**4. Determining the Mueller matrix**

260 Soybean – Genetics and Novel Techniques for Yield Enhancement

Where *Ex* and *Ey* are the electric field vectors along x and y direction and is angle. After normalizing the Stokes parameters by the irradiance *I, Q* describes the amount of light polarized along the horizontal (*Q*= +1) or vertical (*Q*= -1) axes, U describes the amount of light polarized along the +45° (*U*= +1) or -45° (*U*= -1) directions, and V describes the amount

A Mueller matrix, a 4 x 4 matrix, is a mathematical description of how an optical sample interacts or transforms the polarization state of an incident light beam and given

> 11 12 13 14 21 22 23 24 31 32 33 34 41 42 43 44

*S MS out system in* (4)

(3)

*mmmm mmmm*

*mmmm mmmm*

where, M is the 4 x 4 Mueller matrix of the media or sample and can be experimentally measured through the application of various incident polarization states and then by analyzing the state of polarization of the light leaving the sample. Since a Mueller matrix contains 16 elements (*mij*) of the matrix M and reconstruction requires 49 independent polarization measurements according to different polarizer and wave plate orientation as shown in table.1 and Fig.1. [31] The Mueller matrix can be thought of as the "optical fingerprint" of a sample. This matrix operates directly on an input or incident Stokes vector, thus resulting in an output 4 x 1 Stokes vector that describes the polarization state of the light leaving the sample. This is described mathematically by the equation given as

where [*Sout*] the output Stokes vector, [*Sin*] the Stokes input vector and [*Msystem*] is the Mueller

[ ] [ ][ ][ ][ ][ ] *Msystem QW A M QW P M M* (5)

The output stokes vector [*Sout*] can be calculated by relation in Eq. 4, putting the values of the Mueller matrix for optics and the Mueller matrix of the system. [33] The complete characterization of the polarization state of light by means of the Stokes parameters permits

*QUV S S S DOP*

For purely polarized light, the degree of polarization is unity i.e. 1, and the Stokes parameters obey the equality I2=Q2+U2+V2, while for partially polarized light, the degree of polarization is smaller than unity, leading to I2Q2+U2+V2. An input beam can be decomposed into purely polarized beams. After propagation through an optical system, the Stokes parameters of the purely polarized beam components are added to give the Stokes

222 2 2 2

*I S*

123 2 0

(6)

of right (*V*= +1) or left (*V*= -1) circularly polarized light.

*M*

matrix representing the entire experimental optical system given as

the calculation of the degree of polarization (DOP), defined as

parameters for the original input beam. [34]

as [30]

[32]

If the Mueller matrix is not known, all the elements can be determined experimentally. It can be shown that 49 intensity measurements with various orientations of polarizers and analyzers are necessary to obtain the 16 elements of the Mueller matrix. [25]

$$\begin{aligned} m\_{11} &= I\_{0o} \\ m\_{12} &= I\_{0o} - I\_{-o} \\ m\_{13} &= I\_{-o} - I\_{-o} \\ m\_{14} &= I\_{0o} - I\_{-o} \\ m\_{21} &= I\_{0o} - I\_{0o} \\ m\_{22} &= (I\_{+h} + I\_{-v}) - (I\_{-h} + I\_{sc}) \\ m\_{23} &= (I\_{+h} + I\_{-v}) - (I\_{-h} + I\_{+v}) \\ m\_{24} &= I\_{1o} + I\_{-v} \\ m\_{31} &= I\_{0o} + I\_{-v} \\ m\_{31} &= I\_{1o} + I\_{-v} \\ m\_{33} &= (I\_{+h} + I\_{-v}) - (I\_{-v} + I\_{-v}) \\ m\_{41} &= I\_{0i} - I\_{-v} \\ m\_{41} &= I\_{1o} - I\_{-v} \\ m\_{41} &= I\_{1o} - I\_{0o} \\ m\_{42} &= (I\_{l} + I\_{-v}) - (I\_{-l} + I\_{+v}) \\ m\_{50} &= (I\_{l} + I\_{-v}) - (I\_{-l} + I\_{+v}) \\ m\_{63} &= (I\_{l} + I\_{-v}) - (I\_{-l} + I\_{+v}) \\ \end{aligned} \tag{7}$$

Where the first term represents the input polarization state while the second the output polarization state of light. The states are defined as: h = horizontal, v = vertical, + = +45°, - = -45°, r = right circular and l = left circular. Once all 16 elements of the matrix are obtained, the medium is completely described in terms of its optical properties.


*5.2. Risk Factors*

In a prospective study of 173 patients with COVID-19, evidence of pulmonary fibrosis was observed in 90 patients (52%) at 3-month CT follow-ups [26]. Risk factors were pulmonary consolidation (odds ratio [OR] = 2.84), severe disease (OR 2.40), and a higher CT severity score (OR 1.10) at admission [26]. Of 62 patients who underwent chest CT scans again at 6 months of follow-up, in 41 patients (66.1%) the fibrosis remained unchanged, whereas in 21 patients (33.9%) a radiological improvement was documented [26]. In addition, older age, cigarette smoking, high-dose systemic corticosteroid use, and longterm mechanical ventilation were risk factors in another study [27]. The study of Han et al. showed that in older patients, high-dose systemic corticosteroid use and mechanical ventilation are risk factors as well [28]. Aging may cause a shift to a more profibrotic and irreversible senescent phenotype of fibroblasts [29].

Other risk factors were higher C-reactive protein (CRP) and lower lymphocyte counts [30–32].

### *5.3. Radiology*

The fibrotic changes include traction bronchiectasis, honeycombing, parenchymal bands, and interlobar septal thickening (IST) [26]. Nabahati et al. did not see any patients with progressive pulmonary fibrosis at the 6-month chest CT follow-up [26]. For this study cohort, a longer follow-up has not yet been published.

Two other studies demonstrated fibrotic abnormalities in the 6-month chest CT followup of 35% and 32% of patients, respectively [27,28]. Importantly, studies are difficult to compare because of different patient characteristics at diagnosis. It has been suggested that PCPF's course could be similar to other well-documented forms of postviral pulmonary fibrosis, such as those occurring after SARS, MERS, or influenza H1N1 infections [33]. Although in SARS patients, fibrosis could be demonstrated in more than 50% of patients after an average of 37 days, only 5% of patients continued to show fibrotic changes after a 15-year follow-up [33].

### *5.4. Pathophysiology*

The key site of SARS-CoV-2 infection is the angiotensin converting enzyme 2 (ACE2). To enter the human host cell, the virus uses the spike "S" protein, which results in binding to ACE2. In human cells, the ACE2 gene expression is the highest in nasal epithelial cells and decreases throughout the lower respiratory tract, including epithelial cells of the trachea, bronchi, and alveolar cells. In normal human cells, the function of ACE2 is converting angiotensin II (Ang II) into angiotensin 1–7 (Ang 1–7), in order to regulate the cardiovascular system and blood pressure. Ang II has a fibrotic effect by upregulating the level of a pro-fibrotic cytokine named transforming growth factor-1β (TGF-β), which transforms fibroblasts into myofibroblasts and promotes extensive collagen deposition [24,34]. On the contrary, Ang 1–7 has an anti-fibrotic effect. When the spike "S" protein of the SARS-CoV-2 down-regulates the level of ACE2, it increases the level of Ang II and decreases the level of Ang 1–7, resulting in promoting inflammation and pulmonary fibrosis [24].

#### **6. How Should We Treat COVID-19-Related End Stage Lung Disease?**

The cause of COVID-related end stage lung disease still remains unclear, and more studies are needed to build our understanding of why some patients develop lung fibrosis, and other patients do not. This question is important, as these patients will require longterm medical care and the number of patients is considerable. Moreover, the long-term behavior of fibrotic changes is still unknown, as most studies have a relatively short follow-up period, which is an inherent problem when dealing with a relatively new virus. Although the long-term pulmonary consequences of COVID-19-related pulmonary fibrosis remain speculative, the large number of individuals affected by COVID-19-related fibrosis could lead to a worldwide healthcare challenge of unprecedented magnitude [35]. Another unanswered question is whether other variants of SARS-CoV-2 will influence disease severity. The omicron variant of SARS-CoV-2 (PANGO lineage B.1.1.529) was reported on

24 November 2021 and has been associated with a more proximal adherence in the airways, in contrast to the delta variant, which has a more distal distribution in the airways. The omicron variant replicates significantly less efficiently than other SARS-CoV-2 variants in both nasal turbinates and lungs and induces substantially attenuated lung pathology [36].

Currently, there is no consensus on the use of antifibrotics in patients with COVID-19-related end stage lung disease. Only two drugs (pirfenidone and nintedanib) are used to treat idiopathic pulmonary fibrosis (IPF). Both drugs have been approved by both the European Medicines Agency (EMA) and the United States Food and Drug Administration (FDA) and can decrease the rate of pulmonary fibrosis progression. Pamrevlumab (FG-3019), a new antifibrotic drug for intravenous use, is currently being investigated in phase 2 trials and has shown promising results in patients with IPF in the phase 2 PRAISE trial [37]. It is a monoclonal antibody against connective tissue growth factor (CTGF) in IPF. However, in COVID-19-associated pulmonary fibrosis, there are no published studies on pamrevlumab yet.

Pirfenidone is an orally administered pyridine with combined anti-inflammatory, antioxidant, and antifibrotic properties. The mechanism of action includes the inhibition of fibroblast proliferation, but details have not yet been fully determined. Nintedanib is an inhibitor of multiple receptor tyrosine kinases and was initially developed as an antiproliferative and anti-angiogenic drug for cancer treatment.

IPF is defined as a spontaneously occurring (idiopathic) specific form of chronic fibrosing interstitial pneumonia associated with a pattern of Usual Interstitial Pneumonia (UIP) on imaging or histology [38]. It has been shown that both pirfenidone and nintedanib also show efficacy in non-IPF patients with progressive fibrosis [39]. This was shown both for nintedanib (INBUILD) and for pirfenidone [40,41]. The INBUILD study showed a reduction in FVC decline of about 60% compared to placebo. For both antifibrotic drugs, pirfenidone and nintedanib, a meta-analysis showed a reduced decline in the forced vital capacity (FVC) in patients with idiopathic pulmonary fibrosis (IPF) and also non-IPF patients [39]. As suggested by Wells et al., there may be a common pathway in non-IPF disease with IPF-like disease progression [42]. Based on this hypothesis, antifibrotics might be effective in COVID-19-associated lung fibrosis in an early stage. The similar cytokine profile in both IPF and COVID-19 also suggests a common pathway [43]. Although it is currently uncertain to which extent COVID-19-associated lung fibrosis will be progressive, the early treatment seems prudent considering the autopsy study in ARDS patients, showing that a longer disease duration led to a higher risk of lung fibrosis [44]. Using a score to assess the risk of progression to severe disease may help in the timing of treatment escalation [45]. Another argument for antifibrotic treatment early in the disease course is that previous coronavirus outbreaks have been associated with substantial postviral lung fibrosis and physical impairment [46]. In one interventional study of 30 patients with COVID-19 requiring mechanical ventilation, the patients were treated with nintedanib, showing no significant differences in 28-day mortality compared to the control group without nintedanib, but it showed significantly shorter lengths of mechanical ventilation and lower percentages of high-attenuation in computed tomography volumetry [47].

The results of four ongoing trials investigating antifibrotic drugs in COVID-19-associated lung fibrosis probably will shed some light on this important and urgent question [48–51]. Interestingly, the mTOR inhibitor rapamycin, a well-known drug in transplant medicine, has been suggested to have both antiviral action and antifibrotic properties. The latter is also known for the treatment of lymphangioleiomyomatosis [46].

In severe lung fibrosis, only lung transplantation as the ultimate therapy could be an option for selected patients. The number of COVID-19-related lung transplantations is relatively small but increasing. In the United States, a query of the United Network for Organ Sharing showed that as of 30 April 2021, only 78 lung transplantations had been performed (50 for CARDS and 28 for PCPF) [14], increasing to 299 (183 for CARDS and 107 for PCPF) as of 31 January 2022 [52]. In Europe, the Eurotransplant consortium reported only 21 patients undergoing lung transplantation for COVID-19-related end stage lung disease [14]. In the scientific literature, only a limited number of case reports and case series have been published so far with limited follow-up data. The difficult question is whether patients benefit from lung transplantation in COVID-19-related end stage lung disease and how their perioperative and long-term disease course differs in comparison to lung transplant experiences with other fibrotic lung diseases. Since COVID-19 frequently affects multiple organ systems the short-term and long-term outcomes may differ. ECMO is being used increasingly for COVID-19 patients with improved outcomes [53–55]. In the case reports and case series, early outcomes have been acceptable for these marginal lung transplant recipient candidates, although Cypel et al. warned about publication bias in the context of a new disease [56]. The ideal "transplant window" is transplantation that is not too early (e.g., in patients that may recover spontaneously or due to medical therapy) and not too late (e.g., in extremely debilitated patients with muscle wasting, critical illness polyneuropathy, and myopathy, lacking good potential for recovery and rehabilitation, relevant pulmonary hypertension with right-sided heart failure, or even with multi-organ failure). Some potential for recovery has been described in patients with severe COVID-19-related ARDS [55]. Unfortunately, regarding the risk factors for COVID-19-associated fibrosis, many patients will be excluded from being transplant candidates due to comorbidities, and/or secondary complications such as renal dysfunction, muscle wasting, or multiple organ failure while in the intensive care unit (ICU) [56]. Importantly, the patients should be completely free of the coronavirus infection to prevent harboring the virus and developing a relapse of the infection later, especially during chronic triple immunosuppressive therapy after lung transplantation. Another important issue is that there is a general consensus described in the latest consensus document of the International Society of Heart and Lung Transplantation (ISHLT), that patients should be awake and able to discuss the lung transplantation and provide consent for the procedure [57,58]. The enormous impact of lung transplantation on quality of life should be fully understood. Waking up after intubation for an acute viral illness and being informed that lung transplantation has been performed, implicating a life with chronic triple immunosuppression and potential complications ahead, can lead to a severe psychotrauma, that can be too difficult to deal with [56]. The ten considerations that should be carefully evaluated when assessing a patient with COVID-19-associated lung transplantation have been published in 2020. In very experienced high-volume centers some criteria may be disregarded on a case-by-case basis. As mentioned by Lepper et al., the case series of lung transplantation for COVID-19 described by Bharat et al. showed that many patients had a complicated intraoperative and postoperative course, including mass transfusions, continued extracorporeal support, re-thoracotomy, primary graft dysfunction, and prolonged postoperative stay in the ICU [54,59]. However, in this case series, patients did not necessarily fulfill all requirements as mentioned by some other authors [54,56]. Defining the ideal recipient in this situation remains difficult, in which increased in-hospital SARS-CoV-2 transmission, and depleting healthcare resourses should also be taken into account. Those patients who have a risk of imminent death and those who are on mechanical ventilation or ECMO should have priority if there are no absolute or important relative contraindications or major risk factors.

An unanswered question is whether patients with pre-existing lung fibrosis should be vaccinated with the SARS-CoV-2 vaccination. Prevention of COVID-19 seems to be the best strategy. However, acute exacerbations of idiopathic pulmonary fibrosis after the SARS-CoV-2 vaccination have been described, and this could be underreported as many cases will not necessarily be published in the medical literature [60]. Another study described the development of pulmonary fibrosis in a previously healthy patient after the SARS-CoV-2 vaccination [61]. VigiAccess, the global pharmacovigilance database of the World Health Organization (WHO), reports 679 patients with ARDS, 346 patients with lung consolidation, 159 patients with organizing pneumonia, and 280 patients with pulmonary fibrosis, related to Comirnaty (Pfizer) COVID-19 vaccination. In relation to the Spikevax (Moderna) vaccination, ARDS was reported in 679 patients, interstitial lung disease in 678 patients, idiopathic pulmonary fibrosis in 56 patients, acute lung injury in 12 patients, and idiopathic interstitial pneumonia in 3 patients [62].


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

Emmanouil G. Thrapsaniotis

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/67774

#### 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 [1–7]. 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.

© 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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,

$$H = H\_e + H\_f + H\_I.\tag{1}$$

He has the form

$$H\_{\ell} = \frac{1}{2}\overrightarrow{p}^2 + V(\overrightarrow{r}),\tag{2}$$

where Vðr !Þ is the atomic potential. The photonic field has the Hamiltonian

$$H\_f = a a^+ a,\tag{3}$$

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

Atoms-Photonic Field Interaction: Influence Functional and Perturbation Theory http://dx.doi.org/10.5772/67774 153

$$H\_I = -\vec{e}\,\vec{r}\cdot\dot{E}\_f(\vec{r},\tau). \tag{4}$$

E ! fðr !, τÞ is the field operator of the photonic pulse given by the expression

$$\overrightarrow{E}\_f(\overrightarrow{r},\tau) = \frac{1}{\sqrt{V}} \text{il}(\omega) \mathfrak{g}(\tau) \left[ \widehat{\varepsilon} a \overline{e}^{i\overrightarrow{\tilde{K}}\_{ph}\cdot\overrightarrow{r}} - \widehat{\varepsilon}^\* a^+ e^{-i\overrightarrow{\tilde{K}}\_{ph}\cdot\overrightarrow{r}} \right]. \tag{5}$$

℘ðτÞ is the pulse's envelope function. In Eq. (5) lðωÞ ¼ ffiffiffiffiffiffiffiffiffi 2πω p is a real frequency function, ε \_ is the polarization vector, ω is the pulse's carrier frequency, k ! ph is the radiation wave vector and V is a large volume. Then HI has the form

$$H\_I = \mathbf{g}(\tau)a + \mathbf{g}^\*(\tau)a^+. \tag{6}$$

We have set

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

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

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.

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

þ Vðr

He ¼ 1 2 p !2

!Þ is the atomic potential. The photonic field has the Hamiltonian

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

H ¼ He þ Hf þ HI: ð1Þ

Hf ¼ ωaþa, ð3Þ

!Þ, ð2Þ

obtain a closed angularly decomposed expression of the whole path integral.

atoms (see Refs. [6, 7, 9, 10]).

152 Recent Studies in Perturbation Theory

2. System Hamiltonian and path integration

photonic field with the electron HI .that is,

He has the form

where Vðr

$$g(\tau) = -\frac{1}{\sqrt{V}} \text{id}(\omega) \otimes (\tau) \stackrel{\frown}{\varepsilon} \cdot \overrightarrow{r}(\tau) e^{i\overrightarrow{\tilde{K}}\_{\text{ph}} \cdot \overrightarrow{r}^{\circ}(\tau)}.\tag{7}$$

Now we combine the photonic field variables in the term

$$H\_0(a^+,a;\tau) = H\_f + H\_I = a a^+ a + g(\tau)a + g^\*(\tau)a^+. \tag{8}$$

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:

$$\begin{split} \mathcal{K}(\boldsymbol{a\_{f}},\overrightarrow{\boldsymbol{r}}\_{f},\boldsymbol{t}\_{i};\boldsymbol{a\_{i}},\overrightarrow{\boldsymbol{r}}\_{i};\boldsymbol{t}\_{i}) &= \int \boldsymbol{D}\overrightarrow{\boldsymbol{r}}(\boldsymbol{\tau}) \frac{\boldsymbol{D}\overrightarrow{\boldsymbol{p}}(\boldsymbol{\tau})}{\left(2\pi\right)^{3}} \times \\ \exp\left[\begin{matrix} \overset{\scriptstyle l}{i} \\ \overset{\scriptstyle l}{i} \end{matrix} \bigg| \boldsymbol{\pi}\left(\overrightarrow{\boldsymbol{p}}(\boldsymbol{\tau}) \cdot \overset{\scriptstyle \dot{\boldsymbol{r}}}{\overset{\scriptstyle \boldsymbol{r}}{\boldsymbol{r}}}(\boldsymbol{\tau}) - \frac{\overrightarrow{\boldsymbol{p}}^{2}(\boldsymbol{\tau})}{2} - \boldsymbol{V}(\overset{\scriptstyle \boldsymbol{r}}{\boldsymbol{r}}(\boldsymbol{\tau})) \right) - \overset{\scriptstyle l}{i} \bigg| \boldsymbol{\pi}\boldsymbol{g}(\boldsymbol{\tau})\boldsymbol{Z}(\boldsymbol{\tau},\boldsymbol{t}\_{i}) - \\ \frac{1}{2} \left(|\boldsymbol{a\_{f}}|^{2} + |\boldsymbol{a\_{i}}|^{2}\right) + \boldsymbol{Y}(\boldsymbol{t\_{f}},\boldsymbol{t\_{i}})\boldsymbol{a\_{i}^{\*}}\boldsymbol{\alpha\_{i}} + \boldsymbol{Z}(\boldsymbol{t\_{f}},\boldsymbol{t\_{i}})\boldsymbol{\alpha\_{f}^{\*}} - \boldsymbol{i}\boldsymbol{\alpha\_{i}}\boldsymbol{X}(\boldsymbol{t\_{f}},\boldsymbol{t\_{i}}) \bigg], \end{split} \tag{9}$$

where Yðtf , tiÞ, Xðtf , tiÞ, and Zðtf , tiÞ read:

$$Y(t\_{\not{t}}, t\_{\!i}) = \exp\left[-i\not{t}\_{\not{t}}^{\not{t}} d\tau \omega(\tau)\right] = \exp\left(-i\omega(t\_{\not{t}} - t\_{\!i})\right),\tag{10}$$

$$X(t\_{f'}, t\_i) = \int\_{t\_i}^{t\_f} d\tau g(\tau) Y(\tau, t\_i) \,\tag{11}$$

$$Z(t\_f, t\_i) = -i \int\_{t\_i}^{t\_f} d\tau g^\*(\tau) \exp\left[-i \oint\_{\tau} d\tau' \omega(\tau')\right]. \tag{12}$$

The propagator in Eq. (9) with diagonal field variables (αi ¼ αf ¼ α) can be written as

$$K(\mathbf{z}, \overrightarrow{r}\_f, t\_f; \mathbf{z}, \overrightarrow{r}\_\nu, t\_i) = \int D\overrightarrow{r}(\tau) \frac{D\overrightarrow{p}(\tau)}{(2\pi)^3} \exp\left[i \int\_l^t d\tau \left[\overrightarrow{p}(\tau) \cdot \overrightarrow{\overrightarrow{r}}(\tau) - \frac{\overrightarrow{p}^2(\tau)}{2} - V(\overrightarrow{r}(\tau))\right]\right]$$

$$+ A - B|\alpha|^2 + D\_1\alpha + D\alpha^\*\right].\tag{13}$$

The parameters are given as follows:

$$A(t\_f, t\_i) = -\frac{1}{V} e^2 \vec{l}^2(\omega) \oint\_{\substack{t\_i \\ t\_i}} \stackrel{\scriptstyle \tau}{\int} d\rho \otimes \rho(\tau) \stackrel{\scriptstyle \alpha}{\cdot} \cdot \stackrel{\scriptstyle \overrightarrow{r}}{r}(\tau) e^{i\vec{k}\_{ph}\cdot\overrightarrow{r}\cdot(\tau)} \otimes (\rho) \stackrel{\scriptstyle \alpha}{\cdot}^\* \cdot \stackrel{\scriptstyle \overrightarrow{r}}{r}(\rho) e^{-i\vec{k}\_{ph}\cdot\overrightarrow{r}\cdot(\rho)} e^{-i\omega(\tau-\rho)},\qquad(14)$$

$$B(t\_f - t\_i) = 1 - Y(t\_{f'}, t\_i) = 1 - e^{-i\omega(t\_f - t\_i)},\tag{15}$$

$$D(t\_f, t\_i) = \frac{1}{\sqrt{V}}el(\omega) \int\_{t\_i}^{t\_f} d\tau \otimes (\tau) \stackrel{\frown}{\varepsilon}^\* \cdot \vec{r}(\tau) e^{-i\overrightarrow{K}\_{\text{ph}} \cdot \overrightarrow{r}(\tau)} e^{-i\omega(t\_f - \tau)}\,. \tag{16}$$

$$D\_1(t\_{\not\prime}, t\_i) = -\frac{1}{\sqrt{V}}el(\omega)\oint\_{t\_i}^{t\_{\not\prime}} d\tau \otimes (\tau)\stackrel{\frown}{\varepsilon} \cdot \overrightarrow{r}(\tau)e^{i\overrightarrow{k}\_{\not\prime}\cdot\overrightarrow{r}(\tau)}e^{-i\omega(\not\equiv t\_i)}.\tag{17}$$

In the case of a field transition between an initial photonic state jΦ1〉 and a final one jΦ2〉, the reduced propagator of finite time takes the form

$$
\tilde{K}(\overrightarrow{r}\_{f}, t\_{f}; \overrightarrow{r}\_{i}, t\_{i}) = \int \frac{d^{2}\alpha}{\pi} e^{|\alpha|^{2}} \langle \Phi\_{2} | \alpha \rangle \mathcal{K}(\alpha, \overrightarrow{r}\_{f}, t\_{f}; \alpha, \overrightarrow{r}\_{i}, t\_{i}) \langle \alpha | \Phi\_{1} \rangle. \tag{18}
$$

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

$$\begin{split} \overset{\circ}{K}(\overrightarrow{r}\_{f},t\_{\circ};\overrightarrow{r}\_{i\circ},t\_{i}) &= \mathsf{C}(t\_{f}-t\_{i})\mathsf{K}\_{0}(\overrightarrow{r}\_{f},t\_{\circ};\overrightarrow{r}\_{i\circ},t\_{i}) \\ &= \mathsf{C}(t\_{f}-t\_{i}) \Big{[} \Big{]} \overset{\circ}{D}\overrightarrow{r}(\pi) \frac{\overrightarrow{Dp}(\pi)}{\left(2\pi\right)^{3}} \exp\left\{i\mathsf{S}\_{\text{tot}}[\overrightarrow{p},\overrightarrow{r},\pi] \right\}, \end{split} \tag{19}$$

where

#### Atoms-Photonic Field Interaction: Influence Functional and Perturbation Theory http://dx.doi.org/10.5772/67774 155

$$\mathcal{C}(t) = \frac{\exp\left(\frac{\beta\gamma^\*}{B(t)} - \frac{1}{2}|\beta|^2 - \frac{1}{2}|\gamma|^2\right)}{B(t)}.\tag{20}$$

The action is

Zðtf , ti޼�i

þA � Bjαj

dρ℘ðτÞε

ffiffiffiffi V p elðωÞ

> ffiffiffiffi V p elðωÞ

> > ð d2 α π e jαj 2

i, tiÞ ¼ Cðtf � tiÞK0ðr

¼ Cðtf � tiÞ

\_ � r !ðτÞe i k ! ph�r

> ð tf

ti

Bðtf � tiÞ ¼ 1 � Yðtf , tiÞ ¼ 1 � e

dτ℘ðτÞε \_∗ � r !ðτÞe �i k ! ph�r !ðτÞ e �iωðtf �τÞ

dτ℘ðτÞε

In the case of a field transition between an initial photonic state jΦ1〉 and a final one jΦ2〉, the

Here we consider that we have a field transition from an initial coherent state jβ〉 to a final one jγ〉. So we can integrate to obtain the following reduced propagator for the motion of the

> ! f , tf ; r ! i, tiÞ

D r!ðτÞ

ð ð

〈Φ2jα〉Kðα, r

! f , tf ; α, r !

D p!ðτÞ ð2πÞ

3 exp fiStot½p

!, r

\_ � r !ðτÞe i k ! ph�r !ðτÞ e �iωðτ�tiÞ

ð tf

ti

Kðα, r ! f , tf ; α, r ! i, tiÞ ¼ ð D r!ðτÞ

154 Recent Studies in Perturbation Theory

Aðtf , ti޼� 1

The parameters are given as follows:

V e 2 l 2 ðωÞ ð tf

ti dτ ð τ

Dðtf , tiÞ ¼ 1

D1ðtf , ti޼� 1

reduced propagator of finite time takes the form

K~ ðr ! f , tf ; r ! i, tiÞ ¼

K~ ðr ! f , tf ; r !

electron,

where

ti

ð tf

dτg∗ðτÞ exp �i

ð tf

3 7

" # 2

: ð13Þ

5: ð12Þ

!ðτÞÞ

, ð15Þ

i, tiÞ〈αjΦ1〉: ð18Þ

!, τ�g, ð19Þ

, ð16Þ

: ð17Þ

, ð14Þ

2 6 4

τ dτ0 ωðτ0 Þ

ti

D p!ðτÞ ð2πÞ

The propagator in Eq.  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

$$\lim\_{t \to \infty} \frac{1}{B(t)} = \lim\_{t \to \infty} \frac{1}{1 - \exp\left(-i(\omega - i0)t\right)} = 1. \tag{26}$$

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

$$
\overrightarrow{r}(\rho) = \overrightarrow{r}(\pi) + (\rho - \pi)\dot{\overrightarrow{r}}(\pi) + \dots \tag{27}
$$

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 Vðr !Þ. So the part of the Lagrangian that interests us reads

$$\begin{split} L &= \frac{\stackrel{\cdot}{\cdot}^{2}(\mathsf{r})}{2} + \frac{1}{\sqrt{V}} \mathsf{el}(\omega) \Big( \beta \chi(\mathsf{r}) \stackrel{\cdot}{\bar{\varepsilon}} \cdot \overline{\bar{r}}(\mathsf{r}) \stackrel{\cdot}{e^{i\vec{K}\_{\mathrm{ph}} \cdot \vec{\mathcal{T}}^{\prime}(\mathsf{r})}} + \mathsf{y}^{\*} \chi^{\*}(\mathsf{r}) \stackrel{\cdot}{\bar{\varepsilon}}^{\*} \cdot \overline{\bar{r}}(\mathsf{r}) e^{-i\vec{K}\_{\mathrm{ph}} \cdot \vec{\mathcal{T}}^{\prime}(\mathsf{r})} \Big) \\ &+ \frac{1}{V} \mathrm{e}^{2} l^{2}(\omega) \mathsf{g} \rho(\mathsf{r}) \Big[ \mathsf{d} \rho \otimes \rho \big] \Big[ \mathrm{i} \frac{e^{i\omega(\mathsf{r}-\rho)}}{\dot{e}^{i\omega(\mathsf{j}-\mathsf{k})}} - \mathrm{i} \Big( \stackrel{\cdot}{\bar{\varepsilon}}^{\*} \cdot \overline{\bar{r}}(\mathsf{r}) \mathrm{e}^{-i\vec{K}\_{\mathrm{ph}} \cdot \vec{\mathcal{T}}^{\prime}(\mathsf{r})} \Big) \Big( \stackrel{\cdot}{\bar{\varepsilon}} \cdot \overline{\bar{r}}(\rho) \mathrm{e}^{i\vec{K}\_{\mathrm{ph}} \cdot \vec{\mathcal{T}}^{\prime}(\rho)} + \mathrm{c.c.} \Big], \end{split} \tag{28}$$

and has equation of motion

$$
\ddot{\vec{r}}\,(\tau) = O\left(\frac{1}{\sqrt{V}}\right). \tag{29}
$$

Therefore we can set,

$$
\overrightarrow{r}(\rho) = \overrightarrow{r}(\tau) + O\left(\frac{1}{\sqrt{V}}\right). \tag{30}
$$

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

$$K\_0 = T + TVT + TVTVT + \dots \tag{31}$$

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

$$\begin{split} \mathbb{S}\_{\text{tot}} \left[ \overrightarrow{p}, \overrightarrow{r}, \tau \right] &= \int\_{\vec{\imath}}^{\vec{\imath}} \left[ \overrightarrow{p}(\tau) \cdot \dot{\overrightarrow{r}}(\tau) - \frac{\overrightarrow{p}^{2}(\tau)}{2} - V(\overrightarrow{r}(\tau)) \right] d\tau \\ &+ i \frac{1}{\sqrt{V}} el(\omega) \Big|\_{\vec{\imath}}^{\vec{\imath}} d\tau \Big( \beta \chi(\tau) \overset{\leftrightarrow}{\varepsilon} \cdot \overrightarrow{r}(\tau) e^{i \overrightarrow{k}\_{\text{ph}} \cdot \overrightarrow{r}(\tau)} + \upgamma^{\*} \chi^{\*}(\tau) \overset{\leftrightarrow}{\varepsilon}^{\ast} \cdot \overrightarrow{r}(\tau) e^{-i \overleftarrow{k}\_{\text{ph}} \cdot \overrightarrow{r}(\tau)} \Big) \\ &+ \frac{1}{V} \varepsilon^{2} l^{2}(\omega) \Big| \underbrace{\overleftarrow{\imath}}\_{\vec{\imath}} \left( \text{d}\tau \nu(\tau) \Big| \overset{\leftrightarrow}{\varepsilon} \cdot \overrightarrow{r}(\tau) \Big)^{2} , \end{split} \tag{32}$$

where

limt!∞ 1 BðtÞ

> r !ðρÞ ¼ r

\_ � r !ðτÞe i k ! ph�r

dρ℘ðρÞ i eiωðτ�ρÞ

eiωðtf �tiÞ � 1

€ r

r !ðρÞ ¼ r

propagator in Eq. (19) with respect to the potential term. That is,

Taylor expansions

156 Recent Studies in Perturbation Theory

in the absence of Vðr

1 ffiffiffiffi V

and has equation of motion

In the case of the presence of Vðr

Therefore we can set,

action

p elðωÞ βχðτÞε

ð τ

ti

L ¼ \_ r !2 ðτÞ 2 þ

> þ 1 V e 2 l 2 ðωÞ℘ðτÞ

¼ limt!∞

1 � exp

1

r

� iðω � i0Þt

� ¼ 1: ð26Þ

,

ð28Þ

!ðτÞ þ …, ð27Þ

: ð29Þ

: ð30Þ

�

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

!ðτÞþðρ � τÞ \_

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

!Þ. So the part of the Lagrangian that interests us reads

� ε \_∗ � r !ðτÞe �i k ! ph�r !ðτÞ �� ε \_ � r !ðρÞe i k ! ph�r !ðρÞ � þ c:c:

!ðτÞ ¼ O 1

ffiffiffiffi V p � �

> ffiffiffiffi V p � �

!Þ we perform a full order perturbation expansion of the full

K0 ¼ T þ TVT þ TVTVT þ …: ð31Þ

!ðτÞ þ O 1

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

!ðτÞ þ γ∗χ∗ðτÞε

!

\_∗ � r !ðτÞe �i k ! ph�r !ðτÞ

" #

$$\nu(\tau) = \gcd\_{\vec{t}\_i}^{\tau} \left[ \wp(\rho) \xi(\tau - \rho) d\rho,\tag{33}$$

$$\xi(\tau-\rho) = \csc\left[\frac{\omega(t\_f-t\_i)}{2}\right] \cos\left[\omega(\tau-\rho) - \frac{\omega(t\_f-t\_i)}{2}\right].\tag{34}$$

Finally, we notice that in the long wavelength approximation we can set ei k ! ph�r ! ffi 1. So we obtain the following expression

$$\begin{split} \mathcal{S}\_{\text{tot}}\left[\overrightarrow{p},\overrightarrow{r},\tau\right] &= \bigcap\_{t\_i}^{t\_f} \left[\overrightarrow{p}(\tau)\cdot\overrightarrow{\dot{r}}\left(\tau\right) - \frac{\overrightarrow{p}^2\left(\tau\right)}{2} - V(\overrightarrow{r}\left(\tau\right))\right] d\tau + \\ \leq \inf\_{t\_i}^{t\_f} el(\omega) \left[d\tau \left[\beta\chi\left(\tau\right)\widehat{\boldsymbol{\varepsilon}}\cdot\overrightarrow{\boldsymbol{r}}\left(\tau\right) + \boldsymbol{\gamma}^\*\chi^\*\left(\tau\right)\widehat{\boldsymbol{\varepsilon}}^\*\cdot\overrightarrow{\boldsymbol{r}}\left(\tau\right)\right] + \\ \leq \frac{1}{V} e^2 l^2(\omega) \left[d\tau\nu\left(\tau\right)\left|\widehat{\boldsymbol{\varepsilon}}\cdot\overrightarrow{\boldsymbol{r}}\left(\tau\right)\right|^2\right. \end{split} \tag{35}$$

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

$$
\stackrel\frown{\varepsilon} = \stackrel\frown{\varepsilon}\_x \cos\left(\frac{\xi}{2}\right) \pm i \stackrel\frown{\varepsilon}\_y \sin\left(\frac{\xi}{2}\right),
\tag{36}
$$

where ε \_ x and ε \_ y 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 Kξ 0ðr ! f , tf ; r ! i, tiÞ of Eq. (19) with the above polarization vector ε \_ has the discrete form

$$K\_0^\sharp(\vec{r\_f}, t\_{\vec{r}}; \vec{r\_{\nu}}, t\_{\vec{\nu}}) = \prod\_{n=1}^N \left[ \int\_{-\infty}^\infty d\vec{r\_n} \right] \prod\_{n=1}^{N+1} \left[ \int\_{-\infty}^\infty \frac{d\vec{p\_n}}{(2\pi)^3} \right]$$

$$\times \exp\left\{ i \sum\_{n=1}^{N+1} \left[ \vec{p\_n} \cdot (\vec{r\_n} - \vec{r\_{n-1}}) - \varepsilon \left( \frac{\vec{p\_n}^2(\tau)}{2} + V(\vec{r\_n}) \right) \right. \\ \tag{37}$$

$$+ i \sqrt{\frac{2\pi\omega}{V}} \varepsilon (\beta \chi\_n \hat{\varepsilon} \cdot \vec{r\_n} + \gamma^\* \chi\_n^\* \hat{\varepsilon}^\* \cdot \vec{r\_n}) + \frac{2\pi\omega}{V} \varepsilon \nu\_n \left| \hat{\varepsilon} \cdot \vec{r\_n} \right|^2 \right] \right\}.$$

All the functions with index n are evaluated at time τn ¼ nε þ ti where ε ¼ tf �ti Nþ1. χn and νn have the form (see Eqs. (22) and (33))

$$\chi\_n = \wp(\tau\_n) \frac{e^{-i\omega \tau\_n}}{e^{-i\omega t\_i} - e^{-i\omega t\_f}},\tag{38}$$

$$
\nu\_n = \nu(\tau\_n). \tag{39}
$$

Additionally, we note that we have set r ! 0 ¼ r ! i and r ! Nþ1 ¼ r ! f . Now we insert delta functions in Eq. (37) to get the expression

$$\begin{split} K\_{0}^{k}(\vec{r}\_{f},t\_{f};\vec{r}\_{\boldsymbol{n}},t\_{i}) &= \\ &\prod\_{n=1}^{N} \left[ \int\_{-\infty}^{\infty} d\vec{r}\_{n} \right] \prod\_{n=1}^{N+1} \left[ \int\_{-\infty}^{\infty} \frac{d\vec{p}\_{n}}{(2\pi)^{3}} \prod\_{n=1}^{N+1} \left[ \int\_{-\infty}^{\infty} d^{2}w\_{n} \right] \prod\_{n=1}^{N+1} \left[ \delta^{(2)}(w\_{n}-\hat{\boldsymbol{\xi}}\cdot\vec{r}\_{n}) \right] \\ &\times \exp\left\{ i\sum\_{n=1}^{N+1} \left[ \vec{p}\_{n}\cdot\left(\vec{r}\_{n}-\vec{r}\_{n-1}\right) - \varepsilon \left(\frac{\vec{p}\_{n}^{2}\,\textrm{(\tau)}}{2} + V(\vec{r}\_{n})\right) \right. \\ &\left. + i\sqrt{\frac{2\pi\omega}{V}} \varepsilon \left(\beta\chi\_{n}w\_{n}+\gamma^{\*}\chi\_{n}^{\*}w\_{n}^{\*}\right) + \frac{2\pi\omega}{V}\varepsilon v\_{n} \left| w\_{n} \right|^{2} \right\} \right\}. \end{split} \tag{40}$$

We have defined δð2Þ ðzÞ ¼ δðzÞδðz∗Þ. Moreover wn ¼ wxn þ iwyn. The delta functions have the representation

$$\begin{split} \delta^{(2)}\left(w\_{n}-\hat{\boldsymbol{\varepsilon}}\cdot\overrightarrow{\boldsymbol{r}\_{n}}\right) &= \\ \delta^{(2)}\left(w\_{n}-\boldsymbol{r}\_{n}\left[\sin\vartheta\_{n}\left(\cos\left(\frac{\xi}{2}\right)\cos\varphi\_{n}\pm i\sin\left(\frac{\xi}{2}\right)\sin\varphi\_{n}\right)\right]\right) &= \frac{1}{\left(2\pi\right)^{2}} \\ \times \int\_{-\infty}^{\infty}\int\_{-\infty}^{\infty}\hat{d}^{2}\lambda\_{n}\exp\left[i\frac{1}{2}\left(\lambda\_{n}w\_{n}+\lambda\_{n}^{\*}w\_{n}^{\*}\right)-i\frac{1}{2}\lambda\_{n}\left(\hat{\boldsymbol{\varepsilon}}\_{x}\cos\left(\frac{\xi}{2}\right)\pm i\hat{\boldsymbol{\varepsilon}}\_{y}\sin\left(\frac{\xi}{2}\right)\right)\cdot\overrightarrow{\boldsymbol{r}\_{n}}\right] \\ -i\frac{1}{2}\lambda\_{n}^{\*}\left(\hat{\boldsymbol{\varepsilon}}\_{x}\cos\left(\frac{\xi}{2}\right)\mp i\hat{\boldsymbol{\varepsilon}}\_{y}\sin\left(\frac{\xi}{2}\right)\right)\cdot\overrightarrow{\boldsymbol{r}\_{n}}\right] &= \frac{1}{\left(2\pi\right)^{2}}\int\_{-\infty}^{\infty}\hat{d}^{2}\lambda\_{n} \\ \times\exp\left[i\lambda\_{m}w\_{m}-i\lambda\_{\eta n}w\_{\eta n}-i\lambda\_{m}\cos\left(\frac{\xi}{2}\right)\hat{\boldsymbol{\varepsilon}}\_{x}\cdot\overrightarrow{\boldsymbol{r}\_{n}}\pm i\lambda\_{\eta n}\sin\left(\frac{\xi}{2}\right)\hat{\boldsymbol{\varepsilon}}\_{y}\cdot\overrightarrow{\boldsymbol{r}\_{n}}\right]. \end{split} \tag{41}$$

We have set λn ¼ λxn þ iλyn.  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,

$$e^{i\overrightarrow{\mathbf{k}\cdot\mathbf{r}}\cdot\overrightarrow{r}} = 4\pi \sum\_{l=0}^{\infty} \sum\_{m=-l}^{l} i^l j\_l(|\overrightarrow{\mathbf{k}}|r) Y\_{\text{lm}}^{\ast}(\mathfrak{d}\_{\mathbf{k}\prime}\,\boldsymbol{\varrho}\_{\mathbf{x}}) Y\_{\text{lm}}(\mathfrak{d}\_{\mathbf{}}\,\boldsymbol{\varrho}),\tag{42}$$

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

$$\delta^{(2)}(w\_n - \stackrel{\frown}{\varepsilon} \cdot \vec{r}\_n) = \sum\_{l\_n=0}^{\infty} \sum\_{m\_n=-l\_n}^{l\_n} g\_{l\_n m\_n}(w\_n', r\_n) \sqrt{4\pi} Y\_{l\_n m\_n}(\mathfrak{G}\_{n\nu} \varphi\_n), \tag{43}$$

where

�j ln ðjλnjrnÞ exp ð�imnϕλnÞ,

where ε \_ x and ε \_

form

The propagator Kξ

158 Recent Studies in Perturbation Theory

polarization while the lower one to right one.

i, tiÞ ¼ Y N

> p ! n � ðr ! n � r !

εðβχn ε \_ � r !

Now we insert delta functions in Eq.  f .

 Moreover wn ¼ wxn þ iwyn. The delta functions have the

�∞

ε βχnwn þ γ∗χ∗

� � þ

0ðr ! f , tf ; r !

Kξ 0ðr ! f , tf ; r !

� exp

r

þ i

Additionally, we note that we have set r

Y N

ð ∞

2 4

�∞ d r ! n

> ( i N X þ1

ffiffiffiffiffiffiffiffiffi 2πω V

n¼1 p ! n � r ! n � r ! n�1 � �

n¼1

� exp

r

þi

Kξ 0ðr ! f , tf ; r ! i, tiÞ ¼

We have defined δð2Þ

representation

the form (see Eqs. (22) and (33))

( i N X þ1

ffiffiffiffiffiffiffiffiffi 2πω V

n¼1

y are the unit vectors along the x- and y-axis. The upper sign corresponds to left

\_ has the discrete

Nþ1. χn and νn have

ð37Þ

ð40Þ

i, tiÞ of Eq. (19) with the above polarization vector ε

ð ∞

2 4

d p ! n ð Þ 2π 3 3 5

V ενn ε

\_ � r ! n

νn ¼ νðτnÞ: ð39Þ

� � � � �

, ð38Þ

� � � � �

�∞

!2 nðτÞ 2 þ Vðr ! nÞ

nÞ þ 2πω

$$g\_{l\_{\rm u}m\_{\rm u}}(w\_{\rm n}',r\_{\rm n}) = (-i)^{l\_{\rm u}} \frac{O\_{l\_{\rm u}m\_{\rm u}}}{\left(2\pi\right)^{2}} \int\_{-\infty}^{\infty} \int d^{2}\lambda\_{\rm n} \exp\left[i\lambda\_{\rm xn}w\_{\rm xn}' - i\lambda\_{\rm yn}w\_{\rm yn}'\right] \tag{44}$$

$$\begin{split} O\_{l\_{n}m\_{n}} &= \sqrt{(2l\_{n}+1)\frac{(l\_{n}-m\_{n})!}{(l\_{n}+m\_{n})!}}P\_{l\_{n}}^{m\_{n}}(0) \\ &= \sqrt{(2l\_{n}+1)\frac{(l\_{n}-m\_{n})!}{(l\_{n}+m\_{n})!}}\frac{\sqrt{\tau}2^{m\_{n}}}{\Gamma\left(\frac{l\_{n}-m\_{n}}{2}+1\right)\Gamma\left(\frac{-l\_{n}-m\_{n}+1}{2}\right)}. \end{split} \tag{45}$$

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

$$w\_{\mathbf{x}n} = w\_{\mathbf{x}n}' \cos\left(\frac{\xi}{2}\right) = |w\_n'| \cos\left(\varphi\_{w'n}\right) \cos\left(\frac{\xi}{2}\right),\tag{46}$$

$$w\_{\rm yn} = w\_{\rm yn}' \sin\left(\frac{\xi}{2}\right) = |w\_n'| \sin\left(\varphi\_{w'n}\right) \sin\left(\frac{\xi}{2}\right),\tag{47}$$

and

$$
\hbar \mathbf{w}'\_n = \mathbf{w}'\_{\infty} + i \mathbf{w}'\_{\text{yn}} = |\mathbf{w}'\_n| e^{i\boldsymbol{\wp}\_{\mathbf{w}'\mathbf{w}}}.\tag{48}
$$

On integrating over ϕλn we get

$$\begin{split} g\_{l\_{\nu}m\_{\boldsymbol{n}}}(\boldsymbol{w}'\_{\boldsymbol{n}},\boldsymbol{r}\_{\boldsymbol{n}}) &= (-\mathop{\rm l}{\mathop{\rm l}}{\mathop{\rm l}}{\mathop{\rm l}}{\mathop{\rm l}}\_{\nabla\boldsymbol{n}} \exp\left( \mathop{\rm im}{\mathop{\rm l}}{\mathop{\rm l}}\_{\boldsymbol{n}} \exp\left( \mathop{\rm j}{\mathop{\rm l}}\_{\boldsymbol{n}'\boldsymbol{n}} \left( \boldsymbol{\rho}\_{\boldsymbol{n}'\boldsymbol{n}} + \frac{\boldsymbol{\tau}}{2} \right) \right) \\ & \quad \times \begin{cases} d\boldsymbol{\rho}\_{\boldsymbol{\lambda}\_{\boldsymbol{n}}} \boldsymbol{\rho}\_{\boldsymbol{\lambda}\_{\boldsymbol{n}}} \boldsymbol{j}\_{\boldsymbol{l}\_{\boldsymbol{n}}} (\boldsymbol{\rho}\_{\boldsymbol{\lambda}\_{\boldsymbol{n}}} \boldsymbol{r}\_{\boldsymbol{n}}) \boldsymbol{I}\_{\boldsymbol{m}\_{\boldsymbol{n}}} (\boldsymbol{\rho}\_{\boldsymbol{\lambda}\_{\boldsymbol{n}}} | \boldsymbol{w}'\_{\boldsymbol{n}}|) \,. \end{split} \tag{49}$$

ρλn ¼ jλnj 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

$$\begin{aligned} K\_0^\xi(\vec{r\_f}, t\_f; \vec{r\_i}, t\_i) &= \\ \frac{1}{r\_f r\_i} \sum\_{l=0}^m \sum\_{m=-l}^l \sum\_{q=0}^m \sum\_{p=-q}^q K\_{\text{lmq}}^\xi(r\_f, t\_f; r\_i, t\_i) \sqrt{4\pi} Y\_{\text{lm}}(\mathfrak{S}\_f, \mathfrak{g}\_f) Y\_{\text{qp}}(\mathfrak{S}\_f, \mathfrak{g}\_f) Y\_{\text{qp}}^\*(\mathfrak{S}\_i, \mathfrak{g}\_i), \end{aligned} \tag{50}$$

where after standard manipulations [11] on the angular parts of the atomic system Kξ lmqðrf , tf ; ri, tiÞ takes the form

$$\begin{split} K\_{\text{Im}q}^{\psi}(\boldsymbol{r}\_{f},t\_{f};\boldsymbol{r}\_{i},t\_{i}) &= \prod\_{n=1}^{N} \left[ \int\_{0}^{\omega} d\boldsymbol{r}\_{n} \right] \prod\_{n=1}^{N+1} \left[ \int\_{-\infty}^{\omega} \frac{d\boldsymbol{p}\_{n}}{2\pi} \right] \prod\_{n=1}^{N+1} \left[ \iint\_{|\boldsymbol{w}\_{n}^{\prime}| < \boldsymbol{r}\_{n}} d^{2}\boldsymbol{w}\_{n}^{\prime} \right] \prod\_{n=1}^{N} \left[ g\_{00}(\boldsymbol{w}\_{n,\boldsymbol{r}\_{n}}^{\prime}\boldsymbol{r}\_{n}) \right] \\ & \quad \times g\_{\text{Im}}(\boldsymbol{w}\_{N+1}^{\prime},\boldsymbol{r}\_{N+1}) \exp\left\{ i \sum\_{n=1}^{N+1} \left[ p\_{n}(\boldsymbol{r}\_{n} - \boldsymbol{r}\_{n-1}) - \varepsilon \left( \frac{p\_{n}^{2}}{2} + \frac{q(q+1)}{2r\_{n}^{2}} + V(\boldsymbol{r}\_{n}) \right) \right. \\ & \left. + i \sqrt{\frac{2\pi\omega}{V}} \varepsilon (\beta\boldsymbol{\chi}\_{n}\boldsymbol{w}\_{n} + \boldsymbol{\gamma}^{\*}\boldsymbol{\chi}\_{n}^{\*}\boldsymbol{w}\_{n}^{\*}) + \frac{2\pi\omega}{V} \varepsilon \boldsymbol{v}\_{n} |\boldsymbol{w}\_{n}|^{2} \right] \right\}. \end{split} \tag{51}$$

Further we observe that

$$\begin{split} & \lim\_{N \to \infty} \prod\_{n=1}^{N} \left[ \iint\limits\_{|w'\_n| < \tau\_n} d^2 w'\_n \right] \prod\_{n=1}^{N} \left[ g\_{00} (w'\_n, r\_n) \right] \\ & \times \exp\left\{ i \frac{t\_f - t\_i}{N+1} \sum\_{n=1}^{N} \left[ i \sqrt{\frac{2\pi\omega}{V}} (\beta \chi\_n w\_n + \gamma^\* \chi\_n^\* w\_n^\*) + \frac{2\pi\omega}{V} \nu\_n |w\_n|^2 \right] \right\} \\ &= \exp\left\{ i \frac{2\pi\omega}{3V} \int\_{t\_i}^{t\_f} d\tau \left[ \nu(\tau) r^2(\tau) \right] \right\}. \end{split} \tag{52}$$

So Eq. (51) becomes

$$\begin{split} K\_{\text{lmq}}^{\vec{\epsilon}}(r\_{f}, t\_{f}; r\_{i}, t\_{i}) &= F\_{\text{lm}}(r\_{f}) \left[ \iint D r(\tau) \frac{Dp(\tau)}{2\pi} \\ \times \exp\left\{i \int\_{t\_{i}}^{t\_{f}} \left[p\dot{r} - \left(\frac{p^{2}}{2} + \frac{q(q+1)}{2r^{2}} + V(r)\right)\right] + i \frac{2\pi\omega}{3V} \int\_{t\_{i}}^{t\_{f}} \nu(\tau)r^{2}(\tau)d\tau \right\}, \end{split} \tag{53}$$

where

We notice that if ln þ mn is odd then Olnmn is zero. Moreover jλnj, ϕλn are the polar coordinates

¼ jw0

¼ jw0

xn þ iw0

ln Olnmn

ρλn ¼ jλnj and Jmn are Bessel functions. In the appendix we give results for the expression in

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

lmqðrf , tf ; ri, tiÞ ffiffiffiffiffiffi

where after standard manipulations [11] on the angular parts of the atomic system

2 6 4

ðð

d2 w0 n 3 7 5 Y N

pnðrn � rn�1Þ � ε p2

V ενnjwnj

n¼1

2 #) :

jw0 nj<rn

nÞ þ 2πω

nj cos ðϕw0

nj sin ðϕw0

nje

2π exp imn ϕw0

yn ¼ jw0

nÞ cos

nÞ sin ξ 2 � �

> n þ π 2

> > jw0 njÞ:

4π p Ylmðϑf ,ϕfÞYqpðϑf ,ϕfÞY∗

g00 w0 ð Þ n, rn � �

> 2 þ qðq þ 1Þ 2r2 n

n

� � �

� � � �

rnÞJmn ðρλn

ξ 2 � �

, ð46Þ

, ð47Þ

ð49Þ

qpðϑi,ϕi Þ,

þ VðrnÞ

ð50Þ

ð51Þ

iϕw0n : ð48Þ

of λn on the x-y plane. We have set

160 Recent Studies in Perturbation Theory

On integrating over ϕλn we get

and

Eq. (49).

Kξ 0ðr ! f , tf ; r ! i, tiÞ ¼

Kξ

Kξ

wxn ¼ w0

wyn ¼ w0

glnmn ðw0

1 rfri

N

ð ∞

2 4

0 drn

ffiffiffiffiffiffiffiffiffi 2πω V

n¼1

�glmðw0

r

þi

lmqðrf , tf ; ri, tiÞ takes the form

lmqðrf , tf ; ri, tiÞ ¼ Y

Further we observe that

X∞ l¼0

X l

X∞ q¼0

X q

p¼�q Kξ

> ð ∞

2 4

dpn 2π

3 5 N Y þ1

( i N X þ1

n¼1

n¼1

nw∗

�∞

m¼�l

3 5 N Y þ1

n¼1

Nþ1, rNþ1Þ exp

εðβχnwn þ γ∗χ∗

xn cos

w0 n ¼ w0

n, rnÞ ¼ ð�iÞ

� ð ∞

0 dρλn ρλn j ln ðρλn

yn sin ξ 2 � �

ξ 2 � �

$$\begin{split} F\_{\rm lm}(r\_f) &= \iint\_{|w\_f'| < r\_f} d^2 w\_f' g\_{\rm lm}(w\_{f'}, r\_f) \times \\ &\exp\left\{ -\sqrt{\frac{2\pi\omega}{V}} \varepsilon (\beta \chi w\_f + \gamma^\* \chi^\* w\_f^\*) + i \frac{2\pi\omega}{V} \varepsilon \nu |w\_f|^2 \right\}. \end{split} \tag{54}$$

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

$$\begin{split} K\_{1}^{\xi}(\overrightarrow{r\_{f}},t;\overrightarrow{r}\_{i},0) &= \frac{1}{r\_{f}r\_{i}}\delta(r\_{f}-r\_{i})\sum\_{l=0}^{\infty}\sum\_{m=-l}^{l}\sum\_{q=0}^{\infty}\sum\_{p=-q}^{q}Y\_{\text{qp}}(\mathfrak{F}\_{f},\boldsymbol{\varrho}\_{f})Y\_{\text{qp}}^{\*}(\mathfrak{F}\_{l},\boldsymbol{\varrho}\_{i}) \\ &\times \sqrt{4\pi}Y\_{\text{lm}}(\mathfrak{F}\_{f},\boldsymbol{\varrho}\_{f})F\_{\text{lm}}(r\_{f})\exp\left[i\frac{2\pi\omega}{3V}r\_{f}^{2}\bigg]\nu(\tau)d\tau\right]. \end{split} \tag{55}$$

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

$$V(\vec{r}) = -\frac{1}{r}.\tag{56}$$

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

$$\Psi\_i(\stackrel{\frown}{r},t) = \Psi\_i(\stackrel{\frown}{r})e^{-i\varepsilon\_i t} = R\_{1i}(r)Y\_{00}(\theta,\phi)e^{-i\varepsilon\_i t} = 2e^{-r}Y\_{00}(\theta,\phi)e^{-i\varepsilon\_i t} \tag{57}$$

where εi ¼ �1=2 is the energy of the ground H(1s) state.

The final state of the ionized electron with wave vector k ! ¼ kð sin ϑk cosϕk, sin ϑk sinϕk, cos ϑkÞ is

$$\begin{split} \Psi\_{\vec{f}}^{\vec{k}}(\vec{r},t) &= \\ \Psi\_{\vec{f}}^{\vec{k}}(\vec{r})e^{-i\epsilon t} &= \exp\left(\frac{\pi}{2k}\right)\Gamma\left(1+\frac{i}{k}\right)e^{i\vec{k}\cdot\vec{r}}\,\_1F\_1\left(-\frac{i}{k};1;-i\vec{k}\cdot\vec{r};\vec{r}\right)e^{-i\epsilon t}. \end{split} \tag{58}$$

It has energy

$$
\varepsilon = \mathbf{^k}/\_{\mathfrak{d}'} \tag{59}
$$

and partial wave expansion

$$\Psi\_f^{\vec{k}}(\vec{r}) = \frac{2\pi}{k} \sum\_{s=0}^{\
\bullet} \vec{r}^s e^{-i\delta\_s} R\_s^k(r) \sum\_{t=-s}^s Y\_{\rm st}^\*(\mathfrak{d}\_\cdot \phi) Y\_{\rm st}(\mathfrak{d}\_{k\cdot} \phi\_k). \tag{60}$$

$$\begin{split} R\_s^k(r) &= \frac{\sqrt{8\pi k}}{\sqrt{1 - \exp\left(-2\pi/k\right)}} \prod\_{y=1}^s \left( \sqrt{y^2 + \frac{1}{k^2}} \right) \frac{1}{(2s+1)!} \\ &\times (2kr)^s e^{-ikr} \,\_1F\_1\left(\frac{i}{k} + s + 1, 2s + 2, 2ikr\right) \end{split} \tag{61}$$

is the radial function and δs ¼ arg Γ 1 � i k þ 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

$$A\_{\rm fi} = \langle \Phi\_f^-(t) | \Phi\_i^+(t) \rangle,\tag{62}$$

where Φ� f ðr !, tÞ and Φþ i ðr !, tÞ are exact solutions of the present system's time-dependent Schrodinger equation subject to the asymptotic conditions

Atoms-Photonic Field Interaction: Influence Functional and Perturbation Theory http://dx.doi.org/10.5772/67774 163

$$\Phi\_f^-\left(\overrightarrow{r},t\right)\_{t\to+\infty} \Psi\_f^{\overrightarrow{\vec{k}}}\left(\overrightarrow{r},t\right),\tag{63}$$

$$
\Phi\_i^+ (\overrightarrow{r}, t) \underset{t \to -\infty}{\longrightarrow} \Psi\_i (\overrightarrow{r}, t). \tag{64}
$$

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

$$\begin{aligned} A\_{\text{fl}} &= \\ \frac{1}{2} \lim\_{t\_1 \to \dots \to} \left\langle \Psi\_f^{\vec{k}} \middle| \mathcal{U}^0(t\_2)^+ \exp\left(-i \Big| \mathcal{H}\_{\text{eff}}(t\_2, \rho) d\rho + i \Big| \mathcal{H}\_{\text{eff}}(t\_1, \rho) d\rho\right) \mathcal{U}^0(t\_1) \right\vert \mathcal{W}\_i \right\rangle. \end{aligned} \tag{65}$$

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

$$H\_{\rm eff} = H\_{\rm c} - \mathrm{i}\,\frac{1}{\sqrt{V}}\mathrm{el}(\boldsymbol{\omega})(\boldsymbol{\beta}\chi\stackrel{\frown}{\boldsymbol{\varepsilon}}\cdot\stackrel{\frown}{r} + \boldsymbol{\beta}^\*\chi^\*\stackrel{\frown}{\boldsymbol{\varepsilon}}^\*\cdot\stackrel{\frown}{r}) - \frac{1}{V}\mathrm{e}^2\mathrm{l}^2(\boldsymbol{\omega})\boldsymbol{\nu}|\stackrel{\frown}{\boldsymbol{\varepsilon}}\cdot\stackrel{\frown}{r}|^2. \tag{66}$$

Moreover

4. Application and results

162 Recent Studies in Perturbation Theory

Ψiðr

Ψ k ! f ðr !, tÞ ¼

Ψ k ! f ðr

and partial wave expansion

It has energy

where Φ�

f ðr

is

!, tÞ ¼ Ψiðr

!Þe�iεt ¼ exp

Ψ k ! f ðr

Rk sðrÞ ¼

is the radial function and δs ¼ arg Γ 1 � i

!, tÞ and Φþ

i ðr

Schrodinger equation subject to the asymptotic conditions

!Þ ¼ 2π k X∞ s¼0 i s e �iδs Rk sðrÞ Xs t¼�s Y∗

�ð2krÞ s e�ikr 1F1 i k

where εi ¼ �1=2 is the energy of the ground H(1s) state.

The final state of the ionized electron with wave vector k

Proceeding to an application of the present theory we apply the above formalism to the case of

!޼� 1 r

�iεit ¼ 2e

!

1F1 � i k �r

; 1; � ikr � i k

� �

! � r !

=2, ð59Þ

ð2s þ 1Þ!

� � ð61Þ

k þ s � � a phase. Then the transition amplitude from the

stðϑ, φÞYstðϑk, φkÞ: ð60Þ

i ðtÞ〉, ð62Þ

: ð56Þ

�iεit

¼ kð sin ϑk cosϕk, sin ϑk sinϕk, cos ϑkÞ

e�iεt :

, ð57Þ

ð58Þ

Y00ðϑ,φÞe

Vðr

�iεit ¼ R1sðrÞY00ðϑ, φÞe

the ionization of hydrogen. In that case the potential is given as

!Þe

π 2k � �

Γ 1 þ i k � �

ffiffiffiffiffiffiffiffi 8πk p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � exp ð�2π=kÞ p Ys

ei k ! �r !

ε ¼ k2

y¼1

initial state i at t ! �∞ to the final continuum state f at t ! +∞ may be evaluated at any time t; it is

f ðtÞjΦþ

Afi ¼ 〈Φ�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi y2 þ 1 k2 ! r 1

!, tÞ are exact solutions of the present system's time-dependent

þ s þ 1, 2s þ 2, 2ikr

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

$$
\mathcal{U}^0(t) = e^{-iH\_rt}.\tag{67}
$$

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 1 BðtÞ factor in Eq. (20) and the identity in Eq. (25). At the times introduced by the delta functions the propagator Kξ 1ðr ! f , τ; r ! i, 0Þ(see below) becomes zero. Moreover the exponential in Eq. (20) is one as limt!∞ 1 BðtÞ ¼ 1 and β = γ.

Now to proceed we set t2 ¼ �t1 ¼ t and take into account the PT invariance of the whole system as the Hamiltonian Eq. (66) is PT invariant.  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

$$\begin{split} A\_{\rm fi} &= \left\langle \Psi\_{\vec{f}}^{\vec{k}} \,|\Psi\_{i} \right\rangle + \\ &+ \Big\| \frac{\stackrel{\scriptstyle \vec{\tau}}{\mathcal{V}} d\tau \Big\langle \Psi\_{\vec{f}}^{\vec{k}} \,|\mathcal{U}^{0}(\tau)^{+} \exp \left( -i \Big[ H\_{\rm eff}(\tau,\rho) d\rho + i \Big[ \stackrel{0}{\!\!F}\_{\rm eff}(\rho) d\rho \Big] \right. \\ &\times \Big\langle -\frac{1}{\sqrt{V}} el(\omega) (\not\!\chi \stackrel{\scriptstyle \vec{\tau}}{\mathcal{r}} \cdot \vec{r} + \not\!\!\!^{s} \chi^{s} \stackrel{\scriptstyle \vec{\tau}}{\sim} \vec{\tau}) + i \frac{1}{V} \epsilon^{2} \hat{l}^{2}(\omega) \nu |\hat{\vec{\varepsilon}} \cdot \vec{r}|^{2} \Big\vert \mathcal{U}^{0}(\tau) \Big| \mathcal{W}\_{i} \Big\rangle. \end{split} \tag{68}$$

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

$$\begin{split} A\_{\mathrm{fi}} &= \int\_{0}^{\zeta} d\tau \iint \vec{r'}\_{f} d\vec{r}\_{i} \frac{1}{r\_{i}^{2}} \boldsymbol{\varepsilon}^{j(\varepsilon-\varepsilon\_{i})\tau} \left(\boldsymbol{\Psi}\_{f}^{\mathrm{F}}(\overrightarrow{\boldsymbol{r}}\_{f})\right)^{\*} \boldsymbol{K}\_{1}^{\zeta}(\overrightarrow{\boldsymbol{r}}\_{f}, \tau; \overrightarrow{\boldsymbol{r}}\_{i}, \textbf{0}) \\\\ &\times \Big(-\sqrt{\frac{2\pi\omega}{V}} \Big(\beta\chi(\tau)\widehat{\boldsymbol{\varepsilon}}\cdot\overrightarrow{\boldsymbol{r}}\_{i} + \mathrm{c.c.}\Big) + i\frac{2\pi\omega}{V} |\widehat{\boldsymbol{\varepsilon}}\cdot\overrightarrow{\boldsymbol{r}}\_{i}|^{2} \nu(\tau)\bigg) \Psi\_{i}(\overrightarrow{\boldsymbol{r}}\_{i}) .\end{split} \tag{69}$$

We have used the prior form of the transition amplitude. Kξ 1ðr ! f , τ; r ! i, 0Þ 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 � Ψ k ! f ðr !Þ �PT ¼ � Ψ k ! f ðr !Þ �∗ .

Here we want to study two-photon ionization processes. They are of order 1 V 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

$$\begin{split} \mathcal{A} &= \left[ \det \int \prod \vec{r}\_f d\vec{r}\_i \frac{1}{r\_i^2} \mathbf{e}^{i(\varepsilon - \varepsilon)\tau} \left( \Psi\_f^{\vec{k}}(\vec{r}\_f) \right)^\* \\\\ &\times \left( -\sqrt{\frac{2\pi\omega}{V}} \tilde{S}\_{l=1}^\zeta(\vec{r}\_f, \tau; \vec{r}\_i, 0) \Big| \mathcal{J} \chi(\tau) \hat{\boldsymbol{\varepsilon}} \cdot \vec{r}\_i + c.c. \right) + i \frac{2\pi\omega}{V} S\_{l=0}^\zeta(\vec{r}\_f, \tau; \vec{r}\_i, 0) |\hat{\boldsymbol{\varepsilon}} \cdot \vec{r}\_i|^2 \nu(\tau) \right) \Psi\_i(\vec{r}\_i). \end{split} \tag{70}$$

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

$$\begin{split} S\_{l=1}^{\mathbb{F}}(\overrightarrow{r\_{f}},\mathbbm{z};\overrightarrow{r\_{i}},0) &= \frac{1}{r\_{f}r\_{i}}\delta(r\_{f}-r\_{i})\sum\_{q=0}^{\alpha}\sum\_{p=-q}^{q}Y\_{\text{qp}}(\mathbb{S}\_{f},\boldsymbol{\rho}\_{f})Y\_{\text{qp}}^{\*}(\mathbb{S}\_{i},\boldsymbol{\rho}\_{i}) \\ &\times \left[ -\sqrt{\frac{2\pi\omega}{V}} \langle \overset{\scriptstyle\pi}{\boldsymbol{\varepsilon}}\cdot\overset{\scriptstyle\pi}{\boldsymbol{r}}\_{f}\boldsymbol{\beta} \Big|\mathrm{d}\boldsymbol{\rho}\chi(\boldsymbol{\rho})+\text{c.c.}} \right] \exp\left[i\frac{2\pi\omega}{3V}r\_{f}^{2}\right] \nu(\boldsymbol{\rho})d\boldsymbol{\rho} \Big], \end{split} \tag{71}$$

and

#### Atoms-Photonic Field Interaction: Influence Functional and Perturbation Theory http://dx.doi.org/10.5772/67774 165

$$\begin{split} & \mathcal{S}\_{l=0}^{\ell}(\overrightarrow{r\_{f}},\overrightarrow{r\_{i}},\overrightarrow{r\_{i}},0) = \frac{1}{r\_{f}r\_{i}}\delta(r\_{f}-r\_{i})\sum\_{q=0}^{\alpha}\sum\_{p=-q}^{q}Y\_{\text{qp}}(\mathcal{S}\_{f},\rho\_{f})Y\_{\text{qp}}^{\*}\left(\mathcal{S}\_{l\nu}\,\rho\_{l}\right) \\ & \times \left\{1+\frac{2\pi\omega\nu}{3V}r\_{f}^{2}\left(\left.i\right|\nu(\rho)d\rho+\left.\beta\right|\int\_{0}^{\tau}d\rho\chi(\rho)\right|^{2}+\cos\xi\text{Re}\left[\left(\beta\left[d\rho\chi(\rho)\right]^{2}\right]\right]\right\}\right\} \\ & \times \exp\left[i\frac{2\pi\omega}{3V}r\_{f}^{2}\left|\nu(\rho)d\rho\right|\right]. \end{split} \tag{72}$$

Finally, we obtain the second-order transition probability

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

!

As the present theory is PT symmetric we have to use PT symmetric quantum mechanics. So

same transitions the vacuum fluctuations term contributes to the same order. So we take it into

!

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

X∞ q¼0

X q

Yqpðϑf ,ϕfÞY∗

2πω 3V r 2 f ð τ

0

qpðϑi,ϕi Þ

νðρÞdρ

3 5,

p¼�q

3

5 exp i

2 4

δðrf � riÞ

dρχðρÞ þ c:c:

Ψiðr ! iÞ:

i, 0Þ is given by Eq. (55).

V or higher.  (68) becomes

164 Recent Studies in Perturbation Theory

Afi ¼ ð ζ

� �

account.

Here we want to study two-photon ionization processes. They are of order 1

We have used the prior form of the transition amplitude. Kξ

our equations take their final form according to the fact that

i, 0Þ ¼ 1 rfri

0

� 2

$$\frac{\partial P}{\partial \varepsilon} = \frac{1}{4\pi^2}k \int |A|^2 d\Omega\_{\overrightarrow{\phantom{0}}}.\tag{73}$$

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

$$\mathcal{sp}(\tau) = \begin{cases} 1 & 0 \le \tau \le \zeta \\ 0 & \text{otherwise} \end{cases} \tag{74}$$

In Figure 1, we plot the second-order term ∂P ∂ε 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

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 ¼ π 2 (solid) ξ ¼ π 3 (dashed) ξ ¼ π 20 (dotted). We use ω ¼ 0:4275a.u., β = 1 and V ¼ 107.

2ω ¼ 0:855 a:u: Within the range 0 ≤ ξ ≤ π 2 the larger the ξ the smaller the transition probability. ξ ¼ π 2 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].

## 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.

 Conclusions

Appendix

ξ ¼ π

2 the larger the ξ the smaller the transition probability.

2 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

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

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

π 2

2 þ 1 � �

∞

0

ðρλrÞJmðρλjw0

Γðm þ 1Þ

q

Θðr � jw0

1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 � jw0 j 2

dρλρλj l

jÞ

jÞ

ðρλrÞJmðρλjw0

jÞ

ð75Þ

� � � � ð

there have appeared terms due to the electromagnetic vacuum fluctuations.

quantum mechanics of one-electron atoms interacting with radiation.


**3. Role of biochar use in wastewater treatment process**

Biochar could be used at different stages of wastewater treatment (**Figure 1**) to improve the treatment efficiency and recovery of value-added byproducts. Biochar application in wastewater treatment could be governed by the mechanism of adsorption, buffering, and immobilization of microbial cells. If used on the treated effluents, suitably modified biochar could efficiently adsorb nutrients like nitrogen and phosphorus, which can later be used as a nutrient-enriched material for soil remediation. When used in the activated sludge treatment process, biochar could play a role for improving the treatment and settling ability of the sludge by adsorption of inhibitors and toxic compounds or provide a surface for immobilization of microbes. Addition of biochar in the biological system could eventually help to improve the soil amendment properties of the biosolid as well. As interest grows in the use of biochar in soil applications, its use in wastewater treatment could expand the value chain and create additional economic benefits [66]. The following section will discuss the role of biochar for various applications in the wastewater treatment plant.

### **3.1 Organic pollutant removal**

In recent years, significant amount of research has been done to examine the application of biochar for removal of various organic compounds from water, which includes agrochemicals, antibiotics/drugs, polycyclic aromatic

**227**

**Figure 2.**

*Biochar-Assisted Wastewater Treatment and Waste Valorization*

hydrocarbons (PAHs), volatile organic compounds or (VOCs), cationic aromatic dyes [67–70]. Similarly, removal of organic compounds present in specific waste streams such as estrogen compounds in animal manure and sewage, inhibitory compounds of biomass degradation (furfural, hydroxymethylfurfural, phenolic compounds), and toxic organic compounds in landfill leachate has been studied using biochar [71, 72]. **Figure 2** schematically shows different interactions of the

Biochar produced at higher pyrolysis temperature is found better for removal of nonpolar organic compounds due to higher surface area and microporosity [30, 73]. In contrast, biochar produced at a temperature below 500°C contains more O- and H-containing functional groups; thus, they are likely to have a high affinity to polar organic compounds [26]. For example, rice husk and soybean-derived biochar (600–700°C) facilitates removal of nonpolar carbofuran (pesticide) and trichloromethylene (VOC) from contaminated water [26]. Efficient removal of pyrimethanil and diesopropylatrazine (fungicide/pesticide) was observed with red-gum wood chips and broiler litter-derived biochar at temperature >700°C, whereas the same biochar at temperature <500°C was inefficient [74, 75]. On the other hand, removal of polar insecticide and herbicide like 1-naphthol, norflurazon, and fluridone was observed with biochar produced at <300°C, due to interaction of pollutant and the functional groups of biochar [76, 77]. Likewise, higher sorption of aromatic cationic dyes like methyl-violet and methyl-blue was observed with biochar containing more O- and H-functional groups (<400°C) but the mechanism was highly dependent on pH [70, 78]. The sorption of polar antibiotic sulfamethazine (SMZ) by hardwood/softwood-derived biochars (produced at 300–700°C) has pH-dependent

*Biochar interaction with organic and inorganic compounds in wastewater (adapted from Ahmad et al. [33]).*

*DOI: http://dx.doi.org/10.5772/intechopen.92288*

organic pollutant with biochar.

### **Figure 1.**

*Use of biochar at different stages of wastewater treatment.*

*Biochar-Assisted Wastewater Treatment and Waste Valorization DOI: http://dx.doi.org/10.5772/intechopen.92288*

*Applications of Biochar for Environmental Safety*

wastewater treatment.

treatment plant.

**3.1 Organic pollutant removal**

Thus, the selection of biochar and modification methods for the application in wastewater treatment requires a considerable understanding of the biochar properties and mechanism by which it supports the treatment process at different stages of

Biochar could be used at different stages of wastewater treatment (**Figure 1**) to improve the treatment efficiency and recovery of value-added byproducts. Biochar application in wastewater treatment could be governed by the mechanism of adsorption, buffering, and immobilization of microbial cells. If used on the treated effluents, suitably modified biochar could efficiently adsorb nutrients like nitrogen and phosphorus, which can later be used as a nutrient-enriched material for soil remediation. When used in the activated sludge treatment process, biochar could play a role for improving the treatment and settling ability of the sludge by adsorption of inhibitors and toxic compounds or provide a surface for immobilization of microbes. Addition of biochar in the biological system could eventually help to improve the soil amendment properties of the biosolid as well. As interest grows in the use of biochar in soil applications, its use in wastewater treatment could expand the value chain and create additional economic benefits [66]. The following section will discuss the role of biochar for various applications in the wastewater

In recent years, significant amount of research has been done to examine the application of biochar for removal of various organic compounds from water, which includes agrochemicals, antibiotics/drugs, polycyclic aromatic

**3. Role of biochar use in wastewater treatment process**

**226**

**Figure 1.**

*Use of biochar at different stages of wastewater treatment.*

hydrocarbons (PAHs), volatile organic compounds or (VOCs), cationic aromatic dyes [67–70]. Similarly, removal of organic compounds present in specific waste streams such as estrogen compounds in animal manure and sewage, inhibitory compounds of biomass degradation (furfural, hydroxymethylfurfural, phenolic compounds), and toxic organic compounds in landfill leachate has been studied using biochar [71, 72]. **Figure 2** schematically shows different interactions of the organic pollutant with biochar.

Biochar produced at higher pyrolysis temperature is found better for removal of nonpolar organic compounds due to higher surface area and microporosity [30, 73]. In contrast, biochar produced at a temperature below 500°C contains more O- and H-containing functional groups; thus, they are likely to have a high affinity to polar organic compounds [26]. For example, rice husk and soybean-derived biochar (600–700°C) facilitates removal of nonpolar carbofuran (pesticide) and trichloromethylene (VOC) from contaminated water [26]. Efficient removal of pyrimethanil and diesopropylatrazine (fungicide/pesticide) was observed with red-gum wood chips and broiler litter-derived biochar at temperature >700°C, whereas the same biochar at temperature <500°C was inefficient [74, 75]. On the other hand, removal of polar insecticide and herbicide like 1-naphthol, norflurazon, and fluridone was observed with biochar produced at <300°C, due to interaction of pollutant and the functional groups of biochar [76, 77]. Likewise, higher sorption of aromatic cationic dyes like methyl-violet and methyl-blue was observed with biochar containing more O- and H-functional groups (<400°C) but the mechanism was highly dependent on pH [70, 78]. The sorption of polar antibiotic sulfamethazine (SMZ) by hardwood/softwood-derived biochars (produced at 300–700°C) has pH-dependent

### **Figure 2.**

*Biochar interaction with organic and inorganic compounds in wastewater (adapted from Ahmad et al. [33]).*

interactions [79]. It can be said that pH is the most important factor for biochar interactions and removal of polar organic pollutants.

## **3.2 Inorganic pollutant removal**

Inorganic pollutant in wastewater includes heavy metals (Cr, Cu, Pb, Cd, Hg, Fe, Zn, and As ions) and compounds like nitrate (NO3), nitrite (NO2), ammonium (NH4), phosphorus (P), and hydrogen sulfide (H2S) that cause significant risk to public health and environment [80]. Biochar produced at lower pyrolysis temperature (<500°C) has properties that are better suited for removal of inorganic compounds. The chemical composition and the morphological structure play an important role in the sorption nature of biochar [81]. **Figure 2** summarizes the interaction methods for inorganic pollutant and biochar.

### *3.2.1 Heavy metals*

Biochar with high organic carbon content (at non-carbonized fraction), specific porous structure, and numerous functional groups interacts with heavy metals in many ways [82]. The sorption of heavy metals by biochar is mainly by surface interaction through ion exchange and complexation between biochar functional groups (e.g., OH, COOH, R-OH) and heavy metal ions [83, 84], moreover formation of metal precipitates with inorganic constituents [83–85] and coordination of metal ions with π electrons (C〓C) of biochar [74]. The physiochemical properties of biochar affect the adsorption throughout its matrix and are dependent on pyrolysis temperature, feedstock type, pH, and application rate. Cu2+ showed high affinity toward COOH▬ and OH▬ groups of hardwood and crop-derived biochars with dependency on pH and feedstock types [86]. Similarly, sida hermaphrodita-, guayule shrub-, soybean straw-, and wheat straw-derived biochars were effective for removal of Cd2+, Ni2+, and Zn2+ along with Cu2+ [87]. The higher efficiency of the above-mentioned biochar was due to high C and O contents, high O/C molar ratio, and polarity index, which were mainly regulated by pH [88, 89]. Alkaline biochars derived from various agricultural residues (e.g., soybean straw, corncob, cocoa husk, corn stover, switchgrass) and manure were efficient for Hg2+ removal. Animal manure-derived and cocoa husk biochar was highly effective for Hg2+ removal due to high sulfur (SH groups and sulfate) to precipitate 90% of Hg2+ as Hg(OH)2 or HgCl2 mainly through coprecipitation with anions (Cl, O, S) of biochar [73, 90].

For Cd2+, Zn2+, Pb2+, and Cu2+ dosage of biochar also affects the removal of heavy metals. The higher removal efficiency is observed with increasing biochar loading in the aqueous system, due to increased pH and surface area with biochar addition [54, 91].

## *3.2.2 Nitrogen and phosphorus*

The high surface charge density allows biochars to retain cations by cation exchange and the high surface area, internal porosity, and presence of both polar and nonpolar surface sites on biochar enable it to adsorb nutrients [92]. In the limited studies carried out without soil, biochar has shown the absorption NH4 −, NO3 −, and PO4 3− despite the different charges and properties of these nutrients [93]. Some examples include digested sugar beet tailing biochar pyrolyzed at 600°C that adsorbed PO4 ions most likely in binding sites contained in colloidal and nano sized MgO particles on the biochar surface [94]. Also, orange peel biochars pyrolyzed between 250 and 700°C removed between 8 and 83% of phosphate from solution

**229**

biosolids.

**3.4 Anaerobic digestion**

*Biochar-Assisted Wastewater Treatment and Waste Valorization*

[95]. NH4 was adsorbed to biochars produced from rice husk [96] and a mixture of tree trunks and branches [97], albeit weakly, as the partitioning coefficients between water and biochar were low (Freundlich coefficients of 0.251 mg g−1

One of the most utilized systems for treatment of municipal wastewater is biological treatment process like activated sludge system (ASS) because of its cost-effectiveness and comparatively more straightforward operation to advance systems. Activated sludge process is a suspended growth treatment where aerobic microorganism decomposes the organic matter in wastewater, which eventually settles as solids by gravity. Currently, increasing concerns are being raised about the presence of various micro-pollutants from pharmaceuticals, personal care products (PCPs), pesticides, disinfectants, and antiseptic in domestic and municipal wastewaters. These pollutants are alien to the biota in the system, and the conventional treatment process often leads to inadequate removal of these compounds. Correspondingly, discharge requirements are currently being stringent for protection of receiving waters from possible contamination and public health hazard. There have been several modifications and changes in the activated sludge system to address the problem. One such method is AS-PACT (Activated Sludge with Powdered Activated Carbon Treatment) where powdered activated carbon is added to the aeration basin of activated sludge system. The larger surface area of carbon provides various benefits including adsorption of toxic substances such as pharmaceuticals and industrial chemicals, immobilization of bacteria, and increased sedimentation of activated sludge [99, 100]. Such system, however, requires a

Despite the benefits, the higher cost of activated carbon limits its use in municipal wastewater treatment [101]. The biochar could be a low-cost substitute to activated carbon [102], but its merits are less known. The addition of biochar to a biological treatment system, such as within the aeration tank, could result in increased process stability by (a) adsorption of inhibitors (heavy metals, polycyclic aromatic hydrocarbon), (b) increasing the buffering capacity of the system, and (c) immobilization of microbial cells [103]. Limited studies done on the use of biochar in the aeration tank showed increased settling ability of activated sludge [104]. Dissolved organic matter in the biochar could also provide additional carbon to promote denitrification [105]. The availability of organic matter, however, depends on the type of biomass and pyrolysis conditions used for producing biochar. Furthermore, the cascading benefits of using biochar in activated sludge treatment could also be seen on anaerobic digestion of the sludge and in the final quality of the

In the case of anaerobic digestion, the addition of biochar has shown increases in the rate and amount of biogas production [106–108]. This is attributed to the buffering properties of biochar, promoting methanogenesis for higher biogas yield [109, 110]. Several studies have suggested increases in microbial metabolism and growth because of the support provided by the biochar [107, 111]. The biochar could also play a significant role in reducing the mobility or availability of the inhibitors like heavy metals, pesticides, antibiotics, and other organic compounds by binding them in its porous structure and maintain proper microbial activity for the

− has been adsorbed to bamboo charcoal biochar in the concentration

).

*DOI: http://dx.doi.org/10.5772/intechopen.92288*

[98].

continuous makeup of fresh carbon [101].

Similarly, NO3

range of 0–10 mg L−1

**3.3 Activated sludge treatment**

[95]. NH4 was adsorbed to biochars produced from rice husk [96] and a mixture of tree trunks and branches [97], albeit weakly, as the partitioning coefficients between water and biochar were low (Freundlich coefficients of 0.251 mg g−1 ). Similarly, NO3 − has been adsorbed to bamboo charcoal biochar in the concentration range of 0–10 mg L−1 [98].


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## Meet the editor

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book projects. One of her books was selected to be in the Web of Science™ Core Collection. Her publication (*Phys. Chem. Chem. Phys.*, 2014) is rated #1 and two of her publications have been selected for Longuet-Higgins Early Career Researcher Prizes. She is a reviewer for many of the leading journals and is rated at 0.01% of the most read authors by Academia.edu.

Contents

Microelectronics *by Steven H. Voldman*

Nanoporous Silicon

*by Javier Martinez*

to Medical Work *by Klaudia Freire*

*and Sujatha Lakshminarayanan*

Optical Tweezers in Biotechnology *by Zhiyong Gong and Yuchao Li*

**Preface III**

**Chapter 1 1**

**Chapter 2 11**

**Chapter 3 27**

**Chapter 4 43**

**Chapter 5 61**

**Chapter 6 73**

Electrostatic Discharge, Electrical Overstress, and Latchup in VLSI

Development and Characterization of Poly Ethylene-Co-Vinyl Acetate

*by Rashmi Aradhya, Madhu Bilugali Mahadevaswamy and Poornima*

A Novel Sensing Method for VOCs Using Nanoparticle-Coated

*by Selvakumar Varadarajan Subramani, Suganthi Selvakumar*

Nanolithography by Scanning Probes for Biorecognition

Biophotonics in Africa Powered by Light Technology Applied

(PEVA) Hybrid Nanocomposite Encapsulates for Solar PV

## Contents

Preface

The new generation of nanotechnological devices presents increasingly formidable multidisciplinary challenges at the most fundamental level, which result in an urgent need for long-term research based on a scientific approach. These technologies are forcing changes to the way we view and use communications and information. The new technologically innovative devices need understanding of underlying physical mechanisms, and removal of present technological limitations. With a strong emphasis on experimentation and fabrication, the applications span the areas of nano-devices, optical communications, sensors, and biomedical devices. This book is all about the optoelectronic and biomedical applications of micro and nanotechnologies, in which authors from all over the world have contributed their research.

Chapter 1 deals with the issues of electrostatic discharge (ESD), electrical overstress (EOS), and latchup for VLSI microelectronics with a discussion on physical models, failure mechanisms, and design solutions. The author has highlighted solutions to the problems. I hope that this chapter will help the reader by providing useful insight on the ESD design and chip development mechanism and will overcome the number of research challenges in order to speed up technological innovation for

In Chapter 2, the synthesis and characterization of poly ethylene-co-vinyl acetate (PEVA) is presented, which plays an important role in solar materials, such as PV modules and agricultural greenhouse polymer sheets. The morphology and miscibility of fabricated nanocomposite films was analyzed and investigated experimentally. It was observed that these nanocomposites showed increased thermal stability, glass transition temperature, high transmittance, and optimum

The structural aspects of VOCs using porous silicon coated with a layer of ZnO (PS-ZnO), which is essential in sensing Volatile Organic Compounds (VOCs) is discussed in Chapter 3. Further variations in several parameters were measured and

Optical tweezers have the potential for application in nanostructure assembly, cancer cell sorting, targeted drug delivery, single-molecule studies, and biosensing. In Chapter 4, a discussion about nano-optical manipulation techniques is presented to solve the problem of difficult integration and diffraction limitations of conventional optical tweezers and holographic optical tweezers, which can be

Chapter 5 is about the most innovative and reliable probe nanolithography techniques based on the spatial confinement of a chemical reaction within a nanometric size frame, which can be applied for controlled deposition of molecules with high precision or nanotransistors that can be used as sensors for biorecognition

Welcome to the emerging field of micro and nanoelectronics!

nanoelectronics in the near future.

UV-shielding efficiency.

analyzed by experimental means.

used in biophotonic and biomedical applications.


*4.1.1. Poor life adaptation*

To individuals, transitioning from a familiar and comfortable space to an unfamiliar space requires a series of physical and mental adjustments. For example, climate conditions, standard of living, language ability, culture shock, and so forth are sufficient causes of anxiety and fear in working holiday travelers. For instance, a WHM who went to Japan mentioned, *"My English was not very good and I did not know Japanese. So, language was what I was most afraid I would not adapt to when I went over" (J3, 2014)*. A WHM who went to Canada was worried about problems with life adaptations and adopted an adjustment strategy: *"It was winter when I departed, and I was not sure whether I would be able to adapt. So, I set a bottom line. If in three months' time I had used up my money, could not find a job, and had not adapted to the country, I would go home" (C3, 2014).*

The issues of psychological anxiety faced by WHMs not only involve adapting to life in the travel destination but also include life adaptations for work continuity upon returning to their home country. As working holidays may last up to 1 year, WHMs need to consider the delay in entering the workplace compared to their peers. Some travelers were concerned about this form of psychological stress. For example, one participant stated, *"The time between the application and departure was quite difficult because I had quit my job. However, everyone around me was going to interviews, finding jobs, and even starting to work. I felt slight panic because I was worried that I would be one step behind other people in the future" (J1, 2014)*.


*Review* **Arboviruses: How Saliva Impacts the Journey from Vector to Host**

**Christine A. Schneider 1 , Eric Calvo 2 and Karin E. Peterson 1,\***

**Abstract:** Arthropod-borne viruses, referred to collectively as arboviruses, infect millions of people worldwide each year and have the potential to cause severe disease. They are predominately transmitted to humans through blood-feeding behavior of three main groups of biting arthropods: ticks, mosquitoes, and sandflies. The pathogens harbored by these blood-feeding arthropods (BFA) are transferred to animal hosts through deposition of virus-rich saliva into the skin. Sometimes these infections become systemic and can lead to neuro-invasion and life-threatening viral encephalitis. Factors intrinsic to the arboviral vectors can greatly influence the pathogenicity and virulence of infections, with mounting evidence that BFA saliva and salivary proteins can shift the trajectory of viral infection in the host. This review provides an overview of arbovirus infection and ways in which vectors influence viral pathogenesis. In particular, we focus on how saliva and salivary gland extracts from the three dominant arbovirus vectors impact the trajectory of the cellular immune response to arbovirus infection in the skin.

**Keywords:** viral infection; skin; immune enhancement; mosquito


**3.2 Monomethyl ether of hydroquinone/4–0 methoxyphenol**

This compound is a phenol derivative and is also known as p-hydroxyanisole (HA) or mequinol [1].

### *3.2.1 Mechanism of action*

*Depigmentation*

*3.1.3 Precautions*

tions while using MBEH.

on the eyelids.

*3.1.4 Side effects*

adverse reaction, the patient can continue with the application of the cream on the areas of top priority and then move in stages for low priority areas. To avoid contact dermatitis, different concentrations of MBEH can be used. MBEH can be diluted to 5% for use on the neck, 10% on the face, and 20% on the arms and legs. In patients who fail to respond to 20% MBEH over a course of 3 to 4 months, the concentration of MBEH can be increased to 30% and then further to 40%. Concentrations of 30 and 40% MBEH have been used primarily on the extremities, especially the elbows

It takes anywhere between 4 and 12 months for gradual depigmentation [8]. It is to be noted that depigmentation is mostly irreversible and histologically associated

Patients should always be informed and well instructed about certain precau-

1.Application of MBEH at one site can lead to loss of pigment at distant body sites, i.e., application of MBEH to the arm may result in loss of pigment on the

2.Application of MBEH to the eyelids is not advised [8] because of risk of

ochronosis. It may lead to pigmentation of the conjunctiva if MBEH is applied

3.Avoid skin-to-skin contact on a continuous basis with another person as it can cause a decrease in pigmentation at the site of contact in the other person.

4.The use of sunscreens with a high-sun protection factor (SPF) is essential. This

also helps to prevent repigmentation as well as sunburn reactions [4].

5.Follicular repigmentation may occur spontaneously upon sun exposure [8]. This happens mainly because MBEH only destroys epidermal melanocytes

Irritant contact dermatitis and common allergic reactions can develop [9]. In which event, application of MBEH is stopped, and open wet dressings are applied to the affected area along with topical steroids. Once the dermatitis has subsided, MBEH can be restarted at a lower concentration of 5% [8]. Other side effects include exogenous ochronosis [10], unmasking of telangiectasias and phlebectasias on the lower extremities [8], pruritus, xerosis, erythema, rash, edema, conjunctival

Risk of carcinogenesis with MBEH has not been reported but cannot be ruled out, and hence it is banned from the European Union since 2001 in cosmetics [11].

All-trans retinoic acid (RA), which is a vitamin A derivative primarily employed in the treatment of acne, is shown to serve as a weak depigmenting agent when used

and knees. Concentrations greater than this are not recommended [8].

face [4]. Moreover, it can also reactivate a stable disease.

with loss of melanosomes and melanocytes [1].

keeping follicular melanocytes intact.

melanosis, and distant depigmentation [4].

*3.1.5 Combination therapy*

for several weeks.

**58**

Mequinol acts in the similar way as MBEH acts. This compound usually acts via a dose-dependent response manner. It can be used as monotherapy or in conjunction with a Q-switched ruby laser.

## *3.2.2 Administration of treatment*

The compound is used in a 20% concentration in an oil/water cream base. As with MBEH, cream is applied on an initial test patch to observe for any allergic reactions. If there are no reactions, the patient is advised to apply cream twice daily until complete depigmentation is observed [16]. The effectiveness of 4-MP has been correlated with the duration of the use of the cream; the longer the cream was used, better the results that were obtained [1].

A combination product of 2% 4-hydroxyanisole (mequinol) and 0.01% tretinoin was tested in a double-blind multicentric study and was found to significantly improve solar lentigines and related hyperpigmented lesions of the face and hands after a twice-daily application of up to 24 weeks [1].

### *3.2.3 Side effects*

Side effects include mild burning or itching, irregular leukoderma, contact dermatitis, ochronosis, and risk of carcinogenesis cannot be ruled out [11]. Protection from sunlight is necessary or repigmentation risk is high [1, 11].

## **3.3 Phenol solution (88%)**

Phenol is an inexpensive peeling agent having medium-depth capability and used for treatment of photodamage or rhytids. The toxicity of phenol toward melanocytes is well documented. Phenol has the ability to penetrate deeper into the tissue up to the upper reticular dermis.

#### *3.3.1 Mechanism of action*

Phenol is involved in melanogenesis, inducing coagulation of protein in the epidermis. The melanocytes lose their capacity to synthesize melanocytes normally. This property of phenol is different than that of MBEH and hydroquinone wherein they destroy the melanocytes [17]. Hence, 88% phenol can be used as therapeutic option to eliminate residual normally pigmented lesions in patients.

#### *3.3.2 Administration of treatment*

The area to be treated is cleaned with spirit/alcohol. Application of phenol is done with the help of a swab soaked with phenol until cutaneous frosting occurs. There might be a burning sensation experienced by the patient for approximately 60 seconds, which gradually decreases in intensity but can last from minutes to hours. In a case study reported by Zanini and Machado Filho, they reported the use of 88% phenol on a 62-year-old female patient. Post 2 sessions, with a gap of 45 days, total elimination of residual pigmentation was achieved [17].


*11.5.1 Australia*

Despite previous countrywide bans on both euthanasia and PAS, the state of Victoria passed a law in 2017 that allowed doctors to prescribe lethal drugs for selfadministration by terminally ill patients; the law also allowed doctors to administer the medications if the patient is unable to do so themselves. Western Australia followed in 2019, followed in 2021 by Tasmania, Queensland, and South Australia. Parliament will be under pressure to permit legalization of assisted dying in the Northern Territory and in the Australian Capital Territory.