Multiphoton Processes: Floquet Theory

Abstract

The study of the interaction of intense laser fields with atoms and molecules has attracted considerable attention in recent years. In particular, the availability of increasingly intense lasers has made possible the observation of a wide variety of multiphoton processes, including multiphoton ionization, laser-assisted electron–atom collisions and harmonic generation. Also, in the case of molecules, the loss of spherical symmetry and the degrees of freedom associated with the nuclear motion give rise to additional computational difficulties and new effects including multiphoton dissociation. Many reviews of these processes have been written and we mention here comprehensive overviews by Gavrila [365], Burnett et al. [194], Mason [639], Protopapas et al. [759] and Joachain et al. [504] where the emphasis is on atomic multiphoton processes, and by Bandrauk et al. [50, 51], Giusti-Suzor et al. [378] and Posthumus [751], where the emphasis is on molecular multiphoton processes.

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The study of the interaction of intense laser fields with atoms and molecules has attracted considerable attention in recent years. In particular, the availability of increasingly intense lasers has made possible the observation of a wide variety of multiphoton processes, including multiphoton ionization, laser-assisted electron–atom collisions and harmonic generation. Also, in the case of molecules, the loss of spherical symmetry and the degrees of freedom associated with the nuclear motion give rise to additional computational difficulties and new effects including multiphoton dissociation. Many reviews of these processes have been written and we mention here comprehensive overviews by Gavrila [365], Burnett et al. [194], Mason [639], Protopapas et al. [759] and Joachain et al. [504] where the emphasis is on atomic multiphoton processes, and by Bandrauk et al. [50, 51], Giusti-Suzor et al. [378] and Posthumus [751], where the emphasis is on molecular multiphoton processes. In our discussion of these processes we observe that the atomic unit of electric field strength experienced by an electron in the ground state of atomic hydrogen $\epsilon_a\approx 5.1\times 10^9\; \mathrm{V}/\mathrm{cm}$ corresponds to a laser intensity $I_a\approx 3.5\times 10^{16}\;\mathrm{W}/\mathrm{cm}^{2}$. Lasers delivering pulses with intensities much larger than this are now available using the “chirped pulse amplification” (CPA) scheme, in which the laser pulses are stretched, amplified and then compressed [891]. Consequently, many new processes have been observed as a result of exposing atoms and molecules to intense laser fields which require a fully non-perturbative approach for their analysis going beyond the first-order perturbation theory treatment of photoionization considered in Chap. 8. Finally, we mention the fourth-generation light sources which are now coming online which will address new fundamental scientific challenges through their ultra-fast and ultra-bright nature [338].

We commence our discussion of multiphoton processes by considering in this chapter atomic R-matrix–Floquet (RMF) theory and applications, reserving a discussion of time-dependent R-matrix theory, necessary to treat the interaction of ultra-short laser pulses with atoms and ions, until Chap. 10 and R-matrix–Floquet theory of molecular multiphoton processes until Chap. 11. In Sect. 9.1 we consider atomic RMF theory, based on the Floquet–Fourier ansatz [218, 325, 632, 874], which was first formulated by Burke et al. [183, 184] and by Dörr et al. [264] and has since been applied to a wide range of atomic multiphoton processes, including multiphoton ionization, laser-assisted electron–atom collisions and harmonic generation. This is an ab initio theory, which is fully non-perturbative and is applicable to arbitrary multi-electron atoms and atomic ions, allowing an accurate description of electron–electron correlation effects. In principle this theory is confined to treating laser pulses involving many cycles of the field, typically exceeding tens of femtoseconds (10⁻¹⁵ s) and we will discuss applications of this theory to multiphoton ionization, laser-assisted electron–atom collisions and to harmonic generation. We also discuss an extension of this theory using multistate non-hermitian Floquet dynamics [253, 473, 746], which has enabled detailed calculations to be carried out for shorter laser pulse interactions which are in good agreement with fully time-dependent calculations.

Finally, in Sect. 9.2 we present the results of some recent R-matrix–Floquet calculations of multiphoton processes which illustrate the theory presented earlier in this chapter.

1 R-Matrix–Floquet Theory

In this section we describe an ab initio R-matrix–Floquet theory of atomic multiphoton processes where we consider the interaction of an intense laser field with an atom or atomic ion, which we assume has $N + 1$ electrons and nuclear charge number Z.

1.1 Introduction

We consider the following three processes: first, multiphoton ionization

$$n{\mathrm {h}}\nu + A_i\rightarrow A_j^+ +\mathrm{e}^-,$$

((9.1))

where the target atom or ion A _i and the residual ion A _j ⁺ may be in their ground or excited states; second, laser-assisted electron–atom collisions

$$n{\mathrm {h}}\nu +\mathrm{e}^-+ A_i\rightarrow A_j+\mathrm{e}^-,$$

((9.2))

where again the target atom or ion A _i and the final atom or ion A _j may be in their ground or excited states; and third, harmonic generation

$$n{\mathrm {h}}\nu + A_i\rightarrow A_i+{\mathrm {h}}\nu^{\prime},$$

((9.3))

where A _i may be an atom or an ion and the frequencies ν and ν ^′ are related by $\nu^{\prime}=n\nu$. We mainly consider processes where there is at most one ejected or scattered electron. However, as in Chap. 6, two electrons in the continuum can be treated by including pseudostates in the R-matrix expansion.

In R-matrix–Floquet theory the laser field, which is treated classically, is usually assumed to be monochromatic, monomode, linearly polarized and spatially homogeneous, and its wavelength is assumed to be large compared with the size of the target atom. The corresponding electric field vector can then be written as

$${\mbox{\boldmath${\cal E}$}}(t)=-\frac{1}{{c}}\frac{{\mathrm {d}}}{{\mathrm {d}} t}\textbf{A}(t) =\hat{{\mbox{$\boldsymbol\epsilon$}}}{\cal E}_0\cos \omega t,$$

((9.4))

where $\hat{{\mbox{$\boldsymbol\epsilon$}}}$ is a unit vector along the laser polarization direction as in Fig. 8.1, ${\cal E}_0$ is the electric field strength and ω is the angular frequency. The corresponding vector potential $\textbf{A}(t)$ is then given by

$$\textbf{A}(t)=\hat{{\mbox{$\boldsymbol\epsilon$}}} A_0 \sin \omega t,$$

((9.5))

where $A_0 =-{c} {\cal E}_0/\omega$ and where we have adopted the Coulomb gauge such that $\mathrm{div}\textbf{A}=0$. Neglecting relativistic effects, the atomic system in the presence of the external laser field is described by the time-dependent Schrödinger equation

$$\left[H_{N+1}+\frac{1}{{c}}\textbf{A}(t)\cdot \textbf{P}_{N+1} +\frac{N+1}{2{c}^2}\textbf{A}^2(t)\right]\widetilde{\Psi}(\textbf{X}_{N+1},t) ={\mathrm {i}}\frac{\partial}{\partial t}\widetilde{\Psi}(\textbf{X}_{N+1},t),$$

((9.6))

where $H_{N+1}$ is the non-relativistic Hamiltonian of the ($N+1$)-electron atomic system in the absence of the laser field defined by (5.3) and

$$\textbf{P}_{N+1}= \sum_{i=1}^{N+1}\textbf{p}_i$$

((9.7))

is the total electron momentum operator. Also in (9.6) and later equations in this chapter, the tilde on the time-dependent wave function $\widetilde{\Psi}$ distinguishes it from time-independent wave functions Ψ which we consider later in our analysis.

In accordance with multichannel R-matrix theory of electron–atom collisions discussed in Chap. 5 we partition configuration space into three regions as illustrated in Fig. 9.1. We see that the same partitioning is used as in electron–atom collisions, illustrated in Fig. 5.1. Also the same physical criteria for defining the boundaries a ₀ and a _p between the three regions, described in Sect. 5.1, are adopted. Having divided configuration space into three regions, we must solve the time-dependent Schrödinger equation (9.6) in each region. Since the laser field, defined by (9.4), has constant amplitude ${\cal E}_0$ and angular frequency ω, we can represent the wave function $\widetilde{\Psi}(\textbf{X}_{N+1},t)$ in each of the three regions by a Floquet–Fourier expansion [218, 325, 632, 874] in terms of time-independent wave functions $\Psi_n(\textbf{X}_{N+1})$ as follows:

$$\widetilde{\Psi}(\textbf{X}_{N+1},t)=\exp(-{\mathrm {i}} Et)\sum_{n=-\infty}^\infty \exp(-{\mathrm {i}} n\omega t)\Psi_n(\textbf{X}_{N+1}).$$

((9.8))

Fig. 9.1

Partitioning of configuration space in R-matrix–Floquet theory

After substituting (9.8) into (9.6) and equating the coefficients of $\exp[-{\mathrm {i}}(E+n\omega)t]$ to zero, we obtain an infinite set of coupled time-independent equations for the functions Ψ _n , where in practical calculations a finite number of positive and negative terms are retained in the expansion over n in (9.8). The solutions of these equations in each region are then matched on the boundaries a ₀ and a _p between the regions using the R-matrix.

In the internal region it is convenient and appropriate to use the length gauge since in this gauge the laser–atom interaction tends to zero at the origin and hence the Floquet–Fourier expansion (9.8) converges more rapidly. However, at larger distances the interaction in the length gauge diverges and hence we use the velocity gauge to describe the ejected or scattered electron in the external region. Finally, in the asymptotic region we derive an asymptotic expansion where the ejected or scattered electron is described in the velocity gauge. We also consider an asymptotic expansion where the ejected or scattered electron is described in the acceleration frame of reference [547, 453] which enables the asymptotic boundary conditions to be expressed in a simple way.

1.2 Internal Region Solution

In the internal region in Fig. 9.1 we transform the time-dependent Schrödinger equation (9.6) to the dipole length gauge defined by the unitary gauge transformation

$$\widetilde{\Psi}(\textbf{X}_{N+1},t)=\exp\left[-\frac{{\mathrm {i}}}{{c}} \textbf{A}(t)\cdot\textbf{R}_{N+1} \right]\widetilde{\Psi}^\textbf{L}(\textbf{X}_{N+1},t),$$

((9.9))

where

$$\textbf{R}_{N+1}=\sum_{i=1}^{N+1}\textbf{r}_i,$$

((9.10))

and the boldface superscript L in (9.9) and later equations indicates that the $N + 1$ electrons are described in the dipole length gauge. Substituting (9.9) into (9.6) we find that the wave function $\widetilde{\Psi}^\textbf{L}$ satisfies the time-dependent Schrödinger equation

$$\left(H_{N+1}+ {\mbox{\boldmath${\cal E}$}}(t)\cdot\textbf{R}_{N+1}\right)\widetilde{\Psi}^\textbf{L} (\textbf{X}_{N+1},t) ={\mathrm {i}}\frac{\partial}{\partial t}\widetilde{\Psi}^\textbf{L}(\textbf{X}_{N+1},t).$$

((9.11))

In order to solve this equation we introduce the Floquet–Fourier expansion defined by (9.8) which in this region can be written as

$$\widetilde{\Psi}^\textbf{L}(\textbf{X}_{N+1},t)=\exp(-{\mathrm {i}} Et)\sum_{n=-\infty}^\infty \exp(-{\mathrm {i}} n\omega t)\Psi_n^\textbf{L}(\textbf{X}_{N+1}),$$

((9.12))

where E is the quasi-energy of the corresponding stationary state. Substituting (9.12) into (9.11) and equating the coefficient of $\exp[-{\mathrm {i}}(E+n\omega)t]$ to zero yields the infinite set of coupled time-independent equations

$$(H_{N+1}-E-n\omega)\Psi_n^\textbf{L} +D_{N+1}(\Psi_{n-1}^\textbf{L}+\Psi_{n+1}^\textbf{L})=0,$$

((9.13))

where the dipole length operator

$$D_{N+1}={\frac{1}{2}} {\cal E}_0\hat{{\mbox{$\boldsymbol\epsilon$}}}\cdot\textbf{R}_{N+1}.$$

((9.14))

If we regard the functions Ψ _n ^L as the components of a vector Ψ ^L in photon space, then (9.13) can be written as a matrix equation in this space as

$$\left(\textbf{H}_{\mathrm{F}}^\textbf{L}-E\textbf{I}\right) \boldsymbol\Psi^\textbf{L}=0,$$

((9.15))

where the Floquet–Fourier Hamiltonian $\textbf{H}_{\mathrm{F}}^\textbf{ L}$ is an infinite-dimensional tridiagonal matrix operator in photon space with components

$$\textbf{H}_{\mathrm{F}}^\textbf{L} = \left[\begin{array}{ccccc} \ddots&&&&0\\ &H_{N+1}-(n-1)\omega&D_{N+1}&&\\ &D_{N+1}&H_{N+1}-n\omega&D_{N+1}&\\ &&D_{N+1}&H_{N+1}-(n+1)\omega& \\ 0&&&&\ddots \end{array} \right].$$

((9.16))

Also in (9.15), Ψ ^L is a solution vector with components $\dots ,\Psi_{n-1}^\textbf{L},\Psi_n^\textbf{L},\Psi_{n+1}^\textbf{L}, \dots$ and I is a unit matrix operator.

In order to solve (9.15) in the internal region we expand the solution vector Ψ ^L, in analogy with (5.5), as follows:

$$\boldsymbol\Psi_{jE}^{\textbf{L}\gamma}(\textbf{X}_{N+1})= \sum_k {\mbox{$\boldsymbol\psi$}}_k^{\textbf{L}\gamma}(\textbf{X}_{N+1}) A_{kj}^{\textbf{L}\gamma}(E).$$

((9.17))

In this equation, and in the following equations, we introduce a superscript γ which represents the quantum numbers which are conserved in the laser–atom interaction process discussed below. Also in (9.17), j labels the linearly independent solutions of (9.15), ${\mbox{$\boldsymbol\psi$}}_k^{\textbf{L}\gamma}$ are energy-independent vector basis functions with Floquet–Fourier components $\psi_{nk}^{\textbf{L}\gamma}$ defined by (9.20) and $A_{kj}^{\textbf{L}\gamma}(E)$ are energy-dependent expansion coefficients which depend on the asymptotic boundary conditions satisfied by the solution $\boldsymbol\Psi_{jE}^{\textbf{L}\gamma}$.

The conserved quantum numbers γ depend on the symmetry of the atomic target state and the polarization of the laser field. One example of considerable experimental and theoretical interest is the interaction of linearly polarized laser light with a closed-shell atom such as neon or argon initially in its $\hbox{}\mathrm{^1S_0^e}$ ground state. We show in Fig. 9.2 the allowed transitions from the $\hbox{}\mathrm{^1S_0^e}$ $M_L=0$ state as linearly polarized photons are absorbed or emitted. In this case the conserved quantum numbers represented by γ are given by

$$\gamma \equiv \alpha\; S\; M_S\; M_L\; \pi^{\prime},$$

((9.18))

where S is the total spin angular momentum, M _S and M _L are the total spin and orbital magnetic quantum numbers in the laser polarization direction, π ^′ is defined in terms of the parity π of the target atom by

$$\pi^{\prime}=(-1)^n \pi$$

((9.19))

and α represents any other quantum numbers which are conserved in the collision. Unlike the conserved quantum numbers Γ in non-relativistic electron–atom collisions, defined by (2.58), we see that the total orbital angular momentum L of the target atom or ion is not conserved. Also, since the total orbital magnetic quantum numbers of the target atom and the linearly polarized laser light are zero, see (B.55), it follows from the parity selection rule (A.27), satisfied by the Clebsch–Gordan coefficients, that dipole transitions such as $\mathrm{^1P_0^e}\rightarrow \mathrm{^1P_0^o}$, $\mathrm{^1D_0^e}\rightarrow \mathrm{^1D_0^o}$ are forbidden. This limits the number of LSπ states that are coupled to one-half the maximum number, as illustrated in Fig. 9.2, and results in the conservation of π ^′ defined by (9.19). However, π ^′ would not be conserved for linearly polarized laser light incident on target atoms where $M_L\neq 0$ nor for circularly polarized laser light incident on an arbitrary atom. As a second example, we show in Fig. 9.3 the allowed transitions from the $\hbox{}\mathrm{^1S_0^e}$ $M_L=0$ state as circularly polarized photons are absorbed or emitted, where the subscript on the target states in this figure is the M _L value, which is now not conserved in the transitions.

Fig. 9.2

Allowed transitions for linearly polarized laser light incident on an atom in a $\hbox{}\mathrm{^1S_0^e}$ state. L is the total orbital angular momentum of the atom and n is the number of photons absorbed, or emitted, corresponding to the Floquet–Fourier expansion index in (9.12). The arrowed lines show the transitions allowed by the laser field

Following (5.6) we expand each Floquet–Fourier component of the vector basis functions ${\mbox{$\boldsymbol\psi$}}_k^{\textbf{ L}\gamma}(\textbf{X}_{N+1})$ in (9.17) in a close coupling with pseudostates expansion given by

$$\begin{array}{rcl} \psi_{nk}^{\textbf{L}\gamma}(\textbf{X}_{N+1})&=&{\cal A}\sum_{Lij} \overline{\Phi}_{nLi}^\gamma(\textbf{X}_N;\hat{\textbf{r}}_{N+1} \sigma_{N+1})r_{N+1}^{-1}u_{nLij}^0(r_{N+1}) a_{nLijk}^{\textbf{L}\gamma}\\ &&+\;\sum_{L i}{\mbox{\raisebox{.4ex}{$\chi$}}}_{nLi}^\gamma(\textbf{X}_{N+1})b_{nLik}^{\textbf{L}\gamma},\\ &&k=1,\dots,n_{{t}}, \end{array}$$

((9.20))

where $\overline{\Phi}_{nLi}^\gamma$ are channel functions obtained by coupling the residual atom or ion states in the case of multiphoton ionization (or target atom or ion states in the case of laser-assisted electron–atom collisions) and possibly pseudostates with the spin –angle functions of the ejected or scattered electron, $u_{nLij}^0$ are radial continuum basis orbitals, ${\mbox{\raisebox{.4ex}{$\chi$}}}_{nLi}^\gamma$ are quadratically integrable functions and n _t is the total number of terms retained in the expansion. By reference to Figs. 9.2 and 9.3, we see that, in addition to the usual expansion (5.6) for each LSπ combination which appear in these figures, a summation must also be carried out over the number of photons n absorbed or emitted and over the number of total orbital angular momenta L retained in the expansion of ${\mbox{$\boldsymbol\psi$}}_k^{\textbf{L}\gamma}$ in (9.17). Hence the value of n _t will be very much larger than that arising in electron–atom collisions in the absence of the laser field, defined following (5.6). The coefficients $a_{nLijk}^{\textbf{L}\gamma}$ and $b_{nLik}^{\textbf{ L}\gamma}$ in (9.20) are determined by diagonalizing the matrix operator $\textbf{H}_{\mathrm{F}}^\textbf{L}+{{\mbox{\boldmath${\cal L}$}}}_{N+1}$ in the basis functions ${\mbox{$\boldsymbol\psi$}}_k^{\textbf{L}\gamma}$ as follows:

$$\langle \Psi_k^{\textbf{L}\gamma}|\textbf{ H}_{\mathrm{F}}^\textbf{L}+{{\mbox{\boldmath${\cal L}$}}}_{N+1}| \Psi_{k^{\prime}}^{\textbf{ L}\gamma}\rangle_{\mathrm{int}}= \textbf{E}_k^{\textbf{ L}\gamma}\delta_{kk^{\prime}},\quad k,\;k^{\prime}=1,\dots,n_t,$$

((9.21))

where $\textbf{H}_{\mathrm{F}}^\textbf{L}$ is the Floquet Hamiltonian matrix defined by (9.16) and ${{\mbox{\boldmath${\cal L}$}}}_{N+1}$ is a Bloch matrix operator which has the following form:

$${{\mbox{\boldmath${\cal L}$}}}_{N+1}=\sum_{i=1}^{N+1}{\frac{1}{2}}\delta (r_i-a_0) \left(\frac{{\mathrm {d}} }{{\mathrm {d}} r_i}-\frac{b_0-1}{r_i}\right)\textbf{I},$$

((9.22))

where I is a unit matrix in photon space. It follows from our analysis in Sect. 5.1.2 that $\textbf{H}_{\mathrm{F}}^{\mathrm{L}}+{{\mbox{\boldmath${\cal L}$}}}_{N+1}$ is hermitian in the internal region in the basis of quadratically integrable vector functions in photon space satisfying arbitrary boundary conditions at $r=a_0$.

Fig. 9.3

Allowed transitions for circularly polarized laser light incident on an atom in a $\hbox{}\mathrm{^1S_0^e}$ state. L is the total orbital angular momentum of the atom and n is the number of photons absorbed, or emitted, corresponding to the Floquet–Fourier expansion index in (9.12). The arrowed lines show the transitions allowed by the laser field

We can now solve (9.15) in the internal region to determine $\boldsymbol\Psi_{jE}^{\textbf{L}\gamma}$, defined by (9.17). We rewrite (9.15) as

$$\left(\textbf{H}_{\mathrm{F}}^\textbf{L}+{{\mbox{\boldmath${\cal L}$}}}_{N+1}-E\textbf{I}\right) \boldsymbol\Psi_{jE}^{\textbf{L}\gamma}={{\mbox{\boldmath${\cal L}$}}}_{N+1}\boldsymbol\Psi_{jE}^{\textbf{L}\gamma},$$

((9.23))

which has the formal solution

$$\boldsymbol\Psi_{jE}^{\textbf{L}\gamma}=\left(\textbf{H}_{\mathrm{F}}^\textbf{L}+{{\mbox{\boldmath${\cal L}$}}}_{N+1}-E\textbf{I}\right)^{-1} {{\mbox{\boldmath${\cal L}$}}}_{N+1}\boldsymbol\Psi_{jE}^{\textbf{L}\gamma}.$$

((9.24))

Using the spectral representation of $\textbf{H}_{\mathrm{F}}^\textbf{L}+{{\mbox{\boldmath${\cal L}$}}}_{N+1}$ given by (9.21), we can rewrite (9.24) as

$$|\boldsymbol\Psi_{jE}^{\textbf{L}\gamma}\rangle=\sum_k |\Psi_k^{\textbf{L}\gamma}\rangle \frac{1}{\textbf{E}_k^{\textbf{L}\gamma}-E\textbf{I}} \langle\Psi_{k}^{\textbf{L}\gamma}|{{\mbox{\boldmath${\cal L}$}}}_{N+1}|\boldsymbol\Psi_{jE}^{\textbf{L}\gamma}\rangle.$$

((9.25))

We then project this equation onto the nth component in photon space and onto the channel functions $\overline{\Phi}_{nLi}^\gamma$ included in expansion (9.20). We obtain after setting $r_{N+1}=a_0$

$$F_{nLij}^{\textbf{ L}\gamma}(a_0)=\sum_{n^{\prime}L^{\prime}i^{\prime}} R_{nLin^{\prime}L^{\prime}i^{\prime}}^{\textbf{L}\gamma}(E) \left(a_0\frac{{\mathrm {d}} F_{n^{\prime}L^{\prime}i^{\prime}j}^{\textbf{ L}\gamma}}{{\mathrm {d}} r} -b_0F_{n^{\prime}L^{\prime}i^{\prime}j}^{\textbf{ L}\gamma}\right)_{r=a_0},$$

((9.26))

where we have introduced the R-matrix $R_{nLin^{\prime}L^{\prime}i^{\prime}}^{\textbf{L}\gamma}(E)$ in the length gauge by

$$R_{nLin^{\prime}L^{\prime}i^{\prime}}^{\textbf{ L}\gamma}(E)= \frac{1}{2a_0}\sum_{k=1}^{n_t}\frac{w_{nLik}^{\textbf{ L}\gamma} w_{n^{\prime}L^{\prime}i^{\prime}k}^{\textbf{L}\gamma}} {E_k^\gamma-E},$$

((9.27))

the reduced radial wave functions $F_{nLij}^{\textbf{L}\gamma}(r)$ defined by

$$F_{nLij}^{\textbf{L}\gamma}(r_{N+1})= \langle r_{N+1}^{-1}\overline{\Phi}_{nLi}^\gamma |\Psi_{njE}^{\textbf{L}\gamma} \rangle^\prime$$

((9.28))

and the surface amplitudes $w_{nLik}^{\textbf{L}\gamma}$ by

$$w_{nLik}^{\textbf{L}\gamma} =\langle r_{N+1}^{-1}\overline{\Phi}_{nLi}^\gamma |\psi_{nk}^{\textbf{L}\gamma} \rangle_{r_{N+1}=a_0}^\prime =\sum_ju_{nLij}^0(a_0)a_{nLijk}^{\textbf{L}\gamma}.$$

((9.29))

As in (5.20) and (5.21), the primes on the Dirac brackets in (9.28) and (9.29) mean that the integrations are carried out over the space and spin coordinates of all ${N+1}$ electrons in the internal region except the radial coordinate $r_{N+1}$ of the ejected or scattered electron. Also, $\Psi_{njE}^{\textbf{L}\gamma}$ in (9.28) are the Floquet–Fourier components of the solution vector $\boldsymbol\Psi_{jE}^{\textbf{ L}\gamma}$ defined by (9.17). We note that if the radial continuum basis orbitals $u_{nLij}^0(r)$, retained in expansion (9.20), satisfy homogeneous boundary conditions at $r=a_0$, then a Buttle correction must be added to the R-matrix defined by (9.27), as discussed in Sect. 5.3.2.

Equations (9.26) and (9.27) are the basic equations describing the solution of (9.26) in the internal region, where (9.26) provides the boundary condition at $r=a_0$ for solving (9.6) in the external region considered in the next section.

1.3 External Region Solution

In the external region, shown in Fig. 9.1, the ejected or scattered electron with radial coordinate $a_0\le r_{N+1}\le a_p$ is described using the velocity gauge, while the remaining N electrons with radial coordinates $r_i\le a_0,i=1,\dots,N$, are described using the length gauge. This is possible since the outer electron and the N inner electrons occupy different regions of space and are distinguishable. Hence their wave functions can be transformed independently. The corresponding unitary transformation of the time-dependent Schrödinger equation (9.6) is given by

$$\widetilde{\Psi}(\textbf{ X}_{N+1},t)=\exp\left[-\frac{{\mathrm {i}}}{{c}}\textbf{A}(t)\cdot\textbf{ R}_{N} -\frac{{\mathrm {i}}}{2{c}^2}\int^t\textbf{A}^2(t^{\prime}){\mathrm {d}} t^{\prime}\right] \widetilde{\Psi}^\textbf{V}(\textbf{X}_{N+1},t),$$

((9.30))

where

$$r_i\leq a_0,\quad i=1,\dots,N,\quad r_{N+1}\ge a_0,$$

((9.31))

and where R _N is defined by (9.10) with $N + 1$ replaced by N. Substituting (9.30) into (9.6) then yields the following time-dependent Schrödinger equation satisfied by $\widetilde{\Psi}^\textbf{V}(\textbf{X}_{N+1},t):$

$$\left(H_{N+1}+ {\mbox{\boldmath${\cal E}$}}(t)\cdot\textbf{R}_{N} +\frac{1}{{c}}\textbf{A}(t)\cdot\textbf{p}_{N+1}\right) \widetilde{\Psi}^\textbf{V}(\textbf{X}_{N+1},t) ={\mathrm {i}}\frac{\partial}{\partial t}\widetilde{\Psi}^\textbf{V}(\textbf{X}_{N+1},t),$$

((9.32))

where the boldface superscript V in (9.30) and (9.32) and later equations indicates that the ejected or scattered electron is described in the velocity gauge.

Following our discussion of the internal region solution we now make a Floquet –Fourier expansion of the wave function $\widetilde{\Psi}^\textbf{V}$ as follows:

$$\widetilde{\Psi}^\textbf{V}(\textbf{X}_{N+1},t)=\exp(-{\mathrm {i}} E^\textbf{V}t) \sum_{n=-\infty}^\infty \exp(-{\mathrm {i}} n\omega t)\Psi_n^{\textbf{V}}(\textbf{X}_{N+1}),$$

((9.33))

where E ^V is the quasi-energy in the velocity gauge. The quasi-energy E ^V has a negative imaginary part for multiphoton ionization and harmonic generation, corresponding to Siegert [876] outgoing wave boundary conditions, discussed in Sect. 1.3, and is real for laser-assisted electron–atom collisions. The relationship between E ^V and the quasi-energy E in (9.12) is given by (9.48) and (9.52). Substituting (9.33) into (9.32) and equating the coefficient of $\exp[-{\mathrm {i}}(E^\textbf{V}+n\omega)t]$ to zero yields the following infinite set of coupled time-independent equations:

$$\left(H_{N+1}-E^\textbf{V}-n\omega\right)\Psi_n^{\textbf{V}} +D_{N}\left(\Psi_{n-1}^{\textbf{V}}+\Psi_{n+1}^{\textbf{V}}\right) +P_{N+1}\left(\Psi_{n-1}^{\textbf{V}}-\Psi_{n+1}^{\textbf{V}}\right)=0,\vspace*{4pt}$$

((9.34))

where the dipole length operator D _N is defined by (9.10) and (9.14) with $N + 1$ replaced by N and the dipole velocity operator $P_{N+1}$ is defined by

$$P_{N+1}={\mathrm {i}}\frac{A_0}{2{c}}\hat{{\mbox{$\boldsymbol\epsilon$}}}\cdot\textbf{p}_{N+1}.$$

((9.35))

In analogy with (9.15) the functions $\Psi_n^{\textbf{V}}$ can be regarded as components of a vector Ψ ^V in photon space and hence (9.34) can be written as a matrix equation in this space as

$$\left(\textbf{H}_{\mathrm{F}}^{\textbf{V}}-E^\textbf{V}\textbf{I}\right) \boldsymbol\Psi^{\textbf{V}}=0,$$

((9.36))

where the Floquet Hamiltonian $\textbf{H}_{\mathrm{F}}^{\textbf{V}}$ is an infinite-dimensional tridiagonal matrix operator in photon space.

1.3.1 Derivation of Coupled Differential Equations

In order to solve (9.36) in the external region we adopt the following close coupling expansion of the components $\Psi_n^{\textbf{V}\gamma}$ of the total wave function at energy E, for each set of conserved quantum numbers denoted by γ,

$$\Psi_{njE}^{\textbf{V}\gamma}(\textbf{X}_{N+1})=\sum_{L i} \overline{\Phi}_{nLi}^\gamma(\textbf{X}_N;\hat{\textbf{r}}_{N+1} \sigma_{N+1})r_{N+1}^{-1}F_{nLij}^{\textbf{V}\gamma}(r_{N+1}), \quad r_{N+1}\geq a_0,$$

((9.37))

where the channel functions $\overline{\Phi}_{nLi}^\gamma$ retained in this expansion are the same as those retained in the internal region expansion (9.20). Also in (9.37), $F_{nLij}^{\textbf{V}\gamma}(r)$ are the reduced radial wave functions in the velocity gauge corresponding to $F_{nLij}^{\textbf{L}\gamma}(r)$ in the length gauge defined by (9.28) and j labels the linearly independent solutions of (9.15) and (9.36). We note that (9.37) is not antisymmetrized with respect to the $(N+1)$th electron since, as pointed out above, this electron now occupies a different region of space than the remaining N electrons and hence is distinguishable. Also, the quadratically integrable functions in (9.20), which are confined to the internal region, are not included in (9.37).

After substituting (9.37) into (9.36) and projecting onto the channel functions $\overline{\Phi}_{nLi}^\gamma$ we obtain the following coupled second-order differential equations, satisfied by the reduced radial functions $F_{nLij}^{\textbf{V}\gamma}(r):$

$$\begin{array}{rcl} &&\left(\frac{{\mathrm {d}} ^2}{{\mathrm {d}} r^2}-\frac{\ell_i(\ell_i+1)}{r^2}+\frac{2(Z-N)}{r} +k_{ni}^2\right) F_{nLij}^{\textbf{V}\gamma}(r)\\ &&\quad=2\sum_{n^{\prime}L^{\prime}i^{\prime}} W_{nLin^{\prime}L^{\prime}i^{\prime}}^{\textbf{V}\gamma} (r) F_{n^{\prime}L^{\prime}i^{\prime}j}^{\textbf{V}\gamma}(r),\quad r\geq a_0, \end{array}$$

((9.38))

where ℓ _i is the orbital angular momentum of the ejected or scattered electron and k _ni ² can be expressed in terms of the channel energies $\overline{e}_i$ of the residual N-electron ion by the equation

$$k_{ni}^2=2\left(E^\textbf{V}-f_{ni}\right),$$

((9.39))

where

$$f_{ni}=\overline{e}_i+n\omega.$$

((9.40))

Also in (9.38), $W_{nLin^{\prime}L^{\prime}i^{\prime}}^{\textbf{ V}\gamma}(r)$ is the long-range potential matrix coupling the channels which can be written in matrix notation as¹

$$\textbf{W}^{\textbf{V}\gamma}=\textbf{V}^{E\gamma}+\textbf{V}^{D\gamma}+\textbf{V}^{P\gamma},$$

((9.41))

where V ^Eγ, V ^Dγ and $\textbf{V}^{P\gamma}$ arise, respectively, from $H_{N+1}$, D _N and $P_{N+1}$ and are defined by the following matrix elements:

$$\begin{array}{rcl} V_{nLin^{\prime}L^{\prime}i^{\prime}}^{E\gamma}&=& \langle r_{N+1}^{-1}\overline{\Phi}_{nLi}^\gamma(\textbf{X}_N;\hat{\textbf{ r}}_{N+1} \sigma_{N+1})\left |\sum_{j=1}^N\frac{1}{r_{jN+1}}-\frac{N}{r_{N+1}}\right| \\ &&\times\;r_{N+1}^{-1}\overline{\Phi}_{n^{\prime}L^{\prime}i^{\prime}}^\gamma(\textbf{ X}_N;\hat{\textbf{r}}_{N+1} \sigma_{N+1})\rangle^{\prime} \delta_{nn^{\prime}}, \end{array}$$

((9.42))

$$\begin{array}{rcl} V_{nLin^{\prime}L^{\prime}i^{\prime}}^{D\gamma}&=& \langle r_{N+1}^{-1}\overline{\Phi}_{nLi}^\gamma(\textbf{X}_N;\hat{\textbf{ r}}_{N+1} \sigma_{N+1})\left |D_N\right|r_{N+1}^{-1} \\ &&\times\;\overline{\Phi}_{n^{\prime}L^{\prime}i^{\prime}}^\gamma(\textbf{ X}_N;\hat{\textbf{r}}_{N+1} \sigma_{N+1})\rangle^{\prime} \left(\delta_{nn^{\prime}-1}+\delta_{nn^{\prime}+1}\right) \end{array}$$

((9.43))

and

$$\begin{array}{rcl} V_{nLin^{\prime}L^{\prime}i^{\prime}}^{P\gamma}&=& \langle r_{N+1}^{-1}\overline{\Phi}_{nLi}^\gamma(\textbf{X}_N;\hat{\textbf{ r}}_{N+1} \sigma_{N+1})\left |{\mathrm {i}}\frac{A_0}{2{c}}\hat{{\mbox{$\boldsymbol\epsilon$}}}\cdot\textbf{p}_{N+1}\right| r_{N+1}^{-1}\\ &&\times\;\overline{\Phi}_{n^{\prime}L^{\prime}i^{\prime}}^\gamma(\textbf{ X}_N;\hat{\textbf{r}}_{N+1} \sigma_{N+1})\rangle^{\prime} \left(\delta_{nn^{\prime}+1}-\delta_{nn^{\prime}-1}\right).\end{array}$$

((9.44))

The integrals in these matrix elements are carried out over all $N + 1$ electron space and spin coordinates except the radial coordinate of the ($N + 1$)th electron.

1.3.2 Boundary Condition at ${r=a_0}$

The boundary condition at $r=a_0$ satisfied by the reduced radial functions $F_{nLij}^{\textbf{V}\gamma}(r)$ in (9.38) can be determined by expressing these functions in terms of the functions $F_{nLij}^{\textbf{L}\gamma}(r)$ in (9.26) at $r=a_0$. The relationship between $F_{nLij}^{\textbf{V}\gamma}(a_0)$ and $F_{nLij}^{\textbf{L}\gamma}(a_0)$ can be obtained from (9.9) and (9.30) which gives

$$\widetilde{\Psi}^\textbf{V}(\textbf{ X}_{N+1},t)=\exp\left[\frac{{\mathrm {i}}}{2{c}^2} \int^t\textbf{ A}^2(t^{\prime}){\mathrm {d}} t^{\prime} -\frac{{\mathrm {i}}}{c}\textbf{A}(t)\cdot\textbf{ r}_{N+1}\right] \widetilde{\Psi}^\textbf{L}(\textbf{X}_{N+1},t),$$

((9.45))

where

$$r_i\leq a_0,\quad i=1,\dots,N,\quad r_{N+1}= a_0.$$

((9.46))

We make use of the explicit form of the vector potential $\textbf{A}(t)$, given by (9.5), to rewrite (9.45) as

$$\begin{array}{rcl} \widetilde{\Psi}^\textbf{V}(\textbf{X}_{N+1},t)&=&\exp\left(\frac{{\mathrm {i}} A_0^2}{4{c}^2}t \right)\exp\left[-\frac{{\mathrm {i}} A_0^2}{8\omega \textrm{c}^2}\sin(2\omega t) -\frac{{\mathrm {i}} A_0}{{c}}\hat{{\mbox{$\boldsymbol\epsilon$}}}\cdot\textbf{r}_{N+1}\sin(\omega t)\right] \\ &&\times\; \widetilde{\Psi}^\textbf{L}(\textbf{X}_{N+1},t), \end{array}$$

((9.47))

where we have taken the lower limit of integration in (9.45) to be zero. The first exponential on the right-hand side of (9.47) has the form $\exp({\mathrm {i}} E_{\mathrm{p}} t)$, where

$$E_{\mathrm{p}}=\frac{A_0^2}{4{c}^2}=\frac{{\cal E}_0^2}{4\omega^2}$$

((9.48))

is the ponderomotive energy of the ejected or scattered electron. This energy is the average kinetic energy of a free electron oscillating in a laser field corresponding to the vector potential defined by (9.5) and the electric field vector defined by (9.4). The second exponential in (9.47) can be expanded in a Fourier series using the equation

$$\exp({\mathrm {i}} z \sin\theta)=\sum_{n=-\infty}^\infty J_n(z)\exp({\mathrm {i}} n \theta),$$

((9.49))

which follows from (C.9) by writing $t=\exp({\mathrm {i}}\theta)$, where the $J_n(z)$ are Bessel functions of the first kind. Substituting the Floquet–Fourier expansions for $\widetilde{\Psi}^\textbf{L}$ and $\widetilde{\Psi}^\textbf{V}$, given by (9.12) and (9.33), respectively, into (9.47) then gives

$$\begin{array}{rcl} &&\exp(-{\mathrm {i}} E^\textbf{V}t)\sum_{n=-\infty}^\infty \exp(-{\mathrm {i}} n\omega t)\Psi_n^{\textbf{V}} (\textbf{X}_{N+1})\\ &&\quad=\exp[-{\mathrm {i}}(E- E_{\mathrm{p}})t]\sum_{\ell=-\infty}^\infty f_{\ell}(A_0,\hat{{\mbox{$\boldsymbol\epsilon$}}}\cdot\textbf{r}_{N+1}) \exp(-{\mathrm {i}} \ell\omega t)\\ &&\quad\quad\times\sum_{n^{\prime}=-\infty}^\infty \exp(-{\mathrm {i}} n^{\prime}\omega t)\Psi_{n^{\prime}}^\textbf{L} (\textbf{X}_{N+1}), \end{array}$$

((9.50))

where

$$f_{\ell}(A_0,\hat{{\mbox{$\boldsymbol\epsilon$}}}\cdot\textbf{r}_{N+1})= \sum_{\ell ^{\prime}=-\infty}^\infty J_{\ell ^{\prime}}\left(\frac{A_0^2}{8\omega {c}^2}\right) J_{\ell-2\ell ^{\prime}}\left(\frac{A_0}{{c}}\hat{{\mbox{$\boldsymbol\epsilon$}}}\cdot\textbf{ r}_{N+1}\right).$$

((9.51))

From (9.50) we deduce that the quasi-energies E and E ^V are related by

$$E^\textbf{V}=E-E_{\mathrm{p}}$$

((9.52))

and that

$$\Psi_n^{\textbf{V}}(\textbf{ X}_{N+1})=\sum_{n^{\prime}=-\infty}^\infty f_{n-n^{\prime}}(A_0,\hat{{\mbox{$\boldsymbol\epsilon$}}}\cdot\textbf{ r}_{N+1})\Psi_{n^{\prime}}^\textbf{L} (\textbf{X}_{N+1}).$$

((9.53))

Projecting (9.53) onto the channel functions $\overline{\Phi}_{nLi}^\gamma$ retained in the internal and external regions in expansions (9.20) and (9.37), evaluating the result on the boundary $r_{N+1}=a_0$ and remembering that $F_{nLij}^{\textbf{L}\gamma}$ and $F_{nLij}^{\textbf{V}\gamma}$ are defined by (9.28) and (9.37), respectively, we obtain

$$F_{nLij}^{\textbf{ V}\gamma}(a_0)=\sum_{n^{\prime}L^{\prime}i^{\prime}} C_{nLin^{\prime}L^{\prime}i^{\prime}}^{\gamma}F_{n^{\prime}L^{\prime}i^{\prime}j}^{\textbf{ L}\gamma}(a_0),$$

((9.54))

where we have introduced the matrix elements

$$\begin{array}{rcl} C_{nLin^{\prime}L^{\prime}i^{\prime}}^{\gamma}&=& \langle r_{N+1}^{-1}\overline{\Phi}_{nLi}^\gamma (\textbf{X}_N;\hat{\textbf{ r}}_{N+1}\sigma_{N+1}) |f_{n-n^{\prime}}(A_0,\hat{{\mbox{$\boldsymbol\epsilon$}}}\cdot\textbf{r}_{N+1})|\\ &&\times\; r_{N+1}^{-1}\overline{\Phi}_{n^{\prime}L^{\prime}i^{\prime}}^{\gamma} (\textbf{X}_N;\hat{\textbf{r}}_{N+1}\sigma_{N+1}) \rangle^{\prime}, \quad r_{N+1}=a_0. \end{array}$$

((9.55))

For notational convenience we rewrite (9.54) in matrix notation as

$$\textbf{F}_j^{\textbf{V}\gamma}(a_0)=\textbf{C}\textbf{F}_j^{\textbf{L}\gamma}(a_0),$$

((9.56))

where $\textbf{F}_j^{\textbf{V}\gamma}(a_0)$ and $\textbf{F}_j^{\textbf{L}\gamma}(a_0)$ are column vectors, for each linearly independent solution j, whose dimensions are the number of coupled channels, and C is an orthogonal matrix.

We can determine the required relation between $\textbf{F}_j^{\textbf{V}\gamma}$ and ${\mathrm {d}}\textbf{F}_j^{\textbf{V}\gamma}/{\mathrm {d}} r$ on the boundary $r=a_0$ by first rewriting (9.26) in matrix notation as

$$\textbf{F}_j^{\textbf{L}\gamma}(a_0)= \textbf{R}^{\textbf{L}\gamma}(E)a_0\left. \frac{{\mathrm {d}} \textbf{F}_j^{\textbf{L}\gamma}}{{\mathrm {d}} r}\right|_{r=a_0},$$

((9.57))

where we have set the arbitrary constant $b_0=0$. Taking the derivative of both sides of (9.56) with respect to r and using (9.57) gives

$$\frac{{\mathrm {d}} \textbf{F}_j^{\textbf{V}\gamma}}{{\mathrm {d}} r}=\left\{\frac{{\mathrm {d}} \textbf{C}}{{\mathrm {d}} r} +a_0^{-1}\textbf{C}\left[\textbf{R}^{\textbf{L}\gamma}(E)\right]^{-1}\right\}\textbf{C}^{-1} \textbf{F}_j^{\textbf{V}\gamma}, \quad r=a_0.$$

((9.58))

This equation can be rewritten as

$$\textbf{F}_j^{\textbf{V}\gamma}(a_0)= \textbf{R}^{\textbf{V}\gamma}(E)a_0\left. \frac{{\mathrm {d}} \textbf{F}_j^{\textbf{V}\gamma}}{{\mathrm {d}} r} \right|_{r=a_0},$$

((9.59))

where

$$\textbf{R}^{\textbf{V}\gamma}(E)=a_0^{-1}\textbf{C}\left\{\frac{{\mathrm {d}} \textbf{C}}{{\mathrm {d}} r} +a_0^{-1}\textbf{C}\left[\textbf{R}^{\textbf{L}\gamma}(E)\right]^{-1}\right\}^{-1}.$$

((9.60))

Equation (9.60) enables the R-matrix $\textbf{R}^{\textbf{ V}\gamma}(E)$ in the velocity gauge at $r=a_0$ to be determined in terms of the R-matrix $\textbf{R}^{\textbf{ L}\gamma}(E)$ in the length gauge at $r=a_0$, and hence provides the boundary condition at $r=a_0$ satisfied by the solution of (9.38) in the external region.

1.3.3 Solution of Coupled Differential Equations

In order to solve the coupled second-order differential equations (9.38) in the external region we rewrite these equations in matrix form as follows:

$$\left(\frac{{\mathrm {d}} ^2}{{\mathrm {d}} r^2} +\textbf{P}\frac{{\mathrm {d}} }{{\mathrm {d}} r}+\textbf{Q}\frac{1}{r} +\textbf{V}(r)+\textbf{D}+\textbf{k}^2\right)\textbf{F}^\textbf{V}(r)=0,$$

((9.61))

where the P and Q terms arise from the $\textbf{V}^{P\gamma}$ term in (9.41), the D term arises from the V ^Dγ term in (9.41) and the $\textbf{V}(r)$ term arises from the V ^Eγ term in (9.41) together with the orbital angular momentum and Coulomb terms on the left-hand side of (9.38), as discussed in Sect. D.2. For notational convenience, we omit the superscripts V and γ on the quantities in this and later equations except for the superscript V on $\textbf{F}^\textbf{V}(r)$ and related functions and on E ^V. For example, $\textbf{R}^{\textbf{V}\gamma}(E)$, defined by (9.60), is written as $\textbf{R}(E)$. Also in (9.61), the diagonal matrix k ², defined by (9.39), is written as

$$\textbf{k}^2=2E^\textbf{V}\textbf{I}-2\textbf{f},$$

((9.62))

where I is the unit matrix, the diagonal matrix f is real and the quasi-energy E ^V is complex for multiphoton ionization and real for laser-assisted electron–atom collisions. The matrices P, Q, $\textbf{V}(r)$ and D are shown in Appendix D.2 to have the following properties using the Fano–Racah phase convention:

$$\begin{array}{rcl} \textbf{P} &\mbox{---}&\mbox{pure imaginary, symmetric, antihermitian, \textit{r}-independent}\\ \textbf{Q} &\mbox{---}&\mbox{pure imaginary, antisymmetric, hermitian, \textit{r}-independent}\\ \textbf{V}&\mbox{---}&\mbox{real, symmetric, hermitian, \textit{r}-dependent}\\ \textbf{D} &\mbox{---}&\mbox{pure imaginary, antisymmetric, hermitian, \textit{r}-independent}. \end{array}$$

We consider first the solution of (9.61) in the external region using the R-matrix propagator method described in Appendix E.5, or an equivalent method, where the first derivative term is non-zero. In order to reduce (9.61) to standard form, we first diagonalize the r-independent terms $\textbf{D}\, + \,\textbf{k}^2$. Since D is hermitian and independent of energy and the quasi-energy E-dependent part of k ², defined by (9.39), is a multiple of the unit matrix, then $\textbf{D}+\textbf{k}^2$ can be diagonalized by a unitary, r- and energy-independent matrix U ₁ giving

$$\textbf{U}_1^\dag\left(\textbf{D}+\textbf{k}^2\right)\textbf{U}_1={{\mbox{\boldmath${\cal K}$}}}^2.$$

((9.63))

Equations (9.61) can then be rewritten as

$$\left(\frac{{\mathrm {d}} ^2}{{\mathrm {d}} r^2} +{\mbox{\boldmath${\cal P}$}}\frac{{\mathrm {d}} }{{\mathrm {d}} r} +{\mbox{\boldmath${\cal V}$}}(r)+{{\mbox{\boldmath${\cal K}$}}}^2\right){\mbox{\boldmath${\cal F}$}}^\textbf{V}(r)=0,$$

((9.64))

where

$$\begin{array}{rcl} {\mbox{\boldmath${\cal P}$}}&=&\textbf{U}_1^\dag\textbf{P}\textbf{U}_1,\\ {\mbox{\boldmath${\cal V}$}}(r)&=&\textbf{U}_1^\dag\left[\textbf{Q}\frac{1}{r}+\textbf{V}(r)\right]\textbf{U}_1,\\ {\mbox{\boldmath${\cal F}$}}^\textbf{V}(r)&=&\textbf{U}_1^\dag\textbf{F}^\textbf{V}(r). \end{array}$$

((9.65))

The unitary transformation (9.63) defines a new target basis which is a linear combination of the original basis and corresponds to target states which are “dressed” by the laser field. The elements of the diagonal matrix ${{\mbox{\boldmath${\cal K}$}}}^2$ are the modified kinetic energies of the ejected or scattered electron corresponding to these dressed states and hence the corresponding wave numbers given by the diagonal elements of ${{\mbox{\boldmath${\cal K}$}}}$ are shifted by the laser field from their original values given by k. In practice only a finite number of terms can be retained in the original Floquet–Fourier expansion (9.8). It is then found that while the shifts in the channel wave numbers given by ${{\mbox{\boldmath${\cal K}$}}}$ corresponding to small n in this expansion are small, the shifts become larger as n tends to the upper and lower limits of n retained in expansion (9.8). In some calculations this shift has been neglected. However, it was shown by Day et al. [252], in a model potential study, that while the results obtained neglecting this shift tend to the same limit as those obtained including this shift as the number of terms retained in the Floquet–Fourier expansion (9.33) tends to infinity, convergence is faster for calculations which use the shifted wave numbers. In the remainder of our analysis we will therefore assume that shifted wave numbers are used.

Equation (9.64) is now in a form that can be solved in the external region using the R-matrix propagator method described in Appendix E.5 where the inhomogeneous term is omitted. In this case, the R-matrix is defined by (E.104) where $\textbf{T}(a_{s-1})$ is omitted and s = 1, corresponding to $r=a_0$. Hence we can write

$${\mbox{\boldmath${\cal F}$}}^\textbf{V}(a_0)={\mbox{\boldmath${\cal R}$}}_0(E) a_0\left(\frac{d{\mbox{\boldmath${\cal F}$}}^\textbf{V}}{dr}+{\frac{1}{2}} {\mbox{\boldmath${\cal P}$}}{\mbox{\boldmath${\cal F}$}}^\textbf{V} \right)_{r=a_0},$$

((9.66))

where ${\mbox{\boldmath${\cal F}$}}^\textbf{V}$ and ${\mbox{\boldmath${\cal P}$}}$ are defined by (9.65). In order to determine the R-matrix ${\mbox{\boldmath${\cal R}$}}_0(E)$ defined by (9.66) at $r=a_0$, we observe that the boundary condition satisfied by $\textbf{F}^\textbf{V}(r)$ at $r=a_0$ is given by (9.59), where the R-matrix $\textbf{R}(E)$ in the velocity gauge at $r=a_0$ is defined in terms of the R-matrix in the length gauge at $r=a_0$ by (9.60). We now substitute for ${\mbox{\boldmath${\cal F}$}}^\textbf{V}(a_0)$ and $(d{\mbox{\boldmath${\cal F}$}}^\textbf{V}/dr)_{r=a_0}$ from (9.65) into (9.66) giving

$$\textbf{U}_1^\dag\textbf{F}^\textbf{V}(a_0)={\mbox{\boldmath${\cal R}$}}_0(E)a_0 \left(\textbf{U}_1^\dag\frac{{\mathrm {d}}\textbf{F}^\textbf{V}}{{\mathrm {d}} r} +{\frac{1}{2}}{\mbox{\boldmath${\cal P}$}}\textbf{U}_1^\dag\textbf{F}^\textbf{V}\right)_{r=a_0}.$$

((9.67))

After substituting for $({\mathrm {d}}\textbf{F}^\textbf{V}/{\mathrm {d}} r)_{r=a_0}$ from (9.59) into (9.67) and re-arranging terms we obtain

$${\mbox{\boldmath${\cal R}$}}_0(E)=\left(\textbf{U}_1^\dag\left[\textbf{R}(E)\right]^{-1}\textbf{U}_1 +{\frac{1}{2}} a_0 {\mbox{\boldmath${\cal P}$}}\right)^{-1},$$

((9.68))

which defines the R-matrix ${\mbox{\boldmath${\cal R}$}}_0(E)$ at $r=a_0$ in terms of the R-matrix $\textbf{R}(E)$ at $r=a_0$, obtained from the solution in the internal region using (9.60). ${\mbox{\boldmath${\cal R}$}}_0(E)$ can then be propagated outwards from $r=a_0$ to a _p , using the propagator method described in Appendix E.5, yielding the R-matrix ${\mbox{\boldmath${\cal R}$}}_p(E)$ at $r=a_p$ which satisfies the equation

$${\mbox{\boldmath${\cal F}$}}^\textbf{V}(a_p)={\mbox{\boldmath${\cal R}$}}_p(E) a_p\left(\frac{d{\mbox{\boldmath${\cal F}$}}^\textbf{V}}{dr}+{\frac{1}{2}} {\mbox{\boldmath${\cal P}$}}{\mbox{\boldmath${\cal F}$}}^\textbf{V} \right)_{r=a_p}.$$

((9.69))

Alternatively, we can eliminate the first derivative term in (9.64) and propagate the resultant R-matrix using the R-matrix propagator method described in Appendix E.1 or an equivalent method. Since P in (9.61) is antihermitian and U ₁ is unitary, it follows from (9.65) that ${\mbox{\boldmath${\cal P}$}}$ is also antihermitian and hence can be diagonalized by a unitary r-independent matrix U ₂ as follows:

$$\textbf{U}_2^\dag{\mbox{\boldmath${\cal P}$}}\textbf{U}_2 =2{\mathrm {i}} \textbf{d},$$

((9.70))

where d is a real, diagonal r-independent matrix. We now introduce a new radial function $\overline{{\mbox{\boldmath${\cal F}$}}}^\textbf{V}(r)$ defined in terms of ${\mbox{\boldmath${\cal F}$}}^\textbf{V}(r)$ by the equation

$$\overline{{\mbox{\boldmath${\cal F}$}}}^\textbf{V}(r)=\exp({\mathrm {i}}\textbf{d}r)\textbf{ U}_2^\dag{\mbox{\boldmath${\cal F}$}}^\textbf{V}(r).$$

((9.71))

Substituting this expression for ${\mbox{\boldmath${\cal F}$}}^\textbf{V}(r)$ into (9.64) then yields the following second-order differential equation without first derivative satisfied by $\overline{{\mbox{\boldmath${\cal F}$}}}^\textbf{V}(r):$

$$\left(\frac{{\mathrm {d}} ^2}{{\mathrm {d}} r^2} + {\mbox{\boldmath${\cal W}$}}(r) + \textbf{d}^2\right) \overline{{\mbox{\boldmath${\cal F}$}}}^\textbf{V}(r)=0,$$

((9.72))

where the potential matrix ${\mbox{\boldmath${\cal W}$}}(r)$ is defined in terms of ${\mbox{\boldmath${\cal V}$}}(r)$ and ${{\mbox{\boldmath${\cal K}$}}}^2$ by

$${\mbox{\boldmath${\cal W}$}}(r)=\exp({\mathrm {i}}\textbf{d}r)\textbf{U}_2^\dag\left({\mbox{\boldmath${\cal V}$}}(r)+{{\mbox{\boldmath${\cal K}$}}}^2\right) \textbf{U}_2\exp(-{\mathrm {i}}\textbf{d}r).$$

((9.73))

The final step is to determine the boundary condition satisfied by the function $\overline{{\mbox{\boldmath${\cal F}$}}}^\textbf{V}(r)$ at $r=a_0$ in terms of the boundary condition satisfied by $\textbf{F}^\textbf{V}(r)$ defined by (9.59). It follows from (9.65) and (9.71) that

$$\textbf{F}^\textbf{V}(r)=\textbf{U}_1\textbf{U}_2\exp(-{\mathrm {i}}\textbf{d}r)\overline{{\mbox{\boldmath${\cal F}$}}}^\textbf{V}(r),$$

((9.74))

and taking the derivative of this equation gives

$$\frac{{\mathrm {d}} \textbf{F}^\textbf{V}}{{\mathrm {d}} r}= \textbf{U}_1\textbf{U}_2\exp(-{\mathrm {i}}\textbf{d}r) \left(\frac{{\mathrm {d}} \overline{{\mbox{\boldmath${\cal F}$}}}^\textbf{V}}{{\mathrm {d}} r}-{\mathrm {i}}\textbf{d}\overline{{\mbox{\boldmath${\cal F}$}}}^\textbf{V} \right).$$

((9.75))

We now substitute for $\textbf{F}^\textbf{V}(r)$ and ${\mathrm {d}} \textbf{F}^\textbf{V}/{\mathrm {d}} r$ at $r=a_0$ into (9.59) yielding the following boundary condition satisfied by $\overline{{\mbox{\boldmath${\cal F}$}}}^\textbf{V}(r):$

$$\overline{{\mbox{\boldmath${\cal F}$}}}^\textbf{V}(a_0)=\overline{{\mbox{\boldmath${\cal R}$}}}(E)a_0\left. \frac{{\mathrm {d}} \overline{{\mbox{\boldmath${\cal F}$}}}^\textbf{V}}{{\mathrm {d}} r} \right|_{r=a_0},$$

((9.76))

where the R-matrix $\overline{{\mbox{\boldmath${\cal R}$}}}(E)$ at $r=a_0$ is defined by

$$\overline{{\mbox{\boldmath${\cal R}$}}}(E)= \left(\textbf{I}+{\mathrm {i}} a_0 \overline{\overline{{\mbox{\boldmath${\cal R}$}}}}(E) \textbf{d}\right)^{-1} \overline{\overline{{\mbox{\boldmath${\cal R}$}}}}(E),$$

((9.77))

and the intermediate matrix $\overline{\overline{{\mbox{\boldmath${\cal R}$}}}}(E)$ is defined by

$$\overline{\overline{{\mbox{\boldmath${\cal R}$}}}}(E)=\exp({\mathrm {i}}\textbf{d}a_0)\textbf{U}_2^\dag\textbf{U}_1^\dag \textbf{R}(E)\textbf{U}_1\textbf{U}_2\exp(-{\mathrm {i}}\textbf{d}a_0).$$

((9.78))

Equation (9.76) defines the boundary condition at $r=a_0$ satisfied by the solution $\overline{{\mbox{\boldmath${\cal F}$}}}^\textbf{V}(r)$ of (9.72), where the R-matrix $\overline{{\mbox{\boldmath${\cal R}$}}}(E)$ is defined in terms of $\textbf{R}(E)$ by (9.77) and (9.78). The R-matrix $\textbf{R}(E)$ at $r=a_0$, defined by (9.60), is determined by the solution in the internal region.

Since the potential ${\mbox{\boldmath${\cal W}$}}(r)$, defined by (9.73), is energy dependent the use of the BBM propagator method, discussed in Appendix E.3, is not appropriate. However, the Light–Walker propagator, discussed in Appendix E.1, or any equivalent method of solving coupled second-order differential equations without first derivative, can be used to propagate the R-matrix from $r=a_0$ to a _p .

1.4 Asymptotic Region Solution in the Velocity Gauge

In the asymptotic region, shown in Fig. 9.1, the ejected or scattered electron with radial coordinate $r_{N+1} \ge a_p$ can be described either in the velocity gauge or in the acceleration frame of reference while the remaining N electrons, with radial coordinates $r_i\leq a_0,\;i=1,\dots,N$, are described in the length gauge. We consider first in this section the asymptotic solution in the velocity gauge which involves modifying the asymptotic expansion used in field-free transitions. We then consider in Sects. 9.1.5 and 9.1.6 two approaches when the solution in the asymptotic region is treated in the acceleration frame of reference. In Sect. 9.1.5 the transformation to the acceleration frame of reference is carried out at a relatively small radius a _p and involves a detailed discussion of the transformation from the velocity gauge to the acceleration frame. In Sect. 9.1.6 the transformation is carried out at a much larger radius a _p which simplifies the analysis. In both cases the wave function describing the ejected or scattered electron is described in a frame of reference where the target states are dressed by the laser field, as described in our discussion following (9.63).

We consider the solution of the coupled second-order differential equations (9.64) when the velocity gauge is adopted. It is convenient first to transform these equations by the r-independent unitary matrix U ₂, defined by (9.70), yielding the coupled second-order differential equations

$$\left(\frac{{\mathrm {d}} ^2}{{\mathrm {d}} r^2}+2{\mathrm {i}}\textbf{d} \frac{{\mathrm {d}} }{{\mathrm {d}} r}+\textbf{W}(r) \right)\textbf{G}^\textbf{V}(r)=0,$$

((9.79))

where we have introduced the reduced radial solution matrix

$$\textbf{G}^\textbf{V}(r)=\textbf{U}_2^\dag {\mbox{\boldmath${\cal F}$}}^\textbf{V}(r)$$

((9.80))

and the potential matrix

$$\textbf{W}(r)=\textbf{U}_2^\dag \left({\mbox{\boldmath${\cal V}$}}(r)+{{\mbox{\boldmath${\cal K}$}}}^2\right)\textbf{U}_2.$$

((9.81))

Also ${\mbox{\boldmath${\cal V}$}}(r)$ and ${{\mbox{\boldmath${\cal K}$}}}^2$ in (9.81) are defined by (9.65) and (9.63), respectively. It follows that we can expand W(r) as

$$\textbf{W}(r)=\sum_{\lambda=0}^{\lambda_{\mathrm{max}}} \textbf{W}_\lambda r^{-\lambda},$$

((9.82))

where $\lambda_{\mathrm{max}}$ is determined by the angular momentum triangular relations satisfied by the potential $\textbf{V}(r)$ in (9.61). Also it follows from (9.63), (9.81) and (9.82) that

$$\textbf{W}_0=\textbf{U}_2^\dag\textbf{U}_1^\dag(\textbf{D}+\textbf{k}^2)\textbf{U}_1\textbf{U}_2,$$

((9.83))

where k ² is defined by (9.62). Hence W ₀ is non-hermitian if E ^V in (9.62) is complex, corresponding to multiphoton ionization, and W ₀ is hermitian if E ^V is real, corresponding to laser-assisted electron–atom collisions. Also the W _λ in (9.82) when $\lambda\ge 1$ do not depend on E ^V and are hermitian.

In Appendix F.2 we derive a complete set of asymptotic solutions of the coupled second-order differential equations (9.79), which we assume here are n _t in number. We obtain 2n _t solutions with the following asymptotic form:

$$\textbf{G}_j^\textbf{V}(r)=\sum_{s=0}^\infty r^{-s} \exp\left({\mathrm {i}} p_j r +{\mathrm {i}}\frac{Z_j}{p_j}\ln 2p_j r\right)\textbf{ A}_j^s, \quad j=1,\dots,2n_t.$$

((9.84))

Physical solutions corresponding to multiphoton ionization or laser-assisted electron–atom collisions are obtained by taking linear combinations of these solutions which satisfy R-matrix boundary conditions at $r=a_p$ and the appropriate asymptotic boundary conditions as $r\rightarrow \infty$.

We also show in Appendix F.2 that the effective momenta p _j and the corresponding vector coefficients $\textbf{A}_j^0$ in (9.84) are determined by solving the 2n _t coupled equations

$$\left(\textbf{I}p_j^2+2\textbf{d}p_j - \textbf{W}_0\right)\textbf{ A}_j^0=0, \;\;j=1,\dots,2n_t,$$

((9.85))

where I is the unit matrix and W ₀ is defined by (9.83). These equations have non-trivial solutions when

$$\det\left(\textbf{I}p_j^2+2\textbf{d}p_j - \textbf{W}_0\right)=0.$$

((9.86))

Expanding this determinant yields a set of algebraic equations of order 2n _t which has 2n _t solutions

$$p_j, \;\;j=1,\dots,2n_t.$$

((9.87))

Substituting for each p _j into (9.85) then gives a set of n _t linear simultaneous equations which enable the n _t components of the vector A _j ⁰ to be determined up to an overall normalization factor. Finally, the effective charges $Z_j,\;j=1,\dots, 2n_t$, and the vectors $\textbf{A}_j^s, j=1,\dots, 2n_t$, for $s\ge 1$ in (9.84) are determined from the recurrence relations derived in Appendix F.2.

We consider first the solution of (9.79) corresponding to multiphoton ionization. In this case we have to find the solutions of (9.86) when W ₀ is non-hermitian. We observe that in the limit when the laser field strength is zero then (9.86) reduces to

$$\det\left(\textbf{I}p_j^2- \textbf{k}^2\right)=0,$$

((9.88))

which follows from (9.83) since the unitary matrices U ₁ and U ₂ are both equal to the unit matrix and $\textbf{D}=0$. It then follows that if an element k _j ² of the diagonal matrix k ² satisfies $k_j^2\ge 0$ then the corresponding effective momentum $p_j=\pm k_j$. On the other hand, if an element k _j ² satisfies $k_j^2< 0$ then the corresponding effective momentum $p_j=\pm {\mathrm {i}} |k_j|$. Hence, the solutions of (9.88) either appear in pairs on the real momentum axis or they appear in pairs in the complex momentum plane where one is the complex conjugate of the other. We can then show that when the laser field is switched on the two solutions corresponding to $k_j^2\ge 0$ move off the real momentum axis, one into the upper half complex momentum plane and the other into the lower half complex momentum plane. Also, the two solutions corresponding to $k_j^2< 0$ move in the complex momentum plane but are no longer complex conjugates of each other. Hence, when the laser field is non-zero we find an equal number of solutions of (9.86) in the upper and lower halves of the complex momentum plane. The 2n _t solutions of (9.86) can therefore be written as

$$\begin{array}{rcl} p_j&=&a_j+{\mathrm {i}} b_j,\quad j=1,\dots, n_t,\\ p_j&=&c_j-{\mathrm {i}} d_j,\quad j=n_t+1,\dots, 2n_t, \end{array}$$

((9.89))

where a _j , b _j , c _j and d _j are all real and where b _j and d _j are both positive. The general solution of (9.79) can then be written as a linear combination of the following 2n _t solutions defined by (9.89) with the asymptotic form

$$\begin{array}{rcl} G_{ij}^\textbf{V}(r) &\mbox{\raisebox{-2.mm} {$\stackrel{\sim}{\scriptstyle r\rightarrow \infty }$}}& \exp\left({\mathrm {i}} p_j r +{\mathrm {i}}\frac{Z_j}{p_j}\ln 2p_j r\right)A_{ij}^0 +O(r^{-1}),\\ &&\;\;\quad\quad\quad\quad i=1,\dots,n_t,\;j=1,\dots,2n_t, \end{array}$$

((9.90))

where it is convenient to normalize the solutions by requiring that

$$\sum_{i=1}^{n_t}|A_{ij}^0|^2=1,\quad j=1,\dots,2n_t.$$

((9.91))

It follows from (9.89) that the first n _t solutions $G_{ij}^\textbf{V}(r),\;j=1,\dots,n_t$, correspond to ingoing waves and the last n _t solutions $G_{ij}^\textbf{ V}(r),\;j=n_t+1,\dots,2n_t$, correspond to outgoing wave solutions. The required solution $G_i^{\mathrm{MI}}(r)$ corresponding to multiphoton ionization is then a linear combination of the outgoing wave solutions

$$G_i^{\mathrm{MI}}(r)= \sum_{j=n_t+1}^{2n_t}G_{ij}^\textbf{V}(r)c_j,\quad i=1,\dots, n_t.$$

((9.92))

In order to determine the coefficients $c_j,\;j=n_t+1,\dots,2n_t$, in (9.92) we substitute $G_i^{\mathrm{MI}}(r)$ into the equation obtained by propagating the R-matrix in the velocity gauge, defined at $r=a_0$ by (9.66), from $r=a_0$ to a _p , as described in Appendix E.5. We then find, after using (9.70) and (9.80), that $G_i^{\mathrm{MI}}(r)$ satisfies the equation

$$\textbf{G}^{\mathrm{MI}}(a_p)=\widetilde{{\mbox{\boldmath${\cal R}$}}}_p(E)a_p \left(\frac{{\mathrm {d}}\textbf{G}^{\mathrm{MI}}}{{\mathrm {d}} r}+{\mathrm {i}}\textbf{d}\textbf{G}^{\mathrm{MI}} \right)_{r=a_p},$$

((9.93))

where $\widetilde{{\mbox{\boldmath${\cal R}$}}}_p(E)$ in this equation is related to ${\mbox{\boldmath${\cal R}$}}_p(E)$ resulting from propagating the R-matrix from $r=a_0$ to a _p by

$$\widetilde{{\mbox{\boldmath${\cal R}$}}}_p(E)=\textbf{U}_2^\dag {\mbox{\boldmath${\cal R}$}}_p(E) \textbf{U}_2.$$

((9.94))

Substituting for $\textbf{G}^{\mathrm{MI}}(a_p)$, given by (9.92), into (9.93) then yields a set of n _t linear homogeneous simultaneous equations satisfied by the coefficients $c_j, j=n_t+1,\dots,2n_t$, which will only have a non-trivial solution when the complex quasi-energy E ^V in (9.62) corresponds to a Siegert [876] outgoing wave solution of (9.79). In order to determine this solution, an iterative procedure can be adopted analogous to that used in determining the initial bound-state energy in photoionization, described in Sect. 8.1.2. The complex quasi-energy E ^V, corresponding to the solution of (9.93), can then be written as

$$E^\textbf{V}=E_0+\Delta-{\frac{1}{2}}{\mathrm {i}}\Gamma,$$

((9.95))

where E ₀ is the field-free energy of the target atom, Δ is the dynamic Stark shift and Γ is the total multiphoton ionization rate. The corresponding Siegert outgoing wave solution defined by (9.90) and (9.92) then has the asymptotic form

$$G_i^{\mathrm{MI}}(r) \mbox{\raisebox{-2.mm} {$\stackrel{\sim}{\scriptstyle r\rightarrow \infty }$}} \sum_{j=n_t+1}^{2n_t} \exp\left({\mathrm {i}} p_j r +{\mathrm {i}}\frac{Z_j}{p_j}\ln 2p_j r\right)A_{ij}^0c_j +O(r^{-1}),\;\;i=1,\dots,n_t,\vspace*{4pt}$$

((9.96))

where p _j and Z _j depend on the quasi-energy E ^V. The branching ratios of the ejected electron can then be obtained from the coefficients $A_{ij}^0$ and c _j . Finally, we note that our discussion leading to (9.96) also defines the asymptotic form of the wave function in harmonic generation which we will consider in Sect. 9.1.7.

We consider next the solution of (9.79) corresponding to laser-assisted electron–atom collisions. In this case W ₀ is hermitian and hence if the effective momentum p _j is a solution of (9.85) then the complex conjugate p _j ^* is also a solution. It follows that the solutions of (9.85) are either real or occur in $n_t-n_a$ complex conjugate pairs. We can, therefore, write the effective momenta as follows:

$$p_j=-a_j,\quad j=1,\dots, n_a,$$

((9.97))

$$p_j = c_j-{\mathrm {i}} d_j,\quad j=n_a+1,\dots, n_t,$$

((9.98))

$$p_{n_t+j} = b_j,\quad j=1,\dots, n_a,$$

((9.99))

$$p_{n_t+j} = c_j+{\mathrm {i}} d_j,\quad j=n_a+1,\dots, n_t,$$

((9.100))

where a _j , b _j , c _j and d _j are all real and where a _j , b _j and d _j are positive. We note that this division between real and complex and also between positive and negative solutions follows from the solutions of (9.88) when the laser field is switched on.

We can now write the general solution of (9.79) as a linear combination of the $n_t+n_a$ solutions defined by (9.97), (9.99) and (9.100) where the solutions corresponding to (9.98) are excluded since they diverge asymptotically and hence are non-physical. We can therefore define n _a ingoing wave solutions

$$G_{ij}^{\mathrm{I}}(r)=(-p_j)^{-{1/2}}G_{ij}^\textbf{ V}(r),\;\;i=1,\dots,n_t,\; j=1,\dots,n_a,$$

((9.101))

n _a outgoing wave solutions

$$G_{ij}^{\mathrm{O}}(r)=(p_{n_t+j})^{-{1/2}}G_{in_t+j}^\textbf{ V}(r),\;\;i=1,\dots,n_t,\; j=1,\dots,n_a,$$

((9.103))

and $n_b=n_t-n_a$ decaying wave solutions

$$G_{ij}^{\mathrm{O}}(r)=G_{in_t+j}^\textbf{V}(r),\;\;i=1,\dots,n_t,\; j=n_a+1,\dots,n_t.$$

((9.103))

In (9.101) and (9.102) we have normalized the solutions to unit ingoing and outgoing wave fluxes, respectively. The general solution of (9.79) corresponding to laser-assisted electron–atom collisions can then be written in terms of these solutions in analogy with our discussion of electron–atom collisions in Sect. 5.1.4. We obtain

$$\textbf{G}^{\mathrm{C}}(r)=\textbf{G}^{\mathrm{I}}(r)-\textbf{G}^{\mathrm{O}}(r)\textbf{H},$$

((9.104))

where $\textbf{G}^{\mathrm{I}}(r)$ has dimension $n_t\times n_a$, $\textbf{G}^{\mathrm{O}}(r)$ has dimension $n_t\times n_t$ and H has dimension $n_t\times n_a$. Also, H is defined by

$$\textbf{H}=\left[ \begin{array}{c} \textbf{S}\\ \textbf{M} \end{array}\right],$$

((9.105))

where S is the $n_a\times n_a$-dimensional S-matrix which multiplies the outgoing wave solutions $G_{ij}^{\mathrm{O}}(r)$, defined by (9.102), and M is the $n_b\times n_a$-dimensional subsidiary matrix which multiplies the decaying wave solutions $G_{ij}^{\mathrm{O}}(r)$, defined by (9.103). Hence (9.104) reduces asymptotically to

$$\textbf{G}^{\mathrm{C}}(r) \mbox{\raisebox{-2.mm} {$\stackrel{\sim}{\scriptstyle r\rightarrow \infty }$}} \textbf{G}^{\mathrm{I}}(r)-\textbf{G}^{\mathrm{O}}(r)\textbf{S},$$

((9.106))

where $\textbf{G}^{\mathrm{I}}(r)$ and $\textbf{G}^{\mathrm{O}}(r)$ are the $n_t\times n_a$ asymptotic solution matrices defined by (9.101) and (9.102), respectively. This equation is analogous to (5.48) which defines the S-matrix in electron collisions with atoms and ions.

The final step in determining the S-matrix in (9.106) follows our analysis in multiphoton ionization leading to (9.93). The R-matrix in the velocity gauge is propagated from $r=a_0$ to a _p . The solution $\textbf{G}^{\mathrm{C}}(r)$, defined by (9.104), then satisfies the equation

$$\textbf{G}^{\mathrm{C}}(a_p)=\widetilde{{\mbox{\boldmath${\cal R}$}}}_p(E)a_p \left(\frac{{\mathrm {d}}\textbf{G}^{\mathrm{C}}}{{\mathrm {d}} r}+{\mathrm {i}}\textbf{d}\textbf{G}^{\mathrm{C}} \right)_{r=a_p},$$

((9.107))

where $\widetilde{{\mbox{\boldmath${\cal R}$}}}_p(E)$ is defined by (9.94). We can then determine the S-matrix in (9.106) by substituting (9.104), evaluated at $r=a_p$, into (9.107) which yields a set of n _t coupled linear simultaneous equations with n _a right-hand sides. The matrix H, and hence the S-matrix, is then determined from the solution of these equations, and the solution of (9.79) corresponding to laser-assisted electron–atom collisions is then given by (9.104).

1.5 Asymptotic Region Solution in the Acceleration Frame

The transformation of the wave function describing the ejected or scattered electron from the velocity gauge, adopted in the external region in Sect. 9.1.3, to the acceleration frame of reference is accomplished using the following Kramers–Henneberger transformation [547, 453]:

$$\widetilde{\Psi}^\textbf{V}(\textbf{X}_{N+1},t)= \exp\left[-\frac{{\mathrm {i}}}{{c}}\textbf{p}_{N+1}\cdot\int^t\textbf{ A}(t^{\prime}){\mathrm {d}} t^{\prime}\right] \widetilde{\Psi}^\textbf{A}(\textbf{ X}_{N+1},t),$$

((9.108))

where

$$r_i\leq a_0,\quad i=1,\dots,N,\quad r_{N+1}\ge a_p.$$

((9.109))

The boldface superscript A in (9.108) and later equations indicates that the functions describing the ejected or scattered electron are defined in the acceleration frame of reference. We now substitute (9.108) into the time-dependent Schrödinger equation (9.32) satisfied by the solution in the external region, and multiply this equation on the left by $\exp[{\mathrm {i}}{\mbox{$\boldsymbol\alpha$}}(t)\cdot\textbf{p}_{N+1}]$. After using the operator identity

$$\exp[{\mathrm {i}} {\mbox{$\boldsymbol\alpha$}}(t)\cdot\textbf{p}][f(\textbf{r})g(\textbf{r})]= f[\textbf{r}+{\mbox{$\boldsymbol\alpha$}}(t)]\exp[{\mathrm {i}}{\mbox{$\boldsymbol\alpha$}}(t)\cdot\textbf{p}]g(\textbf{r}),$$

((9.110))

where f(r) and g(r) are analytic functions of r, we obtain the following time-dependent Schrödinger equation satisfied by the wave function $\widetilde{\Psi}^\textbf{A}(\textbf{X}_{N+1},t):$

$$\left[\widetilde{H}_{N+1}^\textbf{A}(t)+ {\mbox{\boldmath${\cal E}$}}(t)\cdot\textbf{R}_{N}\right] \widetilde{\Psi}^\textbf{A}(\textbf{X}_{N+1},t) ={\mathrm {i}}\frac{\partial}{\partial t}\widetilde{\Psi}^\textbf{A}(\textbf{X}_{N+1},t).$$

((9.111))

The time-dependent Hamiltonian $\widetilde{H}_{N+1}^\textbf{A}(t)$ in (9.111) is defined by

$$\widetilde{H}_{N+1}^\textbf{A}(t)=H_N-{\frac{1}{2}}\nabla_{N+1}^2 -\frac{Z}{|\textbf{r}_{N+1}+{\mbox{$\boldsymbol\alpha$}}(t)|} +\sum_{i=1}^N\frac{1}{|\textbf{r}_{N+1}+{\mbox{$\boldsymbol\alpha$}}(t)-\textbf{r}_i|},$$

((9.112))

where the tilde on $\widetilde{H}_{N+1}^\textbf{A}(t)$ indicates that it is time dependent. Also in (9.111) and (9.112) ${\mbox{\boldmath${\cal E}$}}(t)$ is defined by (9.4) and

$${\mbox{$\boldsymbol\alpha$}}(t)=\frac{1}{{c}}\int^t\textbf{ A}(t^{\prime}){\mathrm {d}} t^{\prime} =\hat{{\mbox{$\boldsymbol\epsilon$}}}\alpha(t)=\hat{{\mbox{$\boldsymbol\epsilon$}}}\alpha_0\cos\omega t,$$

((9.113))

where $\alpha_0={\cal E}_0\omega^{-2}$. By expanding the last two terms in (9.112) in powers of α(t) we can write this equation as [184]

$$\widetilde{H}_{N+1}^\textbf{A}(t)=H_{N+1}+\widetilde{A}_{N+1}(t),$$

((9.114))

where

$$\begin{array}{rcl} \widetilde{A}_{N+1}(t)&=&\alpha(t)\left[\frac{Z-N}{r_{N+1}^2}\cos\theta_{N+1} +\frac{1}{r_{N+1}^3}\sum_{i=1}^N r_i\cos\theta_i\right. \\ &&\left. -\;\frac{3}{r_{N+1}^3}\cos\theta_{N+1}\sum_{i=1}^N r_i\cos\phi_i + O\left(r_{N+1}^{-4}\right)\right] \\ &&+\;O\left[\alpha(t)^2r_{N+1}^{-3}\right], \end{array}$$

((9.115))

and where

$$\cos\theta_i=\hat{\textbf{r}}_i\cdot\hat{{\mbox{$\boldsymbol\epsilon$}}}, \quad i=1,\dots,N+1$$

((9.116))

and

$$\cos\phi_i=\hat{\textbf{r}}_i\cdot \hat{\textbf{r}}_{N+1}, \quad i=1,\dots,N.$$

((9.117))

In practice, all terms in $\widetilde{A}_{N+1}(t)$, except the leading term behaving as $\alpha(t)r_{N+1}^{-2}$, can often be neglected. In Sect. 9.1.6 we consider a simplified analysis in the acceleration frame where the propagation in the velocity gauge is carried out to a sufficiently large radius $r=a_p$ so that the leading term $\alpha(t)r_{N+1}^{-2}$ can also be neglected.

Following the above analysis, (9.111) can be written as

$$\left[H_{N+1}+ {\mbox{\boldmath${\cal E}$}}(t)\cdot\textbf{R}_{N} +\widetilde{A}_{N+1}(t)\right] \widetilde{\Psi}^\textbf{A}(\textbf{X}_{N+1},t) ={\mathrm {i}}\frac{\partial}{\partial t}\widetilde{\Psi}^\textbf{A}(\textbf{X}_{N+1},t),$$

((9.118))

which describes the motion of the ejected or scattered electron in a frame of reference in which this electron is oscillating in the laser field, while the remaining N target electrons, which are bound to the nucleus, are described in the length gauge. This representation has the advantage that the time-dependent Hamiltonian $\widetilde{H}_{N+1}^\textbf{A}(t)$, defined by (9.114), (9.115), (9.116) and (9.117), reduces to the field-free Hamiltonian $H_{N+1}$ when the radial coordinate $r_{N+1}$ of the ejected or scattered electron tends to infinity, enabling simple asymptotic boundary conditions for this electron to be introduced.

In order to solve (9.118) in the asymptotic region we proceed as in Sect. 9.1.3 by introducing the Floquet–Fourier expansion

$$\widetilde{\Psi}^\textbf{A}(\textbf{X}_{N+1},t)=\exp\left(-{\mathrm {i}} E^\textbf{A}t\right) \sum_{n=-\infty}^\infty \exp(-{\mathrm {i}} n\omega t)\Psi_n^\textbf{A}(\textbf{X}_{N+1}),$$

((9.119))

where E ^A is the quasi-energy in the acceleration frame. Substituting (9.119) into (9.118) and equating the coefficient of $\exp[-{\mathrm {i}}(E^\textbf{A}+n\omega)t]$ to zero yields the infinite set of time-independent equations

$$\left(H_{N+1}-E^\textbf{A}-n\omega\right)\Psi_n^\textbf{A} +D_{N}\left(\Psi_{n-1}^\textbf{A}+\Psi_{n+1}^\textbf{A}\right) +A_{N+1}\left(\Psi_{n-1}^\textbf{A}+\Psi_{n+1}^\textbf{A}\right)=0,$$

((9.120))

where the acceleration term

$$\begin{array}{rcl} A_{N+1}&=&{\frac{1}{2}}\alpha_0\left[\frac{Z-N}{r_{N+1}^2}\cos\theta_{N+1} +\frac{1}{r_{N+1}^3}\sum_{i=1}^N r_i\cos\theta_i\right. \\ &&\left. -\;\frac{3}{r_{N+1}^3}\cos\theta_{N+1}\sum_{i=1}^N r_i\cos\phi_i + O(r_{N+1}^{-4})\right], \end{array}$$

((9.121))

and we have omitted terms of $O[\alpha(t)^2r_{N+1}^{-3}]$ in (9.10) which only contribute at high laser intensities. Also in (9.120) D _N is defined by (9.10) and (9.14) with $N + 1$ replaced by N. We see that (9.120) has the same general form as the corresponding equation (9.34) obtained using the velocity gauge in the external region except that the last term in (9.34), involving the dipole velocity operator $P_{N+1}$, is now replaced by the last term in (9.120), involving the acceleration term $A_{N+1}$. We then rewrite (9.120) as a matrix equation in photon space as follows:

$$\left(\textbf{H}_{\mathrm{F}}^{\textbf{A}}-E^\textbf{A}\textbf{I}\right) \boldsymbol\Psi^{\textbf{A}}=0,$$

((9.122))

where the Floquet Hamiltonian $\textbf{H}_{\mathrm{F}}^{\textbf{A}}$ is an infinite-dimensional matrix in this space. Also we introduce the following close coupling expansion for the components $\Psi_n^{\textbf{A}\gamma}$ of the total wave function at energy E for each set of conserved quantum numbers denoted by γ

$$\Psi_{njE}^{\textbf{A}\gamma}(\textbf{X}_{N+1})=\sum_{L i} \overline{\Phi}_{nLi}^\gamma(\textbf{X}_N;\hat{\textbf{r}}_{N+1} \sigma_{N+1})r_{N+1}^{-1}F_{nLij}^{\textbf{A}\gamma}(r_{N+1}), \quad r_{N+1}\geq a_p,$$

((9.123))

where the channel functions $\overline{\Phi}_{nLi}^\gamma$ retained in this expansion are the same as those retained in the internal and external region expansions (9.20) and (9.37) and where j labels the linearly independent solutions. After substituting (9.123) into (9.122) and projecting onto the channel functions $\overline{\Phi}_{nLi}^\gamma$ we obtain the following coupled second-order differential equations satisfied by the reduced radial wave functions $F_{nLij}^{\textbf{A}\gamma}(r)$:

$$\begin{array}{rcl} &&\left(\frac{{\mathrm {d}} ^2}{{\mathrm {d}} r^2}-\frac{\ell_i(\ell_i+1)}{r^2}+\frac{2(Z-N)}{r} +k_{ni}^2\right)F_{nLij}^{\textbf{A}\gamma}(r)\\ &&\quad=2\sum_{n^{\prime}L^{\prime}i^{\prime}} W_{nLin^{\prime}L^{\prime}i^{\prime}}^{\textbf{A}\gamma}(r) F_{n^{\prime}L^{\prime}i^{\prime}j}^{\textbf{A}\gamma}(r),\quad r\geq a_p, \end{array}$$

((9.124))

where

$$k_{ni}^2=2\left(E^\textbf{A}-f_{ni}\right),$$

((9.125))

and where f _ni is defined by (9.40). Also in (9.124), $W_{nLin^{\prime}L^{\prime}i^{\prime}}^{\textbf{ A}\gamma}(r)$ is the potential matrix coupling the channels which can be written in matrix notation as

$$\textbf{W}^{\textbf{A}\gamma}=\textbf{V}^{E\gamma}+\textbf{V}^{D\gamma}+\textbf{V}^{A\gamma}.$$

((9.126))

The V ^Eγ and V ^Dγ terms arise, respectively, from the $H_{N+1}$ and D _N terms in (9.120), which are the same as those found in the velocity gauge in (9.41), and the V ^Aγ term arises from the acceleration term $A_{N+1}$ in (9.120) and is defined by the matrix elements

$$\begin{array}{rcl} V_{nLin^{\prime}L^{\prime}i^{\prime}}^{A\gamma}&=& \langle r_{N+1}^{-1}\overline{\Phi}_{nLi}^\gamma (\textbf{X}_N;\hat{\textbf{ r}}_{N+1}\sigma_{N+1})\left|A_{N+1}\right|r_{N+1}^{-1} \\ &&\times\;\overline{\Phi}_{n^{\prime}L^{\prime}i^{\prime}}^\gamma (\textbf{X}_N;\hat{\textbf{r}}_{N+1}\sigma_{N+1})\rangle^{\prime} (\delta_{nn^{\prime}-1}+\delta_{nn^{\prime}+1}). \end{array}$$

((9.127))

Finally, we note that the integrals in (9.127) are carried out over the space and spin coordinates of all $N + 1$ electrons except the radial coordinate of the ($N + 1$)th electron.

1.5.1 Boundary Condition at $r=a_p$

In order to solve (9.124) in the asymptotic region we must determine the boundary condition satisfied by the solution $F_{nLij}^{\textbf{A}\gamma}(r)$ at $r=a_p$. This boundary condition can be determined by matching the solution of (9.38) in the velocity gauge with the solution of (9.124) in the acceleration frame at $r=a_p$. This can be achieved by substituting the Floquet–Fourier expansions for $\widetilde{\Psi}^\textbf{V}(\textbf{ X}_{N+1},t)$ and $\widetilde{\Psi}^\textbf{A}(\textbf{X}_{N+1},t)$ given by (9.33) and (9.119), respectively, into the Kramers–Henneberger transformation (9.108). We obtain

$$\begin{array}{rcl} &&\exp\left(-{\mathrm {i}} E^\textbf{A}t\right)\sum_{n=-\infty}^\infty \exp(-{\mathrm {i}} n\omega t)\Psi_n^\textbf{A}(\textbf{X}_{N+1})\\ &&\quad=\exp\left\{-{\mathrm {i}} \left[E^\textbf{V}t-{\mbox{$\boldsymbol\alpha$}}(t)\cdot \textbf{ p}_{N+1}\right]\right\} \sum_{n^{\prime}=-\infty}^\infty\exp\left(-{\mathrm {i}} n^{\prime}\omega t\right) \Psi_{n^{\prime}}^{\textbf{V}}(\textbf{X}_{N+1}),\quad\quad \end{array}$$

((9.128))

where we have used (9.113) to rewrite the integral in the exponent on the right-hand side of (9.108). We then substitute the close coupling expansions (9.37) and (9.123) for $\Psi_{njE}^{\textbf{V}\gamma}$ and $\Psi_{njE}^{\textbf{A}\gamma}$, respectively, into (9.128) giving

$$\begin{array}{rcl} &&\exp\left(-{\mathrm {i}} E^\textbf{A}t\right)\sum_{nLi}\exp(-{\mathrm {i}} n\omega t) \overline{\Phi}_{nLi}^\gamma(\textbf{X}_N;\hat{\textbf{r}}_{N+1} \sigma_{N+1})r_{N+1}^{-1}F_{nLij}^{\textbf{A}\gamma}(r_{N+1})\\ &&=\exp\left\{-{\mathrm {i}} \left[E^\textbf{V}t-{\mbox{$\boldsymbol\alpha$}}(t)\cdot \textbf{ p}_{N+1}\right]\right\} \sum_{n^{\prime}L^{\prime}i^{\prime}}\exp\left(-{\mathrm {i}} n^{\prime}\omega t\right) \overline{\Phi}_{n^{\prime}L^{\prime}i^{\prime}}^{\gamma}(\textbf{ X}_N;\hat{\textbf{r}}_{N+1}\sigma_{N+1}) \\ &&\quad\times \;r_{N+1}^{-1}F_{n^{\prime}L^{\prime}i^{\prime}j}^{\textbf{ V}\gamma}(r_{N+1}). \end{array}$$

((9.129))

In order to analyse (9.129), we consider the following expression that appears on the right-hand side of this equation:

$$I=\exp[{\mathrm {i}}{\mbox{$\boldsymbol\alpha$}}(t)\cdot \textbf{p}_{N+1}] \overline{\Phi}_{n^{\prime}L^{\prime}i^{\prime}}^{\gamma}(\textbf{ X}_N;\hat{\textbf{r}}_{N+1} \sigma_{N+1})r_{N+1}^{-1}F_{n^{\prime}L^{\prime}i^{\prime}j}^{\textbf{ V}\gamma}(r_{N+1}).$$

((9.130))

We expand the channel function in (9.130) as follows:

$$\begin{array}{rcl} &&\overline{\Phi}_{n^{\prime}L^{\prime}i^{\prime}}^{\gamma}(\textbf{X}_N;\hat{\textbf{r}}_{N+1}\sigma_{N+1})\\ &&=\sum_{M_{L_{i^{\prime}}}m_{\ell_{i^{\prime}}}}(L_{i^{\prime}}M_{L_{i^{\prime}}}\ell_{i^{\prime}}m_{\ell_{i^{\prime}}}|L^{\prime}M_{L^{\prime}}) \:\overline{\Theta}_{n^{\prime}L_{i^{\prime}}M_{L_{i^{\prime}}}i^{\prime}}^{\gamma}(\textbf{X}_N;\sigma_{N+1})\\ &&\quad\times\;Y_{\ell_{i^{\prime}}m_{\ell_{i^{\prime}}}}(\theta_{N+1},\phi_{N+1}), \end{array}$$

((9.131))

where $\overline{\Theta}_{n^{\prime}L^{\prime}M_{L_{i^{\prime}}}i^{\prime}}^{\gamma}(\textbf{ X}_N;\sigma_{N+1})$ are reduced channel functions in which the N-electron state is coupled to the spin state but not to the orbital angular momentum state of the scattered or ejected electron. Equation (9.130) then becomes

$$\begin{array}{rcl} I&=&\exp[{\mathrm {i}}{\mbox{$\boldsymbol\alpha$}}(t)\cdot \textbf{p}_{N+1}] \sum_{M_{L_{i^{\prime}}}m_{\ell_{i^{\prime}}}} (L_{i^{\prime}}M_{L_{i^{\prime}}}\ell_{i^{\prime}}m_{\ell_{i^{\prime}}}|L^{\prime}M_{L^{\prime}}) \overline{\Theta}_{n^{\prime}L_{i'}M_{L_{i^{\prime}}}i'}^{\gamma}(\textbf{X}_N;\sigma_{N+1})\\ &&\times\;Y_{\ell_{i^{\prime}}m_{\ell_{i'}}}(\theta_{N+1},\phi_{N+1}) r_{N+1}^{-1}F_{n^{\prime}L^{\prime}i'j}^{\textbf{V}\gamma}(r_{N+1}). \end{array}$$

((9.132))

We can evaluate this expression for I using the operator identity

$$\exp[{\mathrm {i}}{\mbox{$\boldsymbol\alpha$}}(t)\cdot \textbf{p}]f(\textbf{r})=f[\textbf{r}+{\mbox{$\boldsymbol\alpha$}}(t)] \equiv f(\textbf{r}_\alpha),$$

((9.133))

which follows from (9.110) by setting $g(\textbf{r})$ equal to unity. Hence the action of the operator $\exp[{\mathrm {i}}{\mbox{$\boldsymbol\alpha$}}(t)\cdot \textbf{p}]$ on $f(\textbf{r})$ is to yield the function $f[\textbf{r}+{\mbox{$\boldsymbol\alpha$}}(t)]$ centred on a new origin. This is illustrated in Fig. 9.4 where we have chosen the z-axis to lie along the laser polarization direction α and where we have defined $\textbf{r}_\alpha(t)=\textbf{r}+{\mbox{$\boldsymbol\alpha$}}(t)$. The displaced origin in this figure is denoted by A and the original origin is denoted by O. The two coordinate systems are thus related by the following equations:

$$\begin{array}{rcl} r_\alpha^2=\alpha^2+r^2+2\alpha r\cos\theta, &&r_\alpha\sin\theta_\alpha=r\sin\theta,\\ r_\alpha\cos\theta_\alpha=r\cos\theta+\alpha, && \phi_\alpha = \phi, \end{array}$$

((9.134))

where in this equation and the following equations we observe that α, r _α and θ _α are functions of time t.

Fig. 9.4

Coordinate system representing the action of the operator $\exp({\mathrm {i}}{\mbox{$\boldsymbol\alpha$}}\cdot \textbf{p})$

In order to evaluate the expression I, defined by (9.132), we follow a procedure similar to that adopted by Harris and Michels [442], who determined an expansion for molecular orbitals on a displaced centre expanded about the original centre. We consider an orbital $u_{n\ell m}(\textbf{r}_\alpha)$ centred on A which we wish to expand about O in Fig. 9.4. We assume that this orbital can be written in terms of spherical harmonics, defined in Appendix B.3, as follows:

$$u_{n\ell m}(\textbf{r}_\alpha)=u_{n\ell}(r_\alpha) Y_{\ell m}(\theta_\alpha,\phi_\alpha),$$

((9.135))

where n represents all the quantum numbers except ℓ and m necessary to define the orbital. We then expand $u_{n\ell m}(\textbf{r}_\alpha)$ about O in Fig. 9.4 as follows:

$$u_{n\ell m}(\textbf{ r}_\alpha)=\sum_{\ell^{\prime}=m}^\infty v_{n\ell m\ell^{\prime}}(\alpha,r) Y_{\ell^{\prime} m}(\theta,\phi),$$

((9.136))

which can be inverted, after substituting for $u_{n\ell m}$ from (9.135), giving

$$v_{n\ell m\ell^{\prime}}(\alpha,r)=\int_0^{2\pi}\int_0^\pi Y_{\ell^{\prime} m}^\ast(\theta,\phi)u_{n\ell}(r_\alpha) Y_{\ell m}(\theta_\alpha,\phi_\alpha) \sin\theta {\mathrm {d}}\theta {\mathrm {d}}\phi.$$

((9.137))

The integration over φ can be carried out immediately, giving 2π, and the integration over θ can be carried out numerically remembering, after using (9.134), that

$$u_{n\ell}(r_\alpha)=u_{n\ell} \left[\left(\alpha^2+r^2+2\alpha r\cos\theta\right)^{{1/2}}\right]$$

((9.138))

and

$$\cos\theta_\alpha=\frac{r\cos\theta+\alpha} {(\alpha^2+r^2+2\alpha r\cos\theta)^{{1/2}}}.$$

((9.139))

Hence the radial functions $v_{n\ell m\ell^{\prime}}(\alpha,r)$ in (9.136) can be calculated for all required values of ∓^′, α and r. If α is much smaller than r then the rate of convergence in (9.136) will be very rapid.

It follows from the above analysis that the following operation in (9.132) can be written as

$$\begin{array}{rcl} &&\exp({\mathrm {i}}{\mbox{$\boldsymbol\alpha$}}\cdot \textbf{p}_{N+1}) \left[r_{N+1}^{-1}F_{n^{\prime}L^{\prime}i^{\prime}j}^{\textbf{ V}\gamma}(r_{N+1}) Y_{\ell_{i^{\prime}}m_{\ell_{i^{\prime}}}}(\theta_{N+1},\phi_{N+1})\right]\\ &&\quad=\;r_\alpha^{-1}F_{n^{\prime}L^{\prime}i^{\prime}j}^{\textbf{ V}\gamma}(r_\alpha) Y_{\ell_{i^{\prime}}m_{\ell_{i^{\prime}}}}(\theta_\alpha,\phi). \end{array}$$

((9.140))

The function on the right-hand side of this equation can be expanded about O in Fig. 9.4 in analogy with (9.136) as follows:

$$\begin{array}{rcl} r_\alpha^{-1}F_{n^{\prime}L^{\prime}i^{\prime}j}^{\textbf{ V}\gamma}(r_\alpha) Y_{\ell_{i'}m_{\ell_{i^{\prime}}}}(\theta_\alpha,\phi) &=&\sum_{\ell=m_{\ell_{i^{\prime}}}}^\infty r_{N+1}^{-1} u_{n^{\prime}L^{\prime}i^{\prime}\ell_{i^{\prime}}m_{\ell_{i^{\prime}}}\ell j}^{\textbf{V}\gamma} (\alpha,r_{N+1})\\ &&\times\;Y_{\ell m_{\ell_{i^{\prime}}}}(\theta_{N+1},\phi_{N+1}). \end{array}$$

((9.141))

Then, in analogy with (9.137), we can invert (9.141) giving

$$\begin{array}{rcl} &&r_{N+1}^{-1}u_{n^{\prime}L^{\prime}i^{\prime}\ell_{i^{\prime}}m_{\ell_{i^{\prime}}}\ell j}^{\textbf{V}\gamma}(\alpha,r_{N+1}) \\ &&\quad=\int_0^{2\pi} \int_0^\pi Y_{\ell m_{\ell_{i^{\prime}}}}^{\ast}(\theta_{N+1},\phi_{N+1}) r_\alpha^{-1}F_{n^{\prime}L^{\prime}i^{\prime}j}^{\textbf{ V}\gamma}(r_\alpha) Y_{\ell_{i^{\prime}} m_{\ell_{i^{\prime}}}}(\theta_\alpha,\phi_{N+1})\\ &&\qquad\times\;\sin\theta_{N+1} {\mathrm {d}}\theta_{N+1} {\mathrm {d}}\phi_{N+1}. \end{array}$$

((9.142))

As in (9.137) the integration over $\phi_{N+1}$ can be carried out immediately and the integration over $\theta_{N+1}$ can be carried out numerically enabling $u_{n^{\prime}L^{\prime}i^{\prime}\ell_{i^{\prime}}m_{\ell_{i^{\prime}}}\ell j}^{\textbf{V}\gamma}(\alpha,r)$ to be determined for all ℓ, α and r values of importance.

We now substitute (9.140) into (9.132) and use (9.141). We then obtain

$$\begin{array}{rcl} I&=& \sum_{M_{L_{i^{\prime}}}m_{\ell_{i^{\prime}}}}(L_{i^{\prime}}M_{L_{i^{\prime}}}\ell_{i^{\prime}}m_{\ell_{i^{\prime}}}|L^{\prime}M_{L^{\prime}}) \overline{\Theta}_{n^{\prime}L_{i^{\prime}}M_{L_{i^{\prime}}}i^{\prime}}^{\gamma}(\textbf{X}_N;\sigma_{N+1})\\ &&\times\;\sum_{\ell=m_{\ell_{i^{\prime}}}}^\infty r_{N+1}^{-1} u_{n^{\prime}L^{\prime}i^{\prime}\ell_{i^{\prime}}m_{\ell_{i^{\prime}}}\ell j}^{\textbf{V}\gamma} (\alpha,r_{N+1})Y_{\ell m_{\ell_{i^{\prime}}}}(\theta_{N+1},\phi_{N+1}). \end{array}$$

((9.143))

We next express the reduced channel functions $\overline{\Theta}_{n^{\prime}L_{i^{\prime}}M_{L_{i^{\prime}}}i^{\prime}}^{\gamma}(\textbf{ X}_N;\sigma_{N+1})$ in (9.143) in terms of the original channel functions $\overline{\Phi}_{n^{\prime}L^{\prime}i^{\prime}}^{\gamma}(\textbf{ X}_N;\hat{\textbf{r}}_{N+1} \sigma_{N+1})$ in (9.130) by the equation

$$\begin{array}{rcl} &&\overline{\Theta}_{n^{\prime}L_{i^{\prime}}M_{L_{i^{\prime}}}i^{\prime}}^{\gamma}(\textbf{ X}_N;\sigma_{N+1}) Y_{\ell m_{\ell_{i^{\prime}}}}(\theta_{N+1},\phi_{N+1})\\ &&\quad=\sum_{L^{\prime\prime}} (L_{i^{\prime}}M_{L_{i^{\prime}}}\ell m_{\ell_{i^{\prime}}}|L^{\prime\prime}M_{L'}) \overline{\Phi}_{n^{\prime}L^{\prime\prime}i^{\prime}}^{\gamma}(\textbf{ X}_N;\hat{\textbf{r}}_{N+1}\sigma_{N+1}). \end{array}$$

((9.144))

Substituting this result into (9.143) gives

$$\begin{array}{rcl} I&=& \sum_{M_{L_{i^{\prime}}}m_{\ell_{i^{\prime}}}}\sum_{\ell=m_{\ell_{i^{\prime}}}}^\infty\sum_{L^{\prime\prime}} (L_{i^{\prime}}M_{L_{i^{\prime}}}\ell_{i^{\prime}}m_{\ell_{i^{\prime}}}|L^{\prime}M_{L^{\prime}}) (L_{i^{\prime}}M_{L_{i^{\prime}}}\ell m_{\ell_{i^{\prime}}}|L^{\prime\prime}M_{L^{\prime}})\\ &&\times\;\overline{\Phi}_{n^{\prime}L^{\prime\prime}i^{\prime}}^{\gamma}(\textbf{ X}_N;\hat{\textbf{r}}_{N+1}\sigma_{N+1}) r_{N+1}^{-1} u_{n^{\prime}L^{\prime}i^{\prime}\ell_{i^{\prime}}m_{\ell_{i^{\prime}}}\ell j}^{\textbf{V}\gamma} (\alpha,r_{N+1}). \end{array}$$

((9.145))

In order to evaluate I we expand the function $u_{n^{\prime}L^{\prime}i^{\prime}\ell_{i^{\prime}}m_{\ell_{i^{\prime}}}\ell j}^{\textbf{V}\gamma}(\alpha,r_{N+1})$ as follows:

$$u_{n^{\prime}L^{\prime}i^{\prime}\ell_{i^{\prime}}m_{\ell_{i^{\prime}}}\ell j}^{\textbf{V}\gamma}(\alpha,r_{N+1}) =\sum_{s=-\infty}^\infty \exp(-{\mathrm {i}} s\omega t) u_{n^{\prime}L^{\prime}i^{\prime}\ell_{i^{\prime}}m_{\ell_{i^{\prime}}}\ell sj}^{\textbf{V}\gamma}(r_{N+1}),$$

((9.146))

where we remember that α is a function of t, defined by (9.113). After substituting this result into I, defined by (9.145), we find that (9.129) can be rewritten as

$$\begin{array}{rcl} && \exp(-{\mathrm {i}} E^\textbf{A}t)\sum_{nLi}\exp(-{\mathrm {i}} n\omega t) \overline{\Phi}_{nLi}^{\gamma}(\textbf{X}_N;\hat{\textbf{r}}_{N+1} \sigma_{N+1})r_{N+1}^{-1}F_{nLij}^{\textbf{A}\gamma}(r_{N+1}) \\ &&=\exp(-{\mathrm {i}} E^\textbf{ V}t) \sum_{n^{\prime}L^{\prime}i^{\prime}s} \exp[-{\mathrm {i}} (n^{\prime}+s)\omega t] \sum_{M_{L_{i^{\prime}}}m_{\ell_{i^{\prime}}}} \sum_{\ell=m_{\ell_{i^{\prime}}}}^\infty\sum_{L^{\prime\prime}} (L_{i^{\prime}}M_{L_{i^{\prime}}}\ell_{i^{\prime}}m_{\ell_{i^{\prime}}}|L^{\prime}M_{L^{\prime}})\\ &&\quad\times\; (L_{i^{\prime}}M_{L_{i^{\prime}}}\ell m_{\ell_{i^{\prime}}}|L^{\prime\prime}M_{L^{\prime}}) \overline{\Phi}_{n^{\prime}L^{\prime\prime}i^{\prime}}^{\gamma}(\textbf{ X}_N;\hat{\textbf{r}}_{N+1} \sigma_{N+1})r_{N+1}^{-1}\\ &&\quad\times\; u_{n^{\prime}L^{\prime}i^{\prime}\ell_{i^{\prime}} m_{\ell_{i^{\prime}}}\ell sj}^{\textbf{V}\gamma}(r_{N+1}). \end{array}$$

((9.147))

From the requirement that the time dependence on both sides of (9.147) is the same then

$$E^\textbf{A}=E^\textbf{V}\vspace*{-3pt}$$

((9.148))

and

$$n=n^{\prime}+s.$$

((9.149))

Hence, the summations over n ^′ and s on the right-hand side of (9.147) can be replaced by a single summation over n ^′. Our final step is to project (9.147) onto the channel functions $\overline{\Phi}_{nLi}^{\gamma}$ which gives

$$\begin{array}{rcl} F_{nLij}^{\textbf{ A}\gamma}(r)&=&\sum_{n^{\prime}L^{\prime}\ell}\sum_{M_{L_i}m_{\ell_i}} (L_iM_{L_i}\ell_im_{\ell_i}|L^{\prime}M_L) (L_iM_{L_i}\ell m_{\ell_i}|LM_L)\\ &&\times\;u_{n^{\prime}L^{\prime}i\ell_i m_{\ell_i}\ell s j}^{\textbf{ V}\gamma}(r), \end{array}$$

((9.150))

where $s=n-n^{\prime}$ here and below. This equation, together with (9.142) and (9.146), relates the reduced radial wave function $F_{nLij}^{\textbf{A}\gamma}(r)$ in the acceleration frame to the reduced radial wave function $F_{nLij}^{\textbf{V}\gamma}(r)$ in the velocity gauge.

We can now determine the R-matrix $\textbf{R}^{\textbf{A}\gamma}(E)$ in the acceleration frame of reference at $r=a_p$. We define this R-matrix, in analogy with (9.26), in terms of the reduced radial wave function $F_{nLij}^{\textbf{A}\gamma}(r)$ by

$$F_{nLij}^{\textbf{ A}\gamma}(a_p)=\sum_{n^{\prime}L^{\prime}i^{\prime}} R_{nLin^{\prime}L^{\prime}i^{\prime}}^{\textbf{A}\gamma}(E) \left.a_p\frac{{\mathrm {d}} F_{n^{\prime}L^{\prime}i^{\prime}j}^{\textbf{ A}\gamma}}{{\mathrm {d}} r}\right|_{r=a_p},$$

((9.151))

where for notational convenience we have set the arbitrary constant $b_0=0$. The function $F_{nLij}^{\textbf{A}\gamma}(a_p)$ in (9.151) is given by (9.150) and

$$\begin{array}{rcl} \left.\frac{{\mathrm {d}} F_{nLij}^{\textbf{A}\gamma}}{{\mathrm {d}} r}\right|_{r=a_p}&=& \sum_{n^{\prime}L^{\prime}\ell}\sum_{M_{L_i}m_{\ell_i}} (L_iM_{L_i}\ell_im_{\ell_i}|L^{\prime}M_L) (L_iM_{L_i}\ell m_{\ell_i}|LM_L)\\ &&\times\;\left\{\frac{{\mathrm {d}}}{{\mathrm {d}} r} \left[u_{n^{\prime}L^{\prime}i\ell_i m_{\ell_i}\ell sj}^{\textbf{ V}\gamma}(r)\right] \right\}_{r=a_p}, \end{array}$$

((9.152))

which follows by taking the derivative of (9.150). Also, by inverting (9.146) we find that

$$\begin{array}{rcl} &&\left\{\frac{{\mathrm {d}}}{{\mathrm {d}} r} \left[u_{n^{\prime}L^{\prime}i^{\prime}\ell_{i^{\prime}}m_{\ell_{i^{\prime}}}\ell sj}^{\textbf{V}\gamma}(r)\right]\right\}_{r=a_p} \\ &&\quad=\frac{\omega}{2\pi}\int_0^{2\pi/\omega} \exp[{\mathrm {i}} s\omega t] \left\{\frac{{\mathrm {d}}}{{\mathrm {d}} r} \left[u_{n^{\prime}L^{\prime}i^{\prime}\ell_{i^{\prime}}m_{\ell_{i^{\prime}}}\ell j}^{\textbf{V}\gamma}(\alpha,r)\right] \right\}_{r=a_p}{\mathrm {d}} t, \end{array}$$

((9.153))

where the derivative term on the right-hand side of this equation can be obtained by differentiating (9.142) numerically. Thus both $F_{nLij}^{\textbf{A}\gamma}(a_p)$ and $\left[{\mathrm {d}} F_{nLij}^{\textbf{A}\gamma}/{\mathrm {d}} r\right]_{r=a_p}$ in (9.151) can be calculated and hence the R-matrix $\textbf{R}^{\textbf{A}\gamma}(E)$ at $r=a_p$, which provides the boundary condition for the solution of the coupled differential equations (9.124) in the asymptotic region, can be determined.

Finally, we consider the solution of the coupled second-order differential equations (9.124) in the asymptotic region. We rewrite these equations in matrix form as

$$\begin{array}{rcl} \left(\frac{{\mathrm {d}} ^2}{{\mathrm {d}} r^2} +\textbf{V}(r)+\textbf{V}^A(r) +\textbf{D}+\textbf{k}^2\right)\textbf{F}^\textbf{A}(r)=0, \end{array}$$

((9.154))

where $\textbf{V}(r)$ and D are the same as in (9.61) and $\textbf{V}^A(r)$ is the long-range acceleration potential term, defined by (9.121) and (9.127) and where, for notational convenience, we have omitted the superscript γ representing the conserved quantum numbers. It follows that $\textbf{V}(r)$ and $\textbf{V}^A(r)$ are real and symmetric and can be expanded as summations in inverse powers of r. On the other hand, D is hermitian and r-independent. We therefore follow the procedure adopted in the velocity gauge by diagonalizing $\textbf{D}+\textbf{k}^2$ by a unitary, r- and energy-independent matrix U ₁, defined by (9.63). Equation (9.154) can then be rewritten as

$$\left(\frac{{\mathrm {d}} ^2}{{\mathrm {d}} r^2} +{\mbox{\boldmath${\cal V}$}}(r)+{{\mbox{\boldmath${\cal K}$}}}^2\right){\mbox{\boldmath${\cal F}$}}^\textbf{A}(r)=0,\vspace*{-3pt}$$

((9.155))

where

$$\begin{array}{rcl} {\mbox{\boldmath${\cal V}$}}(r)&=&\textbf{U}_1^\dag\left[\textbf{V}(r)+\textbf{V}^A(r)\right]\textbf{U}_1, \\ {\mbox{\boldmath${\cal F}$}}^\textbf{A}(r)&=&\textbf{U}_1^\dag\textbf{F}^\textbf{A}(r). \end{array}$$

((9.156))

It follows that the potential ${\mbox{\boldmath${\cal V}$}}(r)$ can be expanded in inverse powers of r as follows:

$${\mbox{\boldmath${\cal V}$}}(r)=\frac{2(Z-N)}{r}\textbf{I} +2\sum_{\lambda=1}^\infty \textbf{c}_\lambda r^{-\lambda-1},$$

((9.157))

where we note that, as in (9.64), the unitary transformation (9.156) defines a new target basis corresponding to target states which are “dressed” by the laser field.

Equation (9.155) is now in a form that can be solved in the asymptotic region, using one of the asymptotic expansion methods described in Appendix F.1. In the case of laser-assisted electron–atom collisions we follow the procedure used to describe electron collisions with atoms and ions, discussed in Sect. 5.1.4, yielding a solution matrix ${\mbox{\boldmath${\cal F}$}}^\textbf{A}(r)$ satisfying the asymptotic boundary condition

$${\mbox{\boldmath${\cal F}$}}^\textbf{A}(r) \mbox{\raisebox{-2.mm} {$\stackrel{\sim}{\scriptstyle r\rightarrow \infty }$}} {{\mbox{\boldmath${\cal K}$}}}^{-{1/2}}\left[\sin{\mbox{$\boldsymbol\theta$}}+\cos{\mbox{$\boldsymbol\theta$}}\textbf{K}\right],$$

((9.158))

in the open channels, where K is the K-matrix. We also define a solution matrix satisfying the asymptotic boundary conditions

$${{\mbox{\boldmath${\cal G}$}}}^\textbf{A}(r) \mbox{\raisebox{-2.mm} {$\stackrel{\sim}{\scriptstyle r\rightarrow \infty }$}} {{\mbox{\boldmath${\cal K}$}}}^{-1/2}\left[\exp(-{\mathrm {i}}{\mbox{$\boldsymbol\theta$}})-\exp({\mathrm {i}}{\mbox{$\boldsymbol\theta$}}) \textbf{ S}\right],$$

((9.159))

which is obtained by taking linear combinations of the solutions defined by (9.158), where S is the S-matrix which is defined in terms of the K-matrix by (5.49). The T-matrix and cross sections can then be determined using the procedure described in Sect. 2.5.

In the case of atomic multiphoton ionization we must determine a solution of (9.155) satisfying Siegert outgoing wave boundary conditions [876] defined by

$${\mbox{\boldmath${\cal H}$}}^\textbf{A}(r) \mbox{\raisebox{-2.mm} {$\stackrel{\sim}{\scriptstyle r\rightarrow \infty }$}} \textbf{N}\exp({\mathrm {i}}{{\mbox{\boldmath${\cal K}$}}}r),$$

((9.160))

where N is a normalization vector. It follows from our discussion in Sect. 3.2.1 that this solution corresponds to a pole in the S-matrix lying on an unphysical sheet in the complex energy plane, as illustrated in Fig. 3.4. The corresponding quasi-energy E ^A can then be rewritten as

$$E^\textbf{A}=E_0+\Delta-{\frac{1}{2}}{\mathrm {i}}\Gamma,$$

((9.161))

where E ₀ is the field-free energy of the target state, Δ is the dynamic Stark shift and Γ is the total multiphoton ionization rate. In order to determine the position of this pole, the energy E ^A defined by (9.161) is varied iteratively and the solution in the asymptotic region re-calculated until the Siegert outgoing wave boundary condition (9.160) is satisfied. This iterative procedure is analogous to that used to calculate the initial bound state wave function in photoionization, discussed in Sect. 8.1.2, and to that used in determining the asymptotic solution in the velocity gauge, discussed in Sect. 9.1.4.

1.6 Asymptotic Region Solution: Simplified Analysis

In this section we consider a simplified analysis of the solution in the asymptotic region in the acceleration frame, where the transformation from the velocity gauge to the acceleration frame is carried out at such a large radius $r = a_p$ that the term $\widetilde{A}_{N+1}(t)$ on the right-hand side of (9.114) can be neglected. This approach, which was first considered by Charlo et al. [214], was analysed in detail by Terao-Dunseath and Dunseath [924] and has been developed and applied to a number of laser-assisted electron–atom collision calculations [275, 276, 278, 925].

In this approach the time-dependent Schrödinger equation (9.118) in the acceleration frame reduces to

$$\left(H_{N+1}+ {\mbox{\boldmath${\cal E}$}}(t)\cdot\textbf{R}_{N}\right) \widetilde{\Psi}^\textbf{A}(\textbf{X}_{N+1},t) ={\mathrm {i}}\frac{\partial}{\partial t}\widetilde{\Psi}^\textbf{A}(\textbf{X}_{N+1},t),$$

((9.162))

which corresponds to a free electron moving in the field of the atom or ion where the N target electrons, which are bound to the nucleus, are described in the length gauge. In order to solve (9.162) we follow our discussion in Sect. 9.1.5 leading to (9.124). We first introduce the following Floquet–Fourier expansion in the acceleration frame:

$$\widetilde{\Psi}^\textbf{A}(\textbf{X}_{N+1},t)=\exp\left(-{\mathrm {i}} E^\textbf{A}t\right) \sum_{n=-\infty}^\infty \exp(-{\mathrm {i}} n\omega t)\Psi_n^\textbf{A}(\textbf{X}_{N+1}),$$

((9.163))

where E ^A is the quasi-energy in this frame. Substituting (9.163) into (9.162) and equating the coefficient of $\exp[-{\mathrm {i}}(E^\textbf{A}+n\omega)t]$ to zero yields the infinite set of time-independent equations

$$\left(H_{N+1}-E^\textbf{A}-n\omega\right)\Psi_n^\textbf{A} +D_{N}\left(\Psi_{n-1}^\textbf{A}+\Psi_{n+1}^\textbf{A}\right)=0,$$

((9.164))

where D _N is defined by (9.10) and (9.14) with $N+1$ replaced by N. We then introduce the following close coupling expansion

$$\Psi_{njE}^{\textbf{A}\gamma}(\textbf{X}_{N+1})=\sum_{L i} \overline{\Phi}_{nLi}^\gamma(\textbf{X}_N;\hat{\textbf{r}}_{N+1} \sigma_{N+1})r_{N+1}^{-1}F_{nLij}^{\textbf{A}\gamma}(r_{N+1}), \quad r_{N+1}\geq a_p,$$

((9.165))

using the same notation as (9.123). Substituting (9.165) into (9.164) and projecting onto the channel functions $\overline{\Phi}_{nLi}^\gamma$ then yields the following coupled second-order differential equations:

$$\begin{array}{rcl} &&\left(\frac{{\mathrm {d}} ^2}{{\mathrm {d}} r^2}-\frac{\ell_i(\ell_i+1)}{r^2}+\frac{2(Z-N)}{r} +k_{ni}^2\right)F_{nLij}^{\textbf{A}\gamma}(r)\\ &&\quad=2\sum_{n^{\prime}L^{\prime}i^{\prime}} W_{nLin^{\prime}L^{\prime}i^{\prime}}^{\textbf{A}\gamma}(r) F_{n^{\prime}L^{\prime}i'j}^{\textbf{A}\gamma}(r),\quad r\geq a_p, \end{array}$$

((9.167))

where

$$k_{ni}^2=2\left(E^\textbf{A}-f_{ni}\right),$$

((9.167))

and f _ni is defined by (9.40). We see that (9.166) has the same form as (9.124). However, the potential $W_{nLin^{\prime}L^{\prime}i^{\prime}}^{\textbf{A}\gamma}(r)$ can now be written in matrix notation as

$$\textbf{W}^{\textbf{A}\gamma}=\textbf{V}^{E\gamma}+\textbf{V}^{D\gamma},$$

((9.168))

where V ^Eγ and V ^Dγ are the same as arose in (9.126) and in the velocity gauge in (9.41), but the acceleration term V ^Aγ is now absent.

In order to determine the solution of (9.166) we follow our discussion in Sect. 9.1.3 and rewrite this equation in matrix form as follows:

$$\left(\frac{{\mathrm {d}} ^2}{{\mathrm {d}} r^2} +\textbf{V}(r)+\textbf{D}+\textbf{k}^2\right) \textbf{F}^\textbf{A}(r)=0,$$

((9.169))

where for notational convenience we have omitted the superscript γ representing the conserved quantum numbers. Following our discussion of (9.61) describing the corresponding equation in the velocity gauge, we transform (9.169) by the unitary r- and energy-independent matrix U ₁, defined by (9.63). Equation (9.169) can then be rewritten as

$$\left(\frac{{\mathrm {d}} ^2}{{\mathrm {d}} r^2} +{\mbox{\boldmath${\cal V}$}}(r)+{{\mbox{\boldmath${\cal K}$}}}^2\right){\mbox{\boldmath${\cal F}$}}^\textbf{A}(r)=0,$$

((9.170))

where

$${\mbox{\boldmath${\cal V}$}}(r)=\textbf{U}_1^\dag\textbf{V}(r)\textbf{U}_1$$

((9.171))

and

$${\mbox{\boldmath${\cal F}$}}^\textbf{A}(r)=\textbf{U}_1^\dag\textbf{F}^\textbf{A}(r).$$

((9.172))

Also we order the channels in (9.170) so that

$${\cal K}_1^2\ge{\cal K}_2^2\ge\cdots \ge {\cal K}_{n_t}^2$$

((9.173))

where n _t is the total number of coupled channels, where the first n _a channels are open with ${\cal K}_i^2\ge 0$ and the last n _b channels are closed with ${\cal K}_i^2< 0$ and where $n_a+n_b=n_t$. Following our discussion of electron–atom collisions in Sect. 5.1.4 we define $n_t+n_a$ linearly independent solutions of (9.170) satisfying the asymptotic boundary conditions

$$\begin{array}{rcl} h_{ij}^\textbf{A}(r) &\mbox{\raisebox{-2.mm} {$\stackrel{\sim}{\scriptstyle r\rightarrow \infty }$}}& {\cal K}_i^{-1/2}\exp(-{\mathrm {i}}\theta_i)\delta_{ij},\quad i=1,\dots,n_t,\;\;j=1,\dots,n_a, \\ h_{ij}^\textbf{A}(r) &\mbox{\raisebox{-2.mm} {$\stackrel{\sim}{\scriptstyle r\rightarrow \infty }$}}& {\cal K}_i^{-1/2}\exp({\mathrm {i}}\theta_i)\delta_{ij}, \quad i=1,\dots,n_t,\;\;j=n_a+1,\dots,2n_a, \\ h_{ij}^\textbf{A}(r) &\mbox{\raisebox{-2.mm} {$\stackrel{\sim}{\scriptstyle r\rightarrow \infty }$}}& \exp(-\phi_i)\delta_{ij-n_a},\quad i=1,\dots,n_t,\;\;j=2n_a+1,\dots,n_t+n_a, \\ \end{array}$$

((9.174))

where for neutral atomic targets

$$\begin{array}{rcl} \theta_i &=& {\cal K}_i r - \frac{1}{2}\ell_i\pi,\quad i=1,\dots,n_a, \\ \phi_i&=&|{\cal K}_i|r,\quad i=n_a+1,\dots,n_t, \end{array}$$

((9.175))

with modifications defined by (5.38), (5.39), (5.40) and (5.41) for ionic targets.

In order to determine the S-matrix we transform these $n_t+n_a$ solutions from the acceleration frame to the velocity gauge and we then match these solutions at $r=a_p$ to the R-matrix ${\mbox{\boldmath${\cal R}$}}_p(E)$ obtained by propagating ${\mbox{\boldmath${\cal R}$}}_0(E)$, defined by (9.66), from $r=a_0$ to a _p as described following (9.68). We commence by rewriting the Kramers–Henneberger transformation (9.108) in the form

$$\widetilde{\Psi}^\textbf{V}(\textbf{X}_{N+1},t)= \exp\left[-{\mathrm {i}} {\mbox{$\boldsymbol\alpha$}}(t)\cdot \textbf{p}_{N+1}\right] \widetilde{\Psi}^\textbf{A}(\textbf{X}_{N+1},t),$$

((9.176))

where

$${\mbox{$\boldsymbol\alpha$}}(t)=\frac{1}{{c}}\int^t\textbf{ A}(t^{\prime}){\mathrm {d}} t^{\prime}= \hat{{\mbox{$\boldsymbol\epsilon$}}}\alpha_0\cos\omega t.$$

((9.177))

We then substitute for $\widetilde{\Psi}^\textbf{A}(\textbf{ X}_{N+1},t)$, given by (9.163) and (9.165), into the right-hand side of (9.176) giving

$$\begin{array}{rcl} \widetilde{\Psi}_j^\textbf{V}(\textbf{X}_{N+1},t)&=& \exp\left[-{\mathrm {i}} {\mbox{$\boldsymbol\alpha$}}(t)\cdot \textbf{p}_{n+1}\right] \sum_{n}\exp(-{\mathrm {i}} n\omega t) \sum_{L i}\overline{\Phi}_{nLi}^\gamma(\textbf{X}_N;\hat{\textbf{r}}_{N+1}\sigma_{N+1}) \\ &&\times\;r_{N+1}^{-1}\sum_k(\textbf{U}_1)_{nLik}{\cal F}_{kj}^\textbf{ A}(r_{N+1}), \quad r_{N+1}\ge a_p, \end{array}$$

((9.178))

where we have used (9.172). Also we observe, following our discussion of (9.147), that the time-dependent term $\exp(-{\mathrm {i}} E^\textbf{A}t)$, which appeared on the right-hand side of (9.178), has cancelled the term $\exp(-{\mathrm {i}} E^\textbf{V}t)$ which appeared on the left-hand side of this equation.

We now consider the action of the operator $\exp[-{\mathrm {i}} {\mbox{$\boldsymbol\alpha$}}(t)\cdot \textbf{p}_{n+1}]$ on the function on the right-hand side of (9.178). Following our discussion of (9.133) and Fig. 9.4 we find that

$$\exp(-{\mathrm {i}}{\mbox{$\boldsymbol\alpha$}}\cdot \textbf{p})f(\textbf{r})=f(\textbf{r}-{\mbox{$\boldsymbol\alpha$}}) \equiv f(\textbf{r}_\alpha),$$

((9.179))

which yields in analogy with (9.134)

$$\begin{array}{rcl} r_\alpha^2=\alpha^2+r^2-2\alpha r\cos\theta, &&r_\alpha\sin\theta_\alpha=r\sin\theta,\\ r_\alpha\cos\theta_\alpha=r\cos\theta-\alpha, && \phi_\alpha = \phi. \end{array}$$

((9.189))

Hence in the limit r, and hence r _α , tends to ∞

$$\begin{array}{rcl} r_\alpha&=&r-\alpha_0\cos\omega t\cos\theta+O(r^{-1}),\\ r_\alpha^{-1}&=&r^{-1}+O(r^{-2}),\\ \theta_\alpha&=&\theta+O(r^{-1}),\\ \phi_\alpha&=&\phi, \end{array}$$

((9.181))

where, as in Fig. 9.4, the z-axis is chosen along the laser polarization direction α. It follows that in this limit, the operator $\exp[-{\mathrm {i}} {\mbox{$\boldsymbol\alpha$}}(t)\cdot \textbf{p}_{n+1}]$ in (9.178) modifies the asymptotic forms of the functions ${\cal F}_{kj}^\textbf{A}(r_{N+1})$ while leaving the channel functions $\overline{\Phi}_{nLi}^\gamma(\textbf{X}_N;\hat{\textbf{r}}_{N+1}\sigma_{N+1})$ unmodified to first order. Therefore, we need to only consider the effect of the operator $\exp[-{\mathrm {i}} {\mbox{$\boldsymbol\alpha$}}(t)\cdot \textbf{p}_{n+1}]$ on the asymptotic form of the $n_t+n_a$ functions defined by (9.174). That is we introduce the $n_t+n_a$ functions

$$H_{ij}^\textbf{V}(r)=\exp[-{\mathrm {i}} {\mbox{$\boldsymbol\alpha$}}(t)\cdot \textbf{ p}]h_{ij}^\textbf{A}(r), \quad i=1,\dots,n_t,\;\;j=1,\dots,n_t+n_a,$$

((9.182))

which in the case of neutral targets satisfy the asymptotic boundary conditions

$$\begin{array}{rcl} H_{ij}^\textbf{V}(r) &\mbox{\raisebox{-2.mm} {$\stackrel{\sim}{\scriptstyle r\rightarrow \infty }$}}& {\cal K}_i^{-{1/2}}\exp\left[-{\mathrm {i}}\left({\cal K}_i r - \frac{1}{2}\ell_i\pi\right) \right]\exp\left({\mathrm {i}}{\cal K}_i\alpha_0\cos\omega t\cos\theta\right), \\ &&\;\;i=1,\dots,n_t,\;\;j=1,\dots,n_a, \\ H_{ij}^\textbf{V}(r) &\mbox{\raisebox{-2.mm} {$\stackrel{\sim}{\scriptstyle r\rightarrow \infty }$}}& {\cal K}_i^{-{1/2}}\exp\left[{\mathrm {i}}\left({\cal K}_i r - \frac{1}{2}\ell_i\pi\right) \right]\exp\left(-{\mathrm {i}}{\cal K}_i\alpha_0\cos\omega t\cos\theta\right), \\ &&\;\;i=1,\dots,n_t,\;\;j=n_a+1,\dots,2n_a, \\ H_{ij}^\textbf{V}(r)| &\mbox{\raisebox{-2.mm} {$\stackrel{\sim}{\scriptstyle r\rightarrow \infty }$}}& \exp\left(-\left|{\cal K}_i\right| r\right) \exp\left(\left|{\cal K}_i\right|\alpha_0\cos\omega t\cos\theta\right), \\ &&\;\;i=1,\dots,n_t,\;\;j=2n_a+1,\dots,n_t+n_a, \end{array}$$

((9.183))

with appropriate modifications in the case of ionic targets. In (9.183) it is convenient to expand the following exponentials:

$$\exp\left(\pm{\mathrm {i}}{\cal K}_i\alpha_0\cos\omega t\cos\theta\right)= 1\pm{\frac{1}{2}}{\mathrm {i}}{\cal K}_i\alpha_0[\exp({\mathrm {i}}\omega t)+\exp(-{\mathrm {i}}\omega t)] \cos\theta +O(\alpha_0^2)$$

((9.184))

and

$$\exp\left(\left|{\cal K}_i\right|\alpha_0\cos\omega t\cos\theta\right) =1+{\frac{1}{2}}\left|{\cal K}_i\right|\alpha_0[\exp({\mathrm {i}}\omega t)+\exp(-{\mathrm {i}}\omega t)] \cos\theta +O(\alpha_0^2),$$

((9.185))

where in most applications it is only necessary to retain the first-order terms in α ₀, although the higher order terms in α ₀ can be retained for high laser intensities.

It follows from the above analysis that we can determine $n_t+n_a$ linearly independent asymptotic solutions in the velocity gauge by substituting $H_{kj}^\textbf{V}$ defined by (9.182) and (9.183) for $\exp[-{\mathrm {i}}{\mbox{$\boldsymbol\alpha$}}(t)\cdot \textbf{p}_{N+1}]{\cal F}_{kj}^\textbf{A}(r_{N+1})$ in (9.178). We obtain

$$\begin{array}{rcl} \widetilde{\Psi}_j^\textbf{V}(\textbf{X}_{N+1},t)&=& \sum_{n}\exp(-{\mathrm {i}} n\omega t) \sum_{L i}\overline{\Phi}_{nLi}^\gamma(\textbf{X}_N;\hat{\textbf{r}}_{N+1}\sigma_{N+1}) \\ &&\times\;r_{N+1}^{-1}\sum_k(\textbf{U}_1)_{nLik}H_{kj}^\textbf{V}(r_{N+1},\theta_{N+1},t), \\ &&\;\;j=1,\dots,n_t+n_a, \end{array}$$

((9.186))

where we observe from the definitions of $H_{kj}^\textbf{V}$ that these functions depend on $r_{N+1}$, $\theta_{N+1}$ and t. We now rewrite (9.186) in standard form as

$$\begin{array}{rcl} \widetilde{\Psi}_j^\textbf{V}(\textbf{X}_{N+1},t)&=& \sum_{n}\exp(-{\mathrm {i}} n\omega t) \sum_{L i}\overline{\Phi}_{nLi}^\gamma(\textbf{X}_N;\hat{\textbf{r}}_{N+1}\sigma_{N+1}) \\ &&\times\;r_{N+1}^{-1}\sum_k(\textbf{U}_1)_{nLik}{\cal F}_{kj}^\textbf{V}(r_{N+1}), \\ &&\;\;j=1,\dots,n_t+n_a, \end{array}$$

((9.187))

where ${\cal F}_{kj}^\textbf{V}(r)$ is defined by the last of equations (9.65). It follows, after expressing the functions $H_{kj}^\textbf{V}$ in (9.186) in terms of the functions $h_{ij}^\textbf{A}(r)$ using (9.182) and (9.183) and then projecting (9.186) and (9.187) onto the channel functions $\overline{\Phi}_{nLi}^\gamma(\textbf{X}_N;\hat{\textbf{ r}}_{N+1}\sigma_{N+1})$ and onto the Floquet–Fourier expansion component denoted by n, that we can determine a linear relation between the functions $h_{ij}^\textbf{A}(r)$ in (9.186) and the functions ${\cal F}_{kj}^\textbf{V}(r)$ in (9.187). In principle this relation is exact only if an infinite number of channel functions and Floquet–Fourier components are retained in the expansion. However, rapid convergence in the number of terms retained in these expansions will occur for laser field strengths of most interest. It follows that we can write

$${\cal F}_{kj}^\textbf{ V}(r)=\sum_{i=1}^{n_t}C_{ki}h_{ij}^\textbf{A}(r),\;\; k=1.\dots,n_t+n_a,$$

((9.188))

where the matrix C _ki is the required linear transformation which is independent of r and the solution index j.

We now consider the determination of the S-matrix and hence the scattering amplitudes and cross sections. This is achieved by returning to the solution of (9.170) in the acceleration frame. The required solution has the asymptotic form

$$\begin{array}{rcl} {\mbox{\boldmath${\cal F}$}}^\textbf{A}(r) &\mbox{\raisebox{-2.mm} {$\stackrel{\sim}{\scriptstyle r\rightarrow \infty }$}}& {\mbox{\boldmath${\cal K}$}}^{-{1/2}}\left[\exp(-{\mathrm {i}}\theta)-\exp({\mathrm {i}}\theta)\textbf{S}\right], \quad\mbox{open channels}, \\ {\mbox{\boldmath${\cal F}$}}^\textbf{A}(r) &\mbox{\raisebox{-2.mm} {$\stackrel{\sim}{\scriptstyle r\rightarrow \infty }$}}& 0, \quad\mbox{closed channels}, \end{array}$$

((9.189))

where S is the usual $n_a\times n_a$-dimensional S-matrix. In order to determine the S-matrix we rewrite (9.189) as

$${\mbox{\boldmath${\cal F}$}}^\textbf{A} = \textbf{I}^\textbf{A}(r) - \textbf{O}^\textbf{A}(r) \textbf{M}^\textbf{A},$$

((9.190))

where $\textbf{I}^\textbf{A}(r)$ and $\textbf{O}^\textbf{A}(r)$ are ingoing and outgoing waves whose matrix elements can be expressed in terms of the $n_t+n_a$ linearly independent solutions $h_{ij}^\textbf{A}(r)$ defined by (9.174). We find that

$$\begin{array}{rcl} I_{ij}^\textbf{A}(r)&=&h_{ij}^\textbf{ A}(r),\;\;i=1,\dots,n_t,\;\;j=1,\dots,n_a, \\ O_{ij}^\textbf{A}(r)&=&h_{ij+n_a}^\textbf{ A}(r),\;\;i=1,\dots,n_t,\;\;j=1,\dots,n_a, \\ O_{ij}^\textbf{A}(r)&=&h_{ij+n_a}^\textbf{ A}(r),\;\;i=1,\dots,n_t,\;\; j=n_a+1,\dots,n. \end{array}$$

((9.191))

Also, the matrix M ^A can be written in the form

$$\textbf{M}^{\textbf{A}}=\left[ \begin{array}{c} \textbf{S}^{\textbf{A}}\\ \textbf{N}^{\textbf{A}} \end{array}\right],$$

((9.192))

where S ^A is the $n_a\times n_a$-dimensional S-matrix in (9.189) and N ^A is a subsidiary matrix, with dimensions $(n_t-n_a)\times n_a$, which multiplies decaying wave solutions in the closed channels in (9.189).

We can now determine the S-matrix S ^A by matching the solution in the velocity gauge to the $n_t\times n_t$-dimensional R-matrix ${\cal R}_p(E)$ in the velocity gauge determined at $r=a_p$, as described in Sect. 9.1.3 where we showed that the required solution satisfies (9.69). We now remember that ${\mbox{\boldmath${\cal F}$}}^\textbf{V}$ is related to h ^A and hence to I ^A and O ^A by the linear transformation (9.188) which we can rewrite as

$$\textbf{h}^\textbf{A}(r)=\textbf{C}^{-1}{\mbox{\boldmath${\cal F}$}}^\textbf{V}(r),$$

((9.193))

where C ⁻¹ is the inverse of C defined by (9.188). We then multiply (9.69) on the left by C ⁻¹ and substitute for ${\mbox{\boldmath${\cal F}$}}^\textbf{A}$ in terms of I ^A and O ^A using (9.190) and (9.191). After re-arranging the terms we obtain the following coupled equations:

$$\begin{array}{rcl} &&\left[\left.\textbf{O}^\textbf{A}(a_p) -a_p\textbf{C}^{-1}{\mbox{\boldmath${\cal R}$}}_p(E)\textbf{C}\frac{{\mathrm {d}} \textbf{O}^\textbf{A}}{{\mathrm {d}} r}\right|_{r=a_p} -{\frac{1}{2}}\textbf{C}^{-1}{\mbox{\boldmath${\cal P}$}}\textbf{C}\textbf{O}^\textbf{A}(a_p)\right]\textbf{M}^\textbf{A} \\ &&\;=\textbf{I}^\textbf{A}(a_p) -a_p\textbf{C}^{-1}{\mbox{\boldmath${\cal R}$}}_p(E)\textbf{C}\left.\frac{{\mathrm {d}} \textbf{I}^\textbf{A}}{{\mathrm {d}} r}\right|_{r=a_p} -{\frac{1}{2}}\textbf{C}^{-1}{\mbox{\boldmath${\cal P}$}}\textbf{C}\textbf{I}^\textbf{A}(a_p), \end{array}$$

((9.194))

where the derivatives of I ^A and O ^A, evaluated at $r=a_p$, can be determined from the solution of (9.170) subject to the boundary conditions (9.190) and (9.191). Equations (9.194) are a set of n _t linear simultaneous equations with n _a right-hand sides which can be solved to yield the $n_t\times n_a$-dimensional matrix M ^A. The $n_a\times n_a$-dimensional S-matrix is then determined from (9.192) and hence the corresponding scattering amplitude and cross section for laser-assisted electron–atom collisions determined using the procedure described in Sect. 2.5.

Finally, in order to calculate the atomic multiphoton ionization rate, we must determine a solution satisfying Siegert outgoing wave boundary conditions [876] corresponding to a pole in the S-matrix. This is achieved by an iterative process, as discussed in Sect. 9.1.4, yielding (9.95), and in Sect. 9.1.5, yielding (9.161). In this way we obtain the complex quasi-energy E ^A given by

$$E^\textbf{A}=E_0+\Delta-{\frac{1}{2}}{\mathrm {i}}\Gamma,$$

((9.195))

where E ₀ is the field-free energy of the target atom, Δ is the dynamic Stark shift and Γ is the total multiphoton ionization rate.

1.7 Harmonic Generation

Atoms interacting with an intense laser field can emit radiation at multiples, or harmonics, of the pump laser frequency, as illustrated in (9.3). For an initial target state with a given parity, the harmonic frequency ν ^′ in (9.3) is an odd multiple of the laser frequency ν, i.e. $\nu^{\prime}=n\nu$ where $n=3,5,7,\dots$. This process, called harmonic generation, has attracted considerable interest in recent years with the availability of intense lasers making it possible to observe high harmonics [590–592, 626, 627, 758]. For example, L’Huillier and Balcou [590] observed the emission spectra of various inert gases using a “pump” laser of wavelength $\lambda=1053$ nm and found harmonic frequencies ν ^′ with $n=133$ in neon at an intensity $I=\mathrm {1.5\times 10^{15}\;W/cm^{2}}$. In these observations the harmonic spectrum exhibited characteristic behaviour of a rapid decrease for the first few harmonic yields, followed by a plateau and an abrupt cut-off at high harmonics.

The theoretical treatment of harmonic generation by an intense laser field interacting with a gaseous medium has two aspects, discussed by L’Huillier et al. [591] and Burnett et al. [194]. First, the microscopic, single-atom response to the laser field must be analysed, where the harmonic spectrum emitted by a single atom is calculated by solving the appropriate Schrödinger equation. Second, the single-atom response must then be combined to obtain the macroscopic harmonic fields generated from the coherent emission of all the atoms in the laser focus. This is achieved by using the single-atom response as source terms in Maxwell’s equations [591]. In this section we will be concerned with the single-atom aspect of harmonic generation, which we will treat by solving the time-dependent Schrödinger equation using R-matrix–Floquet theory.

There have been many contributions to the study of harmonic generation by direct numerical solution of the time-dependent Schrödinger equation. For example, this method has been used to study harmonic generation by one-dimensional model atoms [280, 281], by model atoms with three-dimensional delta-function potentials [85, 86] and by realistic three-dimensional atoms in the single active electron approximation [548, 549, 552–554, 589]. In particular, the time-dependent calculations of Krause et al. [548] showed that for low laser frequencies the maximum cut-off energy at the end of the plateau mentioned above is approximated by $I_p+3E_{\mathrm{p}}$ where I _p is the ionization potential of the atom and E _p is the ponderomotive energy of an electron in the laser field defined by (9.48). This simple result can be understood using a semi-classical model proposed by Kulander et al. [555] and Corkum [231], in which the electron first tunnels through the potential barrier formed by the atomic potential and the oscillating laser field. After escaping from the atom, some electrons driven by the oscillating laser field return to the residual ion and emit harmonics by recombining into the atomic ground state. The cut-off energy for the harmonics predicted by this rescattering model is given by $I_p+3.2 E_{\mathrm{p}}$ which is in good agreement with the time-dependent calculations of Krause et al. [548]. Detailed calculations of harmonic generation rates have also been carried out using the Floquet–Fourier ansatz by Potvliege and Shakeshaft [753], who expanded the harmonic components for atomic hydrogen in terms of Sturmian basis functions. In this work they calculated non-perturbative harmonic generation rates in a linearly polarized laser field, where the intensity ranged from 10¹² to $\mathrm {3\times 10^{13}\;W/cm^{2}}$ and the wavelength ranged from 265 to 1064 nm. However, the extension of this approach to multi-electron atoms becomes computationally very demanding.

More recently, a major programme of work going beyond the single active electron approximation has been undertaken by Parker et al. [713] and in later publications. In this work, the time-dependent Schrödinger equation for helium is solved making full allowance for two-electron correlation effects. This work has yielded accurate multiphoton double-electron ionization cross sections and harmonic generation spectra for helium. However, the extension of this work to more than two strongly interacting electrons presents a major computational challenge which we will return to in Sect. 10.1.5 where we discuss time-dependent R-matrix theory.

1.7.1 R-Matrix–Floquet Theory

We now consider R-matrix–Floquet theory of harmonic generation which is non-perturbative and includes electron–electron correlation effects in multi-electron targets. This theory, which is an extension of R-matrix–Floquet theory of multiphoton ionization and laser-assisted electron–atom collisions, discussed in earlier sections in this chapter, was developed by Gȩbarowski et al. [366] and has been applied to multi-electron atomic targets by Gȩbarowski et al. [367] and by Plummer and Noble [744, 745].

We consider an ($N+1$)-electron atomic system A _i in (9.3) in a laser field which is treated classically and which is assumed to be monochromatic, monomode, linearly polarized and spatially homogeneous, where the electric field vector is defined by (9.4) and the corresponding vector potential is defined by (9.5). The rate of spontaneous emission of photons of frequency $\Omega=n\omega$ with a specific polarization $\hat{{\mbox{$\boldsymbol\epsilon$}}}$ in a direction $\hat{\textbf{n}}$ is then given by [753]

$$\frac{{\mathrm {d}} R(\Omega,\hat{{\mbox{$\boldsymbol\epsilon$}}})}{{\mathrm {d}} \hat{\textbf{n}}}= \frac{\Omega^3}{8\pi {c}^3}\left|\hat{{\mbox{$\boldsymbol\epsilon$}}}^{\ast}\cdot \textbf{ D}\right|^2,$$

((9.196))

corresponding to an electric dipole moment ${\mbox{Re\,}}[\textbf{D}\exp(-{\mathrm {i}}\Omega t)]$ oscillating at a frequency Ω, where the quantity D can be related to the oscillating electric dipole moment $\textbf{d}(t)$ of the atom induced by the laser field which is given by

$$\textbf{d}(t)=-\langle\widetilde{\Psi}_{{\mathrm {T}}}(\textbf{X}_{N+1},t)\left|\textbf{R}_{N+1} \right|\widetilde{\Psi}(\textbf{X}_{N+1},t) \rangle.$$

((9.197))

In (9.197) $\widetilde{\Psi}(\textbf{X}_{N+1},t)$ satisfies the time-dependent Schrödinger equation (9.6), $\textbf{ R}_{N+1}$ is defined by (9.10) and $\widetilde{\Psi}_{{\mathrm {T}}}(\textbf{X}_{N+1},t)$ is the time-reversed wave function corresponding to $\widetilde{\Psi}(\textbf{X}_{N+1},t)$, which we discuss below. The wave function $\widetilde{\Psi}(\textbf{ X}_{N+1},t)$ is expressed as a Floquet–Fourier expansion (9.8), where the functions $\Psi_n(\textbf{ X}_{N+1})$ in this equation are the time-independent harmonic components of the wave function and the quasi-energy E, which corresponds to Siegert [876] outgoing wave boundary conditions, can be written as

$$E=E_0+\Delta-{\frac{1}{2}}{\mathrm {i}}\Gamma.$$

((9.198))

In this equation E ₀ is the field-free energy of the target atom, Δ is the dynamic Stark shift and Γ is the total multiphoton ionization rate of the atom.

The reason for using the time-reversed wave function $\widetilde{\Psi}_{{\mathrm {T}}}(\textbf{X}_{N+1},t)$ in (9.197) has been discussed by a number of authors including Potvliege and Shakeshaft [753], Piraux and Shakeshaft [741] and Plummer and McCann [742]. If we adopt a wave function satisfying the Floquet–Fourier ansatz with Siegert outgoing wave boundary conditions, then the norm of the wave function satisfies

$$\langle \widetilde{\Psi}(\textbf{X}_{N+1},t)| \widetilde{\Psi}(\textbf{X}_{N+1},t)\rangle \approx {\mbox{\raisebox{.4ex}{$\chi$}}}(0)\exp(-\Gamma t),$$

((9.199))

where χ(0) is constant in time. However, the norm of the exact wave function satisfies

$$\langle \widetilde{\Psi}(\textbf{X}_{N+1},t)| \widetilde{\Psi}(\textbf{X}_{N+1},t)\rangle =C,$$

((9.200))

where C is a constant and the integral is taken over an expanding region of space where the wave function is non-zero. The reason for this anomaly is that when we make the Floquet–Fourier ansatz, we treat the atom as having a definite complex energy defined by (9.198) whereas, in fact, the atom has a distribution of energies with a width Γ and should be represented by a localized wave packet which is initially the bound state of the atom. The factor $-{\frac{1}{2}}{\mathrm {i}}\Gamma$ in (9.198) then describes the electron loss from the bound state into the continuum, which is correctly treated by the Floquet–Fourier ansatz because the Floquet state vector describes an electron which is not localized.

In order to resolve these contrasting views, Piraux and Shakeshaft [741] observed that if a fixed region of space, rather than an expanding region of space, is adopted in (9.200) then flux will leave this region as the atom expands in the laser field and the norm will decrease. However, they showed that it is possible to define a new norm corresponding to a fixed finite region of space which remains constant in time. This is achieved by introducing the time-reversed wave function $\widetilde{\Psi}_{{\mathrm {T}}}(\textbf{X}_{N+1},t)$ and defining a new norm by

$$\langle \widetilde{\Psi}_{{\mathrm {T}}}(\textbf{X}_{N+1},t)| \widetilde{\Psi}(\textbf{X}_{N+1},t)\rangle_{F} =N,$$

((9.201))

where the integral is taken over the fixed finite region of space denoted by F. In this case N is a constant, because in the time-reversed state flux enters the fixed finite region of space and compensates for the flux leaving this region in the original state.

Returning to (9.197) we represent the wave functions for the induced dipole moment in this expression by the Floquet–Fourier expansion (9.8) giving

$$\textbf{d}(t)=-\sum_{m=-\infty}^\infty\sum_{n=-\infty}^\infty \langle\Psi_m(\textbf{X}_{N+1})\left|\textbf{R}_{N+1} \right|\Psi_{n}(\textbf{X}_{N+1}) \rangle\exp[-{\mathrm {i}}(n-m)\omega t].$$

((9.202))

Writing $q=n-m$ and defining the dipole moments

$$\textbf{d}_{qn}=-\langle\Psi_{n-q}(\textbf{X}_{N+1})\left|\textbf{R}_{N+1} \right|\Psi_{n}(\textbf{X}_{N+1}) \rangle$$

((9.203))

and

$$\textbf{d}_{q}=\sum_{n=-\infty}^\infty\textbf{d}_{qn},$$

((9.204))

we then find that (9.202) becomes

$$\textbf{d}(t)=\textbf{d}_{0}+\sum_{q=1}^\infty[\textbf{d}_{q}\exp(-{\mathrm {i}} q\omega t)+ \mathrm{CC}],$$

((9.205))

where CC is the complex conjugate term. Since $\textbf{ d}_{q}=\textbf{d}_{-q}^{\ast}$ we can write (9.205) in the form

$$\textbf{d}(t)=\textbf{d}_{0}+2\sum_{q=1}^\infty{\mbox{Re\,}}[\textbf{d}_{q}\exp(-{\mathrm {i}} q\omega t)],$$

((9.206))

By comparing (9.196) and (9.206) it follows that the rate for generating photons of frequency $\Omega=q\omega$ and polarization $\hat{{\mbox{$\boldsymbol\epsilon$}}}$ is obtained by putting $\textbf{D}=2\textbf{d}_{q}$ in (9.196). Hence we obtain

$$\frac{{\mathrm {d}} R(\Omega,\hat{{\mbox{$\boldsymbol\epsilon$}}})}{{\mathrm {d}} \hat{\textbf{n}}}= \frac{\Omega^3}{2\pi {c}^2}\left|\hat{{\mbox{$\boldsymbol\epsilon$}}}^{\ast}\cdot \textbf{ d}_q\right|^2.$$

((9.207))

The rate of emission of photons into a given solid angle ${\mathrm {d}}\hat{\textbf{n}}$ is obtained by summing over polarizations of the emitted radiation. Finally, the total emission rate is obtained by integrating over all directions $\hat{\textbf{n}}$.

In order to calculate the quantities d _q in (9.207), required to determine the harmonic generation emission rates, we proceed following our treatment of multiphoton ionization and laser-assisted electron–atom collisions discussed earlier in this chapter. We partition configuration space into three regions, as illustrated in Fig. 9.1. We then solve the time-dependent Schrödinger equation (9.6) by introducing Floquet–Fourier expansions of the wave function in each region, as discussed in Sects. 9.1.2, 9.1.3 and 9.1.4. In the internal region, where the radial coordinate of all $N + 1$ electrons satisfy $r_i\leq a_0$, we describe all $N + 1$ electrons in the length gauge. In the external and asymptotic regions, where the radial coordinate of the ejected electron $r_{N+1}\ge a_0$ and the radial coordinate of the remaining N electrons satisfy $r_i\leq a_0,\;i=1,\dots,N$, we adopt the velocity gauge for the ejected electron and the length gauge for the remaining N inner electrons. Having obtained the solution of (9.6) in each of these three regions, we then fit the radial asymptotic form of the ejected electron to outgoing wave Siegert [876] boundary conditions corresponding to (9.95) or (9.198). This is achieved, as described in Sect. 9.1.4, using an iterative procedure analogous to that used in determining the initial bound state energy in photoionization, described in Sect. 8.1.2. In this way we determine the complex quasi-energy E ^V defined by (9.95), and the corresponding outgoing reduced radial wave solution (9.96) satisfied by the ejected electron up to an overall normalization factor. The Floquet–Fourier expansion of the total wave function describing the $N+1$ electrons, defined by (9.8), is then obtained in all three regions in Fig. 9.1 up to an overall normalization factor.

Finally, in order to calculate the dipole moments d _qn , defined by (9.203), the integrals must be confined to a finite region of space corresponding to (9.201). In most applications the range of integration can be confined to the internal region in Fig. 9.1 since, as pointed out by L’Huillier et al. [592], the main contribution to the dipole matrix elements comes from the region near the nucleus. In this case the wave function is defined by expansion (9.17). However, if necessary, the integration can be extended to part or all of the external region in Fig. 9.1 where the wave function is defined by expansion (9.37). In both cases, the wave function must be normalized to unity in the region included in the evaluation of the matrix elements and the Siegert outgoing wave boundary conditions (9.96) imposed on the wave function in the asymptotic region, to ensure the solution has the correct asymptotic form.

1.8 Non-hermitian Floquet Dynamics

In this section we consider atomic multiphoton ionization in short-pulse laser fields that cannot be accurately described by the Floquet–Fourier ansatz considered in earlier sections of this chapter. As pointed out by Potvliege and Shakeshaft [752], the Floquet–Fourier expansion (9.8) corresponding to a laser field defined by (9.4) and (9.5) contains no information about the way the field is turned on and is only valid if the initial state of the atom is not significantly depopulated during the rise time of the field. In addition, the laser pulse at constant amplitude must be sufficiently long that the multiphoton ionization rate corresponding to this amplitude is accurately defined.

In order to accurately describe atoms in short-pulse laser fields, we consider a coupled dressed state formalism, proposed by Ho and Chu [473] and generalized by Day et al. [253] to non-hermitian outgoing wave Siegert states. This procedure has been applied to multiphoton ionization of atomic hydrogen by Day et al. [253] and to multiphoton ionization of argon, using an R-matrix approach, by Plummer and Noble [746] and yields accurate results when the laser pulse is not so short that the direct R-matrix method for solving the time-dependent Schrödinger equation, considered in Chap. 10, must be used.

The non-hermitian Floquet approach to multiphoton ionization commences from solutions of the time-dependent Schrödinger equation (9.6) given by the following Fourier–Floquet expansion (9.8), which we write here as

$$\widetilde{\Psi}_j(\textbf{X}_{N+1},t)=\exp(-{\mathrm {i}} E_jt)\sum_{n=-\infty}^\infty \exp(-{\mathrm {i}} n\omega t)\Psi_{nj}(\textbf{X}_{N+1}),$$

((9.208))

corresponding to Siegert outgoing wave boundary conditions defined by (9.160) and (9.161). In (9.208) the energies E _j are given by

$$E_j=E_{0j}+\Delta_j-{\frac{1}{2}}{\mathrm {i}}\Gamma_j,$$

((9.209))

where E _0j are the field-free energies of the target states included in the analysis, Δ _j are the dynamic Stark shifts of these states and Γ _j are the total multiphoton ionization rates of these states corresponding to the vector potential field strength A ₀ in (9.5). We described earlier in this chapter how these quantities can be determined using R-matrix–Floquet theory. We now rewrite (9.208) as follows:

$$\widetilde{\Psi}_j(\textbf{X}_{N+1},t)=\exp(-{\mathrm {i}} E_jt) \widetilde{\psi}_j(\textbf{X}_{N+1},t).$$

((9.210))

We then assume that the wave function describing the atom or ion in a short-pulse laser field can be described, to a good approximation, by a finite superposition of Floquet wave functions of decaying dressed bound states defined by (9.210) given by

$$\widetilde{\Psi}(\textbf{X}_{N+1},t)=\sum_{j=1}^n a_j(t) \widetilde{\psi}_j(\textbf{X}_{N+1},t),$$

((9.211))

where $a_j(t)$ are time-dependent coefficients. In order to determine these coefficients we introduce the adjoint or time-reversed state corresponding to $\widetilde{\Psi}_j(\textbf{X}_{N+1},t)$, defined by

$$\widetilde{\Psi}_j^\dag(\textbf{X}_{N+1},t)=\exp(-{\mathrm {i}} E_j^\ast t) \widetilde{\psi}_j^\dag(\textbf{X}_{N+1},t),$$

((9.212))

which satisfies ingoing wave boundary conditions. It therefore follows that both $\widetilde{\Psi}_j(\textbf{X}_{N+1},t)$ and $\widetilde{\Psi}_j^\dag(\textbf{X}_{N+1},t)$ are periodic functions of time with period $T=2\pi/\omega$, where ω is the angular frequency of the field, and reduce to the field-free wave functions in the limit $A_0\rightarrow 0$. Also, pairs of these Floquet wave functions are bi-orthogonal in the sense that for fixed A ₀

$$\langle\langle\widetilde{\psi}_j^\dag|\widetilde{\psi}_k\rangle\rangle \equiv\frac{1}{T}\int_0^T\langle\widetilde{\psi}_j^\dag|\widetilde{\psi}_k\rangle {\mathrm {d}} t=\delta_{jk},$$

((9.213))

where we have imposed the normalization condition $\langle\langle\widetilde{\psi}_j^\dag|\widetilde{\psi}_j\rangle\rangle=1$ on the wave function. Using this result and the time-dependent Schrödinger equation satisfied by $\widetilde{\Psi}(\textbf{X}_{N+1},t)$, Day et al. [253] show that the coefficients $a_j(t)$ in (9.211) satisfy the following coupled differential equations:

$${\mathrm {i}}\frac{{\mathrm {d}} a_j}{{\mathrm {d}} t}=E_j a_j(t) -{\mathrm {i}}\frac{{\mathrm {d}} A_0}{{\mathrm {d}} t}\sum_{k=1}^n \langle\langle\widetilde{\psi}_j^\dag|\widetilde{\psi}_k^{\prime}\rangle\rangle a_k(t), \;\;j=1,\dots,n,$$

((9.214))

where the laser field is treated as stationary when performing the cycle-averaging time integration of the matrix elements. Also in (9.214), $\widetilde{\psi}_k^{\prime}$ denotes the derivative of $\widetilde{\psi}_k$ with respect to A ₀. The coefficients $a_j(t)$ in (9.211) then represent the atomic state probability amplitudes at the beginning and end of the laser pulse.

As an example of this approach, we show in Fig. 9.5 non-hermitian Floquet calculations for multiphoton transitions between the 1s and 3p states in atomic hydrogen compared with results obtained by Day et al. [253] for the direct solution of the time-dependent equations. The laser pulses used in these calculations have a finite duration, with a temporal envelope $A_0(t)=A_{00}\sin^2\Omega t$ and are linearly polarized. The results presented in this figure show that there is a small but non-negligible transition from the 1s ground state to the 3p excited state, where the probability is marked by “Stückelberg oscillations” [892] caused by a Stark-shift-induced three-photon resonance between the 1s and 3p states dressed by the laser field. We see that there is excellent agreement between the non-hermitian Floquet results and accurate fully time-dependent calculations except for ultra-short laser pulses, less than a few cycles of the field, showing the accuracy of the non-hermitian Floquet results for all but the shortest laser pulses.

Fig. 9.5

Probability that a hydrogen atom, initially in the 1s state before a laser pulse, is in the 3p state at the end of a laser pulse of 300 nm wavelength and $\mathrm{8\times 10^{13} \;W/cm^{2}}$ peak intensity plotted against the duration of the pulse. Solid line, two-state (1s, 3p) non-hermitian Floquet results; dotted line, full time-dependent results (Fig. 1 from [253])

Finally, we note that the non-hermitian R-matrix–Floquet calculation carried out by Plummer and Noble [746] for argon has shown how the laser pulse length and shape can be used, together with a knowledge of the Floquet states, to control the atomic populations both during and at the end of each pulse. Future work comparing this and similar calculations for other atoms and ions with results obtained using the direct R-matrix solution of the time-dependent Schrödinger equation considered in Chap. 10 would be of considerable interest.

2 Illustrative Examples

In this section we present the results of some recent R-matrix–Floquet calculations of multiphoton processes which illustrate the theory presented in this chapter.

2.1 Resonances in Multiphoton Ionization

In the presence of an intense laser field, the ionization threshold energy of an atom and the energies of the associated Rydberg states are increased by approximately the ponderomotive energy E _p defined by (9.48). If the electric field strength ${\cal E}_0$ of the laser is sufficiently high then this will result in channel closing which occurs when

$$E_{\mathrm{g}} +n{\mathrm {h}}\nu -E_{\mathrm{p}} <0,$$

((9.215))

where E _g is the energy of the ground state and n is the number of photons required to ionize the atom at low laser intensities. This effect is responsible for peak suppression in above-threshold ionization (ATI) and for resonance-enhanced multiphoton ionization (REMPI).

To illustrate the role of the ponderomotive energy in REMPI we consider the following multiphoton ionization process in helium:

$$\begin{array}{ccccc} n{\mathrm {h}}\nu+&\mathrm{He(1s^2\;^1S)}&\rightarrow& \mathrm{He^+(1s\;^2S)}&+\mathrm{e^-}. \\ &\searrow&& \nearrow&\\ &\multicolumn{3}{c}{{\mathrm {h}}\nu+ \mathrm{He^{\ast}}(\mathrm{1s}n\ell)}& \end{array}$$

((9.216))

We illustrate this process in Fig. 9.6 which shows the energy level diagram of helium with the ionizing transitions that occur in the field of a KrF laser with wavelength 248 nm and for an intensity close to zero and for an intensity of $\mathrm{5\times 10^{14}\;W/cm^{2}}$. When the intensity of the KrF laser is close to zero we see in Fig. 9.6 that five photons are required to ionize helium from its ground state, but that four photons are insufficient to reach the lowest excited state and so no intermediate resonance occurs. As the laser intensity is increased the effective ionization threshold and the energies of the excited states are increased by the ponderomotive energy E _p with the result that six photons are then required to ionize the atom. Also, the excited state energies 1snℓ in (9.216) come into resonance, one at a time as the intensity is increased, with the energy of five photons giving rise to a series of resonances in the six-photon ionization rate. In Fig. 9.6 we see that at a laser intensity of $\mathrm{5\times 10^{14}\; W/cm^{2}}$ the five-photon energy corresponds closely to the energy of the $\mathrm{1s2p\;^1P^o}$ excited state of helium giving rise to a REMPI enhancement in the six-photon ionization rate.

Fig. 9.6

Energy levels of He in a KrF laser field with intensity I close to zero and with intensity $5\times 10^{14}\;{{W/cm}}^{2}$, showing multiphoton ionization in each case. The shaded areas represent the continuum spectrum of He and the horizontal lines represent the ground state and two excited states of He (Fig. 1 from [379])

To explore this situation in greater detail, Glass and Burke [379] carried out R-matrix–Floquet (RMF) calculations for helium multiphoton ionization corresponding to (9.216). In these calculations two approximations were considered. In the first, only the $\mathrm{1s\;^2S}$ ground state of He⁺ was included in the RMF expansions (9.20) and (9.37), corresponding to the internal and external regions, respectively. In the second, $\mathrm{\overline{2s}}$ and $\mathrm{\overline{2p}}$ pseudostates representing electron–electron correlation effects were also included, where these pseudostates spanned the same range as the $\mathrm{1s\;^2S}$ ground state. Also, in order to obtain converged results at the higher laser intensities, up to 18 Floquet blocks (including 15 for absorption) and 10 angular momenta were retained in the RMF expansions.

The results of these calculations, which are presented in Fig. 9.7, illustrate the features mentioned above. At low intensities, five-photon ionization can occur and the behaviour is close to that predicted by the perturbative power law for the dependence of the ionization rate on intensity. Channel closing occurs at an intensity of $\mathrm{6.7\times 10^{13}\; W/cm^{2}}$ while at higher intensities six-photon ionization is the dominant process and the influence of REMPI on the total ionization rate now becomes apparent. Just above the intensity at which channel closing takes place five-photon resonances occur between the ground state and highly excited bound states with odd parity and, as the intensity increases further, the fifth photon sweeps through Rydberg series of these resonances. We note that for intensities close to the channel-closing intensity ($\mathrm {6.7\times 10^{13}\; W/cm^{2}}$) the calculation became more difficult due to the large radial extent of the intermediate high-energy Rydberg states, requiring large propagation distances $r=a_p$ in the external region in Fig. 9.1, so results are not given in this region.

Fig. 9.7

Total ionization rate for He in a 248 nm KrF laser field as a function of laser intensity. The arrow marks the intensity where channel closing takes place. Below this intensity five-photon ionization occurs, while at higher intensities at least six photons are required for ionization (Fig. 2 from [379])

Fig. 9.8

Ionization rate for Ar subjected to 390 nm laser light as a function of laser intensity. RMF calculations (solid line) are compared with ADK calculations (dashed-dotted line). The label 5s indicates the $\mathrm{3s^23p^55s\:^1P^o}$ resonance and the label 3d indicates the $\mathrm{3s^23p^53d\:^1P^o}$ resonance (Fig. 5 from [933])

Multiphoton ionization rates for He have also been calculated by van der Hart et al. [936] for the frequency-doubled Ti:Sapphire laser wavelength of 390 nm using both RMF theory and also by solving the time-dependent Schrödinger equation using direct numerical integration. The two calculations were found to be in excellent agreement for intensities between $\mathrm{1\times 10^{14}}$ and $\mathrm{2.5\times 10^{14}\;W/cm^{2}}$ where the ionization rate is strongly enhanced by resonances. RMF multiphoton ionization calculations have also been carried out for Ar by Plummer and Noble [744], for Ne and Ar by McKenna and van der Hart [624] and by van der Hart [933] and for Ca by McKenna and van der Hart [623]. In the work on Ar by van der Hart [933], wavelengths between 248.6 and 390 nm were considered, where for the frequency-doubled Ti:Sapphire laser wavelength of 390 nm the ionization rates for Ne and Ar were investigated up to $\mathrm{2.5\times 10^{14} \;W/cm^{2}}$. We show in Fig. 9.8 the resonance-enhanced multiphoton ionization rate for Ar subjected to 390 nm laser light as a function of laser intensity using RMF theory compared with ADK calculations. The RMF calculation included the $\mathrm{3s^23p^5\,^2P^o}$ Ar⁺ ground state and retained 21 Floquet blocks (15 absorption and 5 emission) and angular momenta up to $L=10$ in the RMF expansion. In the intensity region shown, between $\mathrm{1.8\times 10^{13}}$ and $\mathrm{2\times 10^{14}\;W/cm^{2}}$, a minimum of six photons need to be absorbed to achieve ionization. Including the $\mathrm{3s3p^6\,^2S^e}$ Ar⁺ excited state in the RMF expansion modifies the off-resonance ionization rates by ∽2% and the resonance peaks then occur at 1% higher intensity. The ionization rate increases rapidly with laser intensity from $\sim1.6\times 10^9\;\textrm{s}^{-1}$ at an intensity of $\mathrm{2\times 10^{13}\;W/cm^{2}}$ to $2.7\times 10^{13}\;\textrm{s}^{-1}$ at an intensity of $\mathrm{1.9\times 10^{14}\;W/cm^{2}}$. However, this is considerably less than the power law I ⁶ would yield. Also, the results are more than two orders of magnitude greater than the ADK tunnelling model [727, 16] at the lower laser intensities due to the exponential decay of the ADK rate compared with the I ⁶ behaviour of the RMF rate.

Recently, there has been increasing interest in resonance effects in multiphoton ionization of negative ions. We conclude this section by discussing R-matrix–Floquet calculations of two-photon detachment of Li⁻ by van der Hart [932]. We note that Li⁻ has also been the subject of R-matrix photoionization studies by Gorczyca et al. [401], mentioned in Sect. 8.2.2.

While most multiphoton detachment studies of negative ions have focused on outer-shell electrons, there is currently increasing interest in inner-shell processes. The simplest negative ion for which a distinction can be made between outer and inner electrons is Li⁻ which has a $\mathrm{1s^2 2s^2\:^1S^e}$ ground state configuration. In the work of van der Hart the following two-photon detachment process was considered:

$$2{\mathrm {h}}\nu +\mathrm{Li^-(1s^2 2s^2\:^1S^e)}\rightarrow \mathrm{Li(1s\: 2s^2\:^2S^e)+e^-},$$

((9.217))

followed by

$$\mathrm{Li(1s\: 2s^2\:^2S^e)}\rightarrow\mathrm{Li^+(1s^2 \:^1S^e)+e^-}.$$

((9.218))

We see that an inner-shell 1s electron is first detached leaving the Li atom in a $\mathrm{1s\: 2s^2\:^2S^e}$ doubly excited state which subsequently autoionizes leaving the Li⁺ ion in its $\mathrm{1s^2 \:^1S^e}$ ground state.

In the R-matrix–Floquet calculations by van der Hart 20 Li states were included in the internal region expansion. These consisted of eight physical states where two electrons were retained in the 1s orbital, six pseudostates where two electrons were retained in the 1s orbital and six physical states where only one electron was retained in the 1s orbital. In order to obtain convergence at the laser intensity $\mathrm{10^{12}\;W/cm^{2}}$ considered, five Floquet blocks were included in the calculation where three corresponded to absorption and one corresponded to emission. This gave 142 coupled channels where 66 channels described detachment. Finally, an internal region radius of 35 a.u. was adopted and the R-matrix then propagated out to 40 a.u. where it was matched to an asymptotic expansion.

Detachment of an inner-shell electron following absorption of two photons becomes possible for photon energies above 28.53 eV, the threshold photon energy for the $\mathrm{1s\: 2s^2\:^2S^e}$ state of Li in (9.217) and (9.218). We show in Fig. 9.9 the results of the calculation for the two-photon detachment rate of ground state Li⁻ leaving the Li atom in the excited $\mathrm{1s\: 2s^2\:^2S^e}$ state. We see that the photodetachment rate rises quickly from the threshold at 28.53 eV to a maximum of $2.6\times 10^6\;\textrm{s}^{-1}$ and then drops rapidly before rising again due to two shape resonances. The first is identified as a $\mathrm{1s\:2s\: 2p^2\:^1D^e}$ shape resonance which occurs just above the $\mathrm{1s\:(2s\: 2p\:^3P^o)\;^2P^o}$ threshold and the second is identified as a $\mathrm{1s\:2s\: 2p^2\:^1S^e}$ shape resonance which occurs just above the $\mathrm{1s\:(2s\: 2p\:^1P^o)\;^2P^o}$ threshold.

Fig. 9.9

Two-photon 1s-electron detachment rates for Li⁻ in a laser field of $\mathrm{10^{12}\;W/cm^{2}}$ as a function of photon energy leaving the residual Li atom in the excited $\mathrm{1s\: 2s^2\:^2S^e}$ state. The total rate (solid line) has been separated into the contribution from the emission of an s-electron (dashed line) and a d-electron (dot-dashed line) (Fig. 1 from [932])

We see from Fig. 9.9 that the $\mathrm{\:^1D^e}$ channel contributes significantly less than the $\mathrm{\:^1S^e}$ channel apart from the contribution from the $\mathrm{1s\:2s\: 2p^2\:^1D^e}$ shape resonance. This can be understood by observing that near the nucleus the d-orbital experiences significant repulsion due to the centrifugal potential barrier while the s-orbital experiences no repulsion. Hence two-photon absorption to the $\mathrm{\:^1D^e}$ continuum only becomes important at higher continuum energies.

In conclusion, this R-matrix calculation has shown the importance of shape resonances in multiphoton ionization of inner-shell electrons in negative ions. It provides a challenge for future calculations and experiments in this field.

2.2 Harmonic Generation

A further example of the importance of resonances in multiphoton processes is their role in harmonic generation. This is illustrated by RMF calculations by Plummer and Noble [745], who considered resonance-enhanced harmonic generation in argon at the KrF fundamental laser wavelength of 248 nm. In this work they considered the following process:

$$3{\mathrm {h}}\nu+\mathrm{Ar(3p^6\,^1S^e)}\rightarrow \mathrm{ Ar^{\ast}(3p^54d\;^1P^o)}\rightarrow \mathrm{ Ar(3p^6\,^1S^e)}+{\mathrm {h}}\nu^{\prime},$$

((9.219))

as well as the process where a further n photons are absorbed by the intermediate excited state of argon

$$n{\mathrm {h}}\nu+\mathrm{Ar^{\ast}(3p^54d\;^1P^o)}\rightarrow \mathrm{ Ar^{\ast\ast}}(\mathrm{3p^5}k\ell\;\mathrm{^1P^o})\rightarrow \mathrm{ Ar(3p^6\;^1S^e)}+{\mathrm {h}}\nu^{\prime\prime},$$

((9.220))

where $\nu^{\prime}=3\nu$ and $\nu^{\prime\prime}=(n+3)\nu$, with n even. At the KrF laser wavelength and low intensities, $\sim \mathrm{7.5\times 10^{12}\;W/cm^{2}}$, there is a three-photon resonance between the $\mathrm{3p^6\;^1S^e}$ ground state and the $\mathrm{3p^54d\;^1P^o}$ excited state of argon. If three photons with this intensity are absorbed by argon in its ground state, then the $\mathrm{3p^54d\;^1P^o}$ state is excited which can decay back to the ground state with the emission of a photon with frequency $\nu^{\prime}=3\nu$, as indicated in (9.219). Alternatively, before decaying, the $\mathrm{3p^54d\;^1P^o}$ excited state can absorb a further n photons as indicated in (9.220). The energy of the resultant argon atom then lies in the continuum and a laser-induced continuum structure (LICS) state², denoted by $\mathrm{ Ar^{\ast\ast}}(\mathrm{3p^5}k\ell\;\mathrm{^1P^o})$ in (9.220), is formed. If the total number of photons n absorbed in (9.219) and (9.220) is odd and the spin and angular symmetry of the LICS state is $\mathrm{\;^1P^o}$, then this state can decay back to the ground state with the emission of a photon with frequency $\nu^{\prime\prime}=(n+3)\nu$.

In the RMF calculations by Plummer and Noble, both the $\mathrm{3s^23p^5\;^2P^o}$ ground state and the $\mathrm{3s3p^6\,^2S^e}$ first excited state of Ar⁺ were included in the RMF expansions (9.20) and (9.37). In addition, between 8 and 10 Floquet blocks corresponding to absorption and between 3 and 5 Floquet blocks corresponding to emission were retained in the expansion. We show in Figs. 9.10 and 9.11 the third and fifth harmonic generation rates as a function of laser intensity. We see that there is a strong resonant enhancement of the rates at a laser intensity of $\mathrm{7.5\times 10^{12}\;W/cm^{2}}$ caused by the relative ponderomotive shifts of the $\mathrm{\;^1S^e}$ ground state and the $\mathrm{\;^1P^o}$ excited state. Also we see that there is a factor of about 600 in the relative heights of the peak rates between these two harmonics. The seventh harmonic, not shown, also has significant enhancement at this laser intensity, although the peak rate is now only $\sim10^{-9}$ of the peak rate of the third harmonic.

Fig. 9.10

Third harmonic generation rate in Ar as a function of laser intensity for the KrF laser wavelength 248 nm (Fig. 2 from [745])

Fig. 9.11

Fifth harmonic generation rate in Ar as a function of laser intensity for the KrF laser wavelength 248 nm (Fig. 3 from [745])

The resonant enhancement of harmonic generation that we have described is a general feature of harmonic generation and similar results are expected for other atoms and ions. For example, RMF calculations by Gȩbarowski et al. [367] have shown strong enhancement of the third harmonic generation in Mg, corresponding to resonant excitation of the $\mathrm{3s3p\;^1P^o}$ autoionizing state and seven-photon resonances are predicted by Plummer and Noble [745] for neon in a frequency-doubled Ti:Sapphire laser field. Also for neon, there is predicted to be a resonance-enhanced boost to the 13th harmonic in a fundamental Ti:Sapphire laser field. Given the stability of the inert gases and the important role played by resonances, we expect inert gases to be stable sources for generating harmonics.

2.3 Laser-Induced Degenerate States

“Laser-induced degenerate states” (LIDS) arise when an autoionizing state in electron–ion collisions and a state lying in the continuum corresponding to an atom dressed by the laser field become degenerate at certain laser intensities and frequencies. To illustrate the LIDS mechanism we consider the following processes:

$$\begin{array}{ccc} \mathrm{e^-}+A^+&\rightarrow & A_1^{\ast\ast}\\ \\ n{\mathrm {h}}\nu+A&\rightarrow & A_2^{\ast\ast} \end{array} \begin{array}{c} \searrow\\ \nearrow \end{array} \;A^++\mathrm{e^-}.$$

((9.221))

The upper process in (9.221) corresponds to an electron–ion collision, which proceeds through an intermediate autoionizing state $A_1^{\ast\ast}$ lying in the continuum with a complex energy E ₁. The lower process in (9.221) corresponds to multiphoton ionization from the ground state A which proceeds through a “Laser-induced continuum structure” (LICS) state $A_2^{\ast\ast}$ (discussed by Knight et al. [540, 541]) which also lies in the continuum with a complex energy E ₂. This is illustrated in Fig. 9.12 where the LICS state $A_2^{\ast\ast}$, formed by four-photon absorption from the ground state, lies close in energy to the autoionizing state $A_1^{\ast\ast}$. By varying both the laser intensity and frequency, the real and imaginary parts of the energies E ₁ and E ₂ in the complex energy plane can be made to coincide, giving rise to a LIDS state corresponding to a double pole in the laser-assisted electron–ion collision S-matrix.

Fig. 9.12

Doubly excited autoionizing state $A_1^{\ast\ast}$, with complex energy E ₁, and laser-induced continuum structure (LICS) state $A_2^{\ast\ast}$, with complex energy E ₂, which gives rise to a laser-induced degenerate state (LIDS)

Latinne et al. [581] and Cyr et al. [241] first observed LIDS in RMF calculations of multiphoton ionization of argon. In these calculations, the $\mathrm{3s^23p^5\;^2P^o}$ ground state and the $\mathrm{3s3p^6\,^2S^e}$ first excited state of Ar⁺ were included in the RMF expansions (9.20) and (9.37). In this way the $\mathrm{3s3p^64p\;^1P^o}$ autoionizing state, corresponding to $A_1^{\ast\ast}$ in (9.221), and the $\mathrm{3s^23p^6\;^1S^e}$ ground state dressed by the laser field, corresponding to $A_2^{\ast\ast}$ in (9.221) were both included in the calculations. The RMF equations were solved in the internal, external and asymptotic regions and the solution fitted to Siegert outgoing wave boundary conditions [876], yielding the complex energy defined by (9.95), (9.161) or (9.195).

We now consider for illustrative purposes the results obtained by Latinne et al. [581] when n = 1 in (9.221) corresponding to one-photon absorption. We show in Fig. 9.13 the resultant trajectories in the complex energy plane of the Floquet energies E ₁ and E ₂ in Fig. 9.12 as the laser intensity is varied for fixed values of the angular frequency ω. Also for illustrative purposes, the Floquet energies E ₁ and E ₂ are both shifted down by the laser angular frequency ω. The zero-field position of the Ar ground state lies on the real axis at $E_g=-0.57816$ a.u., while the zero-field positions of the autoionizing state (denoted by circles in Fig. 9.13) lie at a complex energy of $0.40936-0.00119 {\mathrm {i}}-\omega$ a.u., where the zero-field resonance width $\Gamma_a=2\times 0.00119$ a.u. For each angular frequency there are two trajectories, one originating at the zero-field position of the ground state and the other originating at the shifted zero-field position of the autoionizing state. We see by inspecting Fig. 9.13 that there are two complex energies where the trajectory originating from the ground state and the trajectory originating from the autoionizing state exchange their roles, corresponding to two LIDS. This occurs for laser angular frequencies $\omega\approx 0.98555$ and 0.98945. At large positive or negative detunings from these angular frequencies, the autoionizing state does not move appreciably from its zero-field position, while the width of the ground state increases rapidly from zero with increasing intensity. On the other hand, at intermediate detunings (e.g. $\omega\approx 0.987$) the opposite occurs, with the width of the trajectory connected to the shifted autoionizing state increasing rapidly with intensity, while the ground state trajectory remains “trapped” close to the real energy axis. Finally, we observe that at these critical laser angular frequencies and intensities the two complex energies are degenerate. The corresponding LIDS therefore each results in a double pole in the S-matrix.

Fig. 9.13

Trajectories in the complex energy plane as a function of laser intensity for Ar showing two LIDS. The trajectories correspond to the Floquet energies for the $\mathrm{3s3p^64p\,^1P^o}$ autoionizing state of Ar and the $\mathrm{3s^23p^6\,^1S^e}$ ground state of Ar dressed by one photon (each shifted down by the laser angular frequency ω), for laser intensities varying from 0 to $\mathrm{5\times 10^{13}\;W/cm^{2}}$. The corresponding value of the laser angular frequency is indicated on the trajectories and the dots on the trajectories give the increase in the laser intensity in steps of $\mathrm{9\times 10^{12}\;W/cm^{2}}$ (Fig. 1 from [581])

It has been known for many years that multiple poles in the S-matrix give rise to new phenomena, including a modification of the exponential decay law and the Breit–Wigner resonance profile (e.g. [387, 683]). Also, the physical implications of LIDS have been discussed by Kylstra and Joachain [557]. However, while LIDS occur quite generally in multiphoton processes and have been demonstrated for a number of targets including Ar, He and H⁻, more work needs to be carried out both theoretically and experimentally to reveal the full implications of this interesting phenomenon in atomic and molecular multiphoton collision processes.

2.4 Laser-Assisted Electron–Atom Collisions

In recent years increasing attention has been given to laser-assisted electron–atom collisions defined by (9.2). This process is of fundamental interest as an aspect of laser–atom interactions and is of importance, for example, in the laser heating of plasmas and high-power gas lasers. One of the most interesting features of this process is the possibility of exciting the target atom at electron collision energies below the field-free threshold via the simultaneous absorption of one or more photons. The first experimental investigations of simultaneous electron–photon excitation (SEPE) of atoms were performed by Mason and Newell [640, 641] on helium in the field of a CW CO₂ laser (photon energy 0.117 eV, wavelength 10.6 μm) at intensities from 10⁴ to $\mathrm{10^5\;W/cm^{2}}$. This was followed by studies up to $\mathrm{10^8\;W/cm^{2}}$ using a pulsed CO₂ laser by Wallbank et al. [945–947] and further experiments were carried out by Wallbank and Holmes [942–944] in which angular distributions were measured yielding cross sections which were much larger than those predicted by the low-frequency theory of Kroll and Watson [550]. An extensive review of early work in this field has been given by Mason [639].

In this section we discuss the results of laser-assisted electron–atom collision calculations carried out by Terao-Dunseath et al. [925] using the simplified R-matrix–Floquet analysis in the asymptotic region discussed in Sect. 9.1.6. In this work, the SEPE of helium atoms for exciting the $\mathrm{2\;^3S}$ and $\mathrm{2\;^3P}$ states near threshold was investigated in a linearly polarized Nd-YAG laser (photon energy 1.17 eV) with an intensity of $I=\mathrm{10^{10}\;W/cm^{2}}$, where it was assumed that the colliding electron was incident along the polarization direction of the field. Also in these calculations, the first five target states $\mathrm{1\;^1S}$, $\mathrm{2\;^3S}$, $\mathrm{2\;^1S}$, $\mathrm{2\;^3P^o}$ and $\mathrm{2\;^1P^o}$ of helium were included in the R-matrix expansion, together with eight Floquet components with $-3\leq n\leq 4$. This was sufficient to ensure convergence for the laser intensity and energy range considered. This approximation also gives good agreement, in the absence of the laser field, with the positions and widths of the resonances obtained in electron–helium atom collision calculations and experiments, discussed in Sect. 5.6.2.

We present in Fig. 9.14 cross sections for SEPE of the $\mathrm{2\;^3S}$ state of He from the ground state together with the field-free excitation cross section. Also we note that although in this laser field there is AC Stark mixing between states (e.g. the $\mathrm{1s2s\;^3S^e}$ and $\mathrm{1s2p\;^3P^o}$ states), for simplicity of notation we denote the field-dressed states by the quantum numbers of their dominant component. A prominent feature in Fig. 9.14 is the pronounced isolated resonance in the excitation cross section corresponding to the absorption of one photon. This occurs below the $\mathrm{2\;^3S}$ threshold at an energy corresponding to the $\mathrm{1s2s^2\;^2S^e}$ He⁻ resonance in the field-free elastic cross section, first observed by Schulz [835]. It corresponds to the process where the incoming electron is first captured in the $\mathrm{1s2s^2\;^2S^e}$ He⁻ resonance state. This state is then ionized by absorbing a laser photon leaving the helium atom in its $\mathrm{1s2s\;^3S^e}$ excited state, giving rise to the peak in the corresponding excitation cross section.

The cross section for exciting the $\mathrm{2\;^3S}$ state with no net exchange of photons in Fig. 9.14 is similar in shape to the field-free cross section but is about 40% lower in magnitude. A narrow resonance feature is obtained at an incident electron energy of 0.7492 Hartrees, which corresponds to the position of the $\mathrm{1s2s^2\;^2S^e}$ resonance shifted up by one photon energy. The origin of this resonance feature corresponds to the process where the electron–helium atom system emits one photon and the electron is temporarily captured in the $\mathrm{1s2s^2\;^2S^e}$ resonance state. It then absorbs one photon leading to excitation with no net exchange of photons. This explanation is supported by the fact that the width of this feature is only slightly larger than the one-photon absorption feature discussed above.

We also observe in Fig. 9.14 a small peak in the cross section for excitation of the $\mathrm{2\;^3S}$ state with the absorption of two photons. This corresponds to the process where the incoming electron together with a photon is captured in the $\mathrm{1s2s^2\;^2S^e}$ He⁻ resonance state. This state is then ionized by absorbing a second laser photon leaving the helium atom in its $\mathrm{1s2s\;^3S^e}$ excited state. In this case the resultant peak is much reduced in size since, as well as resulting from a two-photon process, the initial state of the electron plus helium atom is non-resonant. Similar results have also been reported by Terao-Dunseath et al. [925] for laser-assisted excitation of the helium ground state to the $\mathrm{1s2p\;^3P^o}$ excited state.

Fig. 9.14

Cross sections for electron impact excitation of the helium ground state into the He($\mathrm{2\;^3S}$) excited state in the presence of a laser field with angular frequency ω. Light solid line: excitation with no exchange of photons; dashed line: excitation with absorption of one photon; dash-dotted line: excitation with absorption of two photons; dotted line: excitation with emission of one photon; heavy solid line: field-free cross section for exciting He($\mathrm{2\;^3S}$). The field-free excitation thresholds are indicated by vertical bars above the figure (Fig. 2 from [925])

In conclusion we note that the calculations on helium have been extended to CO₂ laser fields and to very low-energy collisions by Dunseath and Terao-Dunseath [275, 276]. Also, general selection rules for differential cross sections for laser-assisted electron–atom collisions have been derived for the geometry in which a linearly polarized laser field is perpendicular to the scattering plane and results presented for electron–helium atom collisions in CO₂ and in Nd-YAG laser fields by Dunseatha et al. [278]. There is currently considerable interest in extending this work to other atoms and ions and to other laser intensities and frequencies.