition
${\hat A}_3,{\hat A}_4,\ldots$ will be are zero. You can also see that if
${\hat A}_2$ does not depend on
$Q_{k_{\sigma }}$ (although it depends on
$\textbf{r}_a$), then the expression
$|\langle n_{k_{\sigma }}| \hat {S}(t)|0_{k_{\sigma }}\rangle |^2$ does not depend on
${\hat A}_2$. As a result, it turned out that the whole evolution can be represented through one term in the Magnus expansion is
${\hat A}_1$, and the evolution operator is
$\hat {S}(t) = exp({\hat A}_1)$.
The result of the action of the $\hat {S}(t)$ operator on such a state is not difficult to calculate given that
(44)$$\hat{S}(t)=e^{i \phi}exp\left({-}i A(t) Q_{k_{\sigma}}\right) exp\left({-}B(t) \frac{\partial ~~}{\partial Q_{k_{\sigma}}}\right) ,$$
where
$\phi$ is an insignificant phase (there is no sense to write it out), since the final result does not depend on it, therefore we will not write it further,
$A(t)=\int ^t_{-\infty }\alpha ^{'}(t_1)dt_1$,
$B(t)=\int ^t_{-\infty }\beta ^{'}(t_1)dt_1$. The result of the last statement in Eq. (
44) is the offset of the coordinate
$Q_{k_{\sigma }}$ by the amount
$-B (t)$ in the function affected by this operator. If the initial and final state of a field with
$n_{k_{\sigma }}$ are photons, respectively,
$|0_{k_{\sigma }}\rangle =(\pi )^{-1/4}e^{-Q^2_{k_{\sigma }}/2}$ and
$|n_{k_{\sigma }}\rangle =(\sqrt {\pi }2^{n_{k_{\sigma }}}n_{k_{\sigma }}!)^{-1/2}e^{-Q^2_{k_{\sigma }}/2}H_{n_{k_{\sigma }}}(Q_{k_{\sigma }})$ is Fock state, then
(45)$$\langle n_{k_{\sigma}}| \hat{S}(t)|0_{k_{\sigma}}\rangle =\frac{1}{\sqrt{\pi 2^{n_{k_{\sigma}}}n_{k_{\sigma}}!}} \int^{\infty}_{-\infty} e^{-\frac{Q^2_{k_{\sigma}}}{2}}H_{n_{k_{\sigma}}}(Q_{k_{\sigma}})e^{{-}i A(t) Q_{k_{\sigma}}}e^{-\frac{1}{2}\left( Q_{k_{\sigma}}-B(t)\right)} d Q_{k_{\sigma}}.$$
The result of integration is quite simple if Eq. (
45) is converted to the well-known integral [
47]
(46)$$\langle n_{k_{\sigma}}| \hat{S}(t)|0_{k_{\sigma}}\rangle =\frac{1}{\sqrt{n_{k_{\sigma}}!}}e^{ -\frac{1}{4}\left( A^2(t)+B^2(t)\right)}\left(\frac{B(t) -i A(t)}{\sqrt{2}} \right)^{n_{k_{\sigma}}} .$$
As a result, we get
(47)$$|\langle n_{k_{\sigma}}| \hat{S}(t)|0_{k_{\sigma}}\rangle |^2 = \frac{|X(t)|^{2n_{k_{\sigma}}}}{n_{k_{\sigma}}!}e^{-|X(t)|^2} ,$$
where
(48)$$X(t)=\sum_a \sqrt{\frac{2\pi}{V \omega}} \int^{t}_{-\infty} e^{i(\omega t - \textbf{k}\textbf{r}_a)}\textbf{u}_{k_{\sigma}} \textbf{P}(t,\textbf{r}_a) dt .$$
In Eq. (
48)
$\textbf{P}(t,\textbf{r}_a)=\psi ^{-1}_{L}(t,\textbf{r}_a) \left ( \hat{\textbf p}_a + \frac {\textbf{A}^{'}_{2,a}}{c} \right ) \psi _{L}(t,\textbf{r}_a)$.
Since we are interested in a solution after the action of USP, it is necessary to search for a solution at $t \to \infty$. As a result, replacing $X(t\to \infty ) = X$, we obtain Eq. (14).
Appendix B
Consider the problem of finding the probability $P_n$ of the birth of $n$ photons with all possible values of the wave vector $\textbf{k}$ and with averaged over polarization $\sigma$. Obviously, if $\langle \psi ,\Phi |\Phi ,\psi \rangle = 1$ (where $\Phi = \Phi (t = \infty )$), then this equations can be represented using (11) and (46)
(49)$$1= \left\langle \psi \left| \prod_{k_{\sigma}} \sum_{n_{k_{\sigma}}} \frac{|X|^{2n_{k_{\sigma}}}}{n_{k_{\sigma}}!}e^{-|X|^2}\right| \psi \right\rangle .$$
Eq. (
49) can be represented in another form
$1= \sum _{n_{\sigma }} P_{n_{\sigma }}$, where the probability
$P_{n_\sigma }$ birth
${n_\sigma }$ photons with all possible values of the wave vector
$\textbf{k}$ and with two possible values of polarization
$\sigma$ in general form will be
(50)$$P_{n_\sigma}= \left\langle \psi \left| \sum_{\substack{n_{k_{\sigma}}\geq 0 \\ n_{1_\sigma}+n_{2_\sigma}+\cdots+n_{\infty_\sigma}={n_\sigma}}}\prod_{k_{\sigma}} \frac{|X|^{2n_{k_{\sigma}}}}{n_{k_{\sigma}}!}e^{-|X|^2}\right| \psi \right\rangle.$$
We use the well-known
multinominal theorem [
47]
(51)$$(x_1+x_2+\cdots+x_m)^n=\sum_{\substack{k_j\geq 0 \\ k_1+k_2+\cdots+k_m=n}} \frac{n!}{k_1! k_2!\cdots k_m!}x^{k_1}_1 x^{k_2}_2 \cdots x^{k_m}_m ,$$
where
$m$ in our case takes an infinitely large value. You can also note that when multiplying
$\prod _{k_{\sigma }}e^{-|X|^2}$ (
$| X |^2$ also depends on
$k_{\sigma }$) in the exponent the sum over all
$k_{\sigma }$ will appear. As a result, we obtain, for a known transition from the sum over
$k_{\sigma }$ to the integral (
$\sum _{k_{\sigma }}=2\frac {V}{(2\pi )^3}\int d^3 k_{\sigma }$, taking into account two possible polarization values
$\sigma$):
(52)$$P_{n_\sigma}=\left\langle \psi \left| \frac{{Z_\sigma}^{{n_\sigma}}}{{n_\sigma}!}e^{{-}Z_\sigma}\right| \psi \right\rangle ,$$
where
(53)$$Z_\sigma=\frac{1}{(2\pi)^2c^3}\int \omega \left| \sum_a \int^{\infty}_{-\infty} e^{i(\omega t - \textbf{k}\textbf{r}_a)}\textbf{u}_{k_{\sigma}}\textbf{P}(t,\textbf{r}_a)\right|^2 d\Omega d\omega .$$
In order to calculate the probabilities
$P_n$ of the birth of
$n$ photons when averaging over polarization
$\sigma$, you must do the same as when summing over the wave vector
${\textbf k}$ and passing to the continuous spectrum. The result is an equation
(54)$$P_n=\left\langle \psi \left| \frac{Z^{n}}{n!}e^{{-}Z}\right| \psi \right\rangle ,$$
where
(55)$$Z=\frac{1}{(2\pi)^2c^3}\int \omega \left| \sum_a \int^{\infty}_{-\infty} e^{i(\omega t - \textbf{k}\textbf{r}_a)}[\textbf{n}\times\textbf{P}(t,\textbf{r}_a)] \right|^2 d\Omega d\omega ,$$
where
$\textbf{n} = \textbf{k}/k$ is the direction of the scattered pulse.
Appendix C
Next, we consider the problem of finding the average energy $E_n$ of scattered $n$ photons with all possible values of the wave vector $\textbf{k}$ and averaged over the polarization $\sigma$. To do this, we first consider the average energy $E_{n_{\sigma }}$ of the birth of ${n_\sigma }$ photons with all possible values of the wave vector $\textbf{k}$ and with possible values of polarization $\sigma$. The average value of the scattered energy of all photons is, by definition, $\bar {E}=\langle \Psi |\bar {E}_\gamma |\Psi \rangle$, where $\bar {E}_\gamma =\langle \Phi |{\hat H}_\gamma |\Phi \rangle - \bar {E}_0$, and $\bar {E}_0$ - vacuum energy, and $|\Phi \rangle$ is determined by the Eq. (11). If we represent ${\hat H}_\gamma =\sum _{k^{'}_{\sigma }}{\hat H}_{k^{'}_{\sigma }}$, then the result of this the operator on $| \Phi \rangle$ will be
(56)$${\hat H}_\gamma|\Phi\rangle = \sum_{k^{'}_{\sigma}} \prod_{k_{\sigma}} \sum_{n_{k_{\sigma}}} e^{{-}i E_{n_{k_{\sigma}}} t} C_{n_{k_{\sigma}}} \omega_{k^{'}_{\sigma}}\left( n_{k^{'}_{\sigma}}+\frac{1}{2}\right)\Phi_{n_{k_{\sigma}}}(Q_{k_{\sigma}}) .$$
It should be added that in Eq. (
56) there are also modes with
${k_{\sigma }}={k^{'}_{\sigma }}$. To make the recording more simple, select these modes and represent Eq. (
56) as
(57)$${\hat H}_\gamma|\Phi\rangle = \sum_{k^{'}_{\sigma}} \prod_{\substack{k_{\sigma} \\ k_{\sigma} \neq k^{'}_{\sigma}}} \sum_{n_{k_{\sigma}}} \sum_{n_{k^{'}_{\sigma}}} \omega_{k^{'}_{\sigma}}\left( n_{k^{'}_{\sigma}}+\frac{1}{2}\right)C_{n_{k^{'}_{\sigma}}} \Phi_{n_{k^{'}_{\sigma}}}(Q_{k^{'}_{\sigma}}) e^{{-}i E_{n_{k^{'}_{\sigma}}} t} C_{n_{k_{\sigma}}} \Phi_{n_{k_{\sigma}}}(Q_{k_{\sigma}})e^{{-}i E_{n_{k_{\sigma}}} t}.$$
As a result, using Eq. (
57) is not difficult to get
(58)$$\bar{E}_\gamma =\sum_{k^{'}_{\sigma}} \prod_{\substack{k_{\sigma} \\ k_{\sigma} \neq k^{'}_{\sigma}}} \sum_{n_{k_{\sigma}}} \sum_{n_{k^{'}_{\sigma}}} \omega_{k^{'}_{\sigma}} n_{k^{'}_{\sigma}}|C_{n_{k^{'}_{\sigma}}}|^2 |C_{n_{k_{\sigma}}}|^2,$$
where, obviously,
$|C_{n_{k^{'}_{\sigma }}}|^2=\frac {|X|^{2n_{{k^{'}_{\sigma }}}}}{k^{'}_{\sigma } !}e^{-|X|^2}$, defined by Eq. (
47). Further, for convenience, we denote
$|X|^2=Y_{k_{\sigma }}$ (
$| X |^2$ also depends on
${k_{\sigma }}$). Representing
$n_{k^{'}_{\sigma }}Y^{n_{k^{'}_{\sigma }}}_{k_{\sigma }}=Y_{k^{'}_{\sigma }}\frac {d}{d Y_{k^{'}_{\sigma }}}Y^{n_{k^{'}_{\sigma }}}_{k^{'}_{\sigma }}$ can be an Eq. (
58) in the form
(59)$$\bar{E}_\gamma =\sum_{k^{'}_{\sigma}}\omega_{k^{'}_{\sigma}} e^{-\sum_{k_{\sigma}} Y_{k_{\sigma}}}Y_{k^{'}_{\sigma}}\frac{d}{d Y_{k^{'}_{\sigma}}} \prod_{k_{\sigma}} \sum_{n_{k_{\sigma}}} \frac{Y_{k_{\sigma}}^{n_{{k_{\sigma}}}}}{k_{\sigma} !} .$$
Similar to how it was done for probability (see (
49) and (
50)) using the multinominal theorem, it can be shown that
(60)$$\bar{E}_\gamma =\sum_{k^{'}_{\sigma}}\omega_{k^{'}_{\sigma}} e^{-\sum_{k_{\sigma}} Y_{k_{\sigma}}} Y_{k^{'}_{\sigma}}\frac{d}{d Y_{k^{'}_{\sigma}}} \sum^{\infty}_{n_\sigma =0} \frac{\left(Y_{1_{\sigma}}+Y_{2_{\sigma}} + \cdots+ Y_{k^{'}_{\sigma}} + \cdots\right)^{n_\sigma}}{n_\sigma !} .$$
Eventually, after taking the derivative, it’s not hard to get
(61)$$\bar{E}_\gamma =\sum^{\infty}_{n_\sigma =0} \sum_{k^{'}_{\sigma}}\omega_{k^{'}_{\sigma}} Y_{k^{'}_{\sigma}} \frac{\left(\sum_{k_{\sigma}} Y_{k_{\sigma}}\right)^{n_\sigma -1}}{(n_\sigma -1) !}e^{-\sum_{k_{\sigma}} Y_{k_{\sigma}}} .$$
As a result, the equation for the average energy can be represented as
$\bar {E}=\langle \Psi |\bar {E}_\gamma |\Psi \rangle =\sum ^{\infty }_{n_\sigma =0} E_{n_\sigma }$, where
(62)$$E_{n_{\sigma}}=\left\langle \psi \left| Z_{\omega_{\sigma}} \frac{Z^{n_{\sigma}-1}}{(n_{\sigma}-1)!}e^{{-}Z}\right| \psi \right\rangle ,$$
where
(63)$$Z_{\omega_{\sigma}}=\frac{1}{(2\pi)^2c^3}\int \omega^2 \left| \sum_a \int^{\infty}_{-\infty} e^{i(\omega t - \textbf{k}\textbf{r}_a)}\textbf{u}_{k_{\sigma}}\textbf{P}(t,\textbf{r}_a)dt \right|^2 d\Omega d\omega .$$
In order to calculate the energy
$E_n$ of scattered
$n$ photons with all possible values of the wave vector
$\textbf{k}$ and with averaged over polarization
$\sigma$, you need to do the same as it is done when summing over the wave vector
${\textbf k}$ and transition to the continuous spectrum. The result is an equation
(64)$$E_n=\left\langle \psi \left| Z_{\omega} \frac{Z^{n-1}}{(n-1)!}e^{{-}Z}\right| \psi \right\rangle ,$$
where
(65)$$Z_{\omega}=\frac{1}{(2\pi)^2c^3}\int \omega^2 \left| \sum_a \int^{\infty}_{-\infty} e^{i(\omega t - \textbf{k}\textbf{r}_a)}[\textbf{n}\times\textbf{P}(t,\textbf{r}_a)]dt \right|^2 d\Omega d\omega .$$
Often a differential characteristic is used, i.e. the amount of dissipated energy in the selected direction and unit frequency range. It should be added that the expression for this characteristic is not trivial, since the summation over
${\textbf k}$ (or
${\textbf k}^{'}$) in Eq. (
61) is present in several places. Despite this, we need the scattered energy in the chosen direction in a given frequency range, and this energy is given by summing over
${\textbf k}^{'}$ in Eq. (
61) (because
${\hat H}_\gamma =\sum _{k^{'}_{\sigma }}{\hat H}_{k^{'}_{\sigma }}$). As a result, it is easy to get that
(66)$$\frac{d^2 E_n}{d\Omega d\omega}=\left\langle \psi \left| Z_{D} \frac{Z^{n-1}}{(n-1)!}e^{{-}Z}\right| \psi \right\rangle ,$$
where
(67)$$Z_{D}=\frac{1}{(2\pi)^2c^3}\omega^2 \left| \sum_a \int^{\infty}_{-\infty} e^{i(\omega t - \textbf{k}\textbf{r}_a)}[\textbf{n}\times\textbf{P}(t,\textbf{r}_a)] dt\right|^2 .$$
Appendix D
Consider the integral
(68)$$\bar{n}=\int \frac{[\textbf{n}\times \textbf{E}_0]^2}{(2\pi)^2c^3}\frac{F^2_{\omega}}{\omega} \left( N \left\lbrace N_e+N_e(N_e-1)F(\omega,\textbf{p})\right\rbrace +N^2_e F(\omega,\textbf{p}) \left( \frac{\sin^2(\textbf{p}\textbf{d}N/2)}{\sin^2(\textbf{p}\textbf{d}/2)} - N \right) \right) d\Omega d\omega .$$
Let us choose the case when the vectors
$\textbf{n}_0$ and
$\textbf{d}$ are parallel and sufficiently large
$N$ (more precisely, for
$\frac {\omega _0 d}{c}N\gg 1$), the parameter
$F_{\omega }=\frac {\sqrt {\pi }}{2\alpha }exp\left ( -\frac {\omega -\omega _0}{2\alpha }\right )^2$ for Gaussian pulse, and we assume
$\omega _0/\alpha \gg 1$. First, we integrate over the frequency
${\omega }$. Under the condition
$\omega _0/\alpha \gg 1$, the main result of integration over frequency
${\omega }$ is concentrated near
${\omega }\to {\omega _0}$ (according to the property of the function
$F_{\omega }$) this means that
$\int F^2_{\omega } G(\omega )d \omega =\int F^2_{\omega } d\omega G(\omega _0)=\frac {\pi ^{3/2}}{\sqrt {2}4\alpha } G(\omega _0)$, where
$G(\omega )$ is a function. As a result, we get
(69)$$\bar{n}=\int \frac{[\textbf{n}\times \textbf{E}_0]^2}{\sqrt{\pi} c^3 2^4 \sqrt{2}\alpha \omega_0} \left( N N_e\left\lbrace 1-F(\omega_0,\textbf{p}_0)\right\rbrace +N^2_e F(\omega_0,\textbf{p}_0) \frac{\sin^2(\textbf{p}_0\textbf{d}N/2)}{\sin^2(\textbf{p}_0\textbf{d}/2)} \right) d\Omega ,$$
where
$\textbf{p}_0=\omega _0/c (\textbf{n}-\textbf{n}_0)$. Consider separately the integral
(70)$$I_2=\int [\textbf{n}\times \textbf{E}_0]^2 F(\omega_0,\textbf{p}_0) \frac{\sin^2(\textbf{p}_0\textbf{d}N/2)}{\sin^2(\textbf{p}_0\textbf{d}/2)} d\Omega .$$
We use the fact that
$\frac {\omega _0 d}{c}N\gg 1$, then
$\frac {\sin ^2(\textbf{p}_0\textbf{d}N/2)}{\sin ^2(\textbf{p}_0\textbf{d}/2)}\to \pi N \sum _n \delta (\textbf{p}_0\textbf{d}/2 \pm \pi n)$, where
$\delta (x)$ is the delta Dirac function, and summation over
$n$ is carried out to the maximum integer value
$|\textbf{p}_0\textbf{d}/(2\pi )|$. It should be added that the transition to the presented delta function is quite obvious if we use the well-known the equation
$\pi \delta (x)=\lim _{N\to \infty }\frac {1}{N}\frac {\sin ^2(x N)}{x^2}$. In addition, it is necessary to take into account that
$\frac {\sin ^2(x N)}{\sin ^2(x)}$ with
$N \to \infty$ has values at the points when
$x = \pm \pi n$ (sine period), then for
$\sin ^2(x)=\sin ^2(x \pm \pi n)$ and
$x \to \pm \pi n$ the equation
$\sin ^2(x \pm \pi n)\to x \pm \pi n$. Also, the value of
$\sin ^2(x N)=\sin ^2((x \pm \pi n) N)$, which leads to a replacement with
$x \to \pm \pi n$ in the form of
$\frac {\sin ^2(x N)}{\sin ^2(x)}\to \frac {\sin ^2((x \pm \pi n) N)}{(x \pm \pi n)^2}$ and the considered Delta function is obtained. As a result, it is not difficult to integrate the equation
$I_2$ over the solid angle
$\Omega$, we get
(71)$$I_2=\frac{4\pi^2 E^2_0 c N}{\omega_0 d}\left\lbrace 1+\frac{1}{2}\sum^{[\frac{\omega_0 d}{c \pi}]}_{n=1}\left( 1+(1-2\pi n\frac{c}{\omega_0 d})^2 \right) \left( \sum^3_{i=1} \frac{A_i \alpha^2_i}{\frac{8\pi \omega_0}{d c} n+\alpha^2_i} \right)^2 \right\rbrace ,$$
where
$[\frac {\omega _0 d}{c \pi }]$ means that summing over
$n$ goes to rounding off the value
$\frac {\omega _0 d}{c \pi }$, so that this value is the minimum integer (for example, [3.2] = 3, [3.7] = 3, [3.95] = 3). When calculating
$I_2$, you can use an approximate the equation (with an error of no more than
$15 \%$), which is easy to get if you analyze the second part of Eq. (
71) and see that it makes a rather small contribution, as a result
(72)$$I_2\approx \frac{4\pi^2 E^2_0 c N}{\omega_0 d}.$$
Next, we consider the integral
$I_1$, which is equal to
(73)$$I_1=\int [\textbf{n}\times \textbf{E}_0]^2 F(\omega_0,\textbf{p}_0) d\Omega .$$
The result of integrating this equation is presented in [
34]. We will write down the final result.
(74)$$\bar{n}=\sqrt{\frac{\pi}{2}}\frac{E^2_0}{6\omega_0 c^3 \alpha}N N_e\left(1-F\left(\frac{\omega_0}{c}\right)+N_e\frac{3\pi}{2}\frac{c}{\omega_0 d}\right),$$
where
$F\left (\frac {\omega _0}{c}\right )$ will be
(75)$$F\left(\frac{\omega_0}{c}\right)=\sum_{i=1}^{3}A^2_{i}I_{i}+2\sum^3_{{\substack{i,j =1 \\ i \neq j}}}A_{i}A_{j}\alpha^2_{i}\alpha^2_{j}I_{i,j},$$
where
(76)$$\begin{aligned} I_{i}&=\frac{3}{2^5}\frac{1}{\left(\frac{\omega_0}{c} \right)^6 }\frac{\alpha^2_{i}\left( \alpha^2_{i}+2\left(\frac{\omega_0}{c} \right)^2\right) }{\left( \alpha^2_{i}+4\left(\frac{\omega_0}{c} \right)^2\right) }\Biggl\{ 4\left(\frac{\omega_0}{c} \right)^2\left( \alpha^2_{i}+2\left(\frac{\omega_0}{c} \right)^2\right)+\\&\qquad\qquad\qquad\qquad +\alpha^2_{i}\left( \alpha^2_{i}+4\left(\frac{\omega_0}{c} \right)^2\right) \ln\left( \frac{\alpha^2_{i}}{\alpha^2_{i}+4\left(\frac{\omega_0}{c} \right)^2}\right) \Biggr\} , \end{aligned}$$
(77)$$I_{i,j}=\frac{3}{2^4\left(\frac{\omega_0}{c} \right)^4 }+ \frac{3}{2^6}\frac{\ln\left( \frac{\alpha^2_{i}}{\alpha^2_{i}+4\left(\frac{\omega_0}{c} \right)^2}\right)}{\left(\frac{\omega_0}{c} \right)^6\left( \alpha^2_{i}-\alpha^2_{j}\right) } \Biggl\{\alpha^4_{i}+4\alpha^2_{i}\left(\frac{\omega_0}{c} \right)^2+8\left(\frac{\omega_0}{c} \right)^4\Biggr\}.$$
Appendix E
If in our consideration we use a gauge in which $\psi _{v}= exp\left \lbrace -\frac {i}{c}\sum _{a}\textbf{A}_{2,a}\textbf{r}_{a} \right \rbrace \psi _L$, then the final the equations for $P_n$ and $E_n$ will obviously be similar except for the vector ${\textbf P}$, which we will represent in this gauge
(78)$$\textbf{P}_v(t,\textbf{r}_a)=\psi^{{-}1}_{v}(t,\textbf{r}_a) \left( \hat{\textbf p}_a + \frac{\textbf{A}_{2,a}}{c} \right) \psi_{v}(t,\textbf{r}_a) .$$
We show that the final the equations for
$P_n$ and
$E_n$ will be the same for different calibrations. To do this, first introduce
(79)$$\hat{\textbf p}_a \psi_{v}(t,\textbf{r}_a) = exp\left\lbrace -\frac{i}{c}\sum_{a}\textbf{A}_{2,a}\textbf{r}_{a}\right\rbrace \left( -\frac{i}{c}(\hat{\textbf p}_a \textbf{A}_{2,a}\textbf{r}_{a})+\hat{\textbf p}_a \right)\psi_{L}.$$
Consider that
$\textbf{A}_{2,a}=\textbf{A}_{2,a}(t-z_a/c)$ (plane wave), i.e.
${\nabla } \textbf{A}_{2,a}\textbf{r}_{a} = -\frac {\textbf{n}_{0}}{c}\frac {\partial }{\partial t} \textbf{A}_{2,a}\textbf{r}_{a} +\textbf{A}_{2,a} = \frac {\textbf{n}_{0}}{c}\textbf{E}\textbf{r}_{a} +\textbf{A}_{2,a}$. As a result, we get
(80)$$\textbf{P}_v(t,\textbf{r}_a) = \psi^{{-}1}_{v}(t,\textbf{r}_a) exp\left\lbrace -\frac{i}{c}\sum_{a}\textbf{A}_{2,a}\textbf{r}_{a}\right\rbrace \left( \hat{\textbf p}_a - \frac{\textbf{E}\textbf{r}_a }{c}\textbf{n}_{0} \right) \psi_{L}(t,\textbf{r}_a).$$
Consider that
$\textbf{A}^{'}_{2,a}=-\textbf{n}_{0}\textbf{E}\textbf{r}_a$, we get that
$\textbf{P}_v(t,\textbf{r}_a)=\textbf{P}(t,\textbf{r}_a)$.
Funding
Council on grants of the President of the Russian Federation (MK-6289.2018.2); Development Fund of the Northern (Arctic) Federal University.
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