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Engineering » Electrical and Electronic Engineering » "Recent Studies in Perturbation Theory", book edited by Dimo I. Uzunov, ISBN 978-953-51-3262-2, Print ISBN 978-953-51-3261-5, Published: June 14, 2017 under CC BY 3.0 license. © The Author(s).

# Atoms-Photonic Field Interaction: Influence Functional and Perturbation Theory

By Emmanouil G. Thrapsaniotis
DOI: 10.5772/67774

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## Overview

Figure 1. Second-order probability ∂ P ∂ ε of ionization as a function of the ε. We set ζ = 100 as. We give curves corresponding to ξ = π 2 (solid) ξ = π 3 (dashed) ξ = π 20 (dotted). We use ω = 0.4275 a.u., β = 1 and V = 10 7 .

# Atoms-Photonic Field Interaction: Influence Functional and Perturbation Theory

Emmanouil G. Thrapsaniotis
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## Abstract

We study the dynamics of one-electron atoms interacting with a pulsed, elliptically polarized, ultrashort, and coherent state. We use path integral methods. We path integrate the photonic part and extract the corresponding influence functional describing the interaction of the pulse with the atomic electron. Then we angularly decompose it. We keep the first-order angular terms in all but the last factor as otherwise their angular integration would contribute infinites as the number of time slices tends to infinity. Further we use the perturbative expansion of the last factor in powers of the inverse volume and integrate on time. Finally, we obtain a closed angularly decomposed expression of the whole path integral. As an application we develop a scattering theory and study the two-photon ionization of hydrogen.

Keywords: path integrals, influence functional, perturbation, coherent state, hydrogen, sign solved propagator, two photons

## 1. Introduction

The study of the interaction of radiation with matter is an area of major importance in physics. The production in laboratories of pulses of various durations and central frequencies has given a further boost in that study. These pulses can be used in the study of various elementary processes such as the excitation or photoionization of atoms [17]. This is possible due to their short time length of the order of a few femtoseconds or of a few hundreds attoseconds. Sub-100-as pulses have been generated as well. Moreover, their photons’ energy may belong in the ultraviolet or extreme ultraviolet and therefore just one or two photons may be enough to cause excitation or ionization.

In the present chapter, we introduce a fully quantum mechanical field theoretical treatment, for the interaction of a pulsed, elliptically polarized ultrashort coherent state with one optically active electron atoms. We use path integral methods. So we integrate the photonic part and extract the corresponding influence functional describing the interaction of the pulse with the atomic electron.

Proceeding we use the discrete form of that influence functional and angularly decompose its expression. We keep first-order angular terms in all but the last factor as otherwise their angular integration would contribute infinites as the number of time slices tends to infinity. Further, we use the perturbative expansion of the last factor in powers of the inverse volume and integrate on time. So we generate a perturbative series describing the action of the photonic field on the electron of the atom. It includes photonic and vacuum fluctuations contributions. Moreover, we manipulate the angular parts of the atomic action via standard path integral methods to finally obtain a closed angularly decomposed expression of the whole path integral.

As an application we develop a scattering theory and we study the two-photon ionization of hydrogen from its ground state to continuum. For the same transitions and to the same order vacuum fluctuation terms contribute as well. In the present application we consider orthogonal pulses. We use the propagator that appears in its sign solved propagator (SSP) form Ref. [8]. Previously, we have considered other kinds of photonic states interacting with one-electron atoms (see Refs. [6, 7, 9, 10]).

The present chapter proceeds as follows. In Section 2, we describe the present system and integrate its photonic part. Then in Section 3, we give the angular decomposition of the propagator in the case of elliptic polarization. In Section 4, we give an application and our conclusions in Section 5. Finally, in the Appendix we give some functions necessary in the evaluation of certain integrals.

## 2. System Hamiltonian and path integration

In the present chapter, we consider a one-electron atom initially in its ground state under the action of a coherent state. Therefore, the system Hamiltonian H can be decomposed into a sum of three terms. The electron’s one He, the photonic field one Hf, and an interaction term of the photonic field with the electron HI .that is,

 H=He+Hf+HI. (1)

He has the form

 He=12p→2+V(r→), (2)

where V(r) is the atomic potential. The photonic field has the Hamiltonian

 Hf=ωa+a, (3)

while the interaction term HI in the Power-Zienau-Woolley formalism takes the form

 HI=−er→⋅E→f(r→,τ). (4)

Ef(r,τ) is the field operator of the photonic pulse given by the expression

 E→f(r→,τ)=1Vil(ω)℘(τ)[ε⌢aeik→ph⋅r→−ε⌢∗a+e−ik→ph⋅r→]. (5)

(τ) is the pulse’s envelope function. In Eq. (5) l(ω)=2πω is a real frequency function, ε is the polarization vector, ω is the pulse’s carrier frequency, kph is the radiation wave vector and V is a large volume. Then HI has the form

 HI=g(τ)a+g∗(τ)a+. (6)

We have set

 g(τ)=−1Viel(ω)℘(τ)ε⌢⋅r→(τ)eik→ph⋅r→(τ). (7)

Now we combine the photonic field variables in the term

 H0(a+,a;τ)=Hf+HI=ωa+a+g(τ)a+g∗(τ)a+. (8)

The propagator between the initial and final states corresponding to the Hamiltonian Eq. (1) can be obtained by integrating on both the space and photonic field variables. At first we integrate the photonic field variables, which appear only in H0 (Eq. (8)). Then we obtain the following path integral of only the spatial variables:

 K(αf,r→f,tf;αi,r→i;ti)=∫Dr→(τ)Dp→(τ)(2π)3×exp[i∫titfdτ(p→(τ)⋅r→˙(τ)−p→2(τ)2−V(r→(τ)))−i∫titfdτg(τ)Z(τ,ti)−12(|αf|2+|αi|2)+Y(tf,ti)αf∗αi+Z(tf,ti)αf∗−iαiX(tf,ti)], (9)

where Y(tf,ti) , X(tf,ti) , and Z(tf,ti) read:

 Y(tf,ti)=exp[−i∫titfdτω(τ)]=exp(−iω(tf−ti)), (10)
 X(tf,ti)=∫titfdτg(τ)Y(τ,ti), (11)
 Z(tf,ti)=−i∫titfdτg∗(τ)exp[−i∫τtfdτ′ω(τ′)]. (12)

The propagator in Eq. (9) with diagonal field variables ( αi=αf=α ) can be written as

 K(α,r→f,tf;α,r→i,ti)=∫Dr→(τ)Dp→(τ)(2π)3exp[i∫titfdτ[p→(τ)⋅r→˙(τ)−p→2(τ)2−V(r→(τ))]+A−B|α|2+D1α+Dα∗]. (13)

The parameters are given as follows:

 A(tf,ti)=−1Ve2l2(ω)∫titfdτ∫tiτdρ℘(τ)ε⌢⋅r→(τ)eik→ph⋅r→(τ) ℘(ρ)ε⌢∗⋅r→(ρ)e−ik→ph⋅r→(ρ)e−iω(τ−ρ), (14)
 B(tf−ti)=1−Y(tf,ti)=1−e−iω(tf−ti), (15)
 D(tf,ti)=1Vel(ω)∫titfdτ℘(τ)ε⌢∗⋅r→(τ)e−ik→ph⋅r→(τ)e−iω(tf−τ), (16)
 D1(tf,ti)=−1Vel(ω)∫titfdτ℘(τ)ε⌢⋅r→(τ)eik→ph⋅r→(τ)e−iω(τ−ti). (17)

In the case of a field transition between an initial photonic state |Φ1 and a final one |Φ2 , the reduced propagator of finite time takes the form

 K˜(r→f,tf;r→i,ti)=∫d2απe|α|2〈Φ2|α〉K(α,r→f,tf;α,r→i,ti)〈α|Φ1〉. (18)

Here we consider that we have a field transition from an initial coherent state |β to a final one |γ . So we can integrate to obtain the following reduced propagator for the motion of the electron,

 K˜(r→f,tf;r→i,ti)=C(tf−ti)K0(r→f,tf;r→i,ti)=C(tf−ti)∫∫Dr→(τ)Dp→(τ)(2π)3exp{iStot[p→,r→,τ]}, (19)

where

 C(t)=exp(βγ∗B(t)−12|β|2−12|γ|2)B(t). (20)

The action is

 Stot[p→,r→,τ]=∫titf[p→(τ)⋅r→˙(τ)−p→2(τ)2−V(r→(τ))]dτ+i1Vel(ω)∫titfdτ(βχ(τ)ε⌢⋅r→(τ)eik→ph⋅r→(τ)+γ∗χ∗(τ)ε⌢∗⋅r→(τ)e−ik→ph⋅r→(τ))+1Ve2l2(ω)∫titfdτ℘(τ)∫tiτdρ℘(ρ)[ieiω(τ−ρ)eiω(tf−ti)−1(ε⌢∗⋅r→(τ)e−ik→ph⋅r→(τ)) (ε⌢⋅r→(ρ)eik→ph⋅r→(ρ))+c.c.], (21)

where χ(τ) has the form

 χ(τ)=℘(τ)e−iωτe−iωti−e−iωtf. (22)

We notice the following identities:

 1B(t)=12−12icot(ωt2)=12−iω∑m=−∞∞1t−2πmω. (23)

On using them and for arbitrary A(t) we can obtain the following formula after a direct Fourier transform,

 ∫−∞∞A(t)B(t)eiftdt=12∫−∞∞A(t)eiftdt+πω∑m=−∞∞A(2πmω)exp(if2πmω). (24)

Finally, upon using an inverse Fourier transform we obtain the following functional identities

 A(t)B(t)=A(t)[12+πω∑m=−∞∞δ(2mπω−t)]=A(t)[12+12∑m=−∞∞δ(m−ωt2π)]. (25)

In the above expressions, the summation is to be performed symmetrically. Identity in Eq. (25) is to be used in Eqs. (19) and (20). The delta functions do not contribute in the final expressions of Section 4 at the specific times introduced by them the photonic influence functional becomes zero. Moreover, the measure of all those times is zero. Further to handle the exponential in Eq. (20) within the scattering theory of Section 4 we use the limit

 limt→∞1B(t)=limt→∞11−exp(−i(ω−i0)t)=1. (26)

Now due to the large volume V, we shall approximate the exact action (21) by neglecting in the Taylor expansions

 r→(ρ)=r→(τ)+(ρ−τ)r→˙(τ)+…, (27)

higher terms than the first one, as they are going to involve powers of higher order in V in the denominator. To demonstrate this we consider the action in Eq. (21) and we derive the equation of motion of the electron by using Lagrange’s equation and the action’s Lagrangian in the absence of V(r) . So the part of the Lagrangian that interests us reads

 L=r→˙2(τ)2+1Vel(ω)(βχ(τ)ε⌢⋅r→(τ)eik→ph⋅r→(τ)+γ∗χ∗(τ)ε⌢∗⋅r→(τ)e−ik→ph⋅r→(τ))+1Ve2l2(ω)℘(τ)∫tiτdρ℘(ρ)[ieiω(τ−ρ)eiω(tf−ti)−1(ε⌢∗⋅r→(τ)e−ik→ph⋅r→(τ)) (ε⌢⋅r→(ρ)eik→ph⋅r→(ρ))+c.c.], (28)

and has equation of motion

 r→¨(τ)=O(1V). (29)

Therefore we can set,

 r→(ρ)=r→(τ)+O(1V). (30)

In the case of the presence of V(r) we perform a full order perturbation expansion of the full propagator in Eq. (19) with respect to the potential term. That is,

 K0=T+TVT+TVTVT+…. (31)

Then the propagator T, in the expansion, will be the one of the electron in the photonic field for which the approximation of Eq. (30) as discussed above is valid. Then, we sum back to obtain the final full propagator, thus maintaining the same approximation for the total propagator as well. Notice that the expansion (31) may converge very slowly but since it is a full order expansion it does not matter. Eventually in the large volume limit we get the action

 Stot[p→,r→,τ]=∫titf[p→(τ)⋅r→˙(τ)−p→2(τ)2−V(r→(τ))]dτ+i1Vel(ω)∫titfdτ(βχ(τ)ε⌢⋅r→(τ)eik→ph⋅r→(τ)+γ∗χ∗(τ)ε⌢∗⋅r→(τ)e−ik→ph⋅r→(τ))+1Ve2l2(ω)∫titfdτν(τ)|ε⌢⋅r→(τ)|2, (32)

where

 ν(τ)=℘(τ)∫tiτ℘(ρ)ξ(τ−ρ)dρ, (33)
 ξ(τ−ρ)=csc[ω(tf−ti)2]cos[ω(τ−ρ)−ω(tf−ti)2]. (34)

Finally, we notice that in the long wavelength approximation we can set eikphr1 . So we obtain the following expression

 Stot[p→,r→,τ]=∫titf[p→(τ)⋅r→˙(τ)−p→2(τ)2−V(r→(τ))]dτ+i1Vel(ω)∫titfdτ[βχ(τ)ε⌢⋅r→(τ)+γ∗χ∗(τ)ε⌢∗⋅r→(τ)]+1Ve2l2(ω)∫titfdτν(τ)|ε⌢⋅r→(τ)|2. (35)

Now we proceed to the angular decomposition of the above expressions.

## 3. Angular decomposition

We intend to perform angular decomposition and evaluate the SSP corresponding to the propagator of Eq. (19) in the long wavelength approximation.

Here we consider elliptic polarization so that the polarization vector takes the form

 ε⌢=ε⌢xcos(ξ2)±iε⌢ysin(ξ2), (36)

where εx and εy are the unit vectors along the x- and y-axis. The upper sign corresponds to left polarization while the lower one to right one.

The propagator K0ξ(rf,tf;ri,ti) of Eq. (19) with the above polarization vector ε has the discrete form

 K0ξ(r→f,tf;r→i,ti)=∏n=1N[∫−∞∞dr→n]∏n=1N+1[∫−∞∞dp→n(2π)3]×exp{i∑n=1N+1[p→n⋅(r→n−r→n−1)−ε(p→n2(τ)2+V(r→n))+i2πωVε(βχnε⌢⋅r→n+γ∗χn∗ε⌢∗⋅r→n)+2πωVενn|ε⌢⋅r→n|2]}. (37)

All the functions with index n are evaluated at time τn=nε+ti where ε=tftiN+1 . χn and νn have the form (see Eqs. (22) and (33))

 χn=℘(τn)e−iωτne−iωti−e−iωtf, (38)
 νn=ν(τn). (39)

Additionally, we note that we have set r0=ri and rN+1=rf .

Now we insert delta functions in Eq. (37) to get the expression

 K0ξ(r→f,tf;r→i,ti)=∏n=1N[∫−∞∞dr→n]∏n=1N+1[∫−∞∞dp→n(2π)3]∏n=1N+1[∫−∞∞d2wn]∏n=1N+1[δ(2)(wn−ε⌢⋅r→n)]×exp{i∑n=1N+1[p→n⋅(r→n−r→n−1)−ε(p→n2(τ)2+V(r→n))+i2πωVε(βχnwn+γ∗χn∗wn∗)+2πωVενn|wn|2]}. (40)

We have defined δ(2)(z)=δ(z)δ(z) . Moreover wn=wxn+iwyn . The delta functions have the representation

 δ(2)(wn−ε⌢⋅r→n)=δ(2)(wn−rn[sinϑn(cos(ξ2)cosϕn±isin(ξ2)sinϕn)])=1(2π)2×∫−∞∞∫−∞∞d2λnexp[i12(λnwn+λn∗wn∗)−i12λn(ε⌢xcos(ξ2)±iε⌢ysin(ξ2))⋅r→n−i12λn∗(ε⌢xcos(ξ2)∓iε⌢ysin(ξ2))⋅r→n]=1(2π)2∫−∞∞∫−∞∞d2λn×exp[iλxnwxn−iλynwyn−iλxncos(ξ2)ε⌢x⋅r→n±iλynsin(ξ2)ε⌢y⋅r→n]. (41)

We have set λn=λxn+iλyn . Now we perform the change of variables λxnλxncos(ξ2) , λynλynsin(ξ2) , wxncos(ξ2)wxn , wynsin(ξ2)wyn . The factor due to the integration on λn is cancelled with the factor due to the integration on wn. Further we expand angularly according to the identity,

 eiκ→⋅r→=4π∑l=0∞∑m=−lliljl(|κ→|r)Ylm∗(ϑκ,ϕκ)Ylm(ϑ,ϕ), (42)

where jl are spherical Bessel functions, and Ylm are spherical harmonics. So for right elliptic polarization we get

 δ(2)(wn−ε⌢⋅r→n)=∑ln=0∞∑mn=−lnlnglnmn(w′n,rn)4πYlnmn(ϑn,ϕn), (43)

where

 glnmn(w′n,rn)=(−i)lnOlnmn(2π)2∫−∞∞∫−∞∞d2λnexp[iλxnw′xn−iλynw′yn]×jln(|λn|rn)exp(−imnϕλn), (44)
 Olnmn=(2ln+1)(ln−mn)!(ln+mn)!Plnmn(0)=(2ln+1)(ln−mn)!(ln+mn)!π2mnΓ(ln−mn2+1)Γ(−ln−mn+12). (45)

We notice that if ln+mn is odd then Olnmn is zero. Moreover |λn| , ϕλn are the polar coordinates of λn on the x-y plane. We have set

 wxn=w′xncos(ξ2)=|w′n|cos(ϕw′n)cos(ξ2), (46)
 wyn=w′ynsin(ξ2)=|w′n|sin(ϕw′n)sin(ξ2), (47)

and

 w′n=w′xn+iw′yn=|w′n|eiϕw′n. (48)

On integrating over ϕλn we get

 glnmn(w′n,rn)=(−i)lnOlnmn2πexp(imn(ϕw′n+π2))×∫0∞dρλnρλnjln(ρλnrn)Jmn(ρλn|w′n|). (49)

ρλn=|λn| and Jmn are Bessel functions. In the appendix we give results for the expression in Eq. (49).

Finally, we replace the delta functions in Eq. (40) with the above angularly decomposed expressions. As N → ∞ and within the range from n = 0 to N we keep first-order angular terms. Higher order angular parts would contribute infinites. Finally, the propagator takes the form

 K0ξ(r→f,tf;r→i,ti)=1rfri∑l=0∞∑m=−ll∑q=0∞∑p=−qqKlmqξ(rf,tf;ri,ti)4πYlm(ϑf,ϕf)Yqp(ϑf,ϕf)Yqp∗(ϑi,ϕi), (50)

where after standard manipulations [11] on the angular parts of the atomic system Klmqξ(rf,tf;ri,ti) takes the form

 Klmqξ(rf,tf;ri,ti)=∏n=1N[∫0∞drn]∏n=1N+1[∫−∞∞dpn2π]∏n=1N+1[∬|w′n|

Further we observe that

 limN→∞∏n=1N[∬|w′n|

So Eq. (51) becomes

 Klmqξ(rf,tf;ri,ti)=Flm(rf)∬Dr(τ)Dp(τ)2π×exp{i∫titfdτ[pr˙−(p22+q(q+1)2r2+V(r))]+i2πω3V∫titf−ν(τ)r2(τ)dτ}, (53)

where

 Flm(rf)=∬|w′f|

We notice that to evaluate the integrals in Eq. (54) we have to take into account the expressions of Eqs. (46) and (47). Then we expand it on parameters of interest and integrate on time.

In the next section, we use the present propagator in its SSP form which appears after the solution of the sign problem. It is

 K1ξ(r→f,t;r→i,0)=1rfriδ(rf−ri)∑l=0∞∑m=−ll∑q=0∞∑p=−qqYqp(ϑf,ϕf)Yqp∗(ϑi,ϕi)×4πYlm(ϑf,ϕf)Flm(rf)exp[i2πω3Vrf2∫0tν(τ)dτ]. (55)

We have dropped the phase due to the atomic Hamiltonian because in the subsequent application of the present chapter, it eventually cancels.

## 4. Application and results

Proceeding to an application of the present theory we apply the above formalism to the case of the ionization of hydrogen. In that case the potential is given as

 V(r→)=−1r. (56)

We use as an initial state, the hydrogen’s ground one with wavefuction,

 Ψi(r→,t)=Ψi(r→)e−iεit=R1s(r)Y00(ϑ,φ)e−iεit=2e−rY00(ϑ,φ)e−iεit, (57)

where εi=12 is the energy of the ground H(1s) state.

The final state of the ionized electron with wave vector k=k(sinϑkcosϕk,sinϑksinϕk,cosϑk) is

 Ψfk→(r→,t)=Ψfk→(r→)e−iεt=exp(π2k)Γ(1+ik)eik→⋅r→1F1(−ik;1;−ikr−ik→⋅r→)e−iεt. (58)

It has energy

 ε=k22, (59)

and partial wave expansion

 Ψfk→(r→)=2πk∑s=0∞ise−iδsRsk(r)∑t=−ssYst∗(ϑ,φ)Yst(ϑk,φk). (60)
 Rsk(r)=8πk1−exp(−2π/k)∏y=1s(y2+1k2)1(2s+1)!×(2kr)se−ikr1F1(ik+s+1,2s+2,2ikr) (61)

is the radial function and δs=argΓ(1ik+s) a phase. Then the transition amplitude from the initial state i at t → −∞ to the final continuum state f at t → +∞ may be evaluated at any time t; it is

 Afi=〈Φf−(t)|Φi+(t)〉, (62)

where Φf(r,t) and Φi+(r,t) are exact solutions of the present system’s time-dependent Schrodinger equation subject to the asymptotic conditions

 Φf−(r→,t)→t→+∞Ψfk→(r→,t), (63)
 Φi+(r→,t)→t→−∞Ψi(r→,t). (64)

According to standard scattering theory we obtain the following form of the transition amplitude

 Afi=12limt1→−∞t2→∞〈Ψfk→|U0(t2)+exp(−i∫0t2Heff(t2,ρ)dρ+i∫0t1Heff(t1,ρ)dρ)U0(t1)|Ψi〉. (65)

The effective Hamiltonian Heff, appearing above and corresponding to the action of Eq. (35) has the form (see Eq. (2))

 Heff=He−i1Vel(ω)(βχε⌢⋅r→+β∗χ∗ε⌢∗⋅r→)−1Ve2l2(ω)ν|ε⌢⋅r→|2. (66)

Moreover

 U0(t)=e−iHet. (67)

We set β = γ. This appears to be a requirement in order the Hamiltonian to be PT (parity–time reversal) symmetric. The one-half factor in Eq. (65) appears due to the initial 1B(t) factor in Eq. (20) and the identity in Eq. (25). At the times introduced by the delta functions the propagator K1ξ(rf,τ;ri,0) (see below) becomes zero. Moreover the exponential in Eq. (20) is one as limt1B(t)=1 and β = γ.

Now to proceed we set t2=t1=t and take into account the PT invariance of the whole system as the Hamiltonian Eq. (66) is PT invariant. So we reverse the time sign of the terms involving the time t1 something that equivalently implies for the position rr , for the momentum pp and for the imaginary unit i → −i. Then we differentiate the operators between the bra and the ket in Eq. (65), with respect to the variable t. Finally, after certain standard manipulations and a subsequent integration we obtain the result

 Afi=〈Ψfk→|Ψi〉++∫0ςdτ〈Ψfk→|U0(τ)+exp(−i∫0τHeff(τ,ρ)dρ+i∫−τ0He(ρ)dρ)×(−1Vel(ω)(βχε⌢⋅r→+β∗χ∗ε⌢∗⋅r→)+i1Ve2l2(ω)ν|ε⌢⋅r→|2)U0(τ)|Ψi〉. (68)

We have supposed that the duration of the pulse is ς, as well as that it begins at time zero. Now in order to proceed we take into account that the asymptotic initial and final states are orthogonal. Further we make use of the path-integral representation of the exponential in Eq. (68) and angularly decompose it. So on making use of the results of the previous section and solving the sign problem [8], Eq. (68) becomes

 Afi=∫0ζdτ∬dr→fdr→i1ri2ei(ε−εi)τ(Ψfk→(r→f))∗K1ξ(r→f,τ;r→i,0)×(−2πωV(βχ(τ)ε⌢⋅r→i+c.c.)+i2πωV|ε⌢⋅r→i|2ν(τ))Ψi(r→i). (69)

We have used the prior form of the transition amplitude. K1ξ(rf,τ;ri,0) is given by Eq. (55). The phase which appears after the solution of the sign problem has cancelled.

As the present theory is PT symmetric we have to use PT symmetric quantum mechanics. So our equations take their final form according to the fact that (Ψfk(r))PT=(Ψfk(r)) .

Here we want to study two-photon ionization processes. They are of order 1V or higher. For the same transitions the vacuum fluctuations term contributes to the same order. So we take it into account. The amplitude takes the form

 A=∫0ζdτ∬dr→fdr→i1ri2ei(ε−εi)τ(Ψfk→(r→f))∗×(−2πωVSl=1ξ(r→f,τ;r→i,0)(βχ(τ)ε⌢⋅r→i+c.c.)+i2πωVSl=0ξ(r→f,τ;r→i,0)|ε⌢⋅r→i|2ν(τ))Ψi(r→i). (70)

Upon expanding to powers of volume the sign solved propagators appearing in Eq. (70) take the form

 Sl=1ξ(r→f,τ;r→i,0)=1rfriδ(rf−ri)∑q=0∞∑p=−qqYqp(ϑf,ϕf)Yqp∗(ϑi,ϕi)×[−2πωV(ε⌢⋅r→fβ∫0τdρχ(ρ)+c.c.)]exp[i2πω3Vrf2∫0τν(ρ)dρ], (71)

and

 Sl=0ξ(r→f,τ;r→i,0)=1rfriδ(rf−ri)∑q=0∞∑p=−qqYqp(ϑf,ϕf)Yqp∗(ϑi,ϕi)×{1+2πω3Vrf2(i∫0τν(ρ)dρ+|β∫0τdρχ(ρ)|2+cosξRe[(β∫0τdρχ(ρ))2])}×exp[i2πω3Vrf2∫0τν(ρ)dρ]. (72)

Finally, we obtain the second-order transition probability

 ∂P∂ε=14π2k∫|A|2dΩk→. (73)

Here we consider the case of an orthogonal pulse of duration ζ. Then

 ℘(τ)={10≤τ≤ζ0otherwise. (74)

In Figure 1, we plot the second-order term Pε as a function of the energy of the injected electron ε for ζ = 100 as and various values of the elliptic polarization parameter ξ. We use 2ω=0.855 a.u. Within the range 0ξπ2 the larger the ξ the smaller the transition probability. ξ=π2 corresponds to circular polarization. We give another approach of this case in [10]. ξ = 0 corresponds to linear polarization. In that case the present approach is degenerate. We give other approaches in [6, 7, 9].

#### Figure 1.

Second-order probability Pε of ionization as a function of the ε. We set ζ = 100 as. We give curves corresponding to ξ=π2 (solid) ξ=π3 (dashed) ξ=π20 (dotted). We use ω=0.4275 a.u., β = 1 and V=107 .

## 5. Conclusions

In the present chapter we have used path-integral methods in the study of the interaction of electrons with photonic states. We have integrated the photonic field and then angularly decomposed the electron—photonic field influence functional. Within those manipulations there have appeared terms due to the electromagnetic vacuum fluctuations.

As an application we have developed a scattering theory and used it in the two-photon ionization of hydrogen. For those transitions, the electromagnetic vacuum fluctuations contribute to the same order. Moreover to handle the path integrals that appear, we have used the relevant propagators in their sign solved propagator (SSP) form. The SSP theory appears in Ref. [8].

Concluding the present method is tractable and can be used in many problems involving the quantum mechanics of one-electron atoms interacting with radiation.

Appendix

In Eq. (49), we have the expression (here we drop the n indices)

 glm(w′,r)=(−i)lOlm2πexp(im(ϕw′+π2))∫0∞dρλρλjl(ρλr)Jm(ρλ|w′|)=Olm2π(−i)limeimϕw′π2r∫0∞dρλρλJl+12(ρλr)Jm(ρλ|w′|)=Olm2πim(−i)leimϕw′π|w′|mΓ(l+m2+1)rm+2Γ(l−m+12)Γ(m+1)×F(l+m2+1,m−l+12;m+1;|w′|2r2)Θ(r−|w′|)=Olm2πim(−i)leimϕw′2mπΓ(l+m2+1)rΓ(l−m+12)1r2−|w′|2×Pl−m(r2−|w′|2r)Θ(r−|w′|), (75)

where Θ(x) is the step function

 Θ(x)={1, x>00, x<0. (76)

We give the following cases:

 g00(w′,r)=12π1rr2−|w′|2Θ(r−|w′|), (77)
 g1±1(w′,r)=∓32e±iϕw′2π|w′|r21r2−|w′|2Θ(r−|w′|). (78)

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